Mathematical Explanation in Science
by
Alexander Koo
A thesis submitted in conformity with the requirements for the degree of Doctorate of Philosophy
Institute for the History and Philosophy of Science and Technology University of Toronto
© Copyright by Alexander Koo 2015
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Mathematical Explanation in Science
Alexander Koo
Doctorate of Philosophy
Institute for the History and Philosophy of Science and Technology
University of Toronto
2015
Abstract
Inspired by indispensability arguments originating from Quine, mathematical realists such as
Alan Baker argue that since mathematics plays a key explanatory role in our best scientific
theories, then the very same reasons which convince us to be scientific realists should lead us to
be mathematical realists as well. Baker’s enhanced indispensability arguments (EIA) makes use
of inference to the best explanation (IBE) to deliver the realist conclusion. Mathematical
nominalists have resisted this argument by asserting that there is no such thing as a genuine
mathematical explanation (GME) where the mathematics is playing an indispensable explanatory
role. In this dissertation I will argue that the nominalist is incorrect and that GMEs do, in fact,
exist. My methodology will be to develop a set of criteria that clearly defines a GME which the
nominalist would gladly accept. From there, a new example of a GME will be advanced that
satisfies all of the criteria. To solidify this result, I will show that Strevens’ kairetic account of
scientific explanation clearly points to mathematics playing an indispensable explanatory role in
our supposed examples of GME. While this all bodes well for mathematical realism, I will
further argue that the EIA still does not lead to mathematical realism. Baker assumes that using
IBE to infer the existence of mathematical objects is unproblematic. I will challenge this
assumption, and without IBE the EIA does not deliver. By no means does this result block the
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realist project. Ultimately, I believe that freeing ourselves from the yoke of traditional
indispensability arguments and focussing on how it is that mathematics can explain physical
facts explain will advance the realist position even further.
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Acknowledgments
I would like to thank my supervisor, Professor James R. Brown, for all the emotional and
intellectual support that he has given me over the years. Jim was the first to introduce me to the
philosophy of mathematics, and I have found inspiration in both his work and his kindness ever
since. I truly could not have completed this project without his guidance. I would also like to
thank my committee members, Professors Joseph Berkovitz and Christopher Pincock, for their
incredibly helpful comments and the many fruitful conversations that we had. All three have
contributed greatly to my development as a philosopher, and I am a better person for it.
Special thanks to Professors Alan Baker and Mark Colyvan whose works inspired my own
research. I am fortunate to have received helpful comments and criticisms from both, as well as
the encouragement to continue my research. Thanks also to Professor Sorin Bangu for originally
introducing me to indispensability arguments, and for being instrumental in my decision to apply
to graduate school.
This dissertation would not have been finished without the love and support of my friends and
family. Thanks to my study partners, Michael, Agnes, Rebecca, and Anita, who were always a
reliable source for motivation, decompression, and for a lunch buddy. To Jimmy for keeping
things in perspective and helping me unwind before the next work day. To Don for being my
biggest fan. A big thanks to my mom, Sunda, for all her love and support, and for all that she has
done and provided for me. Words cannot express my gratitude. Finally, to my wife Rosie, thank
you for putting up with me over all these years. Your support, encouragement, and love got me
through this journey, and continues to get me through each and every day.
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Table of Contents
Abstract ...........................................................................................................................................ii
Acknowledgments.......................................................................................................................... iv
Chapter 1 The Quinean Picture ....................................................................................................... 1
1 Mathematical Realism ......................................................................................................... 4
2 W. V. O. Quine .................................................................................................................... 7
3 The Quinean Indispensability Argument ........................................................................... 13
4 Three Strengths .................................................................................................................. 16
Chapter 2 Enhancing Quine .......................................................................................................... 20
1 Weakness One: The Quinean Picture ................................................................................ 20
2 Weakness Two: Disrespect towards Mathematics ............................................................. 23
3 Weakness Three: Vagueness .............................................................................................. 28
4 Internal and External Mathematical Explanations ............................................................. 42
5 The Enhanced Indispensability Argument ........................................................................ 45
6 What About Naturalism? ................................................................................................... 49
Chapter 3 Genuine Mathematical Explanation ............................................................................. 52
1 Supposed Genuine Mathematical Explanations................................................................ 53
1.1 Geometric Explanations ............................................................................................. 53
1.2 Contrived Explanations .............................................................................................. 55
1.3 Optimization Explanations ......................................................................................... 56
2 The Indexing Argument Revisited ..................................................................................... 59
3 The Indexing Criteria for Genuine Mathematical Explanations ........................................ 71
4 Genuine Mathematical Explanation: Electron Spin .......................................................... 77
5 Blocking the Nominalist’s Response ................................................................................ 84
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5.1 The Unknown Explanation ......................................................................................... 84
5.2 The Ad Hoc Physical Explanation .............................................................................. 87
5.3 No Explanation ........................................................................................................... 95
6 The Honest Conclusion ...................................................................................................... 97
Chapter 4 Scientific Mathematical Explanation ........................................................................... 99
1 Accounts of Scientific Explanation ................................................................................. 101
1.1 The Deductive-Nomological and Pragmatic Accounts ............................................ 102
1.2 The Unification Account .......................................................................................... 105
1.3 The Statistical-Relevance and Counterfactual Accounts .......................................... 107
2 The Kairetic Account ....................................................................................................... 110
2.1 The Kairetic Criterion ............................................................................................... 111
2.2 Adapting the Kairetic Criterion ................................................................................ 113
3 Applying the Kairetic Criterion ....................................................................................... 120
4 Difference-Making Revisited........................................................................................... 126
4.1 Internal Mathematical Explanations ......................................................................... 126
4.2 The Roles of Mathematics in Genuine Mathematical Explanations ........................ 131
5 Takeaway ......................................................................................................................... 139
Chapter 5 Inference to the Best Mathematical Explanation ....................................................... 140
1 The Inference to the Best Explanation ............................................................................. 142
1.1 Problems with Inference to the Best Explanation .................................................... 143
1.2 Types of Inference to the Best Explanation .............................................................. 147
2 Fictionalism: A Better Explanation? ................................................................................ 150
3 Unjustified Inference to the Best Mathematical Explanation .......................................... 157
4 The Not-So-Enhanced Indispensability Argument .......................................................... 166
Chapter 6 Conclusion .................................................................................................................. 172
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1 Moving Forward .............................................................................................................. 174
2 Final Thoughts ................................................................................................................. 186
Bibliography ............................................................................................................................... 187
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Chapter 1 The Quinean Picture
An indispensability argument for mathematical realism aims to convince the typical scientific
realist that the very same reasons that motivate their commitments to the unobservable entities
postulated in our best physical theories, such as positrons and quarks, should inevitably lead
them to be realists regarding the mathematical entities utilized in science, such as numbers and
sets. The key consideration behind the realist position is the indispensable nature of unobservable
entities. For the time being it is sufficient to characterize a scientific realist as believing in the
objective existence of the entities postulated by science. Scientific realists need not commit to
every entity utilized in science, but rather realists believe in the existence of those unobservable
entities that are indispensable for generating the results, predictions, and explanations in our best
scientific theories. It is this indispensable nature that provides the basis for our belief in the
objective existence of said entities. Indispensability arguments make a parallel argument for
mathematical entities. At first glance it surely seems inconceivable to do science without the use
of mathematical entities. These entities are indispensable, so this points to the objective existence
of mathematical entities in the very same way as it does for physical unobservable entities.
Originally popularized by Quine and Putnam, many different forms of indispensability
arguments have since emerged. The most influential of these indispensability arguments is
championed by Mark Colyvan and is deeply inspired by Quine’s work. Recently, Alan Baker
(2005) has put forward a new enhanced indispensability argument (EIA) that claims to have
significant advantages over its predecessors. Baker’s EIA focuses on how mathematics is
indispensable in scientific explanations in order to lead to the realist conclusion. The move to
mathematical realism is achieved by an inference to the best explanation (IBE), which is a
standard inference for the scientific realist. The EIA hinges on two much discussed topics in the
philosophy of science: scientific explanation and IBE. What makes Baker’s argument interesting
is that these two topics are transplanted from a purely scientific domain, which deals with
concrete physical objects and phenomena, into the abstract realm of mathematics. An upshot of
this move is to build up an argument for mathematical realism using concepts that the scientific
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realist is already intimately familiar and comfortable with. No additional concepts or reasoning is
needed to extend one’s ontology over mathematical objects.
The benefits of the EIA are quite strong in that it aligns itself well with the present discourse in
the philosophy of science, but such a move comes at a cost. At present, there are many
unanswered questions which prevent the EIA from being a convincing argument. Is it the case
that using IBE to infer the existence of abstract objects is the same as the standard usage in
which we infer the existence of physical objects? If not, how can we justify such an inference?
Given the contentious state of our understanding of scientific explanations, can we base an
argument on the even less understood idea of mathematical scientific explanations? Are
mathematical scientific explanations well-defined, and do such things even exist? Only when
these questions are addressed will we be able to properly asses the EIA.
The principle aim of this dissertation is to evaluate whether the EIA represents a significant
advancement over other indispensability arguments or not. In order to critically assess the EIA,
the first task will be to understand the development of indispensability arguments for
mathematical realism. Chapter 1 will be dedicated to a brief history of the influence of Quine,
Putnam, and scientific naturalism. Quine’s naturalism is the driving force behind the
development of indispensability arguments that culminates in Colyvan’s presentation of the
Quinean argument. Chapter 2 will present Baker’s enhanced indispensability argument and
demonstrate how it differentiates itself from previous indispensability arguments by improving
on several critical weaknesses that Quine’s argument faces.
Unfortunately, many of the concepts utilized in the debate surrounding the EIA are unclear or
underdeveloped. As mentioned above, the two most critical are those of mathematical
explanation of scientific facts, and the use of IBE for mathematical realism. In order to properly
assess the worth of the EIA it will be important to make precise these key aspects of the
argument. Largely due to the EIA, the topics of mathematical explanation and IBE have been
typically treated together. The most common approach in the literature has been to assume that
the use of IBE is unproblematic, and then focus on determining the explanatory power of
mathematics. Broadly speaking, this has divided philosophers into two predictable camps.
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Realists support the existence of mathematical explanations of physical facts, whereas anti-
realists deny it. I feel that this is unfortunate as it is the attraction or fear of ontological
commitment that is pointing the direction of analysis of mathematical explanations in science.
My methodological approach will be to not assume anything about IBE and to analyze both
mathematical explanations in science and inferring mathematical realism separately and
independently. Understanding the explanatory role of mathematics in science is a worthwhile
endeavour entirely independent of potential ontological commitment. In addition, such an
understanding will facilitate a much better grasp of how the EIA works, and whether or not it is a
good argument for mathematical realism. The first step will be to consider what is meant by
mathematical explanation while remaining agnostic about any sort of inference to mathematical
realism. Chapter 3 will start down this path by developing criteria for exactly what we mean by a
mathematical explanation of a scientific fact. Once these criteria have been laid out we will
analyze many of the standard examples, and ultimately present a new example of a mathematical
explanation of a scientific fact. Chapter 4 will take this process one step further by making use of
theories of scientific explanation. Specifically, we will make use of Michael Strevens’ kairetic
account of scientific explanation to analyze our examples of mathematical explanations. The
goal here is to reinforce our claim that mathematical entities can play a genuine explanatory role
beyond simply presenting examples and criteria that speak to our intuitions. If the existence of
mathematical explanations of physical facts can be corroborated independently by theories of
scientific explanation, then this will be a large step forward in accepting the explanatory power
of mathematics.
The final task will be to consider if IBE can be employed to infer mathematical realism. This
discussion will be performed with the knowledge that mathematical explanations of physical
facts do exist as argued for in chapters 3 and 4. Yet if this is the case, how can such an analysis
remain independent? Anti-realists would certainly need to deny that IBE is a legitimate tool for
inferring the existence of abstract mathematical entities, whereas realists would argue that there
is no problem with using IBE. My position in chapter 5 will be that the problem with these
attacks and defenses of IBE is that they are ad hoc or question begging in nature. What needs to
be analyzed and understood is how IBE actually works, and such an analysis should be as
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independent as possible from potential ontological outcomes. Ultimately, while I do support the
existence of mathematical explanations of physical facts, I will argue that an independent
analysis does not point to IBE leading to mathematical realism.
My ultimate position will be that the EIA does not represent a significant improvement over
previous indispensability arguments as IBE cannot deliver mathematical realism. However, this
is not to say that studying the EIA is not worthwhile. The EIA has thrust the topic of
mathematical explanation to the forefront. Attempting to understand how mathematics can or
cannot explain physical facts adds critically to our understanding of scientific explanation in
general, and also to our picture of the precise roles that mathematics plays in our best scientific
theories. Similarly, examining whether or not the use of IBE can be extended over abstract
mathematical entities clarifies the boundaries and the spirit of arguably the most important
inference for scientific realists.
1 Mathematical Realism
The goal of indispensability arguments is to convince us to be mathematical realists, but what
this means exactly is a bit unclear. Mathematical realism is often described in a variety of
different and conflicting ways. This is because realism is habitually conflated with 'ontological'
platonism and 'epistemological' platonism. This conflation has as much to do with the word
‘Platonism’ itself indicating its historical roots in the works of Plato, as it does with the actual
differences in beliefs. At the bare minimum, realism is the commitment to at least one of the
following three theses:
(1) Existence: Mathematical objects exist,
(2) Abstractness: Mathematical objects are abstract. They are non-
spatio-temporal,
and,
(3) Independence: Mathematical objects and their properties are
mind and language independent.
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It may be difficult to consistently believe a particular combination of the above three, such as
accepting existence and abstractness but denying independence. Regardless, belief in at least one
is sufficient for being a mathematical realist.
Ontological platonism is the belief in all three of the above theses. This is clearly a strong brand
of realism, and it is often the type of realism that people refer to today. For example, Michael
Dummet says:
Platonism is the doctrine that mathematical theories relate to systems of
abstract objects, existing independently of us, and that the statements of
those theories are determinately true or false independently of our
knowledge. (Dummett, 1991, p. 301)
Hartry Field similarly claims that:
A mathematical realist, or platonist, (as I will use these terms) is a person
who (a) believes in the existence of mathematical entities (numbers,
functions, sets and so forth), and (b) believes them to be mind-independent
and language-independent. (Field, 1989, p. 1)
Notice that Field does not insist that mathematical entities are abstract. Most often this is an
implicit assumption based on the affirmation of the existence and independence theses. However,
to be clear, it could be that someone may still deny the abstractness thesis in which case, by my
classification, they would not be an ontological platonist, but simply a mathematical realist of
some other form.
Epistemological platonism maintains the truth of the above three theses, but at the same time
adds to it some sort of epistemological thesis. This thesis states that we somehow come to know
mathematical truths and properties of objects that are abstract and mind-independent. Exactly
how we do this tends to vary, but generally speaking an appeal is made to some sort of intuition
or mental perception that is unlike our regular senses. Kurt Gödel, perhaps the most famous of
modern epistemological platonists, says that:
[Platonism is] the view that mathematics describes a non-sensual reality,
which exists independently both of the acts and [of] the dispositions of the
human mind and is only perceived, and probably perceived very
incompletely, by the human mind. (Gödel, 1995, p. 323)
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Similarly, James R. Brown explicitly admits an epistemological thesis in his description of
platonism above and beyond those required for ontological platonism.
Mathematical entities can be ‘seen’ or ‘grasped’ with ‘the mind’s eye’.
These terms are, of course, metaphors, but I’m not sure we can do better.
The main idea is that we have a kind of access to the mathematical realm
that is something like our perceptual access to the physical realm. (Brown,
1999, p. 13)
If this talk seems vague, that is because, as Brown admits, it is. There simply is nothing better
available to clarify the matter. This sort of talk is what is often pointed to in a condescending
way for people to argue against mathematical realism; however, for our purposes it is enough to
point out that the weakness of the epistemological thesis does not actually hurt mathematical
realism or ontological platonism as neither position subscribes to this extra-sensory connection.
For the remainder of this dissertation whenever I speak of platonism I am referring to ontological
platonism unless otherwise stated. I aim to distinguish between the use of platonism and
mathematical realism in the same way as laid out above. Of course there are other forms of
realism that I have not discussed here, such as plenitudious platonism or structuralism, but as we
shall see, indispensability arguments do not say anything at all about these respective and distinct
brands of realism.
One last thing remains to be clearly defined. As I have laid out the differences between
mathematical realism and platonism, what then is an anti-realist towards mathematics? The
answer to this is somewhat trivial. An anti-platonist is simply someone who denies at least one of
the three realist theses, whereas an anti-realist is someone who denies them all. However, as was
the case for the realist positions, the ‘anti’ positions are often similarly conflated. For our
purposes we will be mainly interested in two anti-platonist positions: nominalism and
fictionalism.
Nominalism and fictionalism are not actually two distinct versions of anti-platonism. Rather,
fictionalism is just a particular brand of nominalism. Nominalism in its traditional philosophical
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usage refers to the view that universals do not exist.1 In the context of mathematics, nominalism
denies the existence and abstractness theses, but remains silent regarding independence.
Certainly once the nominalist position is adopted it would seem natural to also deny the
independence thesis; however, one could still remain a mathematical realist by not doing so,
although the type of realism remaining may be nothing more than lip-service.
Fictionalism is a nominalist position but it adds to it an account of how we can understand the
practice and application of mathematics given that mathematical entities do not exist and are not
abstract in nature. Mathematical entities are merely fictions in the story of mathematics, just as
Bilbo Baggins is a fiction in the story of the Hobbit. Statements such as ‘the number 3 is less
than 5’, are strictly speaking false, just as saying that ‘Bilbo Baggins has hairy feet’ is also false
as 3, 5, and Bilbo do not really exist. We can admit that statements about 3, 5, and Bilbo are true
in their respective stories, but they will never be true in reality. Mathematical fictions are
certainly more useful than other fictions in that they have proven to be incredibly useful in our
best scientific theories, but that is not to say that they are special in any way that necessitates
existence or abstractness. Given the fictionalist account, it is easy to see how it can be extended
to deny the independence thesis and thus turn into a full anti-realist position. These extensions
will vary depending on whose variety of fictionalism one subscribes to.
2 W. V. O. Quine
In the latter half of the 20th century, Quine put forward what is often regarded as the strongest
argument for mathematical realism.2 The Quinean indispensability argument (QIA) is built upon
Quine’s naturalism and his theory of confirmational holism.3 Essentially, Quine believes in
rejecting any attempt to practice ‘first philosophy’. Science, which is nothing more than an
1 See Burgess and Rosen (1997) for an excellent discussion on nominalism.
2 Quine was not the first to make use of some sort of indispensability argument. Gottlob Frege, for example, said
that “it is applicability alone which elevates arithmetic from a game to the rank of science.” (Frege, 1970, p. 187)
3 Quine is also famous for many other theories, such as his views on translation, underdetermination, and semantic
holism. The QIA need not depend on these other positions.
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extension of our natural common sense, is the sole arbiter for our beliefs. More importantly,
science itself requires no external justification. Naturalism is “the recognition that it is within
science itself, and not in some prior philosophy, that reality is to be identified and described,”
(Quine, 1981c, p. 21) and that science is “not answerable to any supra-scientific tribunal, and not
in need of any justification beyond observation and the hypotheticodeductive method.” (Quine,
1981a, p. 72) Any attempt to impose conditions on the practice of science from outside of
science is unnaturalistic.
This does not mean that science is infallible or beyond critique. Our scientific methods and
beliefs are constantly being revised and improved. The salient point is that criticism and
improvement comes from within the practice of science.
As scientists we accept provisionally our heritage from the dim past, with
intermediate revisions by our more recent forebears; and then we continue
to warp and revise. As Neurath has said, we are in the position of a mariner
who must rebuild his ship plank by plank while continuing to stay afloat
on the open sea. (Quine, 1966, p. 210)
One may ask what role the philosopher has to play given the prized position of science in
Quine’s naturalism. Philosophy is still important, but the practice of philosophy is in line with
that of science; “[t]he philosopher and the scientist are in the same boat.” (Quine, 1960, p. 3)
Quine’s naturalism stems from his attempt to rescue empiricism from two large errors
perpetuated by the logical positivists in the early 20th century. In his famous paper Two Dogmas
of Empiricism, Quine rejects the classic distinction between analytic and synthetic statements. He
argues that there is no such thing as analytic statements which are “true by virtue of meanings
and independently of fact.” (Quine, 1961b, p. 21) Quine bases his rejection of analytic statements
on the claim that any attempt to clearly define analytic relies on other ill-defined notions that
leads to circular reasoning. Quine considers the seemingly analytic statement ‘no unmarried man
is married’. A similar statement, ‘no bachelor is married’, is also seemingly analytic as the word
‘bachelor’ can be replaced with ‘unmarried man’ because they are synonymous. In this sense,
analytic statements depend on the idea of synonymous replacement. Although such a move
appears innocuous, Quine claims that the challenge is to clearly explicate the concept of
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synonymy. Appealing to definitions will not work as definitions are dependent on synonymy in
the first place. Instead, Quine looks towards the idea of interchangeability. This too runs into
problems as understanding interchangeability depends on some concept of necessity or
analyticity – the very concept that we were originally hoping to comprehend in the first place.
Thus the notion of analytic statements is circular and ought to be rejected. Without any analytic
truths, Quine is a true empiricist at heart. All of our knowledge is learned through our
experiences. Moreover, the best way to organize and make sense of our knowledge is through
natural science. The traditional concept that philosophy, independent of any experience, can
deliver us truths about the world is simply incorrect. This leads to Quine’s abandonment of ‘first
philosophy’ and his prizing of science as the main arbiter for our beliefs about the world.
The second dogma that Quine repudiates is the dogma of reductionism which states that “each
statement, taken in isolation from its fellows, can admit of confirmation or infirmation at all.”
(Quine, 1961b, p. 41) Quine believes that no hypothesis can ever be tested as a single unit.
Rather, it is our entire body of knowledge and beliefs that are put to the test every time. “[O]ur
statements about the external world face the tribunal of sense experience not individually but
only as a corporate body.” (Quine, 1961b, p. 41) Our knowledge, which is all acquired through
experience, forms a complex network which Quine calls our ‘web of belief’. Science helps us
organize this web such that at the center are beliefs which are central to our understanding of the
world, such as laws, theories, or mathematics and logic. At the edges of our web lie beliefs that
we may be less certain of, or of which less of our other knowledge depends upon. Everything in
the web is interconnected.
One implication of Quine’s ‘web of belief’ is that it is impossible to test or confirm specific
statements in isolation. Instead, Quine argues that experience confirms or denies in a holistic
sense. Confirmational holism asserts that if a prediction from a theory is experimentally verified
then not only has that prediction and theory received confirmation, but all of the auxiliary
scientific and mathematical beliefs that the theory depends on have received it as well. Consider
a hypothesis X that is predicted by theory T. An experiment is run that is meant to test for X and
the result turns out positive. Such an experiment depends on many other scientific factors other
than T, such as the laboratory equipment being used, the methods of analysis, etc. Each factor is
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supported by other theories, T1, ... , Tn, such as theories of optics or electromagnetism, etc. In
addition, there are many auxiliary assumptions maintained and utilized by the scientists, A1, ... ,
An, such as the constant nature of laws, and of course mathematics. The positive test result for X
is actually a product of not just the theory T that predicted X in the first place, but actually of the
collection of theories and auxiliary assumption {T, T1, ... , Tn, A1, ... , An}. What Quine points out
is that the positive result does not single out and confirm T individually as there is no way to
isolate T from the other theories and auxiliary assumptions. At best, all we know is that at least
one of {T, T1, ... , Tn, A1, ... , An} is supported by our positive experimental result. The only
logical conclusion is that such a positive result confirms everything that was required to generate
it; {T, T1, ... , Tn, A1, ... , An} is confirmed holistically. This directly implies that positive results
empirically confirm the mathematics that are employed in our scientific theories as it is one of
the auxiliary assumptions utilized in almost all scientific experiments.
In the case of a theory facing a recalcitrant experience, the theory and all the other theories and
auxiliary assumption may be suspect. If X was not discovered then at best all we know is that at
least one of {T, T1, ... , Tn, A1, ... , An} is false. It may not be the case that T, the theory we were
hoping to test, is actually the culprit. Logically speaking, it could be any combination of our
other theories or auxiliary assumption that were part of our experimental setup. It could be that
some other theory, such as the theory of optics, or some auxiliary assumption, such as
mathematics, is mistaken and is causing our false prediction of X. We may bristle at this notion
by pointing to the historical success of the theory of optics, or even more so of mathematics and
claim that such assumptions surely cannot be false. But Quine famously maintains that nothing is
impervious to revision: “Any statement can be held true come what may, if we make drastic
enough adjustments elsewhere in the system... Conversely, by the same token, no statement is
immune to revision.” (Quine, 1961b, p. 43) Ultimately what guides our decision as to which
statements we elect to revise is purely practical. Quine cites several factors in his writings which
all boils down to “considerations of simplicity plus a pragmatic guess as to how the overall
system will continue to work in connection with experience.” (Quine, 1966, p. 210) Although
there is nothing stopping us from rejecting and revising something central in our ‘web of belief’
like mathematics, such a change comes at a great cost. Mathematics is central in our web as so
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much of our other knowledge depends on it. If we were to alter our mathematics in the face of a
recalcitrant experience, this would then require a massive reorganization of our web in order for
our knowledge to remain consistent, assuming that this is even possible. Conversely, if we were
to reject a belief on the periphery of the web, such as our theory T, this would be much simpler
as far less knowledge depends on it. Such a move is clearly the more pragmatic choice. If,
however, we were continually getting recalcitrant experiences, there is nothing that is stopping
us from looking to alter more central theories and assumptions.
Quine’s naturalism and confirmational holism have important consequences for the objects and
entities that saturate our natural sciences. It seems obvious that our knowledge of medium-sized
physical objects, such as tables or chairs, is different than our knowledge of entities that we
cannot see, such as electrons or protons. On a trivial level they are different in that our
knowledge of things like a chair seems more certain than that of things like a proton. From this
distinction there is a temptation to conclude that this difference is actually a difference in kind.
Knowledge of medium-sized physical objects is a different sort of knowledge than that of
unobservable entities on the basis that the former are accessible by our senses, whereas the latter
are not. Quine argues that this division in kind is a mistake. The physical objects that are utilized
in science, from macroscopic to microscopic, are all, epistemologically speaking, on par. There
is no actual difference in this sense between a chair, a proton, and even the gods of Homer. What
we perceive to be a difference in kind is merely a difference in degree:
[I]n point of epistemological footing the physical objects and the gods
differ only in degree and not in kind. Both sorts of entities enter our
conception only as cultural posits. The myth of physical objects is
epistemologically superior to most in that it has proved more efficacious
than other myths as a device for working a manageable structure into the
flux of experience. (Quine, 1961b, p. 44)
What we see as a difference between our knowledge of medium-sized physical objects and
unobservable entities is solely based on the fact that macroscopic physical objects are more
central and critical in our ‘web of belief’ to understanding the world as we perceive it than
microscopic objects. But what of mathematics, and things like numbers and sets whose use is
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ubiquitous in the sciences? Quine maintains the same position: “[e]pistemoligcally [abstract
entities] are myths on the same footing with physical objects and gods, neither better nor worse
except for differences in the degree to which they expedite our dealings with sense experiences.”
(Quine, 1961b, p. 45)
Quine’s position that all the entities posited in our scientific theories are epistemologically equal
spills over into his views on ontology with some important clarifications. It is easy to admit into
our ontology objects that we have direct sensory experience of, like a table or chair. The
evidence that unobservable physical objects exists is different in that it is indirect. Again, it
seems natural to point to this difference in evidence as justification for inferring that medium-
sized objects and unobservable objects are different in kind. However, the difference here, just as
above, is solely “a matter of degree.” (Quine, 1969, p. 97) Quine extends this perspective over
abstract objects such as numbers and sets as well. What matters for Quine is what role these
objects play in our scientific theories, be them medium-sized, unobservable, or abstract in nature.
All these entities play the same role in our scientific theories; they make sense of and organize
our experiences and knowledge, and this is why they are all on par with one another. How do we
know that they all play the same role? Quine puts forward the criterion of being a bound variable
under an existential quantifier, and associates that with being indispensable for the truth of the
theory. “[A] theory is committed to those and only those entities to which the bound variables of
the theory must be capable of referring in order that the affirmations made in the theory be true.”
(Quine, 1961a, pp. 13–14) Quantification and indispensability leads to accepting an entity into
our ontology, no matter the type of entity in question, and this is “the only way we can involve
ourselves in ontological commitments.” (Quine, 1961a, p. 12)
Quinean naturalism tells us that science is the sole arbiter for making sense of our experiences
and beliefs. The two key principles of his view is his commitment to empiricism – no knowledge
can be gained independently or prior to science – and to confirmational holism – all our
knowledge is the same in kind and faces confirmation and revision as a whole. Everything is a
part of our ‘web of belief’, and all the objects that our scientific theories depend on are
epistemologically and ontologically equal. With this framework in place, Quine naturally
extends his ontology to include the mathematical objects that are utilized in science. He simply
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has to show that mathematics is indispensable to the truth of our scientific theories, and that
entities such as numbers and sets are existentially quantified over, just like tables, chairs, and
electrons are. Quine mostly assumes this to be the case without much justification. In terms of
the indispensable nature of abstract mathematical objects, Quine is “persuaded that one cannot
thus make a clean sweep of all abstract objects without sacrificing much of science.” (Quine,
1981a, p. 69). Quine also states that mathematical entities, such as numbers, functions and sets,
“figure as values of the [quantified] variables in our overall system of the world. The numbers
and functions contribute just as genuinely to physical theory as do hypothetical particles.”
(Quine, 1981b, p. 150) Given that mathematical entities are indispensable to our best scientific
theories, and that they are existentially quantified, then Quine’s conclusion is that mathematical
objects must exist.
3 The Quinean Indispensability Argument
The QIA has proven to be incredibly influential. One of its most famous supporters was Hilary
Putnam who summarized Quine’s argument as follows:
[Q]uantification over mathematical entities is indispensable for science, both
formal and physical; therefore we should accept such quantification; but this
commits us to accepting the existence of the mathematical entities in question.
This type of argument stems, of course, from Quine, who has stressed both the
indispensability of quantification over mathematical entities and the intellectual
dishonesty of denying the existence of what one daily presupposes. (Putnam,
1979a, p. 338)
In perhaps the strongest defense of the QIA, Mark Colyvan (2001a) presents the argument in a
more formal manner which I have altered slightly here.4
(P1) We ought to have ontological commitment to only those entities
that are indispensable to our best scientific theories.
(P2) We ought to have ontological commitment to all those entities
that are indispensable to our best scientific theories.
4 Colyvan treats the first two premises as one, but the content of the argument is identical to his presentation.
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(P3) Mathematical entities are indispensable to our best scientific
theories.
Therefore:
(C) We ought to have ontological commitment to mathematical
entities.
(P1) is Quine’s commitment to science as the sole arbiter of our beliefs, (P2) reflects
confirmational holism, and the conclusion of mathematical realism follows from (P3). Laid out
in this way the QIA seems to be a valid argument for realism. One difference between Colyvan’s
formulation and Quine’s original version is the absence of requiring quantification over entities
for ontological commitment. Colyvan believes that (P1) and (P2) together capture the
quantification requirement. Implicit in (P2) is that the only entities that are indispensable to
science are those that are quantified over. (P1) restricts ontological commitment to things that are
indispensable to science. Thus, only entities that are quantified in our best scientific theories are
under consideration.
The QIA is meant to target scientific realists who at the same time maintain a nominalist position
towards mathematical entities. If you are not a scientific realist to begin with, the QIA, and any
other indispensability argument for that matter, possesses no force as you would certainly reject
both (P1) and (P2), and for that matter any other realist premise that need not depend on Quine’s
views. Given this, for the remainder of this dissertation we are only interested in the
aforementioned nominalist who does count physical unobservable objects posited by our best
scientific theories amongst their ontology. The QIA means to show that the very same reasoning
that the nominalists’ scientific realism depends on inexorably leads to the conclusion that they
should also be mathematical realists. The power of the QIA is that nominalists of this sort are
trapped. If they wish to withhold commitment to the existence of mathematical entities, then they
are guilty of being ‘intellectually dishonest’ as they accept the argument for realism in one case,
but reject it in another.
One way around the QIA is to say that existence means something different when we speak of
mathematical objects. These objects exist, but not in the same sorts of ways that physical objects
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exist. This would allow us to maintain that mathematical objects exist in some trivial way, such
as mental constructions or simply marks on a page. Quine says that such a move is to employ
“double-talk,” (Quine, 1969, p. 99) Existence should mean the same thing no matter what type of
object we are referring to. To have two different meanings would be to maintain a “double-
standard” (Quine, 1961b, p. 45) that is not supported by any of our best scientific theories. There
is no principle or law in science that tells us to treat observable objects differently than
unobservable objects. To maintain that we should have different standards comes from concerns
stemming from a philosophical nature. Nominalists may have reasons to abhor unobservable or
abstract objects and to prefer their ontology to resemble a ‘desert landscape’, but any such reason
is motivated by supra-scientific concerns which runs directly against Quinean naturalism. Issues
of existence should be governed solely by science, and in science there is no explicit cleft
between observable and unobservable objects and their type of existence.
Fictionalism fares no better against the QIA. Putnam states that to believe that mathematical
entities are indispensable but to maintain that they are merely useful fictions is to be rejected as,
it is silly to agree that a reason for believing that p warrants accepting p in
all scientific circumstances, and then to add ‘but even so it is not good
enough’ [for mathematics]. Such a judgment could only be made if one
accepted a trans-scientific method as superior to the scientific method; but
this philosopher, at least, has no interest in doing that.” (Putnam, 1979a, p.
356)
Like Quine, Putnam wants to uphold a naturalist position, and treating mathematical entities as
fictions is unnaturalistic. Putnam does not stop there. He belittles fictionalism for simply being
an untenable position in the first place, regardless of it being unnaturalistic.
It is like trying to maintain that God does not exist and angels do not exist
while maintaining at the very same time that it is an objective fact that God
has put an angel in charge of each star and the angels in charge of each of
a pair of binary stars were always created at the same time! (Putnam,
1979b, p. 74)
It would seem that the QIA is successful in showing that nominalism and fictionalism are not
tenable alternatives to mathematical realism.
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4 Three Strengths
Beyond simply being a powerful tool against the nominalist, the QIA has three important
strengths that have made it one of the strongest arguments for mathematical realism. The first
strength is that the QIA does not presuppose any truth claims of mathematics or any sort of
abstract objects. Traditional arguments for platonism and mathematical realism often make these
very assumptions. From this starting point an argument is made to add mathematical objects into
our ontology as well. Gödel, for example, famously asserted that the axioms of set theory “force
themselves upon us as being true.” (Gödel, 1983b, p. 270) Once we accept the truth of the basic
axioms we can then argue that there exist some mathematical objects.
It seems to me that the assumption of such objects is quite as legitimate as
the assumption of physical bodies and there is quite as much reason to
believe in their existence. They are in the same sense necessary to obtain
a satisfactory system of mathematics as physical bodies are necessary for
a satisfactory theory of our sense perceptions and in both cases it is
impossible to interpret the propositions one wants to assert about these
entities as propositions about the “data,” i.e., in the latter case the actually
occurring sense perceptions. (Gödel, 1983a, p. 220)
Gödel’s argument only makes sense given that he already takes for granted that the axioms of set
theory are true. The issue here is that for a nominalist who rejects the truth of mathematics, these
arguments for mathematical realism never get off the ground. For the nominalist, any argument
that presupposes the truth of mathematics or the existence of some abstract object to argue for
the existence of abstract mathematical objects is simply begging the question.
In stark contrast are indispensability arguments. No assumptions about the truth of mathematical
entities needs to be made at all in order to deliver the realist conclusion. All that is required is an
analysis of science and scientific practice. The QIA is an argument that can engage nominalists
on their own grounds. Its premises are prima facie acceptable by a nominalist who is also a
scientific realist. Field, an ardent nominalist, echoes this view by stating that “[t]he only non-
question-begging arguments I have ever heard for the view that mathematics is a body of truths
all rest ultimately on the applicability of mathematics to the physical world.” (Field, 1980, p.
viii)
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The second and third strengths of the QIA are related to the works of Paul Benacerraf. In two
famous papers, Benacerraf (1965, 1973) raises significant challenges for the platonist position.
First, if mathematical objects such as numbers do actually exist, a basic question is: what are the
properties of these numbers? Assuming that numbers are sets, the difficulty lies in the fact that
numbers can be represented by sets in an infinite number of unique ways. One way, for example,
is with the progression {Ø}, { Ø,{ Ø}}, { Ø,{ Ø},{ Ø,{ Ø}}}, ... , and another is to use {Ø}, {{
Ø}}, {{{ Ø}}}, ..., to represent the natural numbers. A potential question about the properties of
numbers is: “is the number 3 a member of the number 5?” In the first progression the answer is
yes, but according to the second progression the answer is no. Many more questions about the
properties of the natural numbers generate this same contrary scenario. What this all boils down
to is whether, say, the number 3 = Ø,{ Ø},{ Ø,{ Ø}}}, or 3 = {{{ Ø}}}, or some other
representation. One option is to bite the bullet and say that 3 actually equals all of these
representation, but Benacerraf says that such a conclusion is “absurd.” (Benacerraf, 1965, p. 56)
Benacerraf’s belief is that it must be the case that one of these representations is the right one –
only one representation correctly picks out the natural numbers. However, the problem is that
there is no way to tell which unique representation of numbers is the correct one as they are all
functionally equal. The platonist is left in the embarrassing position that either they do not know
which representation is the true one, which means that they do not know the true nature of
numbers, or they pick one representation to be the one but they cannot justify their choice as all
representations produce the same mathematical consequences.5 Neither horn of the dilemma is
attractive, thus the conclusion is that platonism cannot be true.
While this first challenge seems to pose a problem for platonism, it is neatly sidestepped by the
QIA. The QIA is often confused as an argument for platonism, but it is in fact not that strong.
The QIA does not conclude with the acceptance of all three theses that comprise platonism. In
fact, the conclusion is somewhat vague. Surely there is some sort of mathematical realist
conclusion, but it does not pick out a specific realist position. The QIA could, for example, be
5 In this latter case, one potential response that Benacerraf does not consider is that this is simply a case of
underdetermination. The platonist need not be bothered by the fact that our body of mathematics cannot pick out the
true nature of numbers. Thanks to James R. Brown for pointing this out.
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used as a foundation to justify ones belief in mathematical structuralism advanced by Stewart
Shapiro (1997) and Michael Resnik (1997). One feature of structuralism is that it says that our
mathematical theories are not about mathematical objects at all. Rather, mathematics is about
structures and relations, and not the objects that might instantiate the structure. The structuralist
can then answer Benacerraf’s challenge by stating that the number 3 does not actually equal any
of the set theoretic representations at all. 3 is just a place in the structure of the natural numbers
and has no inherent properties other than its relation to other places. Or, the QIA could be used to
conclude a form of semantic realism wherein we need not commit to the existence of actual
mathematical objects. Rather, we only commit to the independence thesis and withhold talk of
existence altogether. I will have more to say about the multiple ways that we can interpret the
conclusion of the QIA below, but for now it is enough to observe that the QIA is not married to
platonism. Hence, even if we grant that Benacerraf’s first challenge is embarrassing for the
platonist, this does little to harm the effectiveness of the QIA.
Benacerraf’s second challenge is the epistemological problem of access. Again, if platonism is
true, mathematical objects exist and have properties. But how can we know these properties, or
the objects at all for that matter, given that mathematical objects are by their very nature abstract,
non-spatio-temporal objects? Benacerraf subscribes to a causal theory of knowledge which has
since fallen on hard times, but the force of his second challenge remains even without the causal
theory. In the absence of direct contact with the abstract realm, the platonist cannot justify their
knowledge without appealing to some sort of extra-sensory or intuitive connection. As we saw,
some realists, such as Gödel and Brown, embrace this move with open arms. This leads to their
brands of epistemological platonism. Yet Benacerraf and many others find such an option
anathema. The appeal to some extra-sensory connection to the abstract realm operates as a
reductio ad absurdum, and Benacerraf argues that this challenge points to the conclusion that
mathematical objects do not exist and that platonism is fundamentally flawed.
The QIA has an elegant response to Benacerraf’s problem of access. Advocates of the QIA also
shun the move by epistemological platonists in accepting an epistemological thesis. They accept
that mathematical realists who believe in abstract mathematical objects need to be able to say
something about how we learn about these objects. The QIA offers an alternative response that
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Benacerraf did not address. We learn about mathematical objects through their applications in
our best scientific theories. Our knowledge is empirical. This is a natural repercussion from
Quine’s critique of analytic statements and his naturalistic attitude. A committed anti-realist
could press the question further and demand an account of how exactly we come to this
empirical knowledge of things such as numbers. There is an easy response for the Quinean. We
acquire such knowledge in the exact same way that we acquire knowledge of unobservable
physical objects as well. The anti-realist is making a mistake in assuming that knowledge of
abstract objects is different in kind than knowledge of physical objects. They are mistaken as
their difference is a matter of degree only. Others, such as Penelope Maddy, have attempted to
answer this further question more directly. Maddy (1990) argues that we can actually see sets in
the physical world, and that our knowledge of sets, which she considers to be the foundations of
mathematics, is empirical.6 Regardless, the QIA is committed to the belief that knowledge of
mathematical objects does not come through some extra-sensory connection to the abstract
realm, but rather solely through our use of mathematics in our best scientific theories.
Benacerraf’s second challenge is a non-issue.
Benacerraf’s challenges are arguably the most influential and powerful critiques of platonism in
the 20th century. The fact that the QIA easily circumvents them is no trivial matter. Add to that
the fact that the QIA is elegant, straightforward, non-question-begging, and that it aggressively
blocks the nominalist position, then what we have is a potent argument for mathematical realism.
This much has been recognized by mathematical realists and nominalists alike.
6 Maddy has since abandoned this view.
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Chapter 2 Enhancing Quine
The fact that the Quinean indispensability argument (QIA) does not beg the question against the
nominalist and successfully deals with Benacerraf’s two challenges makes it one of the strongest,
some say the strongest argument for mathematical realism. This does not mean that the QIA is
without its flaws. Opponents of the QIA have attacked the argument in a wide variety of ways. In
general, the QIA has three weaknesses where the criticisms have been the fiercest. First, the QIA
depends entirely on Quinean naturalism, and in particular on confirmational holism. Second, the
QIA disrespects mathematical practice. Third, the argument is inherently vague. Once we have
explored these weaknesses we can introduce Baker’s enhanced indispensability argument and
see why it is considered to be an improvement over the QIA.
1 Weakness One: The Quinean Picture
The QIA is intimately connected with Quine’s naturalism. In some ways this is a strength. A
believer of Quinean naturalism is well-equipped to fend off certain attacks against adding
mathematical entities into our ontology. The difficulty for the QIA is that it is not obvious why
we should adhere to Quine’s naturalism in the first place. For our purposes, the primary point of
contention surrounds the thesis of confirmational holism. Maddy (1997) launches several
convincing arguments against confirmational holism that stem from the actual practice of
science. Maddy grants that confirmational holism makes sense on a logical level. It may be
perfectly reasonable that experimental success should be taken to confirm the theory as well as
all auxiliary assumptions such as mathematics, but the sticking point for Maddy is that this is
simply not what happens in practice. Instead of accepting confirmational holism based on some
abstract argument, Maddy urges us to look towards the practice of science directly in order to
draw our conclusions. If we look at the practice of science we see that scientists do test
hypotheses in isolation and that they do not treat confirmation in a holistic manner.
Maddy’s main example to refute confirmational holism is the discovery of the atom and its
adoption into chemical and physical theories around the turn of the 20th century. Even though
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atomic theory was in wide use and its power and usefulness had been well established, scientists
still desired a direct verification, or some crucial experiment that would resolve the issue of
existence. Einstein’s 1905 paper on Brownian motion provided the experimental framework,
and Perrin shortly after confirmed the result.7 Only after this crucial experiment had been
performed was the atom confirmed and the issue of existence finally settled. Maddy’s key point
is that prior to 1905, scientists were withholding confirmation of the atom even though it had
proven to be so useful, perhaps even indispensable to many scientific theories. According to
Quine’s criteria, the atom should have been confirmed and added to our ontology by this point.
Yet confirmation only came when Einstein and Perrin devised a way to test the atomic
hypothesis in isolation. Maddy argues that historical cases such as this shows that the actual
behaviour of scientists does not square away with the Quinean picture of confirmational holism.
Scientists do not confirm in a holistic manner, and indispensability is not a sufficient condition
for ontological status. Instead, scientists often look for crucial experiments in order to test
hypotheses in isolation, and only once these are successful does an entity or theory truly gain
acceptance beyond merely instrumental status.
Maddy further alleges that confirmational holism fails to account for the many idealizations and
strictly false assumptions that scientists use every day. Idealizations such as frictionless planes,
or an infinite number of molecules within a fixed volume are seemingly indispensable to our best
scientific theories, but no one would claim that these mathematical idealizations somehow
receive confirmation when our theories are successful. Without a way to account for the use of
such false idealizations within the holistic framework, there is reason to doubt that scientists
confirm in a holistic manner. If we doubt the veracity of confirmational holism, then we must
reject the premise that confirmational holism is true (P2) from the QIA. Without (P2) there is no
way that we can extend any sort of ontological commitment over mathematical identities even
though their use may be indispensable to our best scientific theories.
7 Maddy utilizes this example in many works, but the most detailed account can be found in Maddy (1997) part II,
chapter 6.
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Maddy’s criticisms are convincing. What is worse for the Quinean is that her attacks stem not
from some ‘supra-scientific’ level, but rather from a look at the actual practice of scientists. This
is the practice that Quine takes so seriously and argues that we should accept at face value. In
essence, Maddy is arguing that the thesis of confirmational holism itself is somehow
unnaturalistic according to Quine’s own standards. Colyvan attempts to defend the QIA from
Maddy’s attacks. With regards to the use of false idealizations, Colyvan argues that consistency
trumps all. Any entity that renders a theory inconsistent, no matter how useful, is “unlikely to be
indispensable to that theory... because there exists a better theory (i.e., a consistent theory) that
does not quantify over the entity in question.” (Colyvan, 2001a, p. 99) Colyvan's argument
depends on the idea that any idealization can be de-idealized into a faithful description of the
physical system. In this way, false idealizations are not indispensable. However, this stance is
controversial, and some argue that certain idealizations are not de-idealizable at all.8 Colyvan
also takes issue with Maddy’s atom example. Maddy argues that prior to Perrin’s experiments in
1905, scientists were withholding ontological commitment to the atom, whereas Quine’s
naturalism says that they should have committed already based on its previous success and
indispensable nature. Maddy’s conclusion is that Quine is wrong. Colyvan agrees with Maddy’s
analysis, but instead concludes the opposite. He believes that the scientists were wrong, and that
the philosopher is in a position to make such judgements. There seems to be something fishy
with this move. It is unclear if such a position runs against Quinean naturalism even though
Colyvan claims it does not. Maddy’s example suggests that the scientists had good reason to
withhold commitment from the atom until 1905. They were waiting for a more direct
confirmation of the atom which is perfectly in line with scientific methodology and practice. If
we say that the scientists were wrong, are we also saying that the methodologies and practice of
the scientific community were also misguided? If so, are we applying an external standard to
judge the methods of science? Arguing that the scientists were mistaken is suspiciously
unnaturalistic, and seems to be a classic example of first philosophy in practice. In fact, Colyvan
himself admits that he “appreciate[s] that many would not share my intuitions here.”(Colyvan,
8 Batterman (2008) argues against de-idealizable idealizations. He claims that certain idealizations lose their
explanatory force when they are de-idealized.
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2001a, p. 100) Colyvan does offer some other possible explanations for this historical example,
such as invoking degrees of belief, but these remain as nothing more than undeveloped
suggestions for possible ways around Maddy’s critique. Ultimately, Colyvan’s defense of the
QIA falls short as it either depends on taking an unnaturalistic stance that is in conflict with the
very position that he wishes to defend, or it rests on other underdeveloped and complicated
concepts such as de-idealizable idealizations and degrees of belief.
2 Weakness Two: Disrespect towards Mathematics
The QIA extends our ontology over those mathematical entities that are utilized in our best
scientific theories. But what of the many mathematical entities that have not yet found a way to
be applied in our sciences? According to Quinean naturalism, science is the sole arbiter of
existence. If certain mathematical entities are not indispensable to science, then we cannot grant
them any ontological rights. Putnam makes this clear in discussing the use of set theory in
physics.
When we come to the higher reaches of set theory, however – sets of sets
of sets of sets – we come to conceptions which are today not needed
outside of pure mathematics itself. The case for ‘realism’ being developed
in the present section is thus a qualified one: at least sets of things, real
numbers, and functions from various kinds of things to real numbers
should be accepted as part of the presently indispensable or nearly
indispensable framework of... physical science... and as part of that
existence we are presently committed to. But sets of very high type or very
high cardinality (higher than the continuum, for example), should today be
investigated in an ‘if-then’ spirit. One day they may be as indispensable to
the very statement of physical laws as, say, rational numbers are today;
then doubt of their ‘existence’ will be as futile as extreme [nominalism]
now is. But for the present we should regard them as what they are –
speculative and daring extensions of the basic mathematical apparatus of
science. (Putnam, 1979a, pp. 346–347)
Quine’s position on unapplied mathematics is similar.
So much of mathematics as is wanted for use in empirical science is for
me on a par with the rest of science. Transfinite ramifications are on the
same footing insofar as they come of a simplificatory rounding out, but
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anything further is on a par rather with uninterpreted systems. (Quine,
1984, p. 788)
This ‘rounding out’ adds a bit more set theory than what is needed for the practice of science, but
it is still clear that the QIA creates a cleft within mathematics between the applied and unapplied.
The difficulty with this division is that it seems absurd to many philosophers of mathematics
who hold strong realist positions, such as platonists, and it also may seem ridiculous to practicing
mathematicians regardless of their philosophical leanings. Higher order set theory is based on the
same axioms and principles as the portions of set theory that have found successful application in
science. To the mathematician, there is no obvious dividing line in the discipline of set theory
like the one that the QIA supporter insists on. Quine eventually softens his position by
considering the higher reaches of set theory to be “meaningful because they are couched in the
same grammar and vocabulary that generate the applied parts of mathematics. We are just
sparing ourselves the unnatural gerrymandering of grammar that would be needed to exclude
them.” (Quine, 1990, p. 94) Although this is generous of Quine, it is a hollow victory at best.
While some portions of mathematics are fortunate enough to luck into having ontological status
simply because it is too inconvenient to explicitly rule them out, thus does not change the fact
that there is still a divide within mathematics.
Quine’s position leads one to wonder what practicing mathematicians are actually doing. The
QIA divides mathematical practice into two basic camps. Mathematicians can be working in
areas and on mathematical entities that are already accepted and exist as they have found
indispensable application to our best scientific theories, or mathematicians are working in areas
and on mathematical entities that do not exist as they have not yet found application. For this
latter camp it is unclear how we can make sense of their pursuit. Is it just some formal game? Is
it some mental exercise in logic? This is even harder to understand when we consider historical
examples where mathematical entities were developed and only later found application in
science.
Consider complex numbers. When Cardano and Bombelli first introduced the concept it was met
by staunch resistance throughout the 18th century by most mathematicians. (Kline, 1972, pp.
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592–596) Understanding of complex numbers continually improved until Gauss provided a
geometric interpretation in 1831. The association of a complex number with a point in a plane
led Gauss to believe that the “intuitive meaning of complex numbers [is] completely established
and more is not needed to admit these quantities into the domain of arithmetic.” (as quoted in
Kline, 1972) The mathematical community was confident that complex numbers should be
admitted to their body of knowledge. Later on, complex numbers were found to have vast
application to physics such as in dynamics, electromagnetism, and quantum mechanics to name a
few. According to the QIA, complex numbers did not exist until this later indispensable
application to science was uncovered. From a mathematician’s perspective this is simply
incorrect. What Cardano and Bombelli were discussing was interesting but ultimately
undeveloped and hence unacceptable in their own time. The important event was when Gauss
provided a clear understanding of complex numbers that satisfied the mathematical community.
The scientific application that came later did nothing to change the minds of mathematicians
regarding the nature of complex numbers as it had already been settled according to
mathematical standards.
This example illustrates two major problems for the Quinean. First, the QIA gives no account as
to how it is that recreational mathematics develops without any notion of application in mind, yet
can one day end up one day being indispensable to scientific practice. Surely we do not want to
admit that complex numbers, which was simply a ‘speculative and daring’ extension or
exploration just coincidentally happened to be useful and true.9 Secondly, on Quine's philosophy
the practice of science imposes certain restrictions on how we can interpret and understand
mathematical practice that conflict with how mathematicians wish to interpret and understand
their own discipline.
9 Of course we could actually admit just this - it is a cosmic coincidence. It is perfectly reasonable that our intuitions
do not find such an admission embarrassing. However, accepting coincidences such as these seem to run against the
fundamental intuitions of the scientific realist, and those are the very group which the QIA is meant to target.
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Maddy highlights this second problem through an analysis of the continuum hypothesis and the
search for a new independent axiom for set theory.10 In order to resolve the status of the
continuum hypothesis, a new axiom must be added to the standard ZFC axioms of set theory.11
The axiom under consideration is the axiom of constructibility, often written as V = L.12 Gödel
proved that V = L implies the truth of the continuum hypothesis, but whether or not we should
adopt V = L as a new axiom is still an open question. Quine utilizes a naturalistic analysis to try
to decide on the new axiom.
Further sentences such as the continuum hypothesis and the axiom of
choice, which are independent of those [set theory] axioms, can still be
submitted to the considerations of simplicity, economy, and naturalness
that contribute to the molding of scientific theories generally. Such
considerations support Gödel’s axiom of constructibility, V = L. It
inactivates the more gratuitous flights of higher set theory, and incidentally
it implies the axiom of choice and the continuum hypothesis. (Quine, 1990,
p. 95)
Quine believes that adopting V = L, which implies the truth of the continuum hypothesis, as an
axiom of set theory would be the most advantageous for the practice of science, thus we should
do exactly that. However, Maddy points out that this is the exact opposite conclusion that the set
theoretic community have proposed. V = L restricts the possibility of many potential sets from
our universe. This ‘minimizing’ property has been met with resistance from set theorists who
prefer to ‘maximize’ the universe of sets. Hence, set theorists wish to reject V = L as doing so
would maximize the potential of set theory as a foundational tool, and thus invalidate the
10 The continuum hypothesis was advanced by Georg Cantor in 1878. It states that there is no infinite set that has
cardinality greater than the set of the natural numbers, and less than the set of the real numbers. The continuum
hypothesis is independent of the ZFC axioms of set theory.
11 There are infinitely many candidates that would resolve the status of the continuum hypothesis. Consider the
axiom which simply states that ‘the continuum hypothesis is true’, for example. However, the only axioms with any
serious potential for being adopted would have to satisfy the mathematical community of having certain properties
that qualify it for being an axiom, such as being ‘self-evident’ or ‘obvious’. Of course the picture is much more
complicated than that in practice. See Maddy (1988) for an excellent discussion on the axioms of set theory.
12 This axiom states that the universe of all sets is exactly equal to the universe of all constructible sets.
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continuum hypothesis. Notice that these arguments have nothing to do with scientific practice or
application, but are solely based within the methodology of mathematical practice.
If a mathematician is asked to defend a mathematical claim, she will most
likely appeal first to a proof, then to intuitions, plausibility arguments, and
intra-mathematical pragmatic considerations in support of the assumptions
that underlie it. From the view of the indispensability theorist, what
actually does the justifying is the role of the claim, or of the assumptions
that underlie its proof, in well-confirmed physical theory. In other words,
the justifications given in mathematical practice differ from those offered
in the course of the indispensability defence of realism. (Maddy, 1997, p.
106)
In the case of the continuum hypothesis, not only are the justifications different, but even worse,
so are the conclusions are drawn by the mathematical and scientific communities. Is it reasonable
for us to reject what actual mathematicians believe to be the case in order to satisfy concerns
regarding indispensability? This would certainly be a troubling conclusion for mathematical
practice, and moreover one that mathematicians would certainly ignore.
Maddy finds this conflict between mathematical and scientific practice introduced by the QIA so
problematic that she rejects the QIA and Quinean naturalism altogether. Instead, she advocates
for her own brand of mathematical naturalism which we will discuss in chapter 6. Although she
is correct in pointing out this inconsistency between indispensability concerns and actual
mathematical practice, I think she has missed the mark in her conclusion. Maddy is too quick to
take Quine’s word as the correct application of indispensability concerns when evaluating the
continuum hypothesis. It could certainly be that Quine was incorrect in his attempt to weigh in
on the practical application of V = L, and that he erred in accepting it rather than denying. Of
course, such a claim appears to have an ad hoc flavour to it meant solely to save Quine from the
embarrassing predicament that Maddy has cornered him in.
Regardless of Quine’s reputation, what is really at stake here is the independence of
mathematical practice from that of science. In the face of a conflict between the interest of
scientific and mathematical concerns regarding mathematics, which discipline should win the
day? Siding with scientific practice seems to be somewhat ludicrous. Mathematicians would pay
no heed, and there is no historical evidence that such an approach has ever been fruitful, while at
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the same time there is ample evidence that mathematicians developing mathematics by their own
standards has been incredibly successful indeed. Siding with the mathematicians thus seems to
be the more reasonable choice, but this is troubling for the Quinean naturalist. If mathematicians
govern mathematics with no concerns about applicability towards science, and subsequently
science finds indispensable application of said mathematics, does this mean that science does
answer to some ‘supra-scientific’ tribunal – that of mathematics? We could conceivably get
around this problem by noting that in general mathematical and scientific practice tend to be in
accordance with regards to mathematics. Quine was simply wrong in the V = L case. But now we
are left with having to explain how scientific and mathematical practice are always in line while
at the same time the QIA tells us that only applied mathematics is meaningful and real.
Explaining this could pose to be even more difficult than the original problem.
My point here is that it is not enough to simply condemn the QIA because Quine believes
something that set theorists do not. The bigger issue is how Quinean naturalism and
mathematical practice relate, if at all. Sadly, this question has been all but ignored. There is a
general disrespect towards mathematical practice within this naturalistic framework even though
it acknowledges the indispensable nature of mathematics in our best scientific theories.
Resolving this should be a primary concern for any naturalist.
3 Weakness Three: Vagueness
At the very core of the QIA lie the notions of indispensability and that of existence. An inherent
problem with the QIA is that these notions are somewhat vague. What do we really mean by
indispensable? Is something indispensable to science if we cannot do science without it? Or does
it require some stronger condition that would narrow the field? What does existence or
ontological commitment really imply? Do abstract objects like numbers and sets exist in the
same way that tables and chairs exist? Or do they exist in some other sense, perhaps as a mental
or non-spatio-temporal object? These terms mean different things for different people and
without a clear and common understanding of their usage in the QIA the argument cannot be
followed.
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Quine talks about existence in a very ambiguous way. Even though they are outside space-time,
abstract objects exist in the same sort of way as unobservable physical objects. This is because
abstract objects are posited and utilized in science in the very same way and for the exact same
reasons that physical objects are posited and utilized. Our theories also commit to them in the
same way via quantification. This may be the case, but surely Quine does not mean that objects
like numbers and sets are just like protons and quarks. By most standard accounts of
mathematics, if mathematical objects are to exist at all they would be non-spatio-temporal in
nature. That is certainly different than the way physical objects exist. Quine’s formulation opens
the door to many different mathematical realist positions depending on how one caches out their
understanding of existence.
Putnam’s version of mathematical realism is not the standard platonistic account. Putnam wants
to be,
realist with respect to mathematical discourse without committing oneself
to the existence of ‘mathematical objects’. The question of realism, as
Kreisel long ago put it, is the question of the objectivity of mathematics
and not the question of the existence of mathematical objects. (Putnam,
1979b, pp. 69–70)
This type of 'semantic' realism is inspired by Michael Dummett (1978). Semantic realism
towards a theory maintains only that the sentences of the theory are true, and that their truth is
independent of us – there is no commitment to existence. Putnam only endorses the
independence thesis of mathematical realism. Such a conclusion is certainly compatible with
Quine’s general assertion that mathematical objects exist just like physical objects. For scientific
realists, a true statement about physical objects is true independent of our minds.
Colyvan sees no inherent problem with Putnam’s semantic realism, but for him (and for me), the
most important and interesting question is: “Do mathematical objects exist?” (Colyvan, 2001a, p.
3) His interest is entirely metaphysical, and his formulation of the QIA reflects that. The
conclusion shifts from simple talk of existence to that of ontological commitment. Furthermore,
Colyvan endorses all three of the realist criteria of existence, abstractness, and independence.
There are other mathematical realists who actually deny abstractness while still maintaining
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existence and independence, but these positions are not the norm. Following Colyvan’s lead, I
will specify existence to mean ontological commitment and take the standard interpretation that
mathematical objects exist, are abstract, and are independent.13
The meaning of indispensability is much more difficult to understand. The first to critically
consider the claim that mathematics is indispensable to scientific practice was Field in his book
Science Without Numbers (1980). Recall that Quine associates existential quantification with
indispensability, and he basically assumes the indispensable nature of mathematics to be true.
The problem with this is that it is always possible to restate a theory so that it does not quantify
over any mathematics. We can simply just take our new theory to be the set of all consequences
that do not contain any reference to mathematical entities, or we can perform a Craigian
reaxiomatizaion to get a recursively axiomatized theory. The problem with this trivial approach
is that these new ‘theories’ are hardly of any significant interest as they are entirely parasitic on
the original mathematized theory in question. It does point out though that existential
quantification is not a suitable criterion for indispensability.
Putnam also believes in the importance of quantification and reference, but he adds to it the idea
of actually being able to ‘do’ science.
Now then, the point of the example is that Newton’s law has a content
which, although in one sense is perfectly clear… quite transcends what can
be expressed in nominalistic language. Even if the world were simpler
than it is, so that gravitation were the only force, and Newton’s law held
exactly, still it would be impossible to ‘do’ physics in nominalistic
language. (Putnam, 1979a, p. 338)
If we cannot ‘do’ Newtonian gravitational theory in this simplified world without mathematics,
then surely we cannot ‘do’ physics without mathematics in the actual world where things are
13 Colyvan also maintains that mathematical knowledge is not a priori as is commonly believed, but is actually a
posteriori. This is because the existence of mathematical objects is only established by empirical means through
their indispensable use in science. This leads to the belief that mathematics is contingent – that is, there exist
mathematical objects, but it isn’t necessarily so. These positions are not always associated with indispensability
arguments, and Colyvan admits that many often point to these conclusions to undermine the QIA. For the most part I
will ignore these issues since indispensability arguments do not explicitly commit themselves to mathematics that is
a posteriori or contingent. See Colyvan (2001a) chapter 6 for details.
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significantly messier. Field takes Putnam’s assertion as a challenge. Field advances his brand of
fictionalism by trying to show that it is indeed possible to do and understand Newtonian
gravitational theory without the use, reference, or quantification over any mathematical entity or
property. His hope is that in showing that this portion of physics can be understood and
performed in a purely nominalistic language, it is then reasonable to believe that all of physics
can be nominalized in a similar way. Thus, mathematics is actually dispensable to scientific
practice. This result is important to Field as he accepts the QIA as a valid argument, but at the
same time he wishes to maintain his nominalist beliefs without maintaining a double-standard.
The only way to do so is to show that the QIA is not sound. If he can demonstrate that
mathematics is not indispensable then there is no need to believe in the existence of
mathematical entities.
Field’s position is that strictly speaking mathematics is false and that mathematical entities are
merely useful fictions. When we say statements like ‘the number 2 is the only even prime
number’, this is like saying ‘unicorns have horns’. Both statements are false as neither the
number 2 nor unicorns actually exist. An important question is how is it that this false language
is so useful and important to the practice of science? This reflects a general problem for any
nominalist account. How can we explain what physicist Eugene Wigner (1960) famously calls
the ‘unreasonable effectiveness of mathematics’? Mathematical realists can utilize the claim that
mathematics is true and that mathematical statements genuinely refer to real objects.
Mathematics is so effective at producing true results in science just because mathematics itself is
true and mathematical objects exist. If Field is to deny the existence of mathematical objects,
then there is no recourse to the notion of truth in explaining why and how mathematics is so
effectively employed in our scientific theories.14
14 Colyvan (2001b) argues that the ‘unreasonable effectiveness of mathematics’ is as much of a problem for the
mathematical realist as it is for the nominalist. What is required by any position is an account of how mathematics is
actually applied, regardless of its ontological status. Mathematical realism merely assumes that it has a superior
solution to the problem of applicability without providing any actual account or explanation of how mathematics is
applied. Colyvan argues that this assumption is false.
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Field’s solution is to argue that mathematics is conservative. In order to have a believable
fictionalist position mathematical theories need not be true; they need only be conservative to be
useful. Mathematics is conservative if when it is added to a nominalist scientific theory then
every nominalist consequence from this expanded theory would have followed from the
nominalist theory alone. Field argues that conservativeness follows from the consistency of
mathematics. A consistent theory only allows us to deduce consequences that are logically true.
Hence if a consequence follows from a mathematized theory, then it should also logically follow
from the original nominalist theory as well. Of course it could turn out that mathematics is not
consistent, but this would be a shocking result and one that we would try to overturn in a variety
of ways. If mathematics is indeed conservative, then Field can explain how it is that mathematics
can be false yet still be extremely useful. The fact that it is false is irrelevant as it is consistency
and consistency alone that ensures we deduce true facts about the world. Furthermore, all these
true facts about the world would have followed from a purely nominalistic theory in the first
place. Mathematics, then, is solely a useful deductive tool that is not, in principle, necessary.
Field does not believe that this gives the nominalist license to utilize and quantify over
mathematical objects freely. The nominalist still needs to show that it is actually possible to have
a nominalist theory in the first place that could do the work on its own. Conservativeness
guarantees that “once such a nominalistic axiom system is available, the nominalist is free to use
any mathematics he likes for deducing consequences.” (Field, 1980, p. 14) This sets the stage for
Field’s next move.
Field’s second task is to show that reference to any existential claims regarding mathematical
objects is eliminable from Newtonian gravitational theory. Field makes use of representation
theorems from measurement theory to give a nominalistic version of space-time. From there he
extends his treatment to laws, differentiation, and Newtonian gravitational theory in general. The
details of this approach are not important to our present discussion. What is worth noting is that
Field believes that this nominalist formulation of Newtonian gravitational theory is ‘attractive’. It
is certainly more attractive than the trivial methods presented above, but more importantly Field
claims that this version is attractive as it does not appeal to any arbitrarily selected objects to
serve as particular units of length or basis for a coordinate system, and that it reveals that all
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physical explanations are ‘intrinsic’ explanations. This demonstrates what Field actually believes
to be at the root of indispensability. Something is dispensable not just because it is eliminable. It
must be eliminable, and the resultant theory must be ‘attractive’.
Many commentators remained unconvinced that Field succeeded in his ambitious project.
Objections stem from the use of second-order logic, the utilization of some notions in set theory,
problems with determinism, projected difficulties in nominalizing general relativity, the
treatment of phase-spaces, and more.15 These technical objections are not important to us here,
but it is safe to say that in the face of all these difficulties the general consensus is that Field’s
project of nominalizing physics did not, and is not likely to succeed. What is important to us is
how Field understands the notion of indispensable. Clearly the notion of ‘attractiveness’ of a
nominalized theory is crucial to this understanding, but Field does little to expand on it. He even
admits that actual usefulness is not important. “I do not of course claim that the nominalistic
concepts are anywhere near as convenient to work in solving problems or performing
computations: for these purpose, the usual numerical apparatus is a practical necessity.” (Field,
1980, p. 91) But how can we possibly claim that a nominalized theory is ‘attractive’ yet at the
same time be practically useless compared to its mathematized counterpart? What we need is to
look at other theoretical virtues, such as simplicity, explanatory power, fruitfulness, etc., in order
to facilitate an informed judgment.
In a critique of Field’s argument, Colyvan takes this very route. His understanding of
indispensability is that an entity is dispensable if there exists an alternate theory with the exact
same observational consequences where all mention of the entity is eliminated, and that this
theory must be preferable to the first. (Colyvan, 2001a, p. 77) Not surprisingly Colyvan asserts
that the regular mathematized theory is preferable to a nominalized counterpart mainly due to the
simplicity and explanatory power that the mathematics brings to the table. Colyvan introduces
the practice of theory choice in order to define what it means to be indispensable. His reasoning
is that mathematics is indispensable to science because any alternate theory that eliminates the
15 For a good survey of many criticisms of Field's project see Burgess and Rosen (1997).
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use of mathematics would always be less preferable. The biggest advantage for mathematics is
that it has the virtue of making a theory significantly simpler.
Joseph Melia also agrees that theoretical virtues are important for theory choice, and that
mathematics makes a theory simple, however, he argues that though “such principles may justify
postulating quarks and space-time, it is a mistake to think that these principles justify our
postulation of mathematical objects.” (Melia, 2000, p. 472) He asserts that it is still possible to
‘weasel’ away from commitment to mathematics even if mathematics is indispensable to science.
Melia looks to the actual practice of scientists for his inspiration. He claims that even though
scientists use and quantify over mathematical entities, they still do not believe that mathematical
objects exist. Melia uses a naturalistic argument and states that we should take the practice of
science at face value and assume that scientists are not unreasonable in their beliefs. His goal is
to account for scientists being realists towards physical objects indispensable to science, while at
the same time maintaining a nominalist attitude towards mathematics. What makes Melia’s
position interesting is that he happily accepts both premises of the QIA; most importantly he
agrees that mathematics is indispensable to our best scientific theories according to Colyvan’s
criteria. The crux of Melia’s argument is that mathematics is not indispensable in the right way
such that it leads to ontological commitment.
An unsatisfactory element to Melia’s position is that he offers no actual evidence for his claim
that most scientists have this split attitude towards physical and mathematical entities employed
in science. Although he cites this as a motivating force, it definitely remains to be seen if the
majority of scientists truly hold such a position. Regardless, Melia could easily construct his
argument as a defense of those that do hold this position independent of whether or not they
comprise the majority. It would lose the naturalistic foundation which adds to the credibility of
his position, but would still be sufficient to undermine the validity of the QIA. Another issue
with his approach is the role of philosophy within the naturalistic framework. As we saw earlier,
the practice of philosophy is supposed to be in line with that of science. However, this does not
mean that it is outside of the philosopher’s role to criticize the beliefs and practices of scientists.
Philosophers are not there simply to provide idle justification for these beliefs and practices
simply because the majority of the scientific community holds them. Rather, philosophers aim to
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make sure that the beliefs and practices that scientists maintain truly hold up to the standards set
out by our scientific methodologies. It is one thing to say that beliefs and practice should be
informed and follow from good and accepted methodologies and principles. It is an entirely
different thing to assert that they actually do follow in the appropriate way. Melia seems to
assume that the goals and the actual practice of scientists are one and the same. Thus, the
philosopher’s role is only to justify, not critique. However, this is not necessarily what
naturalism implies. There is certainly a role for the philosopher to contribute meaningfully to the
practice of science by ensuring that scientists live up to the high standards and methodologies of
science, whatever those may be. So even if we grant Melia’s unjustified claim that the majority
of scientists are realists towards physical objects but nominalist towards mathematical entities,
this does not mean that this is the right attitude to have. We as philosophers should not be in the
business of justifying such a stance if it is somehow unnaturalistic.16 This, of course, is the very
position that Quine maintained when he devised the QIA. Melia needs to go further and argue
that scientific realism paired with mathematical nominalism is the appropriate way to divide our
ontology.
All this talk about the majority of scientists is merely posturing meant to enhance the importance
and justification of Melia’s argument. Even if it is not as well motivated as Melia would have us
believe, this does not affect the importance or legitimacy of his attack against the QIA. Melia’s
strategy is two-fold. First he presents the ‘weaseling argument’ which is a method so that we can
make use of mathematics in the practice of science but at the end of the day still maintain a
16 In the previous discussion regarding Maddy’s atom example I was skeptical of Colyvan’s stance that the naturalist
could claim that scientists were wrong for denying the existence of the atom prior to Perrin’s crucial experiments.
Here I endorse the idea that the philosopher can criticize the practice of scientists within the Quinean naturalistic
framework. These positions are not contradictory. In the atom example, what is fishy about claiming that the
scientists were wrong is that they were discussing a physical entity that was posited by their own physical theories as
a useful tool or explanatory device. It was their very own scientific methodologies and principles which caused them
to withhold ontological rights from the atom until it could be more clearly experimentally verified. All of this seems
perfectly within the desired practice of science. In the present example, scientists who deny the existence of
mathematical entities are doing so not for any reasons dictated by scientific methodology or principles. No further
experiment or criteria can ever be satisfied that would change their minds. The reason being is that this prejudice
against mathematical objects originates from ‘supra-scientific’ beliefs. It is this behaviour that the philosopher of
science should question and be critical of.
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nominalist position towards mathematics. Secondly, Melia argues that there is indeed good
reason to employ the weaseling method.
Melia first points out that Field’s attempt to show that mathematics is dispensable to science
fails. His objection is not about the technical work of nominalizing Newtonian gravitational
theory. Melia (2000) argues that it is Field’s claim that mathematics is conservative that is false.
He considers two theories of mereology, T and T*, where T is a nominalist theory and T* is an
extension of T which utilizes and quantifies over functions and sets. Melia shows that T* does
not entail any new sentences in the nominalist theory, T; however, T* does entail the existence of
a special infinite region that T does not guarantee to exist. Such a region, although possible
according to T, is not directly implied by T alone. In essence, T* is a conservative extension of T,
but at the same time it has implications about the physical world that T does not. By rejecting
conservativeness, Melia is admitting that mathematics can help us express certain physical facts
about the world that we could not do with a nominalistic theory alone – mathematics is
indispensable to scientific practice. So, for any nominalist theory T and mathematical theory T*
as above, we may have to use T* to get the facts about the world correct. Regardless, Melia
insists that scientists and philosophers make use of mathematical theories all the time yet still
maintain a nominalist stance towards mathematics. Surely this is inconsistent. As Melia himself
asks, “[h]ow can anyone coherently assert P, know that P entails Q, yet deny that Q is the case?
How could it ever be rational to assert that P whilst denying a logical consequence of P?” (Melia,
2000, p. 466) The answer is that we can weasel ourselves out of such a seemingly incoherent
predicament.
The crux of the weaseling method is that it is perfectly legitimate to say “T* - but there are no
such things as functions or sets.” By doing this we can ‘subtract’ or ‘prune away’ the abstract
mathematical entities that are implied by T*. The idea here is that simply because we assert T*
does not mean we have to believe that every aspect of T* is actually true. In particular, the
mathematics employed in T* could be false. This may seem strange at first, but Melia contends
that we do this regularly in both everyday and scientific language. Consider the statement:
(1) All F’s are G’s, except for Rosie.
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There are two ways in which we can interpret (1). The first is to claim that (1) is contradictory.
What (1) is really saying is that “all F’s are G’s, and Rosie is an F that is not a G.” This is surely
not what is meant by statement (1), and is an uncharitable interpretation. Instead, we can
understand (1) as saying:
(2) All F’s, except for Rosie, are G’s.
There is no paraconsistency in (2) and it is perfectly clear in meaning. Statements such as (1) are
a manner of speaking wherein we can legitimately ‘take back’ certain claims that we made about
the world in a clear and coherent way. Melia also presents a scientific example where we
introduce a three-dimensional sphere in order to define a two-dimensional world by only taking
the surface of the sphere and pruning away everything else. His claim is that these examples in
common-language and in science make it plausible that we can make statements like
(3) There exists a function from space-time to the real numbers – but there
are no such things as functions or numbers,
without being incoherent. We should understand (3) to be a (2)-like statement. In this way we
can weasel away from the mathematical realist conclusion of the QIA.
A disanology with statements like (3) and statements like (1) is that for (1) there is a clear
reformulation in (2), whereas (3) does not have a clear reformulation. An obvious question for
people who utter statements like (1) is why would we ever do this when we could have just said
(2)? (1) is inherently more misleading and unhelpful. Melia says that this may be the case, but
there is no fault in someone simply being ‘longwinded’ so long as their actual meaning is clear.
In addition, sometimes we simply have to say things such as (1) as the clear reformulation is not
available to us. This is exactly the case for statements like (3). Mathematics allows us to say
things about the physical world that we simply cannot without. “Mathematics is the necessary
scaffolding upon which the bridge must be built. But once the bridge has been built, the
scaffolding can be removed.” (Melia, 2000, p. 469) The fact that a clear reformulation of
statements like (3) is unavailable does not matter. We can still weasel away from commitment to
mathematical objects.
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This last assertion has come under attack from Colyvan (2010). Colyvan argues that (1) is clear
and coherent because we have access to (2). Since there is no reformulation of statements like
(3), which Melia admits, then statements like (3) are neither clear nor coherent. Colyvan is
certainly correct that the availability of (2) is a sufficient condition for us understanding (1), but
is it necessary? Melia alleges that it is not necessary, and Colyvan asserts that in some cases it is
a necessary condition. Neither provides a concrete argument, but instead both use analogies to
make their case. As we saw, Melia says that understanding (3) is just like understanding (1),
irrespective of the presence of a nominalized reformulation. This analogy is weak at best. The
very way that Melia convinces us that (1) is okay is by using (2). He provides no example of
understanding a statement in the absence of a clear reformulation, so how can understanding
statements like (3) be just like understanding (1)? The very route that Melia takes to clarify (1) is
unavailable. That is what makes statements like (3) interesting and challenging to understand in
the first place. Nominalists such as Melia may find this analogy convincing as they are naturally
inclined to a nominalistic attitude, but for someone who has no prejudice towards the existence
of mathematical objects, or even worse for someone who leans towards the realist position, the
analogy that Melia presents is unsatisfactory.
Meanwhile, Colyvan accepts that in certain cases where we only weasel away trivial details the
weaseling method may work, but there also are situations where we clearly try to prune away too
much. In these cases, without a clear reformulation we are unable to weasel in a coherent way.
Colyvan’s analogy, like Melia’s, leaves much to the imagination.
We can change the story we are narrating by adding or subtracting minor
details, but we can hardly be thought to be telling a consistent story (or in
some cases, any story at all!) if we take back too much. In short, there are
limits to how much weaseling can be tolerated. J.R.R. Tolkien could not,
for example, late in The Lord of the Rings trilogy, take back all mention
of hobbits; they are just too central to the story. If Tolkien did retract all
mention of hobbits, we would be right to be puzzled about how much of
the story prior to the retraction remains, and we would also be right to
demand an abridged story – a paraphrase of the hobbitless story thus far.
(Colyvan, 2010, p. 10)
Taking back our commitment to mathematical objects in science is supposed to be another
example of subtracting or pruning away too much.
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Colyvan’s analogy is also problematic. If Tolkien had written near the end of The Lord of the
Rings trilogy that ‘oh, and there are no such things as Hobbits in Middle-earth’17, it is certainly
possible that we could make sense of the story. Middle-earth is full of magic, and Tolkien’s
writing is rich with metaphor. I see no reason why such an ending could not in principle be
believable. Colyvan merely states that it would be right to distrust such a bizarre ending.
However, for people who are predisposed to disbelieving in hobbits even prior to reading The
Lord of the Rings, then they would not only understand and accept this retraction of hobbits, they
would probably feel that it was entirely the right thing to do. This is the case for people who are
predisposed to disbelieving in the existence of abstract objects such as numbers or sets. They
find nothing strange at all about the scientist who weasels away from commitment to the
existence of mathematical objects, and moreover, they feel it is the only sensible thing to do. I
grant that there are some who could not understand such an unpredictable and bizarre ending to
The Lord of the Rings, but it is not clear why these people would have the right to demand, in
their opinion, a better tale.
Another problem with Colyvan’s example is the role that hobbits play in The Lord of the Rings.
Arguably, the trilogy is about hobbits. Hobbits are the central focus of the tale and the entire
trilogy is meant to tell their story. Yet no scientist or mathematician would contend that our best
scientific theories are actually about the mathematical entities that permeate them. Taking back
mathematical entities from our ontology after using them in our theories is not as drastic as
Colyvan’s example suggests, as mathematical objects are not the focus of those theories. Rather,
they are secondary to the true object in question: the physical world. One could easily grant that
Colyvan is correct in pointing out that it would be nonsensical to retract the existence of hobbits
in The Lord of the Rings, yet still not acquiesce to Colvan’s main point that weaseling away from
mathematical objects in science is nonsensical as well. The two are simply disanalogous.
17 Middle-earth is the fictional land in Tolkien’s books. I stipulate that Hobbits don’t exist in Middle-earth to remove
the obvious point that Hobbits don’t exist at all as they are fictional characters in a fictional land. Allowing truths of
fictional entities, then it is the case that Hobbits exist in Middle-earth according to Tolkien.
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As it stands, neither Melia nor Colyvan have done enough to show the legitimacy or illegitimacy
of the weaseling argument. Melia has definitely revealed an interesting avenue for the nominalist
to pursue in avoiding the force of the QIA, but it is unclear what the limits the weaseling method
is. As I have suggested, whether or not you believe that the weaseling method can successfully
apply amounts to how predisposed you are to nominalism or mathematical realism in the first
place. This is a problematic conclusion as then the weaseling argument does little to convince
either side of anything. Regardless, the weaseling argument offers a potential solution for the
nominalist who wishes to make use of mathematized scientific theories without committing to
mathematical objects, and without having to pursue a Field like project of nominalizing all our
scientific theories.
The weaseling argument represents a new tool for the nominalist. What Melia does next is
provide reasons for why we should use it. His claim is that although mathematics is
indispensable to science, it is not indispensable in the sort of way that would normally lead to
ontological commitment. The main attraction of using mathematical objects in our scientific
theories is that it makes our theories simpler, albeit at the cost of a complex ontology. Simplicity
is one of several theoretical virtues that are generally used in assessing the merits of scientific
theories. However, Melia claims that mathematics does not make the world a simpler place, but
rather it only makes our theories simpler. Consider any unobservable entity postulated by our
scientific theories. These unobservables are put forward to account for a wide range of other
different kinds of objects or observable phenomena. They work in simplifying our account of the
world. But mathematics does not behave in this way. The world is not the way it is in virtue of
the fact that mathematics exists. In the mereology example, the infinite region that T* guarantees
to exist does not exist because of the mathematical objects employed in T*. Melia believes that
mathematics only serves as an aid in descriptions and in indexing physical facts in our theories.
Mathematics allows for ways of expressing concrete possibilities that we may not be able to
express without it, yet ultimately it has no real role in simplifying the world as we know it.
Claiming that indispensability leads to ontological commitment is a naïve view of
indispensability. Committing to mathematical objects does not make the world a simpler place,
and without this virtue accepting them only comes at a cost to the complexity of our ontology.
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Melia’s argument is that mathematics only serves to index or represent physical facts, and thus is
not indispensable in the right way to lead to ontological commitment. This indexing argument
provides the justification for the nominalist to make use of weaseling to restrict mathematical
entities from his ontology. We will take a critical look at the indexing argument in chapter 3.
Interestingly, although Melia argues that mathematics does not add to the simplicity of our
scientific theories due to its role of indexing physical facts, he does admit that there could be
other theoretical virtues that mathematics do actually enhance.
Of course, there may be applications of mathematics that do result in a
genuinely more attractive picture of the world – but defenders of this
version of the indispensability argument have yet to show this. And
certainly, the defenders need to do more than point to the fact that adding
mathematics can make a theory more attractive: they have to show that
their theories are more attractive in the right kind of way. That a theory
can recursively generate a wide range of predicates, that a theory has a
particularly elegant proof procedure, that a theory is capable of making a
large number of fine distinctions are all ways in which mathematics can
add to a theory’s attractiveness. But none of these ways results in any kind
of increase in simplicity, elegance or economy to our picture of the world.
Until examples of applied mathematics are found that result in this kind of
an increase in attractiveness, we realists about unobservable physical
objects have been given no reason to believe in the existence of numbers,
sets or functions. (Melia, 2000, pp. 474–475)
Colyvan (2002) and Baker (2005, 2009) were quick to take up the challenge by providing
examples of how mathematics does genuinely enhance our explanatory power. For his part,
Melia accepts that this is the best way to refute his own position.
In my view, Colyvan’s strategy is the best way for those who want to defend the
indispensability argument. Were there clear examples where the postulation of
mathematical objects results in an increase in the same kind of utility as that
provided by the postulation of theoretical entities, then it would seem that the
same kind of considerations that support the existence of atoms, electrons and
space-time equally supports the existence of numbers, functions and sets. (Melia,
2002, pp. 75–76)
This has opened the door for mathematical realists so that if it can be shown that mathematics is
indispensable to our best scientific theories in the right way – namely that mathematics is
explanatorily indispensable – then we should commit to the existence of mathematical objects.
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What indispensability actually means is quite controversial and complex. Quine’s view was that
quantification is the key to indispensability. It turns out to be trivial to remove quantification via
Craigian reaxiomatization, so this is not a sufficient way to characterize indispensability. Field
introduces the notion of conservativeness and a non-trivial reformulation that is ‘attractive’ in
some respects. Colyvan argues that this is not enough as any such nominalized theory needs to
be at least as attractive as the one with mathematics. Any mathematized theory has an
overwhelming advantage in the simplicity that it provides. Finally, Melia points out that
simplicity is not enough. Making a theory simpler is not the same as making our understanding
of the world simpler. Mathematics needs to be indispensable in the right way by impacting our
views and understanding of the world, such as by being indispensable in scientific explanation.
Although there are still many points of contention, mathematical realists and nominalists
presently generally agree that the best way in which we should understand the notion of
indispensability in the context of the QIA is that mathematics is indispensable to our best
scientific theories if it is indispensable to scientific explanations.
Since the exchange between Melia and Colyvan, the discussion has almost exclusively focused
on the explanatory role of mathematics in our best scientific theories. Is mathematics
indispensable in scientific explanation? In a trivial sense the answer is most likely yes. As noted
above, the general consensus today is that Field was unsuccessful in showing that mathematics is
dispensable to Newtonian gravitational theory, and thus also unsuccessful in showing that
mathematics is dispensable in our best scientific theories in general. Seeing as how explanations
are such an integral part of science, it seems perfectly reasonable to assume that mathematics
will be indispensable to scientific explanations. However, as per Melia’s line of reasoning,
mathematics may certainly be indispensable to scientific explanation, but the important question
is it indispensable in the right way? Is it explanatorily indispensable? Unfortunately, the present
discussion by the main actors is unsatisfactory as no one has developed an account of what it
would take for mathematics to be considered explanatorily indispensable.
4 Internal and External Mathematical Explanations
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At this point we need to develop a better understanding of what is meant by mathematical
explanation. There are at least two major types of mathematical explanation. We can have a
mathematical explanation of a mathematical fact, called an internal mathematical explanation, or
a mathematical explanation of a physical fact, called an external mathematical explanation.18
Internal explanations somehow explain particular facts in the realm of mathematics, such as why
a particular theorem is true. A proof of a theorem certainly demonstrates that it holds, but the
belief is that there is an important difference between a proof that merely demonstrates the result,
and a proof that genuinely explains it. Internal explanations need not even be a proof at all. It
could simply be a demonstration of how otherwise seemingly disparate mathematical facts are
actually related. We will return to internal explanations in chapter 4.
External explanations are explanations such that the mathematics explains a physical fact about
the world. Although it seems perfectly plausible that internal explanations exist, it is unclear if
external explanations do at all. No one doubts the immense utility and the ubiquitous use of
mathematics in scientific explanations. What we mean here, then, is to distinguish between
standard mathematical scientific explanations, where the mathematics is utilized for its
simplicity or deductive power, and genuine mathematical scientific explanations. Genuine
mathematical scientific explanations are explanations where the mathematics involved actually
confers an indispensable explanatory role to the overall explanation.
The goal is to differentiate between the use of mathematics as simply a representational tool and
the use of mathematics as an explanatory device. For example, suppose I want to explain the
migratory patterns of birds by modeling certain facts about the Earth. I could construct a
mathematical model that incorporates factors such as temperatures, daylight, food availability,
wind patterns, past migration information, etc. From this mathematical model I could deduce,
and thus explain the migration patterns in question. Mathematics is employed extensively, but at
the end of the day no one would say that the mathematics actually explains the behaviour of the
birds. What is really at the core of the explanation are ecological and biological facts – the very
18 See Hafner and Mancosu (2005) for a nice discussion on internal and external mathematical explanation.
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facts that we represented using mathematics. Mathematics was an important, and perhaps
indispensable19 tool for making sense of all of the information and deducing the relevant
conclusion, but it was not in itself explanatory. The original mathematical explanation is not
genuine, but rather is an example of a standard mathematical explanation in science.
So what is required for mathematics to be explanatory? There are two reasonable ways to
understand how mathematics could be explanatorily indispensable. The first is to claim that if we
have an external mathematical explanation, and at the same time we cannot provide an
alternative explanation in purely physical, nominalistic language, then the mathematics in the
explanation is explanatorily indispensable. This approach is unsatisfactory. It is simply parasitic
on the more general claim that mathematics is indispensable to science at large; i.e. there is no
reformulation of science in purely nominalistic terms. Melia and other nominalists would agree
with this claim, but they would still reject the notion that mathematics is genuinely explanatory.
The second way is to assert that there is something about the mathematics itself that explains the
physical fact in question. If we were no longer privy to the mathematics employed then we
would actually lose the explanation entirely. In essence, the mathematics is contributing to the
explanation in an explanatory way. If such an explanation exists, then all parties would concede
that mathematics genuinely explains.
Consider the bird migration patterns example. If we were to somehow not know or forget our
mathematics, would we no longer be able to explain migration patterns? We certainly would not
be able to explain it in the same way as a sophisticated mathematical model could, but we could
definitely cite the same underlying factors that the mathematics represented in the first place.
Our explanation would appeal to the very same ecological and biological factors as before.
Without a doubt we would lose substantial predictive power and other virtues, but we could still
explain the phenomenon at hand. Contrast that with what would happen if we lost all knowledge
of certain ecological or biological factors, such as temperature or sunlight. In this case, even
19 When I say indispensable here I mean that mathematics may be indispensable in expressing certain physical facts
or as a deductive tool. This notion of indispensable is perfectly acceptable to nominalists such as Melia and need not
confer any ontological commitment.
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though we have our mathematical tools available to us we would fail in explaining the migration
patterns. The intuition here is that without knowledge of temperature or sunlight we would lose
the explanation entirely. Hence it is these factors that are genuinely explanatory. The same
cannot be said about the mathematics.
It may be hard to imagine what sort of explanation would depend on mathematics for its
explanatory force. We will look at many supposed examples in chapter 3 when we analyze
mathematical explanations closely. Hence forth, when I speak of standard mathematical
explanations I mean external mathematical explanations in which the mathematics is not
indispensably explanatory. By genuine mathematical explanation (GME) what I am referring to
are external mathematical explanations where the mathematics is explanatorily indispensable; the
mathematics is not playing a representational role, but actually confers explanatory power to the
overall explanation. Finally, any general reference to mathematical explanation is to the external
sort, and not internal.
5 The Enhanced Indispensability Argument
With the focus placed squarely on mathematical explanation, Baker (2005, 2009) puts forward a
new indispensability argument for mathematical realism. His Enhanced Indispensability
Argument (EIA) is meant to be an improved version of the QIA.
(EP1) We ought to rationally believe in the existence of any entity that
plays an indispensable explanatory role in our best scientific
theories.
(EP2) Mathematical objects play an indispensable explanatory role in
science.
(EC) Hence, we ought rationally to believe in the existence of
mathematical objects.
(Alan Baker, 2009, p. 613)
Recall that (P1) and (P2) of the QIA are the naturalist and the holistic premises respectively.
Here they are replaced by a single premise, (EP1), which is implicitly appealing to inference to
the best explanation (IBE). Like its predecessors, the EIA is meant to target scientific realists
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who have not yet extended their realist sympathies over mathematics. The appeal to IBE in (EP1)
reflects this as IBE is generally considered to be the inferential tool that characterizes scientific
realism, and is often rejected or restricted by those who are anti-realist towards science. We will
perform a critical examination of IBE in chapter 5, but for now we will assume that scientific
realists have no problem with IBE as a tool to infer the existence of objects that genuinely
explain physical facts. Given that IBE is the key inference for the EIA, we can now see why
internal mathematical explanations are not considered in the argument. The standard usage of
IBE requires that we treat the explanandum in question as true, and then we infer that the best
possible explanans is also true. In internal explanations the explanandum would be some
mathematical fact. But if we are to assume this to be true, then this immediately implies that
mathematical objects must exist. Seeing as this is the very conclusion that we are trying to
establish in the first place with the EIA, such an assumption entirely begs the question. Thus,
only external mathematical explanations can be considered as this way we need not assume
anything about the nature of mathematical entities.
(EP2) is the indispensability claim. Unlike its Quinean counterpart, here the emphasis is that
mathematics is indispensable in scientific explanations. This move is clearly motivated by the
exchange between Melia and Colyvan. The claim is that GMEs exist, and hence that
mathematics is indispensable to science in the right way. From the first two premises it follows
that we should believe in the existence of mathematical objects.
In many ways the EIA is just a simple extension of the QIA. It leads to the same realist
conclusion, and it possesses the same three strengths that made the QIA so attractive in the first
place: it can face both of Benacerraf’s challenges, and it does not beg the question against the
nominalist. What makes Baker’s argument enhanced is that while it retains the same strengths as
the QIA, it supposedly does not suffer from the same three weaknesses discussed above.
The first two weaknesses of the QIA are directly related to its reliance on Quinean naturalism.
The first weakness is tied to the use of confirmational holism as the key tool for delivering
mathematical realism. As we saw, there are many good reasons to be suspicious of this holistic
thesis, but perhaps the most troubling is simply the fact that it is perfectly reasonable to be a
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scientific realist without also being a supporter of confirmational holism. If this is the case, then
any argument relying on confirmational holism is critically limited. The EIA is able to avoid this
weakness by making IBE the inferential tool. All the criticisms of confirmational holism are
entirely irrelevant to the enhanced argument. Even better, by tying its success to IBE, the EIA
more closely aligns itself with the standard views of scientific realists. The assumption here is
that any scientific realist necessarily endorses IBE. This assumption is unjustified, but it is a far
more believable than claiming that every scientific realist endorses confirmational holism. By
freeing itself from confirmational holism, the EIA is immune to the first weakness of the QIA.
The second weakness of the QIA is that it disrespects the actual practice of mathematics. The
reason for this is based off of its first premise which states that:
(P1) We ought to have ontological commitment to only those entities
that are indispensable to our best scientific theories.
A core principle of Quine’s naturalism is that science is the sole arbiter of our beliefs. This
allows us to reject epistemological or ontological claims from non-scientific disciplines such as
religion or voodoo. However, this also results in the unfortunate situation where mathematical
practice has no say regarding the epistemological or ontological status of mathematical entities.
In particularly embarrassing cases, Quine’s naturalistic views seem to put the claims and beliefs
of the scientific community directly at odds with those of the mathematical community. The EIA
appears to fare better in this regard. (EP1) is significantly weaker in that while it necessarily
asserts that we must commit to the existence of any entity that is explanatorily indispensable to
our scientific practice, it does not at the same time explicitly state that this is the only way in
which we can add to our ontology. This opens the door for mathematics to contribute to
ontological questions regarding its own subject matter.
Although it does appear that the EIA improves on the second weakness in that it respects
mathematical practice more than its Quinean counterpart, it is actually not clear that this is the
case. While it is certainly true that the EIA does not explicitly state that mathematical practice
has no say with regards to the status of its subject matter, this does not mean that the argument
endorses the input from mathematical practice either. Whether or not this is so depends entirely
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on what naturalistic views the EIA is committed to. Consider the case put forward by Maddy
where Quine believes that we should adopt the mathematical axiom V = L as it would benefit the
scientific community, but the set-theoretic community disagrees. Would Baker side with the
practice of science, or the practice of mathematics? Similarly, was the pivotal moment in the
acceptance of complex numbers Gauss’ geometric interpretation, or was it the indispensable use
of complex numbers in disciplines such as quantum mechanics? As it stands the EIA does not
give us guidance on these questions. The best we can say is that while the EIA may actually not
fare any better than the QIA with regards to the disrespect of mathematical practice, it certainly
can fare no worse.
The final weakness of the QIA is its vagueness with regards to the meanings of existence and
indispensability. The conclusion of the EIA is essentially identical to that of the QIA. For us, the
interesting interpretation of existence is that mathematical objects exist as abstract and
independent entities. It is in how the EIA commits to indispensability meaning explanatorily
indispensable that separates itself from the QIA and eliminates the vagueness surrounding the
Quinean argument. Although Baker codified this explicitly in his argument, he was not the first
to suggest that explanatory concerns be at the forefront. Colyvan wrote, “we could easily
construct an argument that relied on quantification over mathematical entities being
indispensable for explanations.” (Colyvan, 2001a, p. 13) Notice that Colyvan is still committed
to Quine’s view that quantification is the sign of indispensability. The EIA frees itself from this
and concentrates instead on the exact role played by the mathematics within a scientific
explanation. The EIA is precise in stating that mathematics plays an indispensable explanatory
role and hence is to be granted ontological rights. In doing say it faces Melia’s arguments head
on by stating that mathematics is explanatorily indispensable in the right way.
It is easy to see why Baker considers his argument to be an enhanced indispensability argument.
Yet for all the apparent improvements there are two significant obstacles to overcome before the
EIA can be considered a truly powerful argument for mathematical realism. The first is that it
must be established that there exist GMEs in science – it must be shown that (EP2) is true.
Almost all of the literature surrounding the EIA is focused on this issue and no consensus has yet
been reached. The second obstacle is showing that IBE is the right tool for the job. It is simply
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assumed that inferring the existence of abstract mathematical objects is the same in kind as
inferring the existence of unobservable physical objects. The justification of this assumption has
not been undertaken, but it is critical to the validity of the overall argument. The remainder of
this dissertation will be dedicated to overcoming these obstacles. Once this is complete, we will
be able to judge whether Baker’s argument truly enhances Quine’s argument or not.
6 What About Naturalism?
Quinean naturalism plays such an important role in the QIA. It helps to motivate the QIA, it is
essential for delivering the mathematical realist conclusion, and it provides a front line of
defense against many would be criticisms. Yet, as demonstrated above, Quine’s specific brand of
naturalism also leads to two critical weaknesses of the QIA. In order for the EIA to improve in
these areas it must be the case that the naturalistic views that the EIA depends on are different
from the Quinean picture. The difficulty here is that it is not an easy task picking out exactly
what type of naturalism the EIA subscribes to. Complicating this is that my present usage of the
term ‘naturalism’ is poor. Generally speaking, Quinean naturalism was defined in 1.2 as the
belief that:
(i) we should take the practice of science at face value,
(ii) science is not answerable to some supra-scientific tribunal,
(iii) science confirms in a holistic manner,
and,
(iv) science is the sole arbiter of our beliefs.
But when we refer to naturalism in general, what does this mean? At a bare minimum, naturalism
would surely include the basic idea of taking the practice of science at face value and thus would
endorse some form of scientific realism. But beyond that it is hard to say what else is necessary.
In his survey of naturalism, David Papineau begins with the statement, “the term ‘naturalism’ has
no very precise meaning in contemporary philosophy… It would be fruitless to try to adjudicate
some official way of understanding the term.” (Papineau, 2009) Maddy remarks that “these days,
it seems there are at least as many strains of naturalism as there are self-professed naturalistic
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philosophers.” (Maddy, 2001, p. 37) What, then, are we actually committed to if we endorse the
EIA?
(EP1) is the scientific realist premise of the argument, and thus we are meeting the bare
minimum requirement of naturalism and endorsing (i) from the Quinean view. In an earlier
discussion on indispensability, Baker gives us more clues as to what naturalistic position is
implied in his EIA.
There is one theme that will surface repeatedly in the subsequent
discussion and that I want to stress at the outset. It derives from the insight
that – given the naturalistic basis of the Indispensability Argument, which
rejects the idea of philosophy as a higher court of appeal for scientific
judgments, – the only sensible way of judging alternatives to current
science is on scientific grounds. If such alternatives are to be adequate,
they must preserve those features of our current scientific theories that are
of value to scientists. Many of these features may also be deemed valuable
from some broader philosophical perspective. But if there is conflict
between the verdicts of the scientist and the philosopher then it is those of
the former that must take precedence. (Alan Baker, 2001, p. 87)
It certainly sounds here as if Baker is endorsing a position similar to (ii). There is to be no supra-
scientific tribunal which judges our best scientific theories. Philosophers should work in line
with the current methodology and practices of the scientific community.
Even though this all sounds much like Quine, the fact that there is no mention of (iii) is what
makes Baker’s naturalism distinct. Confirmational holism is at the root of Quine’s beliefs on
science and ontology. Baker is clear that his position does not depend on confirmational holism
at all. The reason why we grant mathematics ontological rights is not because we are holists
regarding our best scientific theories, but rather because mathematical entities play the exact
same roles as do unobservable physical entities in our scientific theories: they are both
explanatorily indispensable. The very same reasons that lead to scientific realists believing in the
existence of protons and positrons are the very same reasons why we should believe in functions
and sets. More importantly, the very same inference used for adding unobservable physical
entities to our ontology is to be used to add mathematical entities as well. No extra baggage or
views on science are needed for the EIA to lead us to mathematical realism. If there was not a
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distinct departure from confirmational holism, then the EIA would not actually be an enhanced
argument at all. It would simply collapse back to the QIA in an essential way.
Laslty is whether or not the EIA assents to (iv). Maintaining that science is the sole arbiter of our
beliefs leads to the disrespect of mathematical practice. As shown above, it is possible for the
EIA to adopt a more respectful attitude towards mathematical practice based on whether or not
we commit to (iv) or not. At the present moment I am happy to leave this an open issue. The
reason being is that for the majority of the participants in the debate surrounding indispensability
arguments, respect towards mathematical practice is not a make or break issue. There are
certainly those who disagree with this and argue that disrespect towards mathematical practice is
a fatal flaw. I will return to this viewpoint in chapter 6. Regardless, the main claim that I wish to
advance is that even if the premises of the EIA are found to be true, the critical factor in gauging
whether the EIA is truly an enhanced argument over the QIA or not is if the EIA successfully
frees itself from any reliance on the thesis of confirmational holism. Even if the EIA is found to
improve significantly on the second and third weaknesses of the QIA, respect of mathematical
practice and vagueness respectively, dependence on confirmational holism would essentially
reduce the argument to a slightly reworded Quinean argument. This is not enough to consider it
enhanced as its success would still entirely depend on the acceptance of Quine’s holistic views.
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Chapter 3 Genuine Mathematical Explanation
The second premise of the Enhanced Indispensability Argument (EIA) asserts that mathematics
plays an explanatorily indispensable role in our best scientific theories. The veracity of this
statement has been subject to much debate. In this chapter we will take the first steps towards
determining if (EP2) is true or false. The determining factor will be whether or not a genuine
mathematical explanation (GME) exists. Recall that a GME is a mathematical explanation of a
physical fact such that the mathematics is explanatorily indispensable. Supporters of GMEs have
advanced many supposed examples, whereas detractors have been quick to argue that these
examples are not genuine at all, but merely are standard mathematical explanations where the
mathematics is not explanatory. My diagnosis of this difference in opinion is straightforward.
Neither side has put forward a clear and acceptable set of criteria for what makes a mathematical
explanation genuine. Without this there can be no consensus on what it means to be a GME, and
thus it is no surprise that there is no agreement on whether or not supposed examples are genuine
or not.
My methodology for making progress in this debate is threefold. First, the task will be to put
together a set of criteria which all parties can agree on such that, if satisfied, a mathematical
explanation would be considered genuine. These criteria will be developed from a close look at
the present stock of examples put forward by supporters of GMEs, and also by examining the
reasons for their rejection by nominalists. The most powerful argument against the supposed
examples comes in the form of the indexing argument inspired by Melia which we will dissect
carefully. With the criteria in hand I will then present a new example which I claim meets all the
requirements, and hence should be considered a GME. Although it may be tempting at this point
to claim that GMEs exist, I will withhold this conclusion until the third task is completed. What
still needs to be accomplished is corroborating this GME via an analysis using accounts of
scientific explanation. Chapter 4 will be dedicated to this final undertaking. Only once all three
of these tasks are complete will the status of GMEs and (EP2) be decided.
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1 Supposed Genuine Mathematical Explanations
Do examples of GMEs exist? Supporters of GME have put forward many examples in an attempt
to show that mathematics can genuinely explain. I present here a sampling of these examples as
well as some criticisms found in the literature.
1.1 Geometric Explanations
One of the most famous examples of mathematical explanation comes from Graham Nerlich
(1979) long before any of the present discussion on mathematical explanation got off the ground.
Consider a particle in motion along the surface of a sphere. Classical mechanics tells us that the
particle, free from any other forces, would move at a uniform speed along a geodesic.20 Now
consider a cloud of particles moving with the same direction and speed. If such a cloud was
moving along a flat surface of approximate Euclidean space, the size of the particle cloud would
remain the same. But if the same cloud is moving along the surface of a sphere, the size of the
cloud would change. Why is this so? The explanation is that geodesics, which are the paths that
the particles would follow on the surface of the sphere, are not parallel and thus the distances
between any two given geodesics constantly change. Thus the size of the particle cloud also
constantly changes.
This explanation seems straightforward, but what exactly is doing the explaining? Nerlich notes
that this is a “non-causal style of explanation.” (Nerlich, 1979, p. 74) It is the curvature of the
space that explains the changes in size. This explanation is clearly a mathematical one. The full
explanation invokes things such as geodesics, vectors, curvature, etc., and it is certainly
explaining a physical fact. But is it a GME? Is it the mathematics that is genuinely explaining?
The problem with considering it genuine is that the explanation seems to hinge on facts about
geometry. This is problematic as it is unclear if these geometric properties are actually
mathematical properties, or properties of space-time. If space-time naturally has geometric
20 A geodesic is the shortest path between two points on a curved space.
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properties, as many believe it does, then these properties are of a physical nature, and not
mathematical. It is these physical properties that are doing the explaining, and thus this example
is not genuinely mathematically explanatory at all.
Many mathematical explanations share this difficulty in that geometric properties can be seen as
physical and not mathematical in nature. Peter Lipton gives another example of a non-causal
explanation:
Suppose that a bunch of sticks are thrown into the air with a lot of spin so
that they twirl and tumble as they fall. We freeze the scene as the sticks are
in free fall and find that appreciably more of them are near the horizontal
than near the vertical orientation. Why is this? The reason is that there are
more ways for a stick to be near the horizontal than near the vertical.
(Lipton, 2004b, p. 9)
This explanation also relies on geometric properties as well as implicitly making use of some
basic probability, and so is certainly mathematical; yet we cannot consider it genuine in that
these geometric properties are typically considered to be physical rather than mathematical. It is
the physical nature of space that is conferring the explanatory power.
Due to this, such examples are not strong candidates for establishing the existence of GME. It is
difficult to determine if the geometric properties invoked are physical or mathematical. The
natural response of nominalists would be to side with the physical interpretation, so this is even
worse for the prospects of finding a GME of a geometric sort within the context of trying to
convince the nominalist that mathematics can genuinely explain. In criticizing some of
Colyvan’s purported examples of GME, Melia writes that the “explanation is a geometric
explanation... not a mathematical one.” (Melia, 2002, p. 76) Baker also admits that geometric
explanations are problematic. “[N]ominalists often object that geometrical explanations are not
genuinely mathematical... [I]t suggests that we should look elsewhere than geometry for a
convincing case of [genuine] mathematical explanation in science.” (Baker, 2005, p. 228) The
key here is Baker’s admission that it is the responsibility of supporters of GME to present an
example that will convince the nominalist. If the EIA is to have any force against the nominalist
we will have to come up with a GME that is not geometric in nature as these are too easily
dismissed by the nominalist.
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1.2 Contrived Explanations
Colyvan puts forward another example as a candidate for a GME. His antipodal weather patterns
example is as follows:
We discover that at some time t0 there are two antipodal points p1 and p2
on the earth’s surface with exactly the same temperature and barometric
pressure. What is the explanation for this coincidence? Notice that there
are really two coincidences to be explained here: (1) Why are there any
such antipodal points? and (2) Why p1 and p2 in particular? (Colyvan,
2001a, p. 49)
Colyvan notes that the second question can be explained in causal terms by looking at the causal
history of weather patterns to account for the behaviour of p1 and p2 at time t0. What this causal
explanation cannot explain is the first question. For this, the explanation follows from a corollary
of the Borsuk-Ulam theorem, a theorem of algebraic topology. This theorem proves that there
will always be two such antipodal points at any given time, thus explaining question (1). This
explanation makes direct use of things such as continuous functions, and hence is mathematical
in nature.
The problem with this example is that it is entirely contrived. Baker (2005) notes that no such
antipodal points have ever been empirically reported, nor are they likely to be discovered unless
we actually set out to look for them. But it seems that we would only set out to look for them if
we already knew the result from mathematics. It appears, then, that what Colyvan treats as an
explanandum, namely why two antipodal points have the same temperature and barometric
pressure at the same time, is actually not an explanandum at all but rather is a prediction. There
was never any phenomenon that scientists wanted explaining in the first place. If this is so, then
certainly the antipodal weather patterns example is not a GME as it is not even an explanation.
The underlying point here is that we still want to maintain our naturalistic sympathies. If we wish
to tie GME to the practice of science, we must actually look towards science for examples. It is
not naturalistic to artificially manufacture an example that fits the mold of a GME but is not
actually one utilized in scientific practice. We must seek examples that actual scientists are both
interested in and accept as good scientific explanations of physical facts that truly warrant
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explaining. In short, an example of a GME must be endorsed by the scientific community as a
good explanation. To claim otherwise would be equivalent to imposing external standards onto
the practice of science.
1.3 Optimization Explanations
Baker (2005, 2009) presents his widely discussed cicada example which avoids the previous two
pitfalls; it is not a geometrical explanation nor is it contrived. Certain species of North American
cicadas, large fly-like insects, have been discovered to share a peculiar property. These cicadas
exhibit a periodic life-cycle where they lie dormant in the soil for 13 or 17 years, depending on
their region, until they all emerge together as adults. What biologists have found interesting is
that the periodic length of the life-cycle is prime. What needs explaining, then, is the prime-
numbered-year cicada life-cycle lengths. Baker’s explanation of the 17 year life-cycle is as
follows:
(1) Having a life-cycle period which minimizes intersection with
other (nearby/lower) periods is evolutionarily advantageous.
(biological law)
(2) Prime periods minimize intersection (compared to non-prime
periods). (number theoretic theorem)
(3) Hence organisms with periodic life cycles are likely to evolve
periods that are prime. (‘mixed’ biological/mathematical law)
(4) Cicadas in ecosystem-type E are limited by biological
constraints to periods from 14 to 18 years. (ecological
constraint)
(5) Hence cicadas in ecosystem-type E are likely to evolve 17-year
periods.
(Baker, 2009, p. 614)
(1) is motivated by biologists who posit that minimizing intersection would be advantageous for
one of two reasons. First, it could be that at one point in time there were predators of the cicada
that also had a periodic nature. (Goles, Schulz, & Markus, 2001) Minimizing intersection would
minimize exposure to these predators, and this is certainly advantageous. Secondly, it could also
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be that minimizing intersection is advantageous to avoid hybridization with similar subspecies.
(Cox & Carlton, 1988; Yoshimura, 1997) (2) is a straightforward mathematical proof in number
theory that makes use of lowest common denominators of pairs of natural numbers. Finally, (4)
is established by looking at factors such as average sunlight, soil temperatures, etc., of the given
region. In a different region of North America a similar ecological constraint would appear but
with different values, such as between 12 and 16 years, that could be used to explain the life-
cycle duration of 13 years. (1) and (2) entail (3), and from (3) together with (4) we can deduce
(5).
The cicada example is the most discussed example in the current literature on mathematical
explanation. No one doubts that this is an example of a mathematical explanation, but the
important question is whether or not it is actually genuine. This question, as Baker recognizes, is
not easily answered.
What needs to be checked in the cicada example, therefore, is that the
mathematical component of the explanation is explanatory in its own right,
rather than functioning as a descriptive or calculational framework for the
overall explanation. This is difficult to do without having in hand some
substantive general account of explanation. (Alan Baker, 2005, p. 234)
Baker briefly canvases some accounts of scientific explanation to try to show that the cicada
example can be considered genuine, but it leaves much to be desired and is ultimately more
gesturing than it is argument. I will revisit this criticism in chapter 4.
Another explanation similar in form to the cicada example comes from Lyon & Colyvan (2008).
Honeybees make honeycombs to tile an area for the storage of eggs, pollen, and honey. All
honeybees make their honeycombs in the shape of a hexagon. Why is this the case? Why do
honeybees make honeycombs that are always made up of hexagonal cells and not some other
shape, or combination of shapes? I will recast Lyon and Colyvan’s explanation to follow the
form of the cicada example.
(i) Minimizing the amount of materials used to tile an area is
evolutionarily advantageous. (biological law)
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(ii) The hexagon is the most efficient way to tile an area using the
least amount of material. (mathematical theorem (Hales, 2001))
(iii) Hence, organisms that tile an area are likely to evolve to create
hexagonal structures. (‘mixed’ biological/mathematical law)
(iv) Honeybees build honeycombs to tile an area where biological
constraints limit the size and maximum number of possible
sides of their honeycombs to include the hexagon. (ecological
constraint)
(v) Hence honeybees make hexagonal honeycombs.
Honeycombs are made from beeswax which is produced in the glands of worker bees. Bees must
consume honey in order to produce the beeswax. Hence, minimizing the amount of beeswax
required to make honeycombs is advantageous as it minimizes the amount of food and energy
spent, thus implying (i). (ii) was proved mathematically by Thomas Hales in 1999. (iv) is simply
the observation that the hexagon is within the realm of possibility for the honeybee to construct.
The honeycomb example follows the same basic deductive pattern as the cicada example: (i) and
(ii) imply (iii), and (iii) together with (iv) imply (v). Interestingly, for reasons we will see below,
although Lyon and Colyvan present this as an example of mathematical explanation, they do not
think that it is genuine.
The cicada and the honeycomb examples exhibit the same basic structure. I call any such
explanation that follows this pattern an optimization explanation. Optimization explanations
must include the following three requirements.
(a) A scientific law that stipulates the requirement of being efficient
or optimal.
(b) A mathematical theorem that demonstrates the optimality of a
particular state of affairs.
(c) Real world constraints that show the optimal state of affairs is
obtainable.
These three requirements together can derive the specific conclusion that something, from living
things to elementary particles, obtains an optimized state. Optimization examples, or course,
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depend on much more than simply (a), (b) and (c). (a) is dependent on a body of science that we
accept and is able to generate the required law.21 Likewise, (b) requires a mathematical proof to
establish it as a theorem. (c) is most likely a contingent empirical fact that could be learned in a
variety of different ways.
Optimization explanations represent some of the most popular candidates for a GME. There are
definitely many other types of supposed GMEs that have been put forward22, but for now these
examples should suffice in giving us a basic understanding of the type of explanations that we
are considering.
2 The Indexing Argument Revisited
So far we have looked at three different ways in which nominalists have argued against the
existence of GMEs. First, supposed examples of GMEs can be rejected if they are found to be
geometrical or contrived in nature. Secondly, without an account of scientific explanation that
can make sense of mathematical explanations of physical facts it can be very difficult to make a
convincing case that mathematics genuinely explains. This was recognized by Baker as a
significant challenge for supporters of GMEs which we will attempt to address in chapter 4.
Finally, we saw that nominalists such as Melia believe that the sole role of mathematics is to
index physical facts. If this is the case, then even if mathematics is indispensable in scientific
explanations, it is not indispensable in the right sort of way that leads to ontological
commitment. This indexing argument draws a distinction between being mathematics being
explanatorily indispensable, and indispensable to explanations. Nominalists can grant the latter
but reject the former, and by making use of the weaseling argument they can freely utilize
21 I have not mentioned what I take ‘law’ to mean. Certainly I do not mean a strict law of nature, as (i) from the
honeycomb example would be disputed as being a fundamental law of nature. In this discussion I take a very loose
interpretation of ‘law’. If the reader prefers, ‘law’ can be replaced with something less problematic, such as
‘acceptable scientific generalization’.
22 See Pincock (2007, 2011) for his bridges of Königsberg example, Bangu (2012) for a probabilistic example from
economics, and Batterman (2002, 2008, 2010) for many examples of asymptotic mathematical explanation.
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mathematics without having to be mathematical realists. To better understand GMEs we first
have to make sense of how the indexing argument actually works.
The indexing argument has proven to be the most powerful weapon against the existence of
GMEs. Unfortunately, the argument itself has not been clearly defined. There is no explication
for what it means that the sole role of mathematics is to index physical facts, and due to this there
is no clear way to show that the indexing claim is not the case (or, for that matter, to show that it
is the case). My understanding of the indexing argument is that it is actually just an expression of
a core nominalistic intuition. This intuition is that mathematics cannot explain as the role of
mathematics in all mathematical explanations is solely to index or represent physical facts or
properties. In trying to explicate this intuition two notions of indexing have emerged. First,
mathematics solely indexes physical facts as the mathematics utilized in scientific explanation is
always arbitrary. Second, even if the mathematics is not arbitrary, it is still reasonable to believe
that purely physical facts underlie all mathematical explanations. The indexing argument argues
that since the sole role of mathematics is to index physical facts, then it is never the case that
mathematics is genuinely explanatory. Rather, it is the physical facts that the mathematics is
indexing which are truly conferring the explanatory force.
Let us first consider how mathematics indexes when its application is deemed to be arbitrary.
Suppose we have an explanation of some phenomenon that relies on the fact that the length of a
particular rod is 7
11 of a meter. Such an explanation would make use of the number
7
11, but this
number is entirely arbitrary. If we had chosen a different unit of measurement, then the length of
the rod would be a different number. The explanation is not explanatory in virtue of the number
7
11, but rather it is explanatory in virtue of the physical length of the rod. This example is meant
to illustrate Melia’s (2002) suggestion that if the use of mathematics is found to be entirely
arbitrary, then it is not explaining anything. Rather, the mathematics is merely indexing a
physical fact – in this case the actual physical length.
Daly and Langford (2009) borrow Melia’s suggestion and argue that Baker’s cicada example
suffers from this exact form of arbitrariness. Recall that what biologists wish to explain is why
the periodic-life-cycles of the cicada are prime. However, the life-cycles are prime only if we
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take years to be our fundamental unit. If we take months, days, or seasons, then the life-cycle
will be some other composite number instead. Baker (2009) defends the choice of years as being
unarbitrary in a variety of ways, such as appealing to the fact that biologists consider years to be
the proper unit of measurement, and also by pointing out that years do reflect a significant and
real regularity. My goal is not to take sides on this issue. It should be clear that the threat of
arbitrariness does not rear its head in all supposed examples of GME, nor do Daly and Langford
claim this either. In the honeycomb example, for instance, one would be hard pressed to assert
that the number of sides of the honeycomb cell is anything other than 6! The takeaway from this
discussion should be that in the eyes of the nominalist, lots of the usage of mathematics in
science seems to be arbitrary in the sense that Melia describes. It is this observation that leads us
to the intuition that the sole purpose of mathematics is to index physical facts. The supporter of
GME needs to produce an example that will fly in the face of this intuition by having an
explanation where the mathematics is not arbitrary.
The true force of Melia’s argument is that even in cases where the mathematics is not arbitrary it
can still be argued that mathematics is only indexing or representing purely physical facts. This
is the belief reflected in the second sense of indexing. Consider our two leading examples of
GME which are both optimization explanations. A key feature of the optimization explanation is:
(b) A mathematical theorem that demonstrates the optimality of a
particular state of affairs.
Supporters of GME claim that (b) is necessary for an optimization explanation; the explanation
would simply be lost without it. Thus, mathematics is playing an explanatory role. Opponents of
mathematical explanation argue that (b) is actually not necessary at all, or at least not necessary
in an explanatory sense. Instead, the (b) statement is merely a statement that indexes or
represents a purely physical fact. All the nominalist needs to do is show what these purely
physical facts actually are, and then the mathematical explanation can no longer be considered
genuine. For example, in the cicada example, the relevant physical fact would be that units of
time have the property such that 17 years is the most optimal number of years between 14 and 19
such that it would minimize intersection with other years. It is this fact about time that is
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conferring the explanatory power to the optimization explanation, and not the mathematical
theorem itself.
The realist could object that mathematics is still necessary in order to express some of these
physical facts. However, nominalists can simply agree with this and claim that this
fundamentally does not change the fact that in the end the explanatory force is not being supplied
by the mathematical theorem, but rather by the underlying physical fact. Another option for the
realist is to claim that we only know the physical fact to be true via the mathematical theorem
due to its mathematical proof. Juha Saatsi (2010) argues that the representational notion can be
extended to the mathematical proofs as well. In the cicada example, he suggests that we can
deduce that 17 should be the optimal number of years for the cicada life-cycle by using sticks to
represent annual periods of time.
[O]ne could represent periods of time with sticks as follows. Take a bunch
of sticks of 14, 15, 16, 17, and 18 cm. You’ll need fewer than 20 sticks of
each kind. Lay down sticks of each kind one after another to find the least
common multiple (LCM) for each pair... You’ll soon find out that the least
common multiple is almost always clearly longer for the pairs one member
of which comprises 17 cm sticks. (Saatsi, 2010, p. 8)
The point of this example is that the mathematics functions like the sticks. It merely represents
the concept of time and from this purely physical concept we can deduce the relevant physical
fact that is the genuine explanatory factor. To conclude that mathematics is genuinely
explanatory is as ludicrous as it is to conclude that the sticks are genuinely explanatory as well.
The honeycomb example is similar to the cicada example in that the nominalist can argue that it
is not the mathematical theorem that explains the hexagonal shape of the honeycomb, but rather
it is a purely physical fact about approximately Euclidean space. What makes this example
different and slightly more difficult to attack is that the mathematics employed is not as easily
recreated in a purely nominalistic fashion. A quick glance at Hales’ mathematical proof of the
honeycomb theorem reveals that it involves limits, disjoint measureable sets, and more.
Recreating the proof with physical objects like sticks is most likely impossible. Even so, Colyvan
sees a weakness in the honeycomb example.
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[T]he mathematical part of the explanation for the hexagonal structure of
the hive-bee honeycomb comes from the proof of the honeycomb
conjecture – a result in geometry and topology. But Field has shown how
we can speak nominalistically about the geometry of (Newtonian) space-
time, and so it seems likely that a similar result could be proven in his
system. If this is possible, then any explanation involving the nominalist
form of the honeycomb conjecture would arguably be at least as good as
the original form of explanation presented earlier. (Lyon & Colyvan, 2008,
p. 240)
If we take Field’s project seriously then it may be possible to create a nominalistically acceptable
proof that the hexagon is the most efficient shape to tile an approximately flat area. If this is so,
then this fact is no longer solely a mathematical fact, and thus the nominalist could assert that the
mathematics is merely indexing or representing the physical properties of space-time.23 So long
as the nominalist has the possibility of claiming that physical facts are the only explanatory
factors, then the indexing argument appears to be good enough to reject the existence of GMEs.
Colyvan admits this when he says “for an example of mathematical explanation to be of the
ontologically committing type, there must be no matching nominalist explanation.” (Lyon &
Colyvan, 2008, p. 240) In this light, the honeycomb example, and perhaps the cicada as well, are
not GMEs as there exists nominalistically acceptable counterpart explanations.
Colyvan (2010) presents another supposed example of a GME which he believes avoids the
problem of having a nominalistic counterpart explanation, and hence avoids the force of the
indexing argument.
The Kirkwood gaps are localized regions in the main asteroid belt between
Mars and Jupiter where there are relatively few asteroids. The explanation
for the existence and location of these gaps is mathematical and involves
the eigenvalues of the local region of the solar system (including Jupiter).
The basic idea is that the system has certain resonances and as a
consequence some orbits are unstable. Any object initially heading into
such an orbit, as a result of regular close encounters with other bodies
(most notably Jupiter), will be dragged off to an orbit on either side of its
initial orbit. An eigenanalysis delivers a mathematical explanation of both
the existence and location of these unstable orbits (Murray & Dermott,
23 It could also be pointed out that the honeycomb example fails as it is another instance of a geometric explanation.
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2000)... The explanation of this important astronomical fact is provided by
the mathematics of eigenvalues (that is, basic functional analysis).
(Colyvan, 2010, p. 303)
The mathematics of eigenvalues and functional analysis has not been successfully nominalized
by Field. It is the eigenvalue analysis that explains the Kirkwood gaps, and since there is, at
present, no nominalistic counterpart to the mathematical explanation, Colyvan claims that the
mathematics is genuinely explaining the physical phenomenon. Although Colyvan may be right
in his assertion that there is no nominalistic counterpart to the mathematical analysis in the
Kirkwood gaps explanation, his conclusion shows that he has missed the true force of the
indexing argument. The indexing argument does not tie its success to the complexity of a
mathematical proof, or to the explicit availability of a nominalistic counterpart. All it depends on
is the intuition that what is truly doing the explaining are purely physical facts. The nominalist
can easily assert that what explains the Kirkwood gaps are properties of space-time, gravitation,
planetary mass, etc. This is no different than their belief that the cicada and honeycomb
explanations are fundamentally physical as well.24 What makes the Kirkwood gaps different
from the honeycomb and cicada example is that all parties can agree that the mathematics is
indispensable. The mathematical portions of the cicada and honeycomb explanations can be
replaced with nominalistically acceptable counterparts, whereas in the Kirkwood gaps example
this cannot be done. However, all this indicates to the nominalist is that the mathematics is
indispensable in a deductive or representational way. This is not controversial as recall that
nominalists like Melia consent to this fact already. Colyvan has tied genuinely explanatory
mathematics directly to the unavailability of a nominalistic counterpart. Although I agree that
this is a necessary condition to convince the nominalist that GMEs exist, it is not sufficient. So
long as the nominalist is able to assert that there is something purely physical explaining the
phenomenon, they will be able to resist the conclusion that mathematics is genuinely
explanatory.
24 In a recent paper, Bueno (2012) draws similar conclusions.
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In this presentation of the indexing argument I have invoked the idea of ‘purely’ physical facts or
properties. The argument depends on the idea that mathematics indexes these purely physical
facts, and it is these facts that are genuinely explanatory. Naively speaking, by purely physical
fact I mean a fact that can be entirely understood without the use of mathematics – in essence,
they are facts that are solely physical in nature. Properties such as hardness or shape are
supposed to uncontroversially fall under this label. However, other physical properties such as
instantaneous velocity seem to resist our naïve definition. Is it possible to understand
instantaneous velocity without any mathematics?25 Nominalists take the existence of such purely
physical facts for granted, and their intuitive belief is that things such as time and space are
purely physical as well. Realists in this debate also seem to assent to the belief that we can
discuss purely physical properties in an unproblematic way, and I will assume this as well.
However, one important question that is seldom asked is: are there any physical facts or
properties that are not purely physical? If so, what are these non-purely-physical physical facts?
There are three potential answers to these questions. The first is to maintain that there is no
difference between purely physical facts and physical facts; the word ‘purely’ is just a redundant
descriptor. The second is to admit that it is possible for certain physical facts to be not purely
physical. There could be physical-mathematical facts, and what this precisely means would be up
to much debate. Lastly, one could maintain that this is all entirely mistaken, and that there are no
purely physical facts or properties at all as physical facts do not exist. We live in a purely
mathematical world and hence all facts and properties are actually mathematical in nature.26
Regardless of its independent merits or deficiencies, there are reasons pertinent to our present
discussion for rejecting this third position. If we are committed to the belief that there are only
mathematical facts and properties in the universe, then it is a trivial conclusion that mathematics
would genuinely explain the world around us. This entirely begs the question against the
25 See Berkovitz (forthcoming) for an argument against a purely physical understanding of certain physical
properties.
26 See Max Tegmark (2008) for a spirited defence of this position.
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nominalist. By the same token, ought we to reject the first view that claims all physical facts to
be purely physical in nature? It seems that this could be used in the exact same way to reject
even the possibility of a GME. Although I believe this to be a core belief held by nominalists, it
alone is not enough to a priori reject the existence of GME. It is still entirely reasonable to
maintain that mathematical facts or properties can explain purely physical facts. One way is to
assert that mathematical objects have causal power. Another is to take the more traditional view
that mathematical objects are acausal, but so long as we reject an entirely causal account of
explanation there is no inherent conflict in believing that mathematics can explain. So, asserting
that all physical facts are purely physical is compatible with the existence of genuine
mathematical explanation. Lastly, the second position that allows for mixed physical-
mathematical facts is also open to GMEs. The prima facie problem with this position is agreeing
on what we mean by a mixed fact or property. While all three views are open to the possibility of
GME, we will henceforth only consider the first two.
Although it does seem obvious that nominalists would happily assent to the first of these
metaphysical views, it is entirely unclear what supporters of GMEs believe. This lack of
commitment represents a weakness in the realists’ position, and is something that the indexing
argument successfully exploits. If you believe that all physical properties are purely physical,
then even though this alone does not rule out mathematics genuinely explaining physical
phenomena, it is extremely difficult to conceive of how it could possibly work. In my opinion,
this intuitive belief is what fuels the indexing argument. The crux of the problem is that there are
two separate questions being fused together. The first is ‘does mathematics genuinely explain?’,
and the second is ‘how could mathematics genuinely explain?’. Realists have yet to offer an
answer to the latter question, and without one it is proving difficult to answer the former question
in the affirmative.
We will not delve into these issues much further in this chapter. I will return to the much ignored
‘how’ question in chapter 6. Until then, I will help myself to the naïve understanding of purely
physical properties, and simply note that the indexing argument does not hinge on the nature of
all physical properties in general. If we can put forward an example that resists the intuition that
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there is something purely physical doing the explaining, then we will succeed in avoiding the
challenge presented by the indexing argument.
The indexing argument has proven to be the most challenging argument against GMEs.
Somewhat surprisingly, supporters of GMEs have not aggressively challenged the validity of the
argument, and instead have chosen to search for examples that are impervious to the indexing
critique. This will also be the strategy that I will employ for reasons I will detail below.
However, before we proceed it is worth mentioning some of the shortcomings of the indexing
argument.
Despite its name, the indexing argument is not much of an argument at all. It is actually a loose
application of a deep intuition that nominalists possess regarding the role of mathematics in
science. The intuition is that mathematics simply indexes, and this is why mathematics does not
genuinely explain physical facts. A serious problem is that this application of the indexing
intuition is difficult, if not impossible to defeat. The indexing argument essentially expresses the
intuition that mathematics does not genuinely explain because mathematics cannot genuinely
explain due to its sole role as an indexer of physical facts. This is dangerously close to question
begging. Whether or not mathematics can explain, and hence what precise role(s) mathematics
plays in scientific applications, is exactly what is in question. Obviously, if we assume that
mathematics cannot play an explanatory role then we are also assuming that mathematics does
not play an explanatory role, and in this light the indexing ‘argument’ is impossible to refute.
Consider the following from Daly and Langford. They feel that the indexing argument is strong
enough to rule out the possibility of any GME.27
We suggest that the nominalist’s view should be that there could not be
mathematical explanations. In this paper we have tried to show the
resilience of Melia’s indexing strategy. If his strategy works against some
cases of putative mathematical explanations, it works against all possible
27 In a more recent paper Daly and Langford (2011) are less enamoured with Melia’s indexing argument.
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putative mathematical explanations. (Daly & Langford, 2009, pp. 655–
656)
Even if we grant that the indexing argument is powerful enough to resist the force of the cicada,
honeycomb, and Kirkwood gaps example, Daly and Langford’s conclusion is entirely too strong
and unjustified. They perform an inductive generalization based off the conclusion that the
mathematics in the above examples was merely indexing and hence not genuinely explanatory.
But what justifies the generalization that this will be the case for all future possible mathematical
explanations? The only justification is the intuition that the sole role of mathematics is to index,
and nothing more. Daly and Langford are assuming the very issue that is under investigation in
order to motivate their general conclusion. A more charitable interpretation that could justify the
universal conclusion is simply that it is an instance of enumerative induction. So far, all instances
of supposed GMEs have been found to be not genuine based on the discovery that the
mathematics was solely indexing physical facts. It then stands to reason that all supposed
examples of GMEs will suffer the same conclusion, and hence we should believe that
mathematics does not genuinely explain. My phrasing here is weaker than above, and, like all
instances of enumerative induction, is ultimately fallible. Nominalists may be justified in holding
their position until an example is put forward where mathematics is not solely indexing physical
facts. In this light, the nominalist remains open to the possibility that mathematics could be
found to genuinely explain, and that the indexing argument can be defeated.28
The non-question begging nominalist is the one who we are interested in engaging with. This is
the position that most closely matches Melia’s who presented the indexing argument in the first
place. Recall that Melia says,
Of course, there may be applications of mathematics that do result in a
genuinely more attractive picture of the world – but defenders of this
28 This charitable interpretation of Daly and Langford is most likely not what the authors intended. The extended
quotation from above is: “Our topic is mathematical explanation. Baker thinks that there are mathematical
explanations. Is the opposing nominalist’s view simply that there are no mathematical explanations, but that there
could be, and that it is just an unfortunate coincidence that to date no mathematical explanations have been
produced? We suggest that the nominalist’s view should be that there could not be mathematical explanations.”
(Daly & Langford, 2009, p. 655)
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version of the indispensability argument have yet to show this... Until
examples of applied mathematics are found that result in this kind of an
increase in attractiveness, we realists about unobservable physical objects
have been given no reason to believe in the existence of numbers, sets or
functions. (Melia, 2000, pp. 474–475)
His version of the indexing argument is not undefeatable. It is open to the possibility that
mathematics could genuinely explain, and hence is not question begging. However, it still has
the problem that it is not so much an argument as it is an expression of an intuition. One reason
for this is that the concept of indexing is not elaborated on or explained in any detail. A standard
understanding of mathematics indexing physical things is how we can use, say, the natural
numbers to index physical objects or properties. A simple example of this is how the natural
number 2014 is indexing the current year, which is actually just the number of periodic cycles
completed by the Earth revolving around the Sun starting from an arbitrary starting point. We
can perhaps extend this notion of indexing unproblematically to other mathematical systems,
such as the real numbers indexing positions in space-time, assuming that space-time is
continuous in the right sort of way. But once you get into much more complicated or abstract
mathematics such as complex differential equations, eigenvectors, etc., it is entirely unclear how
these mathematic entities actually index physical facts at all. What is needed here is a robust
account of mathematical representation that details how mathematical concepts index physical
things, and how through this indexing mathematics is able to bestow all its benefits and utility to
the practice of science.
Nominalists have not presented any such account of mathematical representation; they simply
assume that the indexing argument is a coherent and defendable position. This assumption is
rooted in their intuitions regarding the sole role of mathematics, and this is why I do not feel that
the indexing argument is much of an argument at all. It remains to be seen if an anti-realist view
of mathematics such that mathematics only represents physical facts can retain all the epistemic
benefits that mathematics brings to the table.29
29 See Pincock (2011) as well as Bangu (2012) for a discussion on the epistemic benefits of mathematical
representation in science.
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One final issue with the concept of indexing is that nominalists such as Melia, Leng, and Daly
and Langford accept that mathematics is an indispensable tool for the indexing of physical facts
in scientific practice. This poses two interesting problems for the nominalist. The first is that
there is no account as to why mathematics is indispensable for indexing. The second is that due
to its indispensable nature, the language of science seems to directly imply the existence of
mathematics. The first problem remains somewhat of a mystery. Perhaps it is a brute fact,
perhaps there is no account at all, or perhaps our demand for an account is unwarranted.
Regardless, something should be said about this issue to either explain the indispensability of
mathematics, or explain why no explanation is needed. The second issue of the language of
science implying the existence of mathematical entities is supposedly solved by Melia’s
weaseling argument. Recall that the weaseling argument allows us to have our scientific
language imply the existence of mathematical entities, but at the same time we are able to weasel
away from any ontological commitment. In section 2.3.2, concerns were raised regarding the
viability of the weaseling argument. It is not clear that the weaseling method is actually strong
enough to apply to the indispensable use of mathematics when indexing physical facts. The
indexing argument critically depends on weaseling out of the apparent commitment to
mathematical objects in scientific language, and insofar as the weaseling argument is found to be
lacking, so too will the indexing argument be found unsatisfactory as well.
As previously mentioned, one potential strategy to argue against the nominalist is to show that
the indexing argument is not coherent due to any of the above objections. This is not the method
I will pursue. The reason for this stems from my critique that the indexing argument functions
less like an argument and more as a statement of the nominalists’ intuitions. Showing that the
present state of the indexing argument is grossly underdeveloped and lacking in rigour does
nothing to demonstrate that the core intuition that mathematics solely indexes physical facts is
false. The ardent nominalist can still, albeit stubbornly, hold on to their beliefs and assert that a
truly comprehensive account of the indexing argument could potentially be developed if needed.
The catch is that there is, at present, no need for this comprehensive account as no purported
example of a GME comes close to pressuring the intuition that mathematics only indexes
physical facts. Until that time comes, the onus is on the supporter of GMEs to come up with a
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superior example, and not on the nominalist to flesh out their intuitions into a full-fledged
position. My strategy of choice is to meet the nominalists’ intuition head on. I will treat the
indexing argument as a legitimate argument against GMEs, and accept the burden of proof by
presenting an example of a GME that is impervious to the intuitions that lie at the crux of the
argument. The first step will be to develop the indexing argument into a bona fide argument.
Only then will we be able to see precisely what the nominalist intuition that mathematics solely
indexes physical facts consists of, and subsequently how to avoid it.
3 The Indexing Criteria for Genuine Mathematical
Explanations
Contra Daly and Langford, the indexing argument that we will consider does not rule out the
possibility of a GME. Our target is the honest nominalist, such as Melia, who does not beg the
question against GME. Their indexing argument moves from the observation that all present
supposed examples of GMEs fall short as the mathematics is found to index physical facts and
hence does not genuinely explain, to the general conclusion that there are no GMEs. What needs
to be made clear are the criteria that are used to determine that mathematics is playing an
indexing role in these explanations. Since the indexing argument does not mention any such
criteria, our task will be to extract them from the various ways in which nominalists have
deployed the indexing argument to reject past examples.
One way that supposed examples of GMEs were rejected was due to their seeming contrived or
not being accepted by the scientific community as a proper mathematical explanation of a
physical fact. This is not actually a part of the indexing argument proper, but rather it is more of
a prerequisite criterion that an external mathematical explanation must meet before we even
consider it. The first actual employment of the indexing argument was in rejecting any
mathematical explanation where the mathematics is geometric in nature. Nominalists asserted
that geometric properties such as geodesics are actually physical properties. If this is the case,
then the mathematics that we use to explain is actually indexing these physical-geometric
properties.
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Next, the indexing argument was used to reject mathematical explanations where the
mathematics was found to be arbitrary. The claim is that if mathematics is arbitrary in a scientific
explanation, then there is nothing special about the particular mathematical entities being
employed as it could have been some other mathematical entity. Seen in this light, the nominalist
can claim that there is nothing explanatory about the mathematical entities or their properties at
all, as they could have been different. Rather, what is explanatory are the physical facts which
are not arbitrary at all. The arbitrary nature of the mathematics points to the conclusion that the
mathematics holds no explanatory significance to the overall explanation, and merely functions
as a tool to index the explanatory physical facts.
Closely related to arbitrariness is the rejection of examples where the mathematical explanation
was found to be not unique. If there exists another nominalistic explanation of the same
explanandum with no reference to mathematical objects or properties, then there is no reason to
believe that mathematics is explanatory. This line of reasoning was utilized even without direct
access to the nominalistic counterpart explanation, such as Lyon and Colyvan’s rejection of their
own honeycomb explanation as a candidate for a legitimate GME. Even though there is no
known nominalistic proof of Hale’s honeycomb theorem that they can point to, their admission
of merely the possibility of a mathematics free explanation of the hexagonal nature of the
honeycomb was enough to lead them to reject the mathematics as genuinely explanatory. Thus,
even if it is unattractive in other ways compared to a mathematical explanation, it seems that if
there exists the possibility of a purely nominalistic explanation of a phenomenon, then the
conclusion must be that the mathematics is not genuinely explanatory. The nominalist is able to
assert that the physical facts which appear, or would appear, in the nominalistic version of the
explanation are conferring the explanatory force.
Up till now, the applications of the indexing argument seems somewhat clear and plausible. The
final use of the indexing argument, however, is more complicated and controversial. Recall that
the Kirkwood gaps example seems to avoid all the above objections. It is not contrived, nor is it
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necessarily geometric.30 The mathematics is not arbitrary, and presently there is no purely
nominalistic explanation of the Kirkwood gaps, nor is there any reason to believe that such a
nominalistic explanation is even possible. However, even with all these points in its favour, I still
argued that Colyvan was incorrect in asserting that the Kirkwood gaps example is a convincing
example of a GME. The reason why is because it does nothing to address the underlying
intuitions of the indexing argument. Nominalists can still point to (or perhaps only vaguely
gesture at) purely physical facts or properties that they claim are the true explanatory factors. In
the case of the Kirkwood gaps, physical facts about space-time, gravity, planets, mass, etc., could
be cited as the true explanatory factors. If this is the case, then once again the indexing argument
can conclude that the mathematics is simply indexing these explanatory physical factors, and
hence the mathematics is not genuinely explaining anything at all.
This final use of the indexing argument is the most powerful as it seems to be strong enough to
apply in every imaginable situation. Even without a nominalistic counterpart proof in hand that
identifies the key physical explanatory factors, the nominalist is still able to assert that the
mathematics is indexing at least one member from a set of possible purely physical explanatory
facts or properties, and it is from these members that the true explanatory factors are found. My
line of attack against the nominalist is to treat the indexing argument as a legitimate argument
and then aim to defeat even the strongest version of it. However, if we allow for this final usage
of the indexing argument, are we granting the nominalist too much? Consider a mathematical
explanation of a physical fact that meets all the above criteria: it is not contrived, not geometric,
not arbitrary, and there exists no actual or possible purely nominalistic counterpart explanation.
Now let P* be a set consisting of all possible purely physical facts or properties in the universe.
The indexing argument essentially says that there is some subset of P*, called P, that represents
the true explanatory factors of the explanandum in question, and that the mathematics indexes
these physical factors. Notice that the nominalist does not (or perhaps cannot) even need to
identify the correct subset of physical factors! In fact, the nominalist need not even narrow P
30 This depends on whether or not we view space-time as purely geometric, and whether or not such a geometric
model is even the appropriate way to understand space. I will avoid this line of analysis and instead will follow the
lead of Colyvan and others by treating the Kirkwood gaps explanation as non-geometric in nature.
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down in any way for the argument to run. I take this usage of the indexing argument to be the
most accurate instantiation of the nominalists’ intuition that mathematics solely indexes physical
facts. It is also this usage which, I believe, motivates nominalists such as Daly and Langford to
claim that GMEs cannot in principle exist. It is easy to see why, as if this is a viable employment
of the indexing argument then how could we ever find an example of a mathematical explanation
of a physical fact that can avoid it?
I am willing to accept this usage of the indexing argument with two subtle but important
constraints. These constraints should be perfectly amenable to the nominalist, and are both
limitations on what the subset, P, made from all possible purely physical facts or properties in
the world can actually consist of. The first constraint is that P does not include all possible
physical facts or properties, but rather it can only consist of all known physical facts or
properties. This distinction is not motivated from any particular metaphysical view of natural
kinds or ontology. Rather, this restriction limits the nominalists to selecting from physical facts
or properties that our present best scientific theories acknowledge as genuine facts or properties
about the world. If we allow all possible physical facts or properties into P, then there is nothing
stopping us from asserting that mathematics is indexing the physical fact of, say, Koo-ness,
which is some physical fact that has yet to be discovered. At first glance this limitation seems to
be somewhat useless. It does not rule out any of the previous employments of the indexing
argument that we have looked at. Moreover, it seems that this restriction on P is implicitly
assented to by any reasonable nominalist. Surely their aim is to point to actual known physical
facts or properties. To cite future or possible physical facts is not scientific or naturalistic at all.
Rather, it would seem somewhat ludicrous and patently question begging against supporters of
GME. Regardless, making such a restriction explicit will prove to be important.
The second constraint on P is that the physical facts or properties in P need to have independent,
physical motivation or justification for their existence. This restriction is meant to rule out the ad
hoc creation of ‘physical’ properties meant to mimic mathematical properties. Consider again the
Kirkwood gaps explanation. Colyvan believes that it is the eigenvalue analysis that is conferring
the explanatory force. If, instead, I claim that it is not eigenvalues, but rather it is ‘p-eigenvalues’
that are genuinely explaining, then I could run the indexing argument successfully. A p-
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eigenvalue is, of course, a purely physical property. They are actually properties of p-
eigenvectors, which are also purely physical. Asserting the existence of p-eigenvalues arguably
passes the first constraint that the physical property must be known. We know that p-eigenvalues
exist because of the seemingly explanatory nature of the mathematical eigenvalues which
function solely as indexing p-eigenvalues. The problem here is obvious. The existence of p-
eigenvalues is entirely ad hoc and parasitic on the existence of a mathematical explanation and
mathematical formalism. There is no independent or purely physical motivation or justification
for believing in the existence of p-eigenvalues whatsoever, nor is there any independent physical
confirmation of the existence of things such as p-eigenvalues. This is entirely different from how
naturalists and scientific realists typically infer that physical facts or properties exist. As above,
the constraint that P only consist of independently or purely physically motivated or justified
physical facts seems trivially obvious. All previous instances of the indexing argument
considered do not violate this constraint, and I take it that any honest nominalist would assent to
this limitation on P. The reason for this is simply that to not accept such a limitation would make
the indexing argument seem ad hoc and question begging against GMEs. Although I formulate
these restrictions on P, the net effect of the two constraints is to force the nominalist to cite
known, physically independently motivated physical facts as those being indexed by
mathematics when employing the indexing argument. This is exactly what has been happening in
every use of the indexing argument so far, and I take it as uncontroversial that the nominalist
would accept these constraints.
Putting our analysis of the indexing argument together, I present here four criteria that a
mathematical explanation of a physical fact must meet in order to be impervious to the indexing
argument and the core nominalist intuition that underlies it. A mathematical explanation of a
physical explanandum can be considered to be a candidate for genuine mathematical explanation
if:
(A) the explanation is not contrived and is accepted by the scientific
community as a good scientific explanation,
(B) the mathematics employed is not arbitrary,
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(C) there are no purely nominalistic explanations of the
explanandum,
and,
(D) there are no known and physically independently motivated
physical facts that could potentially explain the explanandum
such that the mathematics is simply indexing these facts.
If a mathematical explanation of a physical fact meets all four criteria, then the indexing
argument cannot be used against it to show that the explanation is not genuine. Without the
indexing argument, the nominalist must concede that the mathematical explanation in question
opposes their core intuition that the sole function of mathematics is to index physical facts.
Hence, in the eyes of the nominalist, meeting these four criteria is a necessary condition for a
mathematical explanation to be genuine.
Even though the indexing argument is underdeveloped and quite vague in its presentation and
application, in taking it seriously with the most charitable interpretation possible the true worth
of the argument has emerged. The critical contribution of the indexing argument is in its
expression of the core beliefs and intuitions of the nominalist. Once these beliefs were clearly
laid out, we were finally able to arrive at a set of criteria that a mathematical explanation must
satisfy for the nominalist to take it seriously as a candidate for being a GME. This is not a trivial
achievement. So many of the problems surrounding the debate on mathematical explanations
have come from the fact that neither side has ever been clear on what exactly makes an example
genuine or not. Even advocates of GMEs do not agree on what makes an explanation genuine.
Their approach has been to present examples that they feel are genuine. However, without any
criteria to compare them to, all these feelings amount to are more intuitions. It is no surprise
then, that nominalists remain unconvinced as certainly examples that speak to the intuitions of
the realist would not have the same effect on the intuitions of the nominalist. Seeing as how the
target of all indispensability arguments is the nominalist, the inability to precisely define and
identify what criteria a GME needs to satisfy has been crippling to the success of the Quinean
indispensability argument and the enhanced indispensability argument.
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I have presented the four criteria as necessary conditions for a GME. The reason for this is it
reflects how nominalists have been rejected supposed examples. They show that examples are
lacking at least one necessary criterion, and thus cannot be genuine. What would be ideal is if the
four necessary conditions of a GME could be shown to be sufficient as well. Sadly, I do not think
that this is possible. Nominalists can still point to the fact that even if we had an explanation that
satisfied all four criteria, and thus successfully resists the intuition that the mathematics is
indexing physical facts, there is still no account of scientific explanation that corroborates the
claim that the mathematical explanation is a GME. One more criterion needs to be added to our
list:
(E) An acceptable account of scientific explanation must
corroborate the claim that the mathematics is explanatory.
The remainder of this chapter will be dedicated to presenting an explanation that satisfies (A) –
(D). Chapter 4 will take a detailed look at accounts of scientific explanation and attempt to show
that our explanation also satisfies criterion (E). Once this is complete the nominalist will be
forced to conclude that GMEs exist.
4 Genuine Mathematical Explanation: Electron Spin
Based on our understanding of the indexing argument, the strategy for finding a GME is
straightforward: we must find a mathematical explanation of a physical fact that satisfies criteria
(A) through (D). I advance here an example that I believe does the job, and is one that physics
students are intimately familiar with. The Stern-Gerlach experiment is a standard experiment
taught in every introductory quantum mechanics class. In 1922, Otto Stern and Walther Gerlach
conducted an experiment where they measured the deflection of silver atoms.31 Their goal was to
test that the direction of angular momentum is quantized, and hence act as a form of
confirmation for quantization and as a rejection of classical theory. Stern and Gerlach were
31 See Weinert (1995), and Friedrich and Herschbach (2003) for history and analysis of the Stern-Gerlach
experiment.
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famously successful in this, and in addition they were also the first to observe the quantization of
magnetic moment associated with electron spin. It was not until 1925 when Goudsmit and
Uhlenbeck proposed the concept of electron spin.32 This concept was subsequently formalized by
Pauli in 1927 and Dirac in 1928. Spin is now considered a fundamental quantum property that
explains the outcome of not only the Stern-Gerlach experiment, but it also accounts for the
Zeeman effect, it provides the basis for the periodic table of chemical elements, and much more.
The Stern-Gerlach experiment involves a beam of particles passing through an inhomogenous
magnetic field. If the magnetic field is homogenous then the net forces of the magnets will
cancel each other out and the path of the beam will remain unchanged.
An inhomogenous field results in a net force that will deflect the trajectory of the particles. At
the other end of the Stern-Gerlach are detectors to measure the deflection of the particles.
Consider a beam of electrons passing through a Stern-Gerlach oriented in the z-direction. It is
observed that the beam splits into two distinct beams that are the equidistant from the z-axis
when viewed from the side. What this shows is that the electrons have a form of intrinsic angular
momentum. It also serves as a counterexample to classical theory as according to classical
32 Goudsmit and Uhlenbeck are often credited with the discovery but they were not the first to conceive of electron
spin. In 1921, Compton proposed that the electron is “spinning like a tiny gyroscope” (Compton, 1921), and Kronig
also had an unpublished paper at least six months before Goudsmit and Uhlenbeck that explored electron spin.
South Magnet
z-up direction
Electron Beam
z-direction North Magnet
z-down direction
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physics the electrons should be continuously and randomly dispersed. Instead we see only two
trajectories that are deflected up or down by a discrete amount. The explanandum we are
interested in is: why does a beam of electrons oriented in the z-direction split into two distinct
beams that are the same distance from the z-axis when passed through a Stern-Gerlach
apparatus? A standard explanation of this is as follows.
In quantum mechanics, the state of physical systems is represented by the vector ψ defined in a
Hilbert space. ψ contains all the information of specific properties such as position, momentum,
and spin. Each property is represented by operator functions also defined on the Hilbert space.
The operator functions for spin, S, in the x-, y-, and z-directions are written Sx, Sy, and Sz
respectively and are defined as:
Sx = ½ħσx, Sy = ½ħσy, and Sz = ½ħσz.
According to the Pauli spin matrices, σx, σy, and σz for electrons are defined as:
σx = (0 11 0
), σy = (0 −𝑖𝑖 0
), and σz = (1 00 −1
).
Any square matrix has both eigenvectors and eigenvalues associated with it. An eigenvector, v,
of a square matrix, A, is a non-zero matrix that when multiplied by A yields a multiple of A by a
given number, λ. λ is called the eigenvalue of A. In quantum mechanics, the eigenvalues
represent the only possible magnitudes that the property represented by its linear operator can
possess. It is not difficult to deduce that the Pauli spin matrices have two eigenvectors associated
with each matrix. They are:
λx = 1
√2(
11
), 1
√2(
1−1
), λy = 1
√2(
1𝑖
), 1
√2(
1−𝑖
), and λz = (10
), (01
).
Similarly, it is straightforward to calculate that the eigenvalues for each of the Pauli spin
matrices are ½ħ and -½ħ, which for convenience we call spin up and spin down respectively.
Thus, the only possible values for the spin of an electron are spin up and spin down. This, then,
explains why the beam of electrons deflects into exactly two separate beams in the Stern-Gerlach
apparatus. According to the property of spin, there are only two possible values of the electron
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spin, and hence only two possible ways the electron can be deflected by the magnetic field.
Moreover, |½ħ| = |-½ħ|, and this explains why the spin up beam and the spin down beam are both
equidistant from the original plane or orientation.
This presentation of spin is standard in introductory quantum mechanics textbooks. From the
Pauli spin matrices we can make many predictions of what would happen to particles that pass
through multiple joined Stern-Gerlach apparatuses which are popular exercise questions for
physics students. To briefly summarize, the Stern-Gerlach apparatus leads to the observed
deflection of an electron beam in the z-direction into two equidistant discrete beams when
viewed from the side – one above the original beam and one below. The explanation of this
deflection is that electrons have the property of spin, which is defined by the Pauli spin matrices,
σx, σy, and σz. From the Pauli spin matrices we can deduce that there are only two possible spin
values of the electron, spin up and spin down, and that they have the same magnitude. This
explains why the original beam splits into exactly two equidistant beams.
I claim that this spin explanation of the splitting of the electron beam in a Stern-Gerlach
apparatus is an example of a GME. This is certainly a mathematical explanation of a physical
fact that is well accepted by the scientific community. It is not geometric, contrived, or otherwise
problematic, and hence the spin explanation satisfies criterion (A). The mathematics employed is
also not arbitrary in the way that Melia describes. No trivial shift in units would alter the result
that the eigenvalues for the property of electron spin are ½ħ and -½ħ, so (B) is also satisfied. (C)
requires there to be no purely nominalistic explanation of the splitting of the electron beam. This
would require a nominalistic version of quantum mechanics. Save for an interesting attempt from
Mark Balaguer, the standard assumption is that no such nominalized theory is possible. I will
adopt this assumption for now, but will address Balaguer’s view below. With the first three
criteria met, all that remains is to see if the spin explanation also satisfies (D) – there are no
known and physically independently motivated physical facts that could potentially explain the
explanandum such that the mathematics is simply indexing these facts. (D) reflects the core
intuition behind the indexing argument and is without a doubt the most difficult criterion to
satisfy. I will argue that the spin explanation does indeed pass this criterion.
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In my presentation of the spin example it may appear that I cheated. I only provided the
mathematical formalism of spin without ever saying what spin actually is. I did mention briefly
that spin is somehow related to angular momentum. In classical mechanics, angular momentum
can be thought of in two ways. An object’s center of mass can be in motion, called orbital
angular momentum, and there can also be motion about the center of mass, called spin. A classic
example is the Earth revolving around the sun. The actual revolution of the Earth about the sun is
its orbital angular momentum, and we also know that the Earth spins on an axis. This distinction,
though intuitive, is somewhat misleading. The spinning of the Earth about its axis, denoted spinc
to identify it as classical, is actually just a form of orbital angular momentum. So when we say
angular momentum in the classical sense, we actually just mean the orbital angular momentum,
and this includes spinc. Angular momentum in quantum mechanics is significantly different from
the classical case. In quantum mechanics, angular momentum genuinely refers to two separate
quantities. The first is orbital angular momentum. Here the classical definition of angular
momentum can be carried forward. However, unlike in the classical case, spin is its own unique
type of angular momentum. It is not simply a part of the orbital angular momentum like spinc is
for classical mechanics. This indicates that spin is unlike anything we know in classical physics.
This is troubling as in many ways the behaviour of the quantum world is notoriously difficult to
understand. One way to understand quantum mechanical properties is to relate them to their
classical analogue. We may not have a perfect understanding of position in quantum mechanics,
but we have a good enough understanding of position in classical physics to help us get by. This
strategy, although far from perfect, has proven to be quite successful. But if spin is critically
unlike spinc, then how can we understand this property? Of course there is one sense that we do
genuinely understand spin. We understand it in the way presented above – we understand spin as
Pauli spin matrices.
Perhaps I have not motivated the claim that spin is not understandable in terms of spinc enough.
What I have illustrated is that spin and spinc are not exactly the same, as spinc is essentially
reducible to orbital angular momentum, and spin is not reducible to anything at all. This is only
the tip of the iceberg. Essentially what this means is that a particle in quantum mechanics will
always have spin no matter what frame of reference you observe it from. This is entirely unlike
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how we conceive of spinc. Imagine that you are spinning around and holding a video camera at
arm’s length pointed towards your face. When you later watch the video, it does not actually
appear that you are the one spinning, but rather it is the background that is spinning around you.
Of course, to your friends who were sitting there watching you spinc around while filming
yourself, you were certainly spinning in the classical sense. This demonstrates that whether or
not you have spinc depends on your frame of reference. But what does it mean for an electron to
always have spin no matter how we look at it? For this reason, spin is often referred to as
intrinsic angular momentum, and any attempt to understand it in terms of spinc is nonsensical.
Another reason why spin and spinc are incommensurable is that particles in quantum mechanics
are thought of as point particles. If this is the case, then the notion of a point spinning around its
axis is fundamentally incoherent. Pauli makes the distinction between spin and spinc clear when
he said,
Bohr was able to show on the basis of wave mechanics that the electron
spin cannot be measured by classically describable experiments (as, for
instance, deflection of molecular beams in external electromagnetic fields)
and must therefore be considered as an essentially quantum mechanical
property of the electron. (Pauli, 1994, p. 164)
Modern physics textbooks also remark on the distinction. Griffiths states that “[i]t doesn’t pay to
press this analogy [between spin and spinc] too far.” (Griffiths, 2005, p. 171) Similarly,
Townsend writes:
We will see as we go along that such a simple classical picture of intrinsic
spin is entirely untenable and that the intrinsic spin angular momentum we
are discussing is a very different beast indeed. In fact, it appears that even
a point particle in quantum mechanics may have intrinsic spin angular
momentum... [T]here are no classical arguments that we can give to justify
[this]. (Townsend, 2012, p. 2)
In a final appeal to authority, I quote Richard Feynman. With regards to the mathematical laws
of quantum mechanics, he says:
One might still like to ask: “How does it work? What is the machinery
behind the law?” No one has found any machinery behind the law. No one
can ‘explain’ any more than we have just ‘explained’. No one will give
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you any deeper representation of the situation. We have no ideas about a
more basic mechanism from which these results can be deduced.
(Feynman, Leighton, & Sands, 1965, p. 1–10)
Later on, Feynman considers the possibility of trying to account for spin in a purely physical
way.
Until now, it appears that where our logic is the most abstract it always
gives correct results – it agrees with experiment. Only when we try to make
specific models of the internal machinery of the fundamental particles and
their interactions are we unable to find a theory that agrees with
experiment. (Feynman et al., 1965, p. 6–2)
The only way we can explain observed physical facts is through the mathematical formalism of
spin.
Now we can return to the question above: does the spin example satisfy (D)? I believe that it
does. The reason why is clear. If the nominalist asserts that spin is actually some purely physical
property that the Pauli spin matrices simply represents in an indispensable way, the challenge is
that although this may sound coherent, this assertion actually has no content to it. The reason
why such a move worked for the cicada, honeycomb, and Kirkwood gaps examples is because
the nominalist was able to identify, or gesture at, these purely physical properties that the
mathematics is supposedly indexing. Properties such as time, approximately Euclidean space,
gravitation, etc., can be appealed to as the true explanatory factors, and these factors are purely
physical. But what can the nominalist say when we ask what purely physical fact the
mathematics of spin is indexing? The standard route of pointing to some physical fact or
property will not work because, as we have seen, there is no physical property that we can
compare spin to. Without the ability to identify the physical fact(s) in question, this example of
electron spin represents a significant improvement as a candidate for being a genuine
mathematical explanation. There are three possible ways in which the nominalist can still resist
the conclusion that the mathematical explanation of the splitting of the beam of electrons in a
Stern-Gerlach apparatus is a GME. I will show that all three methods are not viable, and that the
nominalist must conclude that the electron spin example meets the necessary requirements of a
GME.
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5 Blocking the Nominalist’s Response
In order to preserve the nominalistic intuition that the mathematical explanation of the Stern-
Gerlach experiment is not genuine, the nominalist can argue either that there is some unknown
physical property that is actually doing the explaining, that there is some known purely physical
property that actually is electron spin, or that there is simply no explanation of the phenomenon
whatsoever. Each of these three possibilities leads to undesirable outcomes, and I argue that to
stubbornly commit to any of them would be ad hoc and ultimately untenable.
5.1 The Unknown Explanation
Consider the first option that there is some physical fact that explains the splitting of the electron
beam in the Stern-Gerlach apparatus which is indexed by the Pauli spin matrices, but we just do
not know what this physical fact is. It could be that we have not yet discovered the true physical
nature of spin, or we could even admit that it is beyond our ability to ever know, but this does not
change the fact that there is some physical fact or facts which are the true explanatory factors. If
this is true, then certainly the mathematical formalism of spin is simply indexing the true
physical nature of spin, whatever that may be. Hence, even though the mathematics may be
indispensable in this representation, it would not be considered explanatory due to the indexing
argument.
The problem with the assertion that there is some unknown physical fact that explains is that it
does not meet the standards set out in criterion (D). We specifically constrained the physical
facts that we can appeal to from possible physical facts to only known physical facts. This was
argued for on the basis that appealing to unknown physical facts is entirely unnaturalistic. There
is no scientific reason to believe in the existence of unknown physical facts.
At the time when this constraint to known physical facts was placed, there was no reason for the
nominalist to object. This constraint did not hurt any of the previous employments of the
indexing argument. In all other cases a known physical fact was cited as the true explanatory
factor that the mathematics was indexing. The nominalist could argue though that now they do
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have legitimate reason to object, and that the prior justification is not sufficient. The claim would
be that we have been guilty of some equivocation. There are actually two senses of explanation.
The nominalist could grant that the Pauli spin matrices explains the splitting of the electron beam
in the Stern-Gerlach apparatus, but it does not really explain it.33 In the first sense of explaining,
this type of explanation is commonly employed when we use things that we do understand to
explain things that we do not understand. I could, for example, explain the behaviour of electrons
by treating them as bullets or billiard balls. Call this type of explaining epistemological
explanation. In the second sense, we explain something that we understand well in terms of
things that we do not actually understand. Call this ontological explanation. So what type of
explanation do we have of spin? The nominalist can assert that the Pauli spin matrices are an
epistemological explanation of the splitting of the electron beam. It may be indispensable to our
understanding, but regardless it is not an ontological explanation. Only ontological explanations
can count as genuine. What is the ontological explanation of spin? We do not know, and in fact
we may never know. But that still does not change the fact that the mathematics explains only in
the epistemological sense and solely represents. The true ontological explanation, even though it
remains hidden to us, is ultimately purely physical.
There are two problems with this reply. First off, it patently begs the question. What the
nominalist is doing here is basically saying that even though mathematics can explain, it cannot
explain in the right way to be considered genuine. They are ruling out the very possibility of
GME to begin with. Of course this would result in their rejection of any supposed example of
GME as they assume that this is impossible right off the bat. The only way to get around this
obvious question begging would be to provide independent reasons for believing that
mathematics cannot explain physical facts in the ontological sense. Any such argument would
necessarily involve metaphysical views on the relationship between mathematics and the
physical world. But ultimately this relationship is the very thing that we are presently
investigating. Brown (2013), for example, maintains the view that mathematics is only
33 I am indebted to James R. Brown for pointing out this objection. See Brown (2013) for his take on quantum
mechanical spin.
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epistemologically explanatory and not ontologically explanatory, but he admits that this stems
directly from his prior belief that mathematics is solely representational. We saw that Daly and
Langford also attempted to suggest that this is the case. They tried to provide independent
reasons that mathematics is solely representational, but their argument was entirely unconvincing
and was more wishful thinking than anything else. It hinged on the generalization that since the
indexing argument is successful against certain supposed examples of GME, that it will be
successful against all possible supposed examples of GME. But this too was entirely question
begging. The target of interest is the honest nominalist who has not a priori decided that
mathematics cannot explain. This nominalist may believe that mathematics solely indexes
physical facts, but this belief comes from their experience of our best scientific theories. If an
acceptable explanation emerges such that the mathematics does not seem to index anything at all,
then the honest response is that it simply is not indexing. If the nominalist chooses to invoke the
distinction between epistemological and ontological explanations and assert that any
mathematical explanation is epistemological, then they are not being honest. For them, the role
of mathematics has been decided a priori, and this begs the question against the possibility of
GME.
The second problem is that all independent arguments that attempt to justify the claim that the
Paul spin matrices are an epistemic explanation fall short. Consider how we know that certain
explanations are epistemological and not ontological. Say we want to explain why we chose a
particular route to take on our road trip. The explanation we give is that when we looked at a
map, the route we chose was the shortest distance on that map from our point of origin to our
destination. This explanation is perfectly acceptable, but it is not ontological. Clearly the route
being the shortest is due not to the distance on the map, but the actual distance in the world. The
map is simply a representation of the physical region. In this case, the reason why we know that
the map explanation is not ontological is in virtue of the fact that we know precisely what the
map is representing. Now consider a scenario where we account for certain behaviour of the
electron by casting an explanation where we treat the electron as a billiard ball. This makes good
sense to do as we understand the nature of billiard balls well, and can thus use this knowledge to
help us understand electron behaviour. This, too, is not an ontological explanation. We know this
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because we know that in some respects, the electron is critically unlike a billiard ball. It is an
empirical fact that electrons at times do not behave like small hard balls, and that any accurate
ontological picture of the electron would have to reflect this.
The point of these two examples is that in each case we have good reason to believe that our
explanation is not ontological. In the first it was due to the fact that we already knew what the
proper ontological explanation was. In the second it was because we knew our explanation was
critically unlike what the actual ontological explanation should be. I claim that in the example of
electron spin, neither the first nor the second reasons present themselves. We are unable to point
to the physical facts that the Pauli spin matrices allegedly represent, nor do we have any reason
to believe that the mathematics somehow conflicts with anything that we know about the
properties of the electron. In the absence of either of these reasons, are we at all justified in
asserting that the mathematical explanation of the Stern-Gerlach experiment is not ontological?
If we are looking for reasons that come from within the practice of science, then clearly the
answer is that we are not justified in maintaining this conclusion. There simply is no scientific
reason to believe that the mathematical explanation is not ontological.
It is now clear that there is no non-question-begging way to maintain that the Pauli spin matrices
are representing some unknown purely physical explanation.
5.2 The Ad Hoc Physical Explanation
A second way around the spin explanation is for the nominalist to claim that the Pauli spin
matrices are actually representing some known physical fact after all. This is the route that
Balaguer (1998) takes. In his defence of fictionalism34, Balaguer argues that contrary to popular
opinion, Field’s project of nominalizing Newtonian gravitational theory was actually a success.
34 Balaguer defends both fictionalism and full-blooded or plenitudinous platonism in his book as being perfectly
viable metaphysical positions. From this he draws the following two conclusions. First, that there will never be an
argument that will settle the dispute over mathematical objects. Secondly, that this is not a limitation on our
philosophical ability, but rather that there is actually no fact of the matter whether or not abstract objects actually
exist or not.
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Even if we suppose this, a problem is that it is unclear if Field’s approach can be extended over
quantum mechanics. David Malament (1982) was the first to raise this objection. Balaguer
recognizes this as a serious concern, and in a similar fashion to Field, he sketches out an
approach to nominalize quantum mechanics.
Balaguer’s scheme for presenting a nominalistically acceptable version of quantum mechanics is
complicated. We will skip over many of the technical details here but will present enough to get
the flavour of Balaguer’s approach. Quantum events are represented by closed subspaces of
Hilbert spaces. It turns out that this representation relationship is quite strong such that the set of
closed subspaces of a Hilbert space is isomorphic to the set of quantum events for a given set of
mutually incompatible observables, such as position and momentum, or spin up and spin down
for spin-1/2 observables. From these two sets it is possible to construct orthomodular lattices.
Call L(H) the orthomodular lattice generated from the set of closed subspaces of a Hilbert space,
and L(E) the orthomodular lattice generated from the set of quantum events for a given set of
mutually incompatible observables. Like their parent sets, L(H) and L(E) are isomorphic.
Malament’s objection is as follows. L(H) is the mathematics used to represent the quantum
events, L(E). What is important to note though is that L(E) is by its very nature abstract; its
members are all possible quantum events, and thus include things that have not occurred which
make them abstract objects. So, even if we are successful in providing a nominalistically
acceptable version of the mathematics in L(H), the very thing that we are representing is still
abstract in nature. As Balaguer puts it, “to replace L(H) with L(E) is just to replace one
platonistic structure with another.” (Balaguer, 1998, p. 120)
Balaguer’s plan to nominalize quantum mechanics has three parts. First, he has to produce a
nominalistic structure that is embedded in L(H). This way the mathematical subspaces of the
Hilbert space just represent this nominalistic structure. Next, he has to get around Malament’s
worry that L(E) is abstract in nature. Balaguer must show that there exists some other ‘set’ which
is both nominalistically acceptable and isomorphic to L(E). Call this new nominalistically
acceptable orthomodular lattice L(P). Finally he has to prove representation theorems between
the mathematics and the physical properties picked out in L(P).
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What I need to do, then, is find a way of taking the closed subspaces of
Hilbert spaces as representing physical phenomena of some sort or other;
if I can do this, I should be able to construct nominalistic structures out of
these physically real things and then prove representations theorems that
enable me to replace the mathematical structures in question – that is, the
orthomodular lattices built up out of closed subspaces of Hilbert spaces –
with these new nominalistic structures. (Balaguer, 1998, p. 120)
We will focus on is Balaguer’s second task: that there are real physical properties of quantum
systems that the mathematics represents. Balaguer is inspired by Field’s approach where Field
argues that for properties such as temperature and length, the mathematics is simply representing
the actual physical properties of temperature and length possessed by the system in question. It is
in the technical details of this approach where most believe that Field failed. Putting aside the
technical details, we see that the fundamental motivation is that properties like temperature and
length are physical in nature and are possessed by the physical object or physical system being
considered. Mathematics is simply an indexing or representational tool for these physical
properties. The problem for Balaguer in using the same approach for looking at quantum
properties is that quantum mechanics is inherently probabilistic. For Field, it is not so
controversial to believe that a physical rod has a purely physical property of some determinate
length that we represent through mathematics. But for, say, quantum mechanical spin, what does
it mean to say that a probability is a physical property?
As we saw in the electron spin example, quantum mechanics tells us that the probability of the
electron having spin-up in the z-axis is 0.5. In order for the indexing argument to run, this
probability would have to be indexing a real physical property. Balaguer’s claim is that the
mathematics of quantum mechanics “represent propensity properties, for example..., the 0.5-
strengthed propensity of a z+ electron to be measured spin-up in the x direction.” (Balaguer,
1998, p. 120) It is propensities that are the physical properties which the probabilities represent.
If this is true, then the Pauli spin matrices are not genuinely explanatory of the behaviour of a
beam of electrons passing through a Stern-Gerlach apparatus. They merely index the true
explanatory facts: the electron possesses the 0.5-strengthed propensity for spin up in the z-axis,
and the 0.5-strengthed propensity for spin down in the z-axis.
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Balaguer’s nominalized quantum mechanics entirely depends on accepting propensities as purely
physical properties. If we do not believe that propensities are nominalistically ‘kosher’, then
Balaguer will have “merely replaced one platonistic structure with another.”(Balaguer, 1998, p.
126) Balaguer does not argue comprehensively for the claim that propensities are
nominalistically acceptable, but he does suggest two ways that he could endeavour to prove it.
The first is “to take quantum propensities... as the basic entities of our nominalistic structures and
simply argue that these things are nominalistically kosher.” (Balaguer, 1998, p. 126) Balaguer
has simply been assuming that this is the case, but he admits that such a view is controversial.
Still, he maintains that we should believe that propensities are physical properties. The
distinction he makes is between physical properties that are properties of particular physical
objects, exist in space-time, and are causally efficacious, and properties-in-abstraction, which
are abstract objects. This distinction is tenuous at best, and Balaguer accepts that he would have
to argue that physical properties can be clearly separated from abstract properties. The most
important step is that even if we grant the demarcation between physical and abstract properties,
Balaguer has to then convince us that propensities fall on the physical properties side, and not the
abstract. His strategy for this is to argue that propensities exist in space-time and are causally
efficacious. Balaguer does not make this argument, but instead suggests that it would run
similarly to how Field argues.
For instance, if we consider a particular particle b, it seems that b’s charge
causes b to move about in certain ways in a magnetic field; but given this,
it seems obvious that b’s charge exists in b (although it might not have any
exact location in b) and it seems almost crazy to say that it exists outside
of spacetime. What would it be doing there? And how could it have causal
influence from there? (Balaguer, 1998, p. 127)
The suggestion here is that propensities are just like charge in this example. They are both
properties of a particular object, exist in space-time, and are causally efficacious. Hence we
should believe that propensities, like charge, are physical properties and not abstract.
There are many problems with Balaguer’s analogy. The critical move is that we are supposed to
believe that abstract properties of particular objects are ‘crazy’ in that existing outside of space-
time leads to two unanswerable questions: where is the property, and how can it have causal
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influence from where it is? In an excellent paper, Baker (2003) convincingly demonstrates that
both these questions are fundamentally flawed. The assumption that Balaguer and most
platonists make is that mathematical objects are abstract which entails that they are non-spatio-
temporal, and are acausal in nature. However, if abstract entities do not exist in space-time, what
does it mean for Balaguer to even ask what they would be doing ‘there’? ‘There’ signifies a
spatiotemporal location, but we are all already assuming that abstract objects are not
spatiotemporal to begin with. As Baker correctly notes,
This sort of looseness is symptomatic of a general tendency to view
abstract objects as akin to ultra-remote, ultra-inert concrete objects... It is
all too easy, for example, to slip from talking of mathematical objects as
being non-spatiotemporal to talking of them as existing outside of space-
time. But ‘outside’ is of course a spatial notion, hence it cannot
legitimately be applied to abstract mathematical objects. (A Baker, 2003,
p. 249)
A similar looseness in reasoning can be seen in Balaguer’s second question. Balaguer is moving
from the assumption that abstract objects are acausal to the objection that they cannot make any
causal difference to the physical world. There is nothing wrong with this move. The problem lies
in Balaguer’s further inference that for abstract objects to make no causal difference means that
abstract objects make no difference to the world at all. This is a leap of reasoning that is not
justified.35 Balaguer is simply assuming that the only way to make a difference to the physical
world is through causation. This assumption is controversial, and also clearly begs the question
against genuine mathematical explanation. Abstract objects may have no causal influence to the
physical world, but that does not mean that they do not make a difference tout court to the
physical world either.
What Baker has shown is that Balaguer’s two unanswerable questions are no good. Without this,
it is not ‘crazy’ to believe that propensities are abstract, and even worse for Balaguer, there is no
reason to believe that propensities are purely ‘physical’ properties which can serve as the
35 Baker (2003) argues that this leap in reasoning critically depends on the argument that mathematical objects exist
‘outside’ or space-time. As we saw, such an argument is fundamentally flawed.
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nominalistic basis for quantum mechanics. Balaguer is aware that we may not be convinced that
propensities in themselves are nominalistically kosher, so he offers a second method in which we
can get around this issue. He asserts that we can nominalize away any commitment to
propensities. Recall that Balaguer believes that Field’s method of nominalizing Newtonian
gravitational theory was successful. Field believes that properties like length are physical
properties possessed by the physical object in question. This belief motivates him to develop a
system that eliminates reference to numbers when referring to the lengths of objects. Balaguer
aims to do the same thing for propensities. Propensities as they are presently formulated are in
the form of an ‘r-strengthed propensity’, such as the electron’s 0.5-strengthed propensity for spin
up in the z-axis. The trick will be to remove reference to the ‘r-strengthed propensity’ and
replace it with something nominalistically acceptable. Balaguer does not go over the technical
details, but he commits himself to employing the exact same method as Field did. A prima facie
problem with this approach is that the general consensus is that Field’s project failed in the
technical details. Balaguer is a notable exception to this consensus, but tying himself to this
approach is unlikely to convince anyone else.
Even if we accept Balaguer’s optimistic outlook that he “does not forsee any real problems”
(Balaguer, 1998, p. 126) eliminating reference of ‘r-strengthed propensities’, there is still a major
flaw in his approach. A strength of Field’s project is that the belief that properties like length and
temperature are purely physical seems perfectly reasonable. On the face of it, it is not difficult to
conceive of temperature and length as existing in space-time, being causally efficacious, being
independent of the mathematical units that we use to express them, and whatever other
conditions we may impose on a property being physical. The reason for this is because we have
independent purely physical experience of these properties. We can see and touch objects that
have length and temperature, and these properties have, or at least we believe they have, a purely
physical basis to them, such as the motion of molecules for temperature. Thus, we have
independent reasons to believe that length and temperature are purely physical properties.
Balaguer takes this belief and tries to appropriate it to the propensity property. At the very start
of his sketch of how to nominalize away reference of ‘r-strengthed properties’, he simply asserts
that “[p]ropensities are just physical properties, like temperature and lengths, and so we can get
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rid of them in the manner of [Field].” (Balaguer, 1998, p. 126) But why should we believe that
propensities are just physical properties? The argument meant to support this claim was shown to
be lacking above. Moreover, Balaguer even admitted that this core belief is controversial to
begin with.
My main criticism with Balaguer ultimately lies in the claim that propensities are just physical
properties like lengths and temperature. Specifically, the objection is that there is a critical
difference between propensities, and lengths or temperatures; the former has no physically
independent motivation for believing it to be purely physical, whereas the latter do. The belief
that propensities are physical properties aligns Balaguer with the propensity interpretation of
probability.36 Roughly speaking, this interpretation is that probabilities are really propensities or
dispositions of physical objects or situations to yield particular outcomes. There are many
challenges that face the propensity interpretation of probability, such as the connection between
propensity and long run relative frequency, or even the precise definition or meaning or
propensity itself. However, the other leading interpretations of probability also face their own
unique challenges. I will not take any sides on the issue here, and will be happy to treat the
propensity interpretation as perfectly viable. The challenge that I will present is the observation
that the propensity interpretation is an interpretation of the Kolmogorov axioms of probability,
and is thus mathematically motivated.37
36 Interestingly, Balaguer states that he is not committed to the propensity interpretation of probability and of
quantum mechanics. He claims that the broad claim that he is committed to is that “quantum systems are irreducibly
probabilistic, or indeterministic.” (Balaguer, 1998, p. 120) This claim is compatible with many different
interpretations of quantum mechanics, excluding hidden variables interpretations. In fact, Balaguer goes on to say
that he need not even commit himself to this claim if some other way to understand quantum mechanics in a
nominalistically acceptable manner surfaces. I find these assertions quite unbelievable. Balaguer seems to
necessarily have to commit to the belief that probabilities index propensities, and that propensities are physical
properties, as simply believing the broad claim does not imply a nominalistically acceptable quantum mechanics.
This is obvious as one could easily be a mathematical realist or believe in that the example of electron spin is a
genuine mathematical explanation of a quantum mechanical physical fact while still believing Balaguer’s broad
claim. I will not argue for this further, but will merely present a version of Balaguer that is committed to the
propensity interpretation of probability and quantum mechanics.
37 The Kolmogorov axioms of probability are not the only way to axiomatize probability calculus, but they have
earned the status of the orthodox interpretation amongst mathematicians. It is also worth noting that some
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The propensity interpretation can be traced back to Peirce (1910), and was advanced
significantly by Popper (1957, 1959). It seems that Balaguer is most influenced by the particular
brand of propensity interpretation put forward by Giere (1973). Popper and Giere were
concerned about how to interpret single case probabilities that present themselves in quantum
mechanics, such as the probability that a particular radioactive atom decaying being equal to 0.5.
Giere believes that statements such as these indicate that the quantum realm is fundamentally
indeterminate. Giere’s solution is to claim that probabilities represent propensities which are
physical properties of particular physical objects, and it is these propensities that lie at the
foundation of quantum mechanics.
At the moment there are no rigorous, widely accepted formalizations of
quantum theory, but it is a good bet that such a formalization will contain
a term with the basic formal characteristics of a probability. The physical
realizations of these quantities will be propensities. (Giere, 1973, p. 476)
In this way, Giere likens propensities in quantum mechanics to how properties such as charge
work in classical physics – they are both physical and not reducible to less theoretical concepts.
This certainly sounds extremely similar to the point of view that Balaguer is espousing.
Christopher Hitchcock presents an interesting challenge for Giere’s interpretation. If
probabilities represent propensities, then propensities necessarily have a mathematical structure.
Giere admits this when he argues that his single case propensity interpretation “provides a
natural interpretation for the whole mathematical theory of probability and statistics since
Kolmogorov.” (Giere, 1973, p. 477) But where does this structure come from? More specifically,
why should we believe that propensities should satisfy the basic laws of probability and statistics
at all? Hitchcock notes that Giere’s analogy of propensities to electric charge is flawed. All our
knowledge of charge is empirical in nature, but out knowledge of propensities is not.
It is surely an empirical matter, e.g., that charge comes in discrete
quantities (one-third the charge of the electron) and in both positive and
negative magnitudes. This cannot be determined a priori. But it is hard to
interpretations of probability do not satisfy the Kolmogorov axioms, however, we will not consider those
interpretations here.
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imagine discovering empirically, e.g., that chances could be greater than 1
or less than 0, or that they are not additive... it is hard to understand what
it could even be to discover empirically that chances can be greater than 1
or less than 0. (Hitchcock, n.d., pp. 10–11)
Hitchcock goes on to illustrate that there is simply no purely physical or empirical motivation for
having propensities model the axioms of probability. However, this assumption is needed in
order for propensities to be utilized as a building block of quantum mechanics.
My challenge to the claim that mathematics indexes propensities is that propensities themselves
are not purely independently physically motivated. The nominalist is helping themselves to
mathematical structure that is motivated solely from the axioms of probability. If this is true,
then applying the indexing argument against the example of electron spin will fail as criteria (D)
states that what the mathematics is allegedly indexing must be physically and independently
supported. Given that propensities as physical properties critically depends on the axioms of
probability, then running the indexing argument will only replace one platonic structure with
another. One way around this is to simply state that propensities follow all the axioms of
probability, but that this is simply a brute fact of nature. In this way it is not influenced by
mathematics, and is arguably physically motivated. However, the problem with this move is that
it is not independently motivated and is entirely ad hoc. This is the very reason why we ruled out
appealing to any facts that are not physically and independently motivated in the first place. Such
an ad hoc explanation is parasitic on the mathematical explanation, and poses no threat to the
genuine nature of the mathematical explanation in question.
5.3 No Explanation
The final option for the nominalist to resist the conclusion that the Pauli spin matrices are a
genuine mathematical explanation for the behaviour of a beam of electrons passing through a
Stern-Gerlach apparatus is to claim that there is simply no explanation at all of this phenomenon.
What we have stumbled upon is an excellent predictive tool in the mathematical formalism, but
this is not an explanation as nothing can explain the nature of the electron. What could it possibly
mean to have a phenomenon that we can accurately predict, but has no explanation? Notice this
is not the same as believing that we just do not yet know the actual explanation, which we
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addressed in 3.5.1. This is the much more controversial position that the explanandum is
fundamentally unexplainable – no possible explanation exists at all.
One way to attack this is to point out that the conclusion is so absurd that it functions as a
reductio to argue in favour of accepting that the mathematical formalism of electron spin is
genuinely explanatory. The conclusion that something exists that is perfectly accounted for and
predictable but is ultimately without explanation is so extreme that it ought to be rejected
immediately. The stubborn nominalist who holds this view would be cutting off their nose to
spite their face. In a last ditch effort to save themselves from admitting the existence of a GME,
the nominalist would be willing to abandon any hope of explaining the behaviour of our most
fundamental entities. This alone is enough to show that such a move is untenable.
While the above rebuttal is perfectly reasonable, I admit that it is not completely airtight. I am
assuming that the sole reason for maintaining that there is no possible explanation of the
behaviour of electrons is because the nominalist is unwilling to admit the mathematical
explanation as being genuinely explanatory. However, there could be entirely independent
reasons for believing that phenomena in the quantum mechanical realm are fundamentally
unexplainable, and if this is the case then such a belief is not as ludicrous as it seems. Our most
basic notion of what a scientific explanation actually is comes from the classical world of
medium-sized objects. Explanations of physical facts almost always cite other physical factors in
a causal relation, or they appeal to deterministic laws of nature. The problem is that the quantum
world is famously unlike the classical picture. On some interpretations of quantum mechanics
causation and deterministic laws are entirely out the window. Given that this is the case, then
perhaps the traditional links that we have become accustomed to between things such as
predictability and understanding to that of explanation do not hold in quantum physics.
This line of reasoning leads to the conclusion that there are two types of explanation: classical
explanation and quantum explanation. Classical explanation we know lots about, but for
quantum explanation we know next to nothing. So when we say that there is no explanation of
the behaviour of electrons, this is due in part to the fact that we do not even know what
explanation means in the quantum realm. Although we have some elements of a classical
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explanation in that the Pauli spin matrices are an excellent predictive tool, we should not confuse
that as having anything to do with quantum explanation. Until we can cache out what quantum
explanations really are, we are justified in saying that there is no explanation of the Stern-
Gerlach experiment.
I have little to say against this objection. If correct, it certainly weakens the force of the electron
spin example that I have advanced as a GME. The problem is that even beginning to explore the
differences between classical and quantum explanation would be an extremely large and difficult
task and is beyond the scope of this dissertation. All I can say is I feel that supporters of such a
radical view carry the burden of proof. If there is such a significant difference between classical
and quantum explanation such that classical factors like prediction and understanding are not
related to explanation in the quantum realm, then it is up to them to establish that this is the case
as such a conclusion is a drastic departure from standard beliefs. In the mean time I am happy to
simply assume that this is not so, and that quantum mechanical phenomena, just like other
classical physical phenomena, have explanations.
6 The Honest Conclusion
Overwhelmingly so, the largest problem in the present debate is that there does not exist any
clear understanding or consensus of what a GME even means. The first half of this chapter was
dedicated to remedying this problem. The strategy was to extract four key criteria from the
various employments of the indexing argument used to block past examples of mathematical
explanation. The remainder of this chapter was aimed to showing that the example of electron
spin does satisfy the four criteria. Satisfying (A) – (C) was not so difficult, but the crux of the
nominalist intuition lies in criterion (D). In order to be considered as a genuine explanation, it
had to be shown that the mathematics of the Pauli spin matrices was not indexing some known,
physically and independently motivated physical fact. This was demonstrated in two ways. First
we demonstrated a fundamental disconnect between quantum mechanical spin and any
classically understood physical property. The standard way of pointing to some other physical
property as being the true explanatory factor is simply unavailable to the nominalist. We also
showed that three other possible responses from the nominalist camp are untenable.
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What has been demonstrated, then, is that the electron spin example satisfies all four criteria laid
out by the indexing argument. The only conclusion for the honest nominalist is that the example
of electron spin is a legitimate candidate for being a GME. This is certainly weaker than the
desired conclusion that the example of electron spin is a GME. In order to move to this stronger
conclusion we must turn our focus towards accounts of scientific explanation.
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Chapter 4 Scientific Mathematical Explanation
Last chapter we developed the following set of five criteria for a genuine mathematical
explanation (GME). A mathematical explanation is a GME if:
(A) the explanation is not contrived and is accepted by the scientific
community as a good scientific explanation,
(B) the mathematics employed is not arbitrary,
(C) there are no purely nominalistic explanations of the
explanandum,
(D) there are no known and physically independently motivated
physical facts that could potentially explain the explanandum
such that the mathematics is simply indexing these facts,
and,
(E) an acceptable account of scientific explanation must
corroborate the claim that the mathematics is explanatory.
The example of electron spin which mathematically explains the behaviour of a beam of
electrons passing through a Stern-Gerlach apparatus was shown to satisfy the first four criteria.
In doing so we are able to avoid the indexing argument, which has been the most potent weapon
for the nominalist. The end result is that the nominalist has no reason to reject the electron spin
example as a candidate for a GME.
The aim of this chapter is to establish that the electron spin example also satisfies criterion (E).
So far, realists have made no attempt at verifying supposed examples of GMEs via accounts of
scientific explanations. The hope has been that simply showing that the nominalist cannot
intuitively resist a mathematical explanation is enough. However, this hope has proven
unrealistic and can be exploited by committed nominalists. Even if an example resists the
nominalists’ intuition that mathematics is solely indexing physical facts, it could still somehow
be the case that the mathematics is not genuinely explanatory. The only way around this is for
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the realist to make the case that according to acceptable accounts of scientific explanation, the
mathematics in our examples of GMEs is indeed genuinely explanatory.
Applying accounts of scientific explanation to mathematical explanations is no easy task. A
prima facie problem is that almost all of the standard accounts of scientific explanation were
never intended to analyze mathematical explanations in the first place. Add to that the
complication that there exists no single accepted account of scientific explanation within the
literature, and it is no wonder why realists have no desire to go down this road. In the last chapter
I made the demand that the nominalist be of an honest variety – one who is open to the
possibility of mathematical explanation. I feel that it is only fair that the realist be honest as well.
The realist needs to have something substantial to say when asked why we should believe that
the mathematics in our examples is explanatory.
Our first task will be to canvas many accounts of scientific explanation. Ultimately, we will
utilize Michael Strevens’ (2008) kairetic account as it is the most likely to be suitable for our
analysis. Next, the kairetic account will have to be adapted such that it will be able to apply to
non-causal, mathematical explanation. Finally, we will use this adapted kairetic account to
analyze some of our examples of GMEs, including the electron spin example. The conclusion
will be that the example of electron spin is corroborated by the kairetic account, and that the
mathematics is found to play the explanatory role of ‘difference-makers’ in our explanation.
Thus, the example of electron spin satisfies all five criteria and will be firmly established as an
example of a GME.
The framework that we will operate under for the bulk of this chapter will be of convincing the
nominalist that the electron spin example is a legitimate GME. Essentially, we will be focussed
on the question of whether or not mathematics can genuinely explain scientific facts. Once this
has been established, we can free ourselves from this restricted perspective and examine what it
is about mathematics in these examples that facilitates a good scientific explanation. We want to
be able to say something to the overlooked question identified in chapter 3: how is it that
mathematics can explain? The hope is to make some inroads into giving a more substantial
answer to what we mean when we say that mathematics is genuinely explaining a physical fact.
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1 Accounts of Scientific Explanation
The literature on explanation is vast and extensive. My aim here is not to assess which accounts
are better than others, but rather to select an account that can help us decide whether or not
mathematics is genuinely explanatory. Three difficulties present themselves in this selection
process. The first difficulty is that there is no widely accepted account of scientific explanation.
All of the classic accounts of scientific explanation have been shown to be fraught with
difficulties that make them each unappealing. In their stead are a collection of newer theories that
differ greatly in their approach and understanding of what an explanation actually is. Given this
landscape, choosing an appropriate account is not a simple or uncontroversial task.
The second difficulty is that many accounts of scientific explanation are exclusively causal
accounts. These theories say that some set of facts, A, explains some physical phenomenon, E,
only if A causes E.38 A causal approach is not surprising as plenty of scientific explanations are
of the causal variety; however, these accounts are not useful for our purposes. Mathematical
explanations by their very nature are noncausal, as by all standard accounts mathematical entities
and their properties are acausal in nature. This greatly restricts the types of accounts of scientific
explanation that we can even consider.
The final difficulty in our selection process is that we have a very particular goal. We do not
want an account of scientific explanation to simply verify that the example of electron spin is a
good scientific explanation. This much is already agreed upon by both realists and nominalists
alike. What is needed is an account that can isolate and identify the contribution of the
mathematics within the explanation. We need an account that can clearly decide whether the
mathematics is playing a genuinely explanatory role or not. In addition, our account must be
sufficiently discriminating in order to satisfy the nominalist. If our account of scientific
explanation trivially decides that all our examples, including the cicada and honeycomb
explanations, have genuinely explanatory mathematics, the nominalist is unlikely to agree that
38 See Salmon (1984) for a classic account of causal explanation.
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this account is the right one to use. Our ideal account of scientific explanation needs to be
sophisticated enough to rule out examples such as the cicada and honeycomb, but at the same
time be able to point to the Pauli spin matrices as conferring an explanatory role.
1.1 The Deductive-Nomological and Pragmatic Accounts
Any exploration into scientific explanation typically begins with Carl Hempel’s (1965)
deductive-nomological (DN) account. The DN account states that the explanans explains the
explanandum if and only if all statements in the explanans are true, there is at least one law of
nature in the explanans, and there is a deduction from the explanans to the explanandum.
Consider the cicada and honeycomb examples. These optimization explanations certainly satisfy
the conditions of truth and deduction, but what of the law of nature? If mathematical theorems
are laws of nature, then we have a good mathematical scientific explanation. If they are not laws
of nature, then we do not have a good explanation. However, there is no way to decide if
mathematical facts are to be laws of nature without entirely begging the question. Understanding
what we mean by laws of nature is complicated, but at the bare minimum laws of nature
fundamentally explain the world around us. If we assume that mathematical theorems are laws of
nature, then this amounts to assuming that they explain physical facts, which is the very thing we
are looking to establish in the first place. Likewise, if we do not grant mathematics law-like
status, then these optimization explanations are not satisfactory on the DN account and hence do
not explain. This too begs the question as not granting mathematics law-like status is equivalent
to claiming that they do not explain the physical world. The problem here, and moreover a
problem in general for the DN model, is that too much hinges on what we consider to be a law of
nature.
The pragmatic account of explanation is most famously championed by Bas van Frassen (1980).
Unlike the DN model, the pragmatic account does not require that an explanation be a formal
deduction. Instead, a good explanation is simply an answer to a why-question. Not any response
to a why-question counts as a good explanation. A why-question must be well-formed in order
for a satisfactory answer to be given. A good why-question has three parts and is fundamentally
contextually based. First, the question must have a topic: the phenomenon that needs explaining.
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There must be a contrast-class which is a set that includes the topic and many alternative
propositions. This contrast-class helps identify the context of the question, and thus the proper
type of answer as well. Lastly, a proper why-question has a relevance-relation which determines
what counts as a possible explanation or explanatory factor. An answer to a good why-question
gives a reason as to why the topic is true as opposed to the other alternatives in the contrast-class
in a way that is relevant to the context of the question.
The pragmatic account is much less formal than the DN account. In applying it to our
optimization explanations it seems obvious that they are good explanations. Why is the
honeycomb hexagonal? The contrast-class implied in the question is that we are asking why,
specifically, is the honeycomb a hexagon and not some other polygon, or even some other
irregular n-sided figure. Also, the type of answer that we are looking for is not simply that they
are hexagons because the honeybees built them that way. We are looking for a reason why all
honeybees build hexagonal honeycombs. The answer is due to Hale’s honeycomb theorem which
states that the hexagon is the optimal number of sides to minimize surface area when tiling an
area. This explanation certainly answers the question in showing why the honeycomb is not
some other n-sided shape.39 Similarly, the electron spin example is an answer to the following
why-question: why does the electron beam split into two distinct beams equidistant from the
center plane? The Pauli spin matrices adequately answers why there are two beams instead of
three or some other number, why they are distinct and not continuous or random, and why they
are equidistant from the center plane as opposed to some other distance.
The present literature goes no further than the DN or pragmatic account in analyzing the
examples of mathematical explanation. Baker (2005, p. 235) seems to think that this is enough to
show that GME’s are confirmed by our accounts of scientific explanation. I find this wholly
unsatisfactory. Even in the best possible scenario where we grant that examples such as the
cicada or honeycomb are indeed good scientific explanations on the DN or pragmatic account,
39 This is just a partial explanation as the full explanation would invoke the other required aspects of an optimization
explanation: appealing to evolutionary biology and ecological constraints.
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the major issue here is that neither account can actually tell us what aspects of the explanation
are genuinely explaining. It could very well be that the mathematics is still not genuinely
explanatory at all even though the overall explanation is mathematical and a good scientific
explanation. This is, in essence, the line of reasoning in the indexing argument.
Recall Nerlich’s example of the cloud of particles that changes in size when moving along a
curved surface. This explanation made use of geodesics, vectors, curvature, etc. If we grant that
Nerlich’s explanation is a good scientific explanation by the DN and pragmatic accounts, what
can we conclude about the mathematics employed in the explanation? Nothing! The reason is
that the role of the mathematics is still unclear even though we grant that the explanation in
general is a good one. Is it the mathematics that is explaining, or is it, as the nominalists contend,
the actual physical properties of curved space-time that explains the behaviour of the particle
cloud? Neither the DN nor the pragmatic account can help us answer this question as neither
account are in the business of actually picking out what the key explanatory contributors are
within an explanation. They are unable to identify the difference-makers to the explanandum.
When trying to establish if GMEs exist or not, what we really need is an account of explanation
that can pick out the proper difference-makers; we need an account that can identify what is
genuinely carrying the explanatory force. It is no surprise that our example explanations seem to
be compatible with the DN and pragmatic account as they are all deductions and answers to why-
questions. But what we are really looking for – the exact role that the mathematics plays – is
beyond the ability of these accounts of explanation to determine.
All that we can take away from this is that, at best, the DN and pragmatic account are open to the
possibility of GMEs, and nothing more. Baker (2005, p. 236) suggests that if we need more
evidence we should look towards our intuitions and those of the biologists who support the
cicada example for an extra nudge in the right direction, but this too is unsatisfactory. Certainly
there are those whose intuitions would lean in the exact opposite direction. For example, one
could have the intuition that mathematics is solely a representational tool; the true explanatory
factors are ultimately physical in nature. This is, of course, the very same intuition that the
nominalist holds. In any case, realists surely want to do better than leaving the status of GMEs
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and accounting for the explanatory role of mathematics to raw intuition, and thus neither the DN
nor the pragmatic account will be our account of choice.
1.2 The Unification Account
The unification account of scientific explanation was advanced by Michael Friedman (1974) and
improved upon by Philip Kitcher (1989). The unification account claims that what makes an
explanation good, or what enhances our understanding of the world, is that it unifies otherwise
seemingly independent different phenomena. Thus, a good scientific explanation leaves us with
less total phenomena that we accept and treat independently than we had before the explanation.
Many examples of such unification exist in the history of science. The kinetic theory of gases
entails the truth of many other laws, such as Boyle’s law, Charles’s law, Avogadro’s law,
Graham’s law of diffusion, and more. Newton’s laws of motion unifies the behaviour of
terrestrial and celestial bodies. This unification is what makes the kinetic theory and Newton’s
laws explanatory. No longer do we have many different, seemingly independent brute facts about
gasses or bodies. Instead we have a unified system where we can derive the exact same results,
and thus explain them.
It does seem that there are many examples in science where mathematics unifies seemingly
disparate phenomena. Colyvan (2002) gives an example of the use of complex numbers in
physics. Consider two differential equations:
(1) 𝑦 − 𝑦′′ = 0,
and
(2) 𝑦 + 𝑦′′ = 0,
where y is a real-valued function of a single variable. (1) describes some physical system
exhibiting growth, and (2) describes certain periodic behaviour. Equation (1) can be solved using
standard real algebra, but (2) requires complex methods to solve. Since complex algebra is a
generalization of real algebra, we can in fact employ the exact same method of solving (2) as we
can for (1). Complex algebra, then, unifies the mathematical theory of differential equations as
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well as the physical theories that make use of differential equations. Without complex numbers
we would have to treat (1) and (2) as disparate and independent phenomena.
Melia (2002) rejects this example. He points out that there is a major difference between
providing a unified account to solving methods and providing a unified account of apparently
independent physical phenomena. One can grant that complex algebra is a genuine example of
unification within mathematics, but this does not mean, contra Colyvan, that complex algebra
also unifies physical phenomena. Melia’s point is that the unification of algebra in mathematics
does lead to a sort of unification in solving methods in physics, thus allowing for a smaller tool
chest for the practicing physicists to work with, but this does not imply anything about the actual
nature of the physical world. The kinetic theory of gas unifies because it explains Boyle’s law,
Charles’s law, and many more laws in virtue of the fact that gas is made up of a large number of
molecules in constant, random motion. These laws hold as all gases are nothing more than the
molecules invoked by the kinetic theory. However, no one would claim that simply because (1)
and (2) have a unified solving method that this implies that they necessarily have a shared
underlying reality.
A more promising example is how the mathematics in the cicada example can be said to unify in
the sense that we use the exact same proof to explain the periodic life-cycle of the 13-year cicada
as for the 17-year cicada. The only difference between the two is that the ecological constraints
are different as they inhabit different regions of North America that possess different weather
patterns. Critically, there is also no appeal made to any sort of unification on the mathematical
end of solving methods in these explanations. It is the exact same mathematical theorem – prime
numbers minimize intersection – that unifies the physical phenomena of the 13 and 17 year
cicada. This avoids the challenge that there is a unification of solving methods on the
mathematical end only, and does seem to more clearly point to the conclusion that there is a
shared underlying nature to the cicada life-cycles, and that this nature is revealed by
mathematics. But herein lies a new but familiar problem. What is doing the unifying – the
mathematics or something else? Is the underlying unifying nature mathematical or purely
physical? Is it even possible for mathematics to unify the physical world?
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What we see is that when we invoke unification we end up with another version of the indexing
argument. Instead of claiming that the mathematics does not confer an explanatory role in
scientific explanations, the nominalist now asserts that mathematics does not confer a unificatory
role. All we have done is shift our attention from explanation to unification – we have solved one
problem by replacing it with another of equal difficulty. A major reason why unification does not
help us is that it is difficult to understand the exact relationship between unification and
explanation. Morrison (2000) makes a strong case that there are examples of unification in
science that are not explanatory. She argues that the unification which is achieved by using
mathematical structures and representations often results in a loss of explanatory power. Pincock
(2011) claims that there are also examples of potentially genuine mathematical explanation in
which their explanatory nature is not due in any part to unification.
All of this points to the fact that we should be reluctant to turn to the unification account in order
to determine the status of GMEs. Even without the major challenges the account faces in light of
Morrison and Pincock, there is still the overarching issue that the unification account cannot
precisely determine the role that mathematics is playing in even the explanations that seem to fit
the unification model best. Just as we saw in the DN and pragmatic accounts, the unanswered
question is whether or not the sole purpose of mathematics in these explanations is to represent
underlying physical facts, and as it stands the best we can do is point to our intuitions in order to
resolve this question.
1.3 The Statistical-Relevance and Counterfactual Accounts
The statistical-relevance account from Salmon (1971) and the counterfactual account popularized
by David Lewis (1973) are both causal accounts of explanation. As stated above, given the
standard assumption that mathematical entities are acausal, any causal account of scientific
explanation will not do the job of analyzing supposed examples of GME as none of these would
be considered a good explanation in the first place. Notwithstanding, there is still an important
benefit in looking at these approaches. These accounts are unlike the DN, pragmatic, and
unification accounts in that they are in the business of identifying the key (causal) factors that
explain an explanandum. The aim of statistical relevance and counterfactuals is to pick out the
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proper difference makers from the many other factors within an explanation. If it is possible to
adapt the statistical-relevance or counterfactual approaches to look at non-causal explanations,
then this could help us in determining if mathematics is genuinely a difference-maker.
The statistical-relevance account states than any factor which is statistically relevant to the
occurrence of the explanandum is explanatory. A good explanation reveals some (or all) of these
factors. To identify if a factor, a, is statistically relevant to our explanandum, E, we turn to
probability. If the probability of E occurring is different from the probability of E occurring
given that we know that a is true or has obtained, then we say that a is statistically relevant to E.
In conditional probability notation this is to say: 𝑃(𝐸|𝑎) ≠ 𝑃(𝐸). But how do we arrive at these
probabilities? For our mathematical explanations, let m be the mathematical theorem that our
explanation depends on. We need to know whether or not 𝑃(𝐸|𝑚) ≠ 𝑃(𝐸) to ascertain if m is an
explanatory factor for E. Even if we are able to assign a value to 𝑃(𝐸|𝑚) in some unproblematic
way, the difficulty is in trying to asses 𝑃(𝐸) alone. That is, we have to determine the probability
of E occurring as if m is false. But m is not like some standard physical fact or state of affairs; it
is a mathematical theorem, and thus on all standard accounts is necessarily true. Is assigning a
value to 𝑃(𝐸) alone even possible?
Closely related is the counterfactual account of explanation. The counterfactual account is rooted
in an account of causation where we can identify if a factor, b, is a cause of an event, E. We do
so by considering counterfactual conditionals. b is a cause of E if and only if the following two
counterfactuals are true: “if b occurs, then E occurs”, and, “if b had not occurred, then E would
not occur.” In the case of explanation, if we are at all considering b to be an explanatory factor
for E then we already know that b and E have occurred or are true and thus the first
counterfactual is trivially true. What needs to be checked is the second counterfactual. If the
second counterfactual is also true, then b is a cause of E and we can conclude that b (in part)
explains E. The greatest challenge for the counterfactual account is determining if the second
counterfactual, “if b had not occurred, then E would not occur”, is true or false. In order to
evaluate this truth value, Lewis uses his theory of possible worlds. Look to the closest possible
world in which b is false, or equivalently where ~b is true, and then determine whether or not E
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occurs. To evaluate closeness we use a relation of comparative similarity between worlds. The
more similar a world is to our actual world, the closer it is to our world.
As above, consider a mathematical explanation where m is the mathematical fact that supposedly
explains E. Running the counterfactual analysis on m to see if it makes a difference to our
explanandum, E, requires us to look towards the closest possible world where ~m is true. But
what would such a world be? In our world, m is logically implied by the axioms of mathematics
which we believe to be consistent.40 In a world where ~m is true, we have one of two options.
The first is that the axioms of mathematics are the same, but somehow we are able to deduce
both m and ~m. This world would be inconsistent. The other option is to have a consistent world
where the mathematical axioms are different than what we have in our world. However, under
the standard interpretation of mathematics this is an impossibility. Even supposing that it is
possible, there would be no way to decide which new set of axioms would represent those in a
closest possible world. In both of these options, the worlds we are looking at are either
inconsistent or are so far removed from our actual world, if they can even exist at all, that
evaluating the status of E in these worlds is impossible.
More sophisticated counterfactual approaches have been developed (cf. Lewis 2000, Woodward
2003); however, all these approaches fail in analyzing mathematical explanations for the same
reasons. Although they may be satisfactory accounts in causal cases, the fundamental problem in
using them to analyze mathematical explanation is that all these accounts depend on asking what
would be the case if a mathematical theorem is somehow false. This works just fine for physical,
actual events, but is essentially incomprehensible when discussing mathematics. If we are to
adapt an account of difference-makers in order to resolve whether or not mathematics is truly
making a difference in scientific explanation we must look beyond any account that requires us
to consider what would happen if a particular mathematical theorem is in fact false.
40 I speak of axioms of mathematics to maintain generality. If m was a theorem of arithmetic, then it would only
need to depend on the axioms of arithmetic, and not, say, of Euclidean geometry. No matter what axioms m actually
depends on, the following argument can still be run.
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2 The Kairetic Account
Strevens (2008) advances a new account of scientific explanation in the tradition of identifying
difference-makers in causal explanations. What makes Strevens’ kairetic account different is that
it frees itself from the use of counterfactuals. Because of this, Strevens suggests that the kairetic
account can be adopted to identify difference-makers in non-causal explanations, and perhaps
even mathematical explanations as well. Strevens uses an approach similar to
Mackie’s (1974) INUS condition whereby the statement ‘c is a cause of E’ is true just in case c is
an insufficient but non-redundant part of an unnecessary but sufficient condition for the
occurrence of E. The key for both the INUS condition and the kairetic account is that a non-
redundant part of a sufficient condition for E is one that cannot be removed from the explanation
without invalidating the entailment of the occurrence of E. The removal process here is different
than in the counterfactual. There is no need to consider ~c, but rather we only remove c from our
picture; that is to say, after our removal of c there is simply no mention of it, negation or
otherwise. This bodes well for mathematical explanation as we no longer need to consider what
happens if a mathematical theorem is false.
Even though the kairetic account avoids counterfactuals and possible worlds, it is still presented
as an account for causal explanations. Given that, how can we apply it to noncausal
mathematical explanations? The answer lies in Strevens’ belief that all theories of explanation
have two fundamental parts.
I will tentatively propose that the complete philosophical theory of
explanation is modular: it consists of two components, a criterion for
explanatory relevance that is the same in every kind of explanation, and a
domain-specific dependence relation. The relevance criterion selects from
the given domain of dependence relations those that must be appreciated
in order to understand the phenomenon to be explained. When the domain
of dependence is causal, the result is a causal explanation. When it is
mathematical, the result is a mathematical explanation. And so on.
(Strevens, 2008, p. 5)
The kairetic account is comprised of two parts. First is the kairetic criterion which is the criterion
for explanatory relevance. This criterion identifies those aspects of an explanation that “make a
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difference to whether or not the phenomenon occurs.” (Strevens, 2008, p. 5) Second is the
domain of explanation. The kairetic account that Strevens presents in his book is a combination
of the kairetic criterion coupled with a causal domain of explanation. But given this modular
style, it seems that if we are able to isolate the kairetic criterion and apply it with the domain of
mathematical explanation, then we should be able to determine if the mathematics is genuinely
making a difference to the physical explanandum.41
2.1 The Kairetic Criterion
The basic idea behind the kairetic criterion is straightforward. Start with a model, M, that entails
the explanandum, E. All the statements within M must be true. In other words, M must be
veridical. Now optimize M through a process of abstraction performed on the individual
statements within M. At each stage of optimization we must verify that M still entails E. Once we
can no longer abstract anything in M without violating the entailment of E, we stop. This
optimized model is called a kernel, K, of E. The claim is that every statement within K is a
difference-maker for E. What difference-makers we come up with may depend on what our
starting model looks like. This is not a problem for the kairetic criterion, but rather a feature. To
not be a difference-maker is to be removed from all models via the optimization procedure, or
equivalently, to not be featured in any kernel that entails E.
The process of abstraction is essential to the kairetic criterion. By abstracting we can remove the
mention of unnecessary and unexplanatory details, and replace them with more general and
explanatorily relevant counterparts. Technically, a model A is an abstraction of a model M just in
case all influences described by A are also described by M, and every proposition in M is implied
by the propositions in A. Practically speaking, we can take specific statements and make them
41 Although Strevens in the quotation above makes reference to mathematical explanation, it is unclear what type of
mathematical explanation he means. In a subsequent section, Strevens returns to mathematical explanation and
seems to mean mathematical explanations of physical facts. However, he makes it clear that in these explanations
the mathematics is simply representing something physical. He also at times seems to refer to mathematical
explanation as mathematical explanation of mathematical facts. At no time does Strevens seem to mean what I call a
genuine mathematical explanation of physical facts where the mathematics is not representing but is instead actually
conferring explanatory power to the overall explanation.
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more general. Consider a 10kg cannonball thrown at a window where the window shatters. The
fact that the ball weighs exactly 10kg should not make a difference to the window breaking. For
any model M that contains the precise weight of the cannonball, we could generate a more
abstract model by replacing the statement ‘the cannonball weighs 10kg’ with a statement that
says, say, ‘the ball’s mass is greater than 1kg’. In doing so we can arrive at an optimized kernel
that yields proper difference-makers. Notice thought that if we try to remove the weight of the
cannonball entirely, or set the weight incredibly low such as greater than 0.1 grams, then the
model would no longer entail the breaking of the window. Hence such an abstraction is not
allowed.
Strevens notes two potential technical problems with the abstraction method. The first is path
dependence. Depending on which statement you optimize first you may arrive at a different
kernel. This is problematic as although many kernels may exist, each particular starting model
should always arrive at the same kernel via the abstraction process. To solve this problem,
Strevens states that there should be a unique end point to the abstraction operation for a
particular starting model. This end point is the maximal abstraction of M while still entailing E,
and we should call this maximally abstract model the kernel, K. It could be that certain starting
models do not have a definitive maximally abstract model, but in these cases Strevens can appeal
to other notions to help him decide on a well-defined set of difference-makers.42
The second issue facing the abstraction process is that of cohesion. The cohesion problem arises
as a disjunction of two models results in a newer, more abstract model. Consider models M and
N such that only one of these models is veridical and entails the event E. Now create a new
model, D, that is the disjunction of M and N. D is a veridical model that entails E and is more
abstract than either M or N on their own. But this implies that the disjunction is a difference-
maker for E, and not anything from simply M or N alone. Strevens writes that “this is not a
42 Strevens suggests that we could look at the facts that all the most abstract models agree on as our difference-
makers. Or, we could use some other possible criteria for comparison, such as notions of generality, in order to
adjudicate between rival kernels generated from the same model. Either way, Strevens states that “[n]othing crucial,
I think, turns on the choice.” (Strevens, 2008, p. 101)
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tolerable conclusion. Even if the disjunction can be said in some extenuated sense to be a
difference-maker, the disjunct... ought to be a difference-maker too.” (Strevens, 2008, p. 102)
Strevens states that models such as D lack cohesion. Although D is, in a sense, a more abstract
model than M alone, its lack of cohesion makes it a bad explanation of E. Strevens defines
cohesion via causal contiguity. “A model is cohesive, I propose, if its realizers constitute a
contiguous set in causal similarity space, or, as I will say, when its realizers are causally
contiguous.” (Strevens, 2008, p. 104) There is a trade-off between a cohesive model and an
abstract model. Strevens makes cohesion and abstraction, then, not strict requirements of the
abstraction process, but rather recommended desiderata. The goal is not to strictly satisfy one or
the other, but rather to maximize combined cohesion and abstractness.
The optimization procedure can be summarized as follows:
the explanatory kernel corresponding to a veridical deterministic causal
model M with target E is the causal model K for E that satisfies the
following conditions:
1. K is an abstraction of M,
2. K causally entails E,
and that, within these constraints, maximizes the following desiderata:
3. K is as abstract as can be (generality), and
4. The fundamental-level realizers of K form a causally contiguous set
(cohesion). (Strevens, 2008, p. 110)
The kairetic criterion states that all statements within a kernel, K, that is arrived at via the
optimization procedure are difference-makers for E.
2.2 Adapting the Kairetic Criterion
Before we are in a position to use the kairetic criterion on mathematical explanations we must
resolve an apparent inconsistency. The reason why Strevens’ kairetic account is appealing in the
first place is due to its two-level approach. The kairetic criterion is supposed to be a method for
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finding difference-makers in all domains of explanation. However, when we take a look at the
definition of the optimization procedure, which is the essential part of the kairetic criterion, we
see that this is not the case. Steps 2 and 4 make reference to causation within the definition of the
procedure. It appears that the kairetic criterion is not independent of causation, and there can be
no two-level approach.
Strevens gives us reason to think that this may be a fixable mistake. He writes,
the difference-making criterion takes as its raw material any dependence
relation of the ‘making it so’ variety, including but not limited to causal
influence. Given a relation by which a state of affairs depends on some
other entities, the kairetic criterion will tell you what facts about those
entities are essential to the dependence relation’s making it so. (Strevens,
2008, p. 179)
An attempt to recast the optimization procedure results in the following:
1. K is an abstraction of M,
2ʹ. K entails E in a ‘making it so’ type of entailment,
and that, within these constraints, maximizes the following desiderata:
3. K is as abstract as can be (generality), and
4ʹ. K is cohesive,
and the ‘making it so’ type of entailment and how we define cohesion
are dependent on the domain of explanation.
The original version of the optimization procedure presented above had already selected causal
explanation for its domain, and thus step 2 defined the ‘making it so’ type of entailment as causal
entailment, and step 4 defined cohesion as causally contiguous. What is now needed is to use
noncausal mathematical explanation as our domain and define steps 2ʹ and 4ʹ accordingly.
In considering how mathematics ‘makes it so’, one place we could start is the deductive power
that mathematics brings to the table. Certainly a key benefit to mathematics is that it functions as
an excellent tool for deductions. Although this is pretty much universally agreed upon, Strevens
does not believe that deductive power is the key contribution of mathematics when it comes to
explanations.
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The derivational conception of mathematics entirely fails to capture the
illumination that mathematical facts bring in the explanations of...
[physical] phenomena. It is not enough to be told that these phenomena
follow mathematically from the relevant laws and boundary conditions.
Somehow, in grasping the way that they follow – in understanding the
mathematics as well as the physics of the scientific treatment of the
explanandum – you come to understand the phenomena better. It is almost
as though, by looking into the mathematical structure of the derivation,
you can see the forces at work in nature itself. (Strevens, 2008, p. 303)
There must be something beyond raw deductive power that mathematics contributes in a GME.
Strevens suggests two ways in which mathematics is more than a deductive tool. First, it is more
than just knowing that the explanandum logically follows from the explanans, it is the way in
which it follows. Second, in understanding the mathematics in itself we can gain insight into our
overall explanation. Beyond merely suggesting these two ways in which mathematics contributes
to scientific explanation, Strevens does nothing to expand or explain what he means. I take his
second claim to refer to mathematical explanations of mathematical facts and how this relates to
mathematical explanations of physical facts. I will analyze this relationship in 4.4.1, but for now
will leave it aside.
What can be said about Strevens’ first suggestion that we must grasp the way in which the
explanandum follows from the mathematics? Unfortunately, not much. Perhaps what we need is
a better understanding of the relationship between mathematics and our best scientific theories.
The problem with this is that any full account of this relationship is sure to have certain
metaphysical assumptions. Among them would be the very things that we are investigating here:
can mathematics genuinely explain, and is mathematical realism true? If we are to assume any
metaphysical stance on these positions it would render the rest of our argument utterly useless.
Fortunately, Strevens believes that it is unnecessary to make any metaphysical assumptions.
The prospect for making sense of mathematics’ explanatory role as
something more than derivational, then, might seem to turn on
foundational questions about the nature of mathematics and its relation to
the world. But it is possible to see how the derivational view of
mathematics’ role in explanation falls short without indulging in any of
this excitement... Mathematical reasoning in explanation is supplying
something more than deductive glue, but you need no metaphysical
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assumptions to spell out its additional contribution to explanatory
goodness. (Strevens, 2008, p. 304)
This sentiment is what makes the kairetic account so appealing to those interested in
mathematical explanation in the first place. We can allegedly use the account independently of
our metaphysical beliefs to analyze and understand how mathematics contributes to scientific
explanations, if at all.
Identifying the exact conditions for ‘making it so’ is a difficult task. The goal is to have
something not so weak as to imply that the bulk of our mathematical explanations are actually
GMEs; the condition needs to be strong enough to rule out explanations that we do not consider
truly genuine. At the same time, the conditions for ‘making it so’ cannot be so strong as to
trivially rule against the existence of GMEs. I propose that we borrow the criteria from the
indexing argument to help us set a condition for ‘making it so’. This makes sense in the present
context as it is the nominalist who we are trying to convince. If we pick a condition that is too
weak by their standards then they will have reason to reject our conclusions. Consider then:
2M. K entails E such that:
(a) there is no purely nominalistic counterpart that also entails
E,
and,
(b) there is no purely physical object or property that the
mathematics is representing.
2M is perfectly in line with the nominalists’ concerns raised within the indexing argument. The
electron spin example was shown to resist the indexing argument in chapter 3. If we tailor 2M in
the above fashion, then we can corroborate the claims from chapter 3 and draw an even stronger
conclusion: not only does the electron spin example resist the indexing argument, but by the
nominalists’ own standards, we can show that the mathematics makes the physical explanandum
so. I am not proposing that 2M is the correct description of how mathematics makes it so in a
GME. In fact, I feel that 2M is actually too strong, but this is exactly what we need to convince
the honest nominalist that GMEs exist. Once it can be established that GMEs exist, then we can
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revisit the issue of trying to pin down more reasonable criteria for how mathematics makes it so
when explaining physical phenomena without having to worry about satisfying the nominalists’
demands.
Now we must define what cohesion means for mathematical explanations. Unfortunately,
making sense of cohesion in noncausal explanations is much more problematic than
understanding the ‘making it so’ type of entailment. The problem of cohesion is strictly a logical
problem. Recall that the cohesion desideratum is meant to block the move of making more and
more abstract models via disjunctions which obfuscate the true difference-makers. Strevens’
solution was to define cohesion via causal contiguity. Clearly this will not work when our
explanations are not purely causal in the first place. A noncausal explanation is not without its
own relevant causal factors. In the electron spin example, the causal factors include the magnetic
field, the detectors, etc. So for the causal components of our explanation we can certainly make
use of causal contiguity as a requirement for cohesion. Unfortunately, this does not save us from
disjoining mathematical sets together that obscure the truly relevant mathematical factors in our
explanation.
In addition to the original presentation of cohesion, I present a new form of the problem
particular to mathematical explanation that has nothing to do with disjoining sets of statements.
Suppose we have a mathematical theorem, T, that our mathematical explanation, M, of some
explanandum, E, seems to depend on. Ideally we would like to conclude that T is the difference-
maker for E. In generating a kernel via optimization we arrive at the following difficulty. M
would necessarily have to include T and also the proof of T from a set of mathematical axioms,
A. Now let us make a more general model, N, from M by removing T and its proof and replacing
them with just the collection of axioms, A. Obviously T is deductively entailed by A, so our new
model N still entails E just as well as our original model. N is more general than M as A implies
many other theorems that T alone may not. We thus arrive at the conclusion that it is the set of
axioms A which are the difference-makers for E, and not T in particular. There appears to be
something wrong with this conclusion, and following Strevens, I claim that N lacks cohesion.
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To summarize, there are two forms of the cohesion problem that present themselves in
mathematical explanations. The first is in creating more general sets via disjunction, and the
second is making a more general set by only including the axioms of particular mathematical
disciplines instead of the specific mathematical theorems in play. Both forms of the problem
result in obscuring what we want to conclude as the true difference-makers in mathematical
explanations. I propose the following definition for cohesion in a mathematical explanation that
will block both of the above problems.
4M. K is cohesive such that,
(a) The fundamental-level realizers of the causal factors in K
form a causally contiguous set,
(b) The entailment of E from K requires no additional
mathematical derivation,
and,
(c) The mathematical statements in K form a maximally small
set.
(a) takes care of the causal factors in the same manner as Strevens. (b) successfully blocks the
new form of the cohesion problem. As above, consider a mathematical theorem, T, that our
mathematical explanation, M, of some explanandum, E, seemingly depends on. In showing that
M entails E, we would not have to do any further mathematical deductions as E depends only on
the mathematical result T which is already in our model. Let N be the more general model of M
that removes T and its proof, and replaces them with the axioms that imply T. In order to show
that N entails E, we would necessarily have to deduce T, but this time the deduction would be
done outside of our kernel. Condition (b) stops us from doing this. T and its proof must remain
within any kernel, and thus we block this form of the cohesion problem.
Finally, condition (c) together with (b) blocks the problem of making mathematical sets via
disjunctions that are more general. Again, let M be the model that is a mathematical explanation
of E. M includes mathematical theorem T and its proof, and thus M satisfies the cohesion
requirement (b). We have two options for generating a new model, D, using disjunction. First,
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we could disjoin T with some other mathematical statement. Call this resultant disjunction S. D
in this case is more general, but we have the unappealing conclusion that S is a difference-maker
for E. Now consider the sets created by taking only the mathematical statements in M and only
the mathematical statements in D. The latter set generated from D has at least one more element
in it than the set generated from M; in particular it includes the additional statement S. So N
violates the condition (c) and is not cohesive.
The other option is quite different. Now let D be a new model that includes any set of axioms, a
true mathematical statement, and its proof. For example, D could contain the axioms of
arithmetic and the statement ‘2+2=4’ along with a simple derivation. In order for D to entail E,
all we need to do is make a disjunction of this true mathematical statement with T. Call this
disjunction S. Now we can see that D entails E as S entails E, and N is more general than M. The
danger with this type of disjunction is that it is possible for the set of mathematical statements in
D to have a smaller cardinality than the set of mathematical statements in M. If this is so then
condition (c) is satisfied. However, what will result from this type of disjunctive explanation is
that condition (b) will be violated. While it is certainly true that S entails E, in order to show that
this is the case we need to derive T as a singular statement. But this mathematical derivation
would be performed outside of our kernel, which is not allowed by the cohesion requirement.
Thus, (b) and (c) together block this second type of disjunctive explanation.
The above discussion on cohesion is technical. All it boils down to are some restrictions on how
we should generalize the mathematics when creating a kernel such that we do not bury the true
difference-makers in strange statements generated by disjunction or by over-generalizing. This
problem will not present itself in my application of the kairetic criterion below. The main
takeaway from all this is simply to motivate the idea that the kairetic criterion can be
appropriated for analyzing noncausal mathematical explanations.
Putting everything together, we now have a kairetic criterion that can operate on mathematical
explanations. Starting with a veridical model, M, that is a mathematical explanation for an
explanandum, E, the corresponding explanatory kernel, K, for E is obtained by the following
optimization procedure:
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1. K is an abstraction of M,
2M. K entails E such that:
(a) there is no purely nominalistic counterpart that also entails
E,
and,
(b) there is no purely physical object or property that the
mathematics is representing,
and that, within these constraints, maximizes the following desiderata:
3. K is as abstract as can be (generality), and
4M. K is cohesive such that,
(a) The fundamental-level realizers of the causal factors in K
form a causally contiguous set,
(b) The entailment of E from K requires no additional
mathematical derivation,
and,
(c) The mathematical statements in K form a maximally small
set.
The kairetic criterion states that all statements found in the explanatory kernel obtained via the
optimization procedure are difference-makers for E. Thus, if mathematical statements appear in
the kernel, then we can conclude that mathematics is a difference-maker for a physical
phenomenon, and hence GMEs exist.
3 Applying the Kairetic Criterion
With a working kairetic criterion in hand, we can now apply it to see if the mathematics in our
many mathematical explanations counts as difference-makers or not. Recall Baker’s
mathematical explanation of the prime numbered life-cycle of the cicada.
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(1) Having a life-cycle period which minimizes intersection with
other (nearby/lower) periods is evolutionarily advantageous.
(biological law)
(2) Prime periods minimize intersection (compared to non-prime
periods). (number theoretic theorem)
(3) Hence organisms with periodic life cycles are likely to evolve
periods that are prime. (‘mixed’ biological/mathematical law)
(4) Cicadas in ecosystem-type E are limited by biological
constraints to periods from 14 to 18 years. (ecological
constraint)
(5) Hence cicadas in ecosystem-type E are likely to evolve 17-year
periods.
This explanation is incomplete as missing from it are the justifications for statements (1), (2),
and (4). The justification for (1) would necessarily include the fundamental laws or assumptions
in evolutionary biology. It needs to be shown that (1) follows from these laws. (2) requires a
mathematical proof in number theory to demonstrate its status as a mathematical theorem.
Statement (4) depends on many areas such as biology, ecology, weather patterns, and so on. A
complete explanation would include all these extra parts.
Let such a complete explanation be a model, M1, which entails the explanandum, E. Attempt to
generate a kernel, K1, from M1 via the optimization procedure for mathematical explanations.
The key to the optimization procedure is to abstract away or remove all explanatorily irrelevant
details while still ensuring that our model logically entails E. What sorts of things would be
abstracted away? It could be that our justifications of (1) and (4) contain superfluous details that
we could remove without violating the entailment. We may abstract away some of the biological
laws and replace them with laws of physics, if such a reduction is possible. Although this is
somewhat vague, what should be clear is that we cannot abstract away any of (1) through (5).
Imagine trying to remove (1) from the explanation. Without (1) there is no reason as to why
minimizing intersection is desirable, and thus the fact that cicada have a prime life-cycle is no
longer entailed by our model. The same problem obtains if we try to remove (2), the
mathematical theorem, from our model. This theorem is essential for entailing the conclusion
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that cicadas should have prime periods. Without (2) we would be unable to entail the
explanandum, and hence (2) cannot be removed from our model. K1, must include all of
statements (1) through (5), as well as the justifications and proofs of statements (1), (2), and (4)
that have been made abstract and general.
Now consider Saatsi’s explanation where we use sticks to explain why cicadas have a 17 year
life-cycle. Many features of this explanation would be the same as in Baker’s mathematical
explanation, such as statements (1) and (4) along with their respective justifications. What is
different is that there is no mention of primeness, and no mathematical theorem. Instead we have
facts about sticks and units. Call this second explanation that has no mathematical theorem the
model, M2, which entails E. Attempt to generate a kernel, K2, by the same process as above. An
interesting part of the process occurs when we run the abstraction procedure on the statements
that involve sticks. Certainly the existence of sticks is not necessary to the overall explanation.
All that is actually needed to perform Saatsi’s nominalistic ‘proof’ are many physical objects that
all have roughly the same length. This shows us that sticks themselves are not difference-makers
for cicada life-cycles, which is of course what we expect. However, what cannot be removed is
the ultimate conclusion of the stick ‘proof’ that 13 and 17 roughly equal length units minimize
intersection with other surrounding equal length units. Without this conclusion we would not be
able to entail the prime life-cycle of the cicada.
Finally, recall the third explanation of the prime-life-cycle of the cicada which does not invoke
mathematical theorems or sticks. Instead, this explanation invokes the physical fact that 13 and
17 units of time minimize intersection, and this is the critical explanatory fact. Call this
explanation the model M3, and attempt to generate a kernel, K3, in the same way as above. It
should be clear that the physical facts of units of time cannot be removed from our explanation
as without it we cannot entail the explanandum.
K1, K2, and K3 all satisfy criterion 1 of the optimization procedure – they are abstractions of their
respective starting models. Although I have not detailed it here, I will simply assume that K1, K2,
and K3 were generated in a way to maximize the desiderata of generality and cohesion. This
assumption is warranted as we are not looking to generate bizarre kernels that involve logical
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tricks, such as disjunction. The only thing that needs to be checked is whether or not K1 satisfies
the making-it-so criterion, 2M. If it does, then K1 is a true kernel for E, and thus all statements in
K1, including the mathematical theorem, are difference-makers. However, it is easy to see that K1
fails to satisfy criterion 2M. The existence of K2, which is a purely nominalistically acceptable
explanation of E, means that our mathematical model, K1, fails to meet 2M(a). K1 also fails to
meet 2M(b); the mathematics can be argued to be simply representing physical units of time
which is evident by the existence of K3. Hence, K1 is not a kernel for E, and the kairetic criterion
does not point to any mathematical difference-makers for the explanation of the cicada’s prime-
life-cycle.
The exact same conclusions can be drawn when analyzing the explanation of the hexagonal
nature of honeycombs. In the honeycomb explanation, the mathematical explanation will fail to
satisfy criteria 2M(a) as there exists a purely nominalistic explanation using a Field type
nominalization of Newtonian space-time. It also fails to meet 2M(b) as the nominalist can claim
that the mathematical honeycomb theorem is representing the physical fact that approximately
Euclidean space is most efficiently tiled by hexagons. We could construct an explanation that
utilizes this physical fact instead of the mathematical theorem that would still entail the
explanandum. Thus the mathematical theorem will not appear in a kernel for the explanandum,
and is not a difference-maker.
In the case of the Kirkwood gaps example, this explanation makes use of an eigenvalue analysis
to explain the existence of gaps in the asteroid belt. In this case, there is no nominalistic
counterpart explanation to the mathematics like there was for the cicada or honeycomb example,
and thus the Kirkwood gaps examples satisfies criterion 2M(a). Yet again, we are unable to
escape the nominalist belief that there is a physical fact that the mathematics is representing. As
discussed in chapter 3, the nominalist could claim that it is in virtue of physical facts such as
gravitation, mass of neighboring planets, space-time, etc., that actually explains the existence of
the Kirkwood gaps. The mathematics is representing these physical facts, and thus we do not
satisfy 2M(b). This blocks the mathematical explanation from being a kernel for the
explanandum, and like the cicada and honeycomb example, the mathematics is not a difference-
maker.
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These negative conclusions should come as no surprise. The making-it-so criterion, 2M, was
designed to reflect the nominalist intuitions raised in the indexing argument which were used
against the cicada, honeycomb, and Kirkwood gaps examples from being a GME. The fact that
the kairetic account corroborates this intuition should add to the credibility of the account in the
eyes of the nominalist. I want to restate that I am not endorsing the results of this section, but
rather I am trying to show that the kairetic criterion motivated by the indexing argument gives us
an account of mathematical explanation that agrees with the way nominalists such as Melia see
the world. Whether or not this is actually the right way to construct the kairetic criterion is not
being considered here. What has been shown is that when the kairetic account is constructed in
this nominalistic friendly way, we find that the mathematics featured in all these explanations are
not difference-makers for their respective physical explananda. This means that they are not
conferring the explanatory power in the explanation, and it follows that these examples are not
GMEs. Even better for the nominalist is that the kairetic account highlights the actual role and
contribution that mathematics provides in these explanations. Mathematics helps represent or
deduce the relevant physical facts, and it is these physical facts that are found to be the true
difference-makers. Up till now, the conclusions of the kairetic account as I have adopted it match
the intuitions of the nominalist in every way.
Now we will analyze the example of electron spin. Electron spin explains the splitting of a beam
of electrons that pass through a Stern-Gerlach apparatus where it splits into two equidistant and
distinct beams. What explains the splitting are the Pauli spin matrices – a mathematical entity.
Let MS be the model that entails the explanandum, E. MS will have to include lots of other
information in addition to what has been mentioned here, such as the theory of quantum
mechanics, magnetism, detectors, etc. Now attempt to generate a kernel, KS, from MS following
the kairetic criterion for mathematical explanations. While lots of information may be removed
during the abstraction process – perhaps we can reduce everything to quantum mechanics, or we
can generalize other facts such as the specific size of the magnets, the exact distance the beam
travels, etc. – one thing that cannot be removed are the Pauli spin matrices. If we try to remove
them from our kernel there would be no possible way for our model to entail the splitting into
two equidistant and distinct beams. Put simply, the Pauli spin matrices are exactly what allows
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our explanation to entail the explanandum. Although I have not precisely said what else is in KS,
I can confidently assert that the Pauli spin matrices will be a member. KS satisfies the
abstractness criterion, and again we assume that that nothing strange has been performed that
would upset the maximizing of abstractness and cohesion. It should also be clear from our
analysis in chapter 3 that KS satisfies 2M(a); there is no nominalistic counterpart to the Pauli spin
matrices that can explain the splitting of the beam.
I take the above analysis of the electron spin example to be perfectly amenable to the nominalist.
All that remains to be answered is: does KS also satisfy 2M(b)? This is where all the other
mathematical explanations critically failed. In each of those cases the nominalist was able to
assert that the mathematics was actually representing some real physical objects or properties
which were the genuine explanatory factors. With spin, however, we have a different story. As
shown in 3.4, there is no purely physical object, property, fact, or anything at all that the Pauli
spin matrices are representing. The nominalist cannot identify anything beyond the mathematics
that constitutes spin. There is no classical counterpart to electron spin that can even begin to
inform some physical understanding of the concept. All that we have is the mathematics, and
nothing more. It is clear that the spin explanation satisfies 2M(b), and hence 2M as well. This
means that KS satisfies all the criteria for being a kernel for E. The kairetic account states that
everything in a kernel is a difference-maker. Thus, the Pauli spin matrices, which are
mathematical objects, are difference-makers for a physical fact.
We have successfully shown that the electron spin example satisfies criterion (E) for GMEs: an
acceptable account of scientific explanation corroborates the claim that the mathematics is
explanatory. This result, together with those from chapter 3, means that the electron spin
example satisfies all five criteria for a GME, and therefore the mathematical explanation of the
splitting of an electron beam into two distinct and equidistant beams when passing through a
Stern-Gerlach apparatus is a genuine mathematical explanation.
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4 Difference-Making Revisited
Up till now we have been exclusively utilizing the standards of the nominalist in order to
examine mathematical explanation. This was done to meet the nominalist on their own ground,
and to show them that according to their very own standards there does exist a GME. Now that
we have demonstrated this, we can free ourselves from this restriction of satisfying the
nominalist and take a closer look at what it means for mathematics to genuinely explain physical
phenomena. We can now attempt to answer the question: how does mathematics explain? What
does it mean for mathematics to be a difference-maker for a physical phenomenon? Borrowing
Strevens’ phrase, we can now analyze how mathematics makes-it-so. I will consider two
approaches to answering these questions here: first by looking at the mathematics internally, and
second will be by closely examining what mathematics brings to our GMEs.
4.1 Internal Mathematical Explanations
Strevens suggested that one way in which mathematics could make a difference in a physical
explanation is through our understanding of the mathematics itself. Strevens is appealing to a
mathematical explanation of mathematical facts, or internal explanations. The idea is that there
is a patent difference between merely demonstrating a mathematical fact and explaining why that
fact holds. Internal explanations can reveal insight into mathematical theorems that otherwise
would have been lacking. Mark Steiner (1978a, 1978b) was one of the first to advance an
account of internal explanations. He claimed that internal explanations had to be in the form of a
mathematical proof that relies on the concepts of ‘characterizing properties’ and ‘deformation’.
A characterizing property is a property unique to a given mathematical object or structure that is
relative to a ‘family’ or domain of similar objects or structures. For example, the number 18 can
be characterized by it being the successor of 17. This characterization is relative to the family of
natural numbers defined with the successor relation. Or, 18 can be characterized by its prime
power expansion, 2 x 32, which is relative to the family of prime factorization of the natural
numbers. Deformation is to change the object or structure in question while holding the
characterizing property steady. Deforming 18 to 15 would change the characterization to be the
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successor of 14 or having the prime power expansion of 3 x 5. For Steiner, an explanatory proof
of a mathematical theorem is one such that the characterizing property is mentioned in the proof,
and that it is evident that the resulting theorem depends on the characterizing property which is
made obvious through deformation.
Steiner advanced an incredibly strong connection between an internal explanatory proof and a
GME. We only have a GME if we have an internal explanatory proof of the key mathematical
theorems being employed. As Steiner puts it, “when we remove the physics, we remain with a
mathematical explanation – of a mathematical truth!... In standard scientific explanations, after
deleting the physics nothing remains.” (Steiner, 1978b, p. 19) Call Steiner’s view the strong
relationship.
What Strevens suggests for how mathematics makes a difference is not nearly as strong as
Steiner. Strevens talks of how “understanding the mathematics as well as the physics of the
scientific treatment of the explanandum... you come to understand the phenomena better.”
(Strevens, 2008, p. 303) In a similar vein, Mancosu speculates that “it is conceivable that
whatever account we will end up giving of mathematical explanations of scientific phenomena, it
won’t be completely independent of mathematical explanation of mathematical facts.”
(Mancosu, 2008, pp. 140–141) The suggestions from Strevens and Mancosu have two major
differences from Steiner’s take. First, Strevens and Mancosu allow for any variety of internal
explanations rather than strictly an explanatory proof. There is good reason to believe that not all
internal explanations need to be mathematical proofs. Diagrams, pictures, or even
axiomatizations could all be explanatory in their own way. Secondly, Strevens and Mancosu do
not claim that for a mathematical explanation of a physical fact to be genuine it must have an
internal explanation within it. Rather, they simply assert that having an internal explanation may
increase the explanatory understanding gained from the external explanation. Having an internal
explanation is neither necessary nor sufficient for how mathematics makes-it-so in an external
explanation, but rather it is just one possible way in which mathematics can make-it-so. Call this
the weak relationship between internal and external explanation.
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There are several reasons why we should be skeptical of Steiner’s strong relationship. A prima
facie problem with this suggestion is that most proofs in mathematics are not explanatory by
Steiner’s own criteria. This results in an incredibly high standard when we are trying to use
mathematics to genuinely explain physical facts as we are limited to the small amount of
theorems that already possess an internal explanation. Even worse is that many GMEs could
depend on several mathematical theorems, all of which would need its own internal proof. This
criticism suggests that the strong relationship sets too high of a standard for GMEs, but there are
further reasons to believe that there is a more fundamental error in linking the existence of a
GME to an internal explanation in the first place. Steiner’s concept of what makes an external
explanation genuine seems to fly in the face of what supporters of GMEs actually believe. All of
the examples that we have looked at pay no heed as to whether or not the mathematical theorems
utilized possess internal explanations. The beliefs of Baker, Colyvan, and myself that our
respective examples are genuine have nothing to with the nature of the proofs behind the
mathematics. At no point did we invoke an internal mathematical explanation as evidence of the
explanatory force of the mathematics. Moreover, nominalists who reject examples such as the
cicada and honeycomb explanations do not do so due to their lack of an internal explanation.
Neither realist nor nominalist seems to attribute any importance to the existence of an
explanatory proof whatsoever. The reason why is because what is truly playing the explanatory
role in a GME is solely the mathematical theorem, and not its proof. By any account of scientific
explanation we would still have the exact same explanatory understanding of the explanandum
regardless of the style of mathematical proof employed. We would have the exact same amount
of predictive power, unificatory power, the same mathematical difference-makers, the same
entailment, and the same answers to why-questions. Steiner’s claim that a GME depends on the
existence of an internal explanation is unsupported and does not reflect how mathematics makes-
it-so in GMEs.
A final criticism of the strong relationship is that it relies on some objective conception of
internal explanation. Michael Resnik and David Kushner (1987) offer a detailed critique of
Steiner’s conception of internal explanatory proofs. They argue that the concepts of
characterizing property and deformation which are necessary for an internal explanation are not
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well-defined. Even worse, the very examples that Steiner uses as models of a good internal
explanation do not seem to satisfy his own criteria. This spells trouble for the strong relationship.
Resnik and Kushner propose their own account of internal explanation that is heavily influenced
by van Frassen’s pragmatic account of scientific explanation, and is drastically different from
Steiner’s. The context of why-questions determines the contrast class and relevance relation, and
thus what counts as an internal explanation is fundamentally relative. Consider the intermediate
value theorem which states that if a real valued function, 𝑓, is continuous on the closed real
interval [𝑎, 𝑏], and if 𝑓(𝑎) < 𝑐 < 𝑓(𝑏), then there is an 𝑥 in [𝑎, 𝑏] such that 𝑓(𝑥) = 𝑐. Someone
could ask, “why is the intermediate value theorem true?” To answer this question, any proof
would suffice. However, in a different context the asker could pose “why is it true for a real
valued function?”, or “why is it true for a continuous function?”, or “why is it true on a closed
interval?” The answers to these questions, and thus the explanations, would be entirely different.
It could be that some answers will be proofs of other facts about things like continuous functions
or the real number line, or it could be that the answer will simply be a counter-example in some
open interval. However, a generic proof of the intermediate value theorem which was
explanatory in the first place would not be explanatory for these more specific questions. What
counts as explanatory depends on the context of the question.
Resnik and Kushner conclude that there is no such thing as an internal mathematical explanation
simpliciter, but we do have explanations relative to the context of the why-question. If this is the
case, then certainly Steiner’s criteria for explanatory proofs are flawed as they do not take into
account the context of the question. Rather, Steiner searches for objective criteria within the
proof that makes it explanatory in its own right. But how does this relative theory mesh with the
intuitive feeling that some proofs are more explanatory than others? Resnik and Kushner write:
We have this intuition, we submit, because we have observed that many
proofs are perfectly satisfactory as proofs but present so little information
concerning the underlying structure treated by the theorem that they leave
many of our why-questions unanswered. In reflecting on this, we tend to
conflate these unanswered why-questions under the one form of words
‘why is this true?’ and thus derive the mistaken idea that there is an
objective distinction between explanatory and non-explanatory proofs. (M.
D. Resnik & Kushner, 1987, p. 154)
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Some proofs simply have the ability to answer more why-questions than others. This does not,
however, make one proof ‘explanatory’ and the other ‘non-explanatory’, as given the proper
why-question it could be that both turn out explanatory, both turn out non-explanatory, or one is
explanatory and the other is not. It all depends on the context of the question.
If we adopt Resnik and Kushner’s account of internal explanation, then the strong relationship
has no content. If we are scientific realists, then we hold an objective notion of scientific
explanation. How could it be, then, that a subjective explanation of the mathematics is somehow
a difference-maker within an objective explanation of a physical fact? To claim such a
relationship is inconsistent. It is certainly the case that if we are like van Frassen and do not
believe in objective scientific explanations, then the strong relationship is trivially true. Either
way, the strong relationship is problematic. On the one hand it is inconsistent, and on the other it
is trivial. Although I am sympathetic to Resnik and Kushner’s approach, I leave it open as to
which conception of internal explanation one should adopt. However, no matter your views on
internal explanations, maintaining that the internal mathematical explanation is a difference-
maker for explanations of physical facts is problematic.
The weak relationship shows more promise, but unfortunately in a trivial way. Advocates of the
weak relationship say that any sort of explanation of the mathematics, not necessarily a Steiner
style explanatory proof, may reveal some insight into the physical explanandum. No sort of
motivation or argument for this relationship is provided beyond the mere suggestion of it. My
issue with this position is that while Steiner’s claim was too strong, this claim is so weak that it is
certainly trivially true. Of course it is possible that an internal explanation would provide us with
some sort of added insight into a physical explanation. This is because supporters of the weak
relationship have not specified what an internal explanation actually is. Steiner was bold enough
to advance a comprehensive account of internal explanations in the form of his deformable
proofs. But for the weak relationship, it seems that anything goes. In this light, the mere
possibility that we gain insight into the physical world from some sort of unspecified type of
internal explanation is enough to make the weak relationship true. Yet this is such a feeble
conclusion that philosophically speaking it is not even worth maintaining. The weak relationship
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is nothing more than idle speculation that can neither be proved nor disproved. It contributes
nothing towards revealing how mathematics makes-it-so in external explanations.
The suggestion that having internal explanations of mathematics is in some way relevant to our
mathematical explanations of the physical world is either misguided or trivially true. If we are to
better understand how mathematics is a difference-maker in our GMEs we have to look
elsewhere. Focusing solely on the mathematics is not the correct approach. Instead we should
focus on what roles the mathematics plays within the explanation itself.
4.2 The Roles of Mathematics in Genuine Mathematical
Explanations
The way we cached out making-it-so above was through a comparative analysis. Criteria 2M of
the kairetic criterion states that mathematics entails a physical explanandum, E, if,
(a) there is no purely nominalistic counterpart that also entails E,
and,
(b) there is no purely physical object or property that the mathematics is
representing.
This idea is that if we cannot cite any purely nominalistic way that makes a physical
explanandum so, then it is reasonable that the mathematics is making-it-so. Although this
satisfies the nominalist and does accurately reflect how they view GMEs, it does very little in
telling us how mathematics actually makes a difference to the physical explanandum. All we can
say from the above nominalist criteria is that mathematical difference-makers make a difference
to physical facts only because we cannot find any non-mathematical difference-makers that can
explain the explanandum. This is not a satisfactory way of understanding how mathematics
makes a difference. We will briefly consider three ways in which we can improve on our
understanding of difference-making in GMEs that focus on what mathematics contributes in
genuine explanations.
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In a recent paper, Marc Lange (2012) correctly recognizes that most philosophers engaged in the
debate surrounding mathematical explanation do not focus on what makes an explanation
genuine, or as he calls it, what makes an explanation ‘distinctively mathematical’. For Lange, in
addition to being non-causal,
[genuine] mathematical explanations in science work by appealing to facts
(including, but not always limited to, mathematical facts) that are modally
stronger than ordinary causal laws—together with contingent conditions
that are contextually understood to be constitutive of the arrangement or
task at issue in the why question. (Lange, 2012, p. 7)
The two key aspects of a GME are that first the mathematics entails the conclusion in a much
stronger way than just the physical facts. The relationship between the explananda and the
explanandum “holds not by virtue of an ordinary contingent law of nature, but typically by
mathematical necessity.” (Lange, 2012, pp. 12–13) The second aspect of a GME is contextual.
The necessary entailment that mathematics brings is not enough to distinguish a regular non-
causal mathematical explanation from a genuine one. What is needed as well is an emphasis on
the role of the mathematical fact(s) being utilized. “[I]t is a matter of degree and of context.
Insofar as mathematical facts alone are emphasized as doing the explaining, the explanation is
properly characterized as distinctively mathematical.” (Lange, 2012, p. 23) Due to this
contextual nature, Lange argues that depending on how you frame explanations such as the
cicada and honeycomb examples, you could in one case conclude that these explanations are not
genuine, and in another case conclude that they in fact are.
Although I feel that Lange is making steps in the right direction in his approach, I find his
dependence on contextual factors largely unsatisfactory. Lange is clearly influenced by van
Frassen’s pragmatic account of scientific explanation. As we saw above, a potential issue with
van Frassen’s approach is that it somewhat trivially implies that GMEs exist as they are certainly
legitimate answers to why-question. This was unappealing as it did not do enough to identify
what role the mathematics was playing, and also because van Frassen’s pragmatic account does
not mesh nicely with scientific realism. Lange supplements the pragmatic approach by adding
the observation that GMEs are modally stronger than regular scientific explanations. My
criticism here is that this observation does not appear to actually add anything of substance at all
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over and above van Frassen’s original approach. Mathematical facts are always going to be
modally stronger than physical facts as on all standard accounts, mathematical facts are true in
every possible world. Once we realize this, Lange’s position essentially collapses into van
Frassen’s pragmatic account where only the contextual factors determine if a mathematical
explanation is genuine or not, as all mathematical explanations will necessarily trivially satisfy
the modally stronger requirement.
Lange’s analysis of the cicada example makes his reliance on contextual factors clear. Baker
phrases his explanandum as “that cicada life-cycle periods are prime” (Alan Baker, 2009, p. 624)
rather than composite in number of years. Baker’s explanation of this involves the idea that
prime periods have been ‘selected for’ for some evolutionarily advantageous reason. Lange, in
line with Sober (1984), argues that explanations that involve ‘selection for’ are classic examples
of causal explanations.
This explanation is also just an ordinary causal explanation. It uses a bit of
mathematics in describing the explanandum’s causal history, but it derives
its explanatory power in the same way as any other selectionist
explanation. Taken as a whole, then, it is not a distinctively mathematical
explanation. (Lange, 2012, p. 15)
Even though there is mathematics in the explanation, and that the mathematics is important and
perhaps even indispensable to the explanation, it is not, according to Lange, a GME. However, if
we rephrase the explanandum we obtain a different result.
But suppose we narrow the explanandum to the fact that in connection with
predators having periodic life-cycles, cicadas with prime periods tend to
suffer less from predation than cicadas with composite periods do. This
fact has a distinctively mathematical explanation. (Lange, 2012, p. 15)
Somehow, this rephrased explanandum is placing the emphasis squarely on the shoulders of the
mathematics, and not some causal selection process.
In a footnote meant to clarify this claim, Lange mentions that “[p]resumably, we would be
prompted to ask for an explanation of this fact only as a result of having used this fact to help
explain why cicada life-cycle periods are prime, an explanation that (I have just suggested) is
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causal.” (Lange, 2012, p. 15) Herein lies the problem with his position. Lange’s rephrasing is
simply not a rephrasing at all. The narrow explanandum is specifically looking to explain why a
prime period would minimize intersection. True, it is couched in language about predators and
cicadas, but ultimately this is asking to explain something about prime periods which is clearly a
mathematical fact. Notice that this is a remarkably different why-question than asking why
cicadas have a prime-life-cycle which is a fact about the physical world. Lange has simply
changed the explanandum to yield an internal mathematical explanation.
Even if we accept this as a legitimate move, a perfectly reasonable question is why would we
ever be prompted to ask for an explanation of Lange’s narrow explanandum? Baker’s
explanation already involves a mathematical proof that primes minimize intersection, which
would be the exact same explanation of the narrow explanandum that supposedly emphasizes the
mathematical factors. Typically, changing the context of a why-question also changes the
explanation that we supply. For example, if I ask ‘why is the traffic in Toronto is so bad?’, an
explanation could involve the poor infrastructure of the city. However, if I ask ‘why is the traffic
in Toronto so bad at 5:00?’, this implies a different sort of explanation entirely. The answer
would have to address my contrast-class of 5:00 rather than, say, 3:00 or noon. Such an
explanation would cite factors such as rush hour, when most office jobs end, location of most
jobs compared to living spaces, etc. The basic idea is that different why-questions are different
because they need different explanations to satisfactorily answer them. In Lange’s presentation
of the cicada example, this is not the case. The explanation of his narrow explanandum is just a
subset, specifically the mathematical subset, of the original explanation. How, then, does this add
anything to our understanding, or explain anything that we did not already know from Baker’s
original explanation? It simply does not. If we want to claim that there is a genuine mathematical
explanation somewhere within the cicada example, rephrasing the explanandum does not seem to
reveal it in any significant way.
Irrespective of my criticisms, I do feel that Lange’s observation of stronger necessity in GMEs is
worth pursuing if we free ourselves from van Frassen’s pragmatism. One way that mathematics
could make a difference in GMEs is that it entails the explanandum with more necessity than a
physical explanation, regardless of context. Pincock (2004, 2011) presents an example of a
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mathematical explanation that demonstrates this necessity nicely. His example seeks to explain
why it is impossible to traverse all the bridges in Königsberg and return to your original starting
point while crossing each bridge only once. We can make a mathematical representation of the
system using a non-Eulerian graph. The vertices of the graph represent the islands or land
masses, and the edges represent the bridges. A key property of non-Eulerian graphs is that there
is no path that starts at any single vertex such that you return to that same vertex while traversing
each edge exactly once. This mathematical property explains the impossibility of the physical
path in Königsberg. Bangu (2012) calls these ‘impossibility results’, and argues that this is an
important benefit that mathematics brings to explanations. Impossibility results are, or course,
just the flip side of mathematical truths being necessary. This necessity tells us that all other
possibilities are necessarily impossible.
It is easy to see that if we analyze the bridges of Königsberg example using the kairetic criterion
above, it will fail to meet the requirements of a GME. This is because it is clear that the non-
Eulerian graph is representing or indexing the actual physical layout of the land masses and
bridges, and it is this physical system that has the relevant explanatory property. This reveals that
the nominalist would most likely not find Pincock’s example compelling. Consider, then, what
happens if we change the making-it-so criteria from the kairetic criterion for mathematical
explanation to reflect impossibility results. Replace criterion 2M with:
2MI. The mathematical explanation, K, entails a physical
explanandum, E, if K shows that all other possibilities are
impossible.
Now we can see that the bridges of Königsberg example will meet the kairetic criterion and
should be considered a GME. This is because the way in which mathematics is making a
difference to the physical explanation is by showing that that it is necessarily impossible for a
return path to actually exist. Note that unlike Lange and the cicada example, I have not changed
the question or the explanandum in order to change the status of the Königsberg example from a
standard mathematical explanation to a GME. What happened is that now that we are free from
solely worrying about indexing or representation, we can focus more closely on what
mathematics actually brings to the table in a GME. By identifying some ways in which
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mathematics can make a difference, we can arrive at a clearer consensus on what counts as a
GME or not.
Applying 2MI to our other examples yields interesting results. One feature of the electron spin
explanation is that it not only explains why the electron beam splits into two distinct beams, in
virtue of the Pauli spin matrices we also know that it is impossible to have anything other than
the two resultant and equidistant beams. There are necessarily only two eigenvalues for the z-
direction Pauli spin matrix, and hence it is impossible for there to be any number of beams other
than two. The same explanation also shows why it is impossible for the beams to be anything
other than equidistant from the original plane. The mathematics is a difference-maker as it
demonstrates that all other possibilities are necessarily impossible, and hence by the standard of
impossibility results, the electron spin example is a GME.
The Kirkwood gaps example has a similar breakdown. Recall that an eigenvalue analysis shows
that it is impossible for the asteroids to settle in certain regions, and these regions represent the
observed gaps. Colyvan notes that it is perfectly possible to give a purely physical and causal
explanation for each individual asteroid that explains why that particular asteroid settled where it
did. However, this is not the same as explaining why no asteroid could ever be in the Kirkwood
gaps. The mathematical explanation delivers the impossibility result we are looking for. So, even
though I was critical of the Kirkwood gaps example in the sections above, that was due to its
arguable representational nature. Now that we are no longer concerned about indexing, we see
that the Kirkwood gaps can count as a GME as the mathematics is a difference-maker in the
context of impossibility results.
It is worth pointing out that there is an important difference between the electron spin and
Kirkwood gaps impossibility results, and the bridges of Königsberg impossibility result. In the
Königsberg example, we could in theory arrive at the very same impossibility result without
utilizing a non-Eulerian graph, or any other mathematics. Imagine someone painstakingly
charting out every possible path and attempting to walk each and every one. Eventually, that
person would realize that it is impossible to return to any starting point whilst crossing each
bridge only once, and this impossibility would be an empirical result. In the Kirkwood gaps and
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electron spin example, an empirical impossibility result is unimaginable. It is surely the case that
we would empirically discover that electrons only deflect into two beams, and they are always
equidistant, but this is not the same as knowing that the alternatives are necessarily impossible.
The former is potentially fallible and the latter is not. The situation is the exact same for the
Kirkwood gaps. Whether or not this difference is significant enough to lead us to rule against the
bridges of Königsberg being a GME I will not address here, but is an interesting open problem
that is worth further examination.
Unfortunately, the cicada and honeycomb examples do not fare as well. In the cicada example,
the key mathematical fact is that primes minimize intersection. In the honeycomb example the
mathematical fact is that a hexagon is the most resource-efficient way to tile an area. The theory
of evolution and ecological constraints provide both explanations with the added biological facts
that it is advantageous to minimize life-cycle intersection and to be resource-efficient
respectively. There is nothing wrong with either of these explanations, but note that neither of
them provides any impossibility results. It is not impossible for cicada to have a composite
number life-cycle. If it were the case that they had composite life-cycles, we could easily account
for this by asserting that cicadas are not maximally evolved. Similarly, it is not impossible for
honeycombs to be some other sided polygon. If so, honeybees would not be maximally resource-
efficient. The mathematics in these explanations is not a difference-maker by impossibility result
standards, and hence are not GMEs. What role are the mathematical theorems playing? In both
cases the mathematics plays the role of informing us what the most optimal conditions actually
are, which is why I call them optimization explanations. If we wish to include optimization
explanations as GMEs, then we need to argue that delivering optimal conditions is a difference-
maker for these explanations of physical facts. I will not make the case here, but will suggest that
although this move is possible, I feel that it will not be successful in the end.
Another way that mathematics could be considered a difference-maker is demonstrated in many
examples from Batterman (2002, 2008, 2010). Batterman’s examples are extremely technical and
I will only consider their features in broad strokes. Batterman is interested in explaining
regularities. He notes that “most (though not all) explanations in physics and applied
mathematics are explanations of patterns or regularities... Nevertheless, despite the fact that
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various details are completely distinct, we witness the same pattern. We would like an
explanation of why.” (Batterman, 2010, p. 20) The fact that the individual details are distinct do
not seem to matter to the overall pattern, which Batterman calls a universal. In fact, often these
individual causal details function as “noise” (Batterman, 2002, p. 4) which hinders our
explanatory efforts. Batterman’s solution is to employ ‘nontraditional’ mathematical
idealizations that are ways in which we can “remove details that distract from [our] focus.”
(Batterman, 2008, p. 430) These idealizations take the form of mathematical limits that allow us
to focus on the relevant explanatory details, and to eliminate the particular causal details of our
system that are irrelevant to the universal regularity. In this way, mathematical limits are a key
difference-maker for these types of GMEs. Pincock also suggests that how mathematics allows
us to focus on relevant properties is a key feature of acausal mathematical representations.
Mathematics can represent novel and explanatorily important features of a system that a faithful
causal representation cannot. Mathematics makes a difference as it “not only captures the feature
of interest but also has as part of its content that various aspects of the system are also irrelevant
to this feature.” (Pincock, 2011, p. 54) Pincock widely agrees with Batterman’s analysis except
that where Batterman is focused exclusively on mathematical limit operations, Pincock is open to
using other mathematical tools in representation. Putting Batterman and Pincock’s suggestions
together, mathematics could be considered a difference-maker in its ability to identify novel non-
causal explanatory features of a physical system, and also in identifying which causal features
are irrelevant.
We have briefly considered three ways in which mathematics can be said to make a difference in
a physical explanation. Mathematics can demonstrate impossibility results, it can demonstrate
optimal conditions, or it can identify both novel non-causal features and irrelevant causal
features. In no way do I see these as exhaustive of all the ways of how mathematics can explain
physical phenomena; nor are these ways exclusive of one another. The way forward in
understanding mathematical explanations should be to precisely cache out these and other roles
that mathematics plays in GMEs.
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5 Takeaway
The present state of affairs has been far too fixated on the problem of indexing and
representation as advanced by the nominalist. Sadly, this has halted progress on more interesting
and challenging issues. Instead of focusing on demonstrating that mathematics does not play
certain roles, we should be looking instead to discover and understand what roles it does play.
The true takeaway of chapters 3 and 4 is not that the electron spin example succeeds at being a
GME where all other examples have failed. This is far from the truth, as we saw above that the
status of a supposed GME rises and falls depending on what criteria we select to characterize
mathematical difference-makers. The electron spin example is special only because it was
carefully selected with a singular purpose in mind: to defeat the intuition that mathematics solely
represents such that we can make progress in actually understanding what it means for
mathematics to explain. My hope is to show that this takeaway is amenable to both the realist
and the nominalist. The fear for the nominalist is that by invoking inference to the best
explanation (IBE) they will have to become realists, and it is this fear that stops them from
seriously considering GMEs and all the benefits that come along with them. This fear is only
warranted if we assume first that IBE can legitimately infer mathematical realism, which I have
explicitly not assumed. With the assumption abandoned, there should be nothing to fear from
analyzing the explanatory contribution of mathematics in science. Clearly though this is nothing
more than a bait and switch if it turns out in the end that IBE is a legitimate inference that should
lead us to be realists. If this is possible, then the nominalist is justified in their aversion to GMEs.
In chapter 5 I will turn to IBE and show that, contra the enhanced indispensability argument, the
nominalist has nothing to fear.
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Chapter 5 Inference to the Best Mathematical Explanation
We have demonstrated that a genuine mathematical explanation (GME) exists. The question now
is whether or not we ought to be mathematical realists. Indispensability arguments are meant to
convince the scientific realists that the very same reasons which lead to their scientific realism
ought to lead them to mathematical realism. This line of reasoning requires two things to be
established. The first is that mathematical entities play the same role in our best scientific
theories as unobservable physical entities, namely that mathematical entities genuinely explain
physical facts just as unobservable physical entities do. The second is that indispensability
arguments must explain the method of inference that scientific realists utilize when they assert
that unobservable objects exist.
Recall Colyvan’s version of the Quinean indispensability argument (QIA):
(P1) We ought to have ontological commitment to only those entities
that are indispensable to our best scientific theories.
(P2) We ought to have ontological commitment to all those entities
that are indispensable to our best scientific theories.
(P3) Mathematical entities are indispensable to our best scientific
theories.
Therefore:
(C) We ought to have ontological commitment to mathematical
entities.
The method of inference in the QIA is the Quinean thesis of confirmational holism which was
found lacking in chapter 1. Arguments boiled down to the claim that actual practicing scientists
simply do not confirm in a holistic manner, and thus there is no reason to believe that
confirmational holism is true. In addition, it seems entirely plausible to be a scientific realist
without endorsing confirmational holism. Unless we can put forward a staunch defence of
confirmational holism and its necessity to scientific realism, the QIA cannot deliver its realist
conclusion.
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Baker’s enhanced indispensability argument (EIA) is meant to solve this problem by avoiding
confirmational holism altogether.
(EP1) We ought to rationally believe in the existence of any entity that
plays an indispensable explanatory role in our best scientific
theories.
(EP2) Mathematical objects play an indispensable explanatory role in
science.
Therefore:
(EC) We ought rationally to believe in the existence of mathematical
objects.
(EP1) ties the argument to the inference to the best explanation (IBE), which is widely
considered to be the standard inference of scientific realists. What makes the EIA compelling is
that at first glance, any scientific realist would gladly assent to (EP1). Now that we have
established that (EP2) is true, then the conclusion of mathematical realism follows more directly
than the route taken by the QIA. We need not justify any additional theses in order to extend our
realism over mathematical objects.
My aim in this chapter is to show that even though we, as scientific realists, endorse IBE,
utilizing this inference to infer mathematical realism is unjustified. There are two main
objections that have been raised in the literature that echo this belief. Some have argued that IBE
is in principle unable to infer the existence of mathematical, or other noncausal entities. I will not
address this objection here as it has already been satisfactorily refuted,43 and I will assume that
there are no fundamental limitations of IBE for inferring the existence of mathematical entities.
43 Steiner (1978b) and Bangu (2008) have argued that any use of IBE to infer the existence of mathematical objects
would necessarily beg the question. Baker (2009) has defended his cicada example, but also correctly points out that
many other examples would not be at risk of begging the question. Bangu (2012) has subsequently abandoned this
criticism. Others such as David Armstrong (1978), Brian Ellis (1990), and Hartry Field (1989) have advanced the
Eleatic Principle which states that IBE should be restricted to only those entities that possess causal powers, and thus
would be in principle unable to lead to mathematical realism. Colyvan (2001a), Lipton (2004a) and Psillos (2012)
have convincingly argued that the Eleatic principle is unjustified and patently begs the question against
mathematical realism.
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A more interesting objection comes from Leng who argues that treating mathematical entities as
fictions is actually the best explanation for GMEs, and hence IBE does not lead to the realist
conclusion. I will argue that Leng’s critique of the use of IBE is deficient, and hence the two
most prominent objections against using IBE for mathematical realism are not compelling.
The objection that I will advance is different from the two above in that I accept both that IBE is
applicable in principle to mathematical entities, and that there exist best explanations of physical
facts that indispensably appeal to mathematical entities which fictionalism cannot account for.
The real problem is that (EP1) is not an accurate presentation of IBE. In particular, (EP1) is too
weak, and does not faithfully reflect the discerning ways in which IBE is employed by the
typical scientific realist. When made precise, it is clear that IBE does not yet warrant inferring
the existence of mathematical objects. I will suggest that there is a way for the EIA to avoid this
new challenge, but it comes at quite a cost. The EIA can make use of IBE only if it at the same
time embraces confirmational holism. However, if this is the case, then the EIA collapses to the
QIA and can hardly be said to be an enhanced argument at all.
1 The Inference to the Best Explanation
The basic pattern of IBE is straightforward. If an explanation is considered to be the best
explanation of the phenomenon in question, then we should infer that the explanation is true.
Psillos (1999) advances the following example. You come down to the kitchen one morning and
hear some strange noises in the walls. You see some mouse-droppings on the counter, and the
bits of cheese that were left out the night before have gone missing. One possible explanation for
this is that your roommate is playing a strange prank on you and is hiding in the walls. Another is
that your kitchen is haunted by a cheese-loving ghost. However, it seems to you that the best
explanation is that there is a mouse in the walls. As this is the best explanation, you infer that this
explanation is true and that you have a mouse problem.
Of course we are not interested in cheese stealing mice. IBE is an extremely important inference
for the scientific realist. Belief in unobservable entities, such as positrons or neutrinos, is not
generated in the same way as our belief in things like tables and chairs; we have no direct
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sensory access to unobservables. We must instead infer their existence by other means, and this
means is IBE.
Suppose, now, that a scientist observes that in the standard account of β-
decay, the principle of the conservation of energy is violated. The energy
of the decaying neutron is not commensurate with the energy of the
emerging proton and electron. What needs to be explained here is not
mouse-droppings, but sure enough something needs to be explained.
Pauli’s positing of the neutrino (a particle with no charge and mass, but
with spin) is just another instance of [IBE]. (Psillos, 1999, p. 212)
There may be other potential explanations for the discrepancy in energy, but scientists feel that
the neutrino, if it exists, explains the observations best. They are justified in believing in the
existence of the neutrino in the very same way that we were justified in believing that we have a
mouse in the kitchen walls. In this way, IBE is the primary method which allows the scientific
realist to add unobservable entities to their ontology.
1.1 Problems with Inference to the Best Explanation
Famously, IBE faces two serious problems. The first is the problem of justification. No one
doubts that we use IBE all the time in our regular lives, but is IBE sufficiently justified in order
to license philosophical conclusions? The second problem is that IBE has been notoriously
difficult to precisely define. In his detailed analysis of IBE, Lipton states that the theory of IBE is
“more a slogan than an articulated philosophical theory.” (Lipton, 2004a, p. 2) This problem of
explication is important to our understanding of IBE, but is also critical for addressing the
problem of justification. The literature surrounding these two issues is vast, but the details are
not much of a concern to us here. Our task is to examine and understand the ways in which
typical scientific realists actually use IBE, which implicitly assumes that the inference is both
justified and reasonably well-defined. Given this assumption, I will only briefly examine
particularly important issues in the problems of justification and explication.
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Although van Fraassen presents several challenges to IBE, one of the most influential is known
as the “best of a bad lot” (van Fraassen, 1989, p. 143) objection.44 IBE allows us to move from
an explanation being the best, to that explanation being true. Van Fraassen notes that this
inference critically requires us to assume that the true explanation is actually within the set of
explanations that we are considering. He argues that this assumption is entirely unjustified.
Without this assumption, our best explanation could simply be the best of a bad lot of
explanations where none of them are true. Hence, IBE cannot reliably lead us to the truth. Van
Fraassen anticipates two potential responses by the committed realist. First, they could argue that
we are in the position to assert that the true explanation is within our set of possible
explanations. The reason we can do this is due to a belief in the “privilege of our genius.” (van
Fraassen, 1989, p. 143) A second potential defense is to soften IBE such that the best explanation
is not true simpliciter. Instead, IBE infers something weaker than truth, such as increasing our
personal probabilities, or potential truth.
Van Fraassen considers the claim that scientific realists have a privileged position that allows
them to claim that the true explanation lies within the available lot to be somewhat absurd. This
notion of privilege seems to be at odds with naturalism, rationalism, and empiricism. However,
Psillos (1996) takes exactly this route when defending against the best of a bad lot argument.
One should observe that the argument from the bad lot works only on the
following assumption: scientists have somehow come up with a set of
hypotheses that entail the evidence - their only relevant information being
that these hypotheses just entail the evidence - and then they want to know
which if any of the hypotheses is true… However… it is at least doubtful
and at most absurd to hold that theory-choice operates in such a
knowledge-vacuum. Rather, theory-choice operates within and is guided
by a network of background knowledge. (Psillos, 1996, p. 38)
Psillos calls this the background knowledge privilege. This privilege furnishes the scientific
realists with two important advantages. First, background knowledge can significantly restrict
44 Van Fraassen (1989) also argues that IBE conflicts with Bayesianism and thus users would be susceptible to a
Dutch book. Other attacks on the justification of IBE come from Lauden (1981) and Fine (1984) who argue that any
justification of IBE is circular, and hence illegitimate.
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the set of possible hypotheses that can potentially explain a given phenomenon. Second,
background knowledge also allows us to look towards explanatory considerations to select the
best explanation among our restricted lot. The key for Psillos is that there is something to be said
about potential explanations that are compatible with already confirmed theories. This
compatibility affects not only the types of explanations that we look at, but also how we
determine the best one. The best explanation is selected by appealing to the same explanatory
virtues that are already entrenched within our background scientific theories. Psillos concludes
that both these aspects of the background knowledge privilege do provide scientists with
significant support for the belief that the best explanation is true. Moreover, scientists are entirely
justified in maintaining the privilege of background knowledge, and thus the best of the bad lot
argument can be circumvented.45 This justification is strengthened by the confirmation that our
background theories accrue over time.46
Whereas Psillos bites the bullet and agrees that scientific realists are in a position of privilege,
others such as Musgrave (1988), and Lipton (1993) have modified IBE by adding other
necessary conditions that an explanation must possess in addition to being the best. Lipton states
that “Inference to the Best Explanation might be more accurately if less memorably called
'Inference to the Best Explanation if the Best is Sufficiently Good.” (Lipton, 1993, p. 92) Later
on, Lipton changes his description of the inference.
So the version of Inference to the Best Explanation we should consider is
Inference to the Loveliest Potential Explanation… This version claims that
the explanation that would, if true, provide the deepest understanding is
the explanation that is likeliest to be true. (Lipton, 2004, p. 63)
The basic idea here is that by adding extra conditions beyond just being the best explanation, we
can avoid the best of the bad lot problem as the extra conditions justify our belief that the true
explanation is within the lot we are considering. Maneuvering in this way leads us to the second
45 For a critical response, see Ladyman, Douven, and van Fraassen (1997).
46 Psillos does acknowledge that this does require us to assume that our background knowledge is approximately
true. While he does not defend against a full-blown skepticism, he does argue that even a constructive empiricist
would agree with accepting the important role of background knowledge.
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main problem with IBE: that of explication. One of the first expressions of IBE comes from
Gilbert Harman.
In making [an inference to the best explanation] one infers, from the fact
that a certain hypothesis would explain the evidence, to the truth of that
hypothesis. In general, there will be several hypotheses which might
explain the evidence, so one must be able to reject all such alternative
hypotheses before one is warranted in making the inference. Thus one
infers, from the premise that a given hypothesis would provide a “better”
explanation for the evidence than would any other hypothesis, to the
conclusion that the given hypothesis is true. (Harman, 1965, p. 89)
Psillos presents the following template for the IBE:
D is a collection of data (facts, observations).
H explains D. (H would, if true, explain D).
No other hypothesis can explain D as well as H can.
Therefore, H is (probably) true. (Psillos, 2007, pp. 442–443)
Although all these versions of detailing IBE seem very closely related, there are important
differences. In order to make sense of any of these expressions of IBE we have to have an
understanding of explanation, truth – be it probable or approximate – as well as a means of
identifying which explanation is the best amongst all candidates. For Lipton, we also need to
make precise the concepts of sufficiently good or loveliness, and its relationship to likeliness. All
these complications leads Psillos to admit that revealing ‘the fine structure of IBE’ may be
impossible. Despite this, scientific realists maintain that they have a good enough understanding
of IBE to legitimize its use, and we will adopt this belief here.
By no means is this section meant to be a comprehensive summary of the main objections and
responses regarding IBE. I have purposefully cherry-picked certain defenses of IBE from two of
its most vocal defenders of the inference, Psillos and Lipton. The reason for this is to motivate
the idea that IBE is not some blanket inference that applies to any explanation that is found to be
the best. Even if van Fraassen’s best of the bad lot argument fails at entirely undermining IBE, it
has at the very least forced the scientific realist to make more precise how the inference is
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actually employed. I will make use of this precision below when I argue that we are not justified
in inferring mathematical realism even though our genuine mathematical explanation is the best
explanation that we presently have.
1.2 Types of Inference to the Best Explanation
We need to distinguish between two typical ways in which IBE is employed: meta-IBE and
local-IBE. In both cases we are inferring the truth of the best explanation. The difference is that
in the meta-IBE case, the best explanation is a thesis, broadly conceived as a collection of
beliefs, which is inferred to be true. In the local-IBE case, a particular explanation is inferred to
be true, but ultimately what is required for this explanation to be true is the actual existence of
some entity or another. Local-IBE is in the business of granting ontological rights onto specific
entities, whereas meta-IBE asserts the truth of a particular theory.47
Within local-IBE lies an important distinction. One type of local-IBE is when a potentially
observable object is postulated in the best explanation of our observations. In the early 19th
century, irregularities in the orbit of Uranus had been well documented. Astronomers Urbain
Leverrier and John Couch Adams both independently began working on an explanation for these
observed irregularities. Leverrier and Adams suggested that an up till then undiscovered eighth
planet would explain the orbit of Uranus due to its gravitational pull. Within a year of their
suggestions the existence of Neptune was confirmed.48 This well-known episode in the history of
science is often used as an exemplar of local-IBE. The best explanation of the observed
irregularities of Uranus’ orbit was that an eighth planet in our solar system existed. This
explanation was treated as the true explanation, and so by local-IBE, astronomers believed that
an eight planet existed. This sparked the search for direct empirical confirmation of the new
planet which ultimately proved successful when Neptune was observed in the sky.
47 This distinction is often not so sharp in practice, but this does not concern us here.
48 See Grosser (1962) for a full account of the discovery of Neptune.
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The second important type of local-IBE is when the entity in question is not potentially
observable, or unobservable. This style of local-IBE was demonstrated above by Psillos’ account
of the discovery of the neutrino. The neutrino is not directly observable the way that Neptune or
a mouse in the wall is. But why does this difference matter? IBE says that the best explanation
should be inferred as true, and in both the Neptune and the neutrino case, these were the best
explanation of the phenomena. One clear difference is that in potentially observable local-IBE
there is a level of confirmation attainable that is not available in the unobservable case. Once
there was a direct confirmation of the existence of an eighth planet in the sky, scientists were
able to confirm the explanation that they previous believed to be true. Moreover, after the
confirmation, scientists no longer need to appeal to any form of IBE as to why they believe
Neptune exists. The reason we now believe that Neptune exists is due to our observation of it.
IBE did play a crucial role in the discovery of the planet, but it is no longer the evidence for how
we know that Neptune exists. This is the case in any instance of potentially observable local-IBE
once the entity in question has been observed. Local-IBE is important to the postulation and
discovery of the entity, but there is another higher level of direct confirmation which scientists
can then seek to attain. When the entity is unobservable, this higher level of direct confirmation
is in principle not available. We cannot go out and acquire any sort of direct observation of
things like neutrinos. For unobservables, there is no higher level of justification beyond local-
IBE.49
Returning to meta-IBE, the most famous employment of the inference is the no-miracles
argument for scientific realism. Putnam states that the “positive argument for realism is that it is
the only philosophy that doesn’t make the success of science a miracle.” (Putnam, 1975, p. 73)
The no-miracles argument claims that if we are not realists about science, then the only way to
account for the immense success of our scientific theories is to consider it a miracle, or a “cosmic
coincidence.” (Smart, 1963, p. 39) A miracle or cosmic coincidence is, of course, an unappealing
49 Some scientific realists would argue that there are other ways to confirm unobservables. Hacking, for example
makes the case that being able to manipulate unobservables is critical to our belief in them. I am not too concerned
with these particular brands of scientific realism mainly because they tend to limit their extra conditions to causal
factors which are not applicable to mathematics.
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explanation compared to the better explanation of scientific realism. Thus, by meta-IBE, we
should be scientific realists. The no-miracles argument is meant to convince anti-realists that
they should be scientific realists. If this does not succeed, at the very least the argument provides
justification for the realists’ set of beliefs.
The no-miracles argument has been criticized in a variety of ways from the antirealist camp.
What is interesting about this meta-IBE is that it has also received criticism from scientific
realists as well. A survey of the literature indicates that scientific realists as a whole, do not all
accept meta-IBE as a legitimate inference. Psillos accepts it and defends the no-miracles
argument, but take, for example, Lipton. Lipton recognizes that there is a difference in the use of
the IBE in the local and meta sense. Although he has aggressively defended the local application
of the inference, he questions whether or not the meta use is justified
Even if the miracles argument is not hopelessly circular, it is still a weak
argument, and weak on its own terms. This is so because the argument is
supposed to be an inference to the best explanation, but the truth of a theory
is not the loveliest available explanation of its predictive success; indeed
it may not be an explanation at all. (Lipton, 2004a, p. 193)
While Lipton rejects the no-miracles argument, he still has hope that there is some form of meta-
IBE that is justified. Other scientific realists such as Musgrave (1988) and Ben-Menahem (1990)
are less optimistic. Their position is that using IBE in the local sense is justified, but using it in
the meta sense, be it the no-miracles argument or some other form, is not.
The EIA aligns itself with IBE as the key inference to motivate mathematical realism. Since
mathematical entities are strictly unobservable, then the closest inference that scientific realists
employ is local-IBE for unobservable entities. It does no good to appeal to a local-IBE for
potentially observable entities, as mathematics can never be directly observed. Moreover, it is
important to note that the EIA is not making a meta-IBE either. The conclusion of the EIA is that
we ought to have ontological commitment to mathematical entities. It is these entities that, if
true, would best explain physical phenomena. The inference is being made to entities, and not to
some general thesis of mathematical realism such as platonism. Another reason why the EIA is
not making use of meta-IBE is that as we just saw, meta-IBE is not a widely accepted inference
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for scientific realists. Tying itself to meta-IBE would make the EIA susceptible to the exact same
arguments and criticisms leveled against meta-IBE, and would thus significantly weaken its
strength. Henceforth, any mention of IBE will refer to unobservable local-IBE unless otherwise
stated.
2 Fictionalism: A Better Explanation?
Leng advances an interesting challenge for using IBE to infer mathematical realism. Leng
accepts that there are mathematical explanations of physical facts in science, and that the use of
mathematics is indispensable in some of these explanations. She identifies herself as a committed
naturalist and scientific realist50, but at the same time is a nominalist with regards to
mathematical objects. Leng realizes that this seems contradictory as being a scientific realist she
gladly endorses the use of IBE. Her route to nominalism is to claim that mathematical realism is
actually not the best explanation for the existence of mathematical explanations in science.
Instead, fictionalism provides just as good an explanation, if not better. If Leng is right, then she
can use the EIA as an argument against mathematical realism as it is fictionalism that is the
superior explanation.
Fictionalism is the belief that mathematical objects do not exist, and that strictly speaking all
mathematical claims are false. They may be ‘true in the fiction of mathematics’ but they are not
true simpliciter. To this end, Leng accepts the burden of proof to provide an explanation for how
mathematics can be explanatory but at the same time be false. Leng’s goal is to show that
fictionalism is a better explanation than realism, but if she merely shows that fictionalism is just
as good, then “Ockham’s razor would counsel adopting the fictionalist alternative.” (Leng, 2010,
p. 218)
In a typical scientific explanation the explanandum is assumed to be true, and then we search for
explanans that explains the explanandum in question. IBE infers that the best explanans is true.
50 In her (2010), Leng identifies scientific realism with also being a mathematical realist. I do not use her definition
of scientific realism here.
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Leng has no problem with this general application of IBE. What she does note is that there are
many good scientific explanations where we actually know that not all the explanans are true. In
fact, some of the explanans are specifically designed to be false, such as false idealizations in
scientific explanations like frictionless planes or oceans having an infinite depth. The fact that
these idealizations are false has no impact on the value of the explanations that utilize them, and
also does not hinder them from being considered the best explanation of their explanandums.
Leng reasons that,
[i]f one is willing to accept that literally false theories can still serve to
provide accurate representations of physical systems, and that these
theories get their value because of conditions they impose on the behaviour
of these systems, then it is plausible that one should hold a similar attitude
to theoretical explanations. Couldn’t mathematical explanation get its
value as an explanation due to the conditions it imposes on concrete, non-
mathematical systems? And couldn’t these conditions be imposed equally
well by a fictional theory as they would be by a literally true one? (Leng,
2005, p. 180)
Leng’s main criticism of the realist is that they have been solely focused on demonstrating
scientific explanations where mathematics is indispensable, but at the same time realists “simply
assume that all good explanations must have true explanans, so that if we drop the assumption
that the mathematical objects posited by our explanations exist, then these ‘explanations’ cease
to explain at all.” (Leng, 2005, pp. 179–180) This assumption is what Leng challenges, and if she
is successful, we cannot use IBE to infer mathematical realism as realism is not the best
explanation.
One way to show that mathematics being false does not affect any of our scientific explanations
would be to take the Field route and produce a nominalized version of our best scientific
theories, but Leng does not believe this path will be successful. We could limit our
nominalization project to just scientific explanations and show that for every mathematical
explanation there exists a non-mathematical explanation of the same phenomenon. However,
Leng is like Melia in that they both grant that mathematics is indispensable to scientific
explanation, so this too will not work. Instead, Leng’s approach will be to argue that we can
retain all of our mathematical explanations even while maintaining that mathematics is false
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since the truth or falseness of mathematics makes no difference to the overall explanation. Again
like Melia, the reason why Leng believes this is because she claims that the sole role of
mathematics is to index or represent physical facts.
Since… mathematical theories are introduced in empirical science in order
to provide models which allow us to represent physical systems as having
particular physical properties, the question of existence of the
mathematical objects posited by these models makes no difference to the
utility of our mathematical theorizing… An understanding of the model as
a theoretical fiction will account for its role in theorizing equally well, and
as such no explanatory power is lost if we suppose that the mathematical
systems made use of in the course of providing mathematical explanations
of physical phenomena are mere fictions.” (Leng, 2005, pp. 167–168)
Leng assumes the indexing argument is correct, and that mathematics is only an indexing or
representational tool, albeit an indispensable tool. From this assumption she claims that all we
need to retain the explanatory force of mathematical explanations is the pretence that
mathematical objects relate to physical objects, and not that they actually do relate in any real
sense. This pretence is adequately supplied under the fictionalist interpretation. She writes,
could it be that the reason that a given mathematical explanation of a non-
mathematical phenomenon is a good one is not that the mathematical
utterances that make up its explanans are true, but rather that they are
fictional in our make-believe of set theory with non-mathematical objects
as urelements? … I will claim, the explanatory value of appeals to
mathematical objects is plausibly not a result of the existence of such
objects, but rather a result of the aptness of the pretence that such objects
are related to non-mathematical objects in the ways our ‘explanations’
suppose. (Leng, 2010, p. 244)
Leng does not advance a general argument for how the pretence of mathematical objects existing
is all that we need for every instance of mathematical explanation. Instead, her strategy is to
show that particular supposed examples of GMEs, such as the antipodal weather and the cicada
examples, can be treated fictionally without any loss of explanatory power. I take no issue with
Leng’s fictionalist interpretation based on the indexing argument here. In fact, it very closely
mirrors my own interpretation of the indexing argument in chapter 3. Leng ultimately concludes
that because the sole role of mathematics is to index or represent, then mathematical
explanations will be explanatory regardless if mathematical objects exist or not. So,
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even though we posit mathematical objects in the context of our best
explanations of empirical phenomena, the role that mathematical posits
play in these explanations is just the same as the role they play in any
theoretical representations of empirical phenomena. That is, what makes
the mathematical explanations good explanations is not that their
mathematical hypotheses are true of a realm of really existing
mathematical objects, but rather, that they allow for good representations
of the non-mathematical objects they model. (Leng, 2010, p. 245)
Of course, this alone does not show that fictionalism is correct, but Leng’s point is that the
fictionalist account of mathematical explanation is just as good as taking mathematical
explanations at face-value which would imply realism. Moreover, since fictionalism has a much
sparser ontology than realism, then by Ockham’s razor, the fictionalist explanation is better.
Thus, IBE does not license an inference to mathematical realism, but we should infer
fictionalism instead.
Leng does advance some cursory arguments on how the pretence of mathematics can still retain
all the explanatory and epistemic benefits that are utilized in scientific practice, but much is left
to be desired. Leng also does little to motivate her appeal to Ockham’s razor. This principle is
notoriously difficult to apply as the notion of simplicity is not clear. Even if we admit that the
fictionalist’s ontology is simpler than the realist, this does not mean that their theory as a whole
is simpler. Realists have the advantage that they can treat science at face value without any
reconstruction or interpretation, and they arguably have a simpler understanding of the world
without having to appeal to relations to fictional objects. In addition, the realist need not worry
about how to explain why fictional objects are so helpful and indispensable to science. Taking
the theories as a whole, it is unclear which one Ockham’s razor would cut. Regardless, the most
egregious error that Leng makes in her argument is that she patently begs the question against the
realist. Even if we assume that the fictionalist position can account for all applications of
mathematics in science adequately, and that the use of Ockham’s razor to rule out realism is
acceptable, Leng’s entire argument for fictionalism stems from her critical assumption that the
sole role of mathematics is to index or represent physical facts. But as we saw in chapter 3,
assuming this is equivalent to assuming that genuine mathematical explanations do not, and
cannot, exist.
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Leng admits that mathematical explanations where the mathematics is indispensable do exist, but
the mathematics is still limited to a representational role. This is what motivates her claim that a
fictional interpretation can account for such an explanation. Baker correctly notes that “her
argument misses the main point of the Enhanced Indispensability Argument, which is precisely
to draw a sharp line between representational and explanatory use of mathematics.” (Alan Baker,
2009, p. 626) Leng is assuming that GMEs do not exist, and from this assumption she argues for
fictionalism. However, as we have gone to great lengths to show, GMEs do exist. In such
explanations, the mathematics is not playing a representational role, and is indeed indispensably
explanatory. Given this, Leng’s argument for fictionalism is a nonstarter as it depends on a false
and unjustified assumption on the role of mathematics in science.
Baker also points out another serious problem with Leng’s argument. Leng, like Baker, is trying
to make use of IBE to motivate her position. The difference between the two is that Leng is
making a meta-IBE. What she is inferring is that the thesis of fictionalism, if true, provides the
best explanation for the success of mathematics in scientific explanations. Baker’s complaint
about this is that this type of argument is very different from claiming that that fictions are
actually doing the explaining.
Note, first, that what Leng is offering here is a kind of ‘second-order’
explanation. She is explaining how a mathematical explanation is possible,
given fictionalism. But she is not using fictionalism about mathematics to
explain a physical explanandum. In this situation it is crucial to distinguish
between acknowledging the possible falsity of the explanans being offered
and actively disbelieving in an explanans while simultaneously putting it
forward as an explanation. (Alan Baker, 2009, p. 627)
Another closely related objection to Leng’s use of a meta-IBE is that, as we saw above, meta-
IBE is not a widely accepted inference for the scientific realist. Many believe that meta-IBE does
not reflect the inference pattern that scientific realists utilize to generate their realism. Given this,
Leng’s argument does not carry much force, and is not one that would compel scientific realists
to adopt the fictionalist position.
The way to rescue Leng’s argument from the above criticisms is to rework it so that it is no
longer a meta-IBE. Doing so would require overcoming two key obstacles. First, we would need
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to revise all mathematical explanations in science such that they make explicit that mathematical
entities are false – that they are fictions. This revisionist project is potentially problematic as it
no longer takes the practice of science at face value. One would also wonder what the motivation
to perform such a reinterpretation of science is as it provides us with no greater understanding of
the physical world. Regardless, I am content with pretending that this obstacle can be overcome.
The second, and larger hurdle, is that the fictionalist would also need to argue that fictions can
genuinely explain physical facts. By genuinely explain, I mean that their explanatory
contribution is not restricted to mere representation or indexing.
The unstated assumption is that fictions can only play a part in an explanation when they are
playing a representational role. This assumption is certainly widely held within the community of
scientific realists, and it seems that Leng herself holds this view. If we believe that mathematics
are fictions, then GMEs where the mathematics is not representing anything run counter to this
assumption, and hence this creates tension between the existence of GMEs and fictionalism. The
committed fictionalist can dissolve this tension by abandoning this assumption and insist that
fictions can genuinely explain even when they are not representing anything.
Recently, Alisa Bokulich (2009, 2011, 2012) has taken up something akin to this position.
Bokulich develops and advances her own account of ‘model explanations’ which she argues can
make sense of fictions being genuinely explanatory. The fictions that she has in mind are
fictional models such as Bohr’s model of the hydrogen atom. Bokulich argues that based on her
model explanation account, Bohr’s fictional model gives us a genuine explanation of many
physical facts, such as the nature of spectral lines. Bokulich admits that “[p]urported
explanations, such as these, that appeal to fictional structures, are not easily accommodated into
any of the canonical philosophical accounts of scientific explanation,” (Bokulich, 2012) which is
what necessitates the development of her new model explanation account. If Bokulich is correct,
then the fictionalist could argue that mathematics explains just like how the Bohr model
explains, and that both are still ultimately fictions.
Without directly criticizing Bokulich’s account of model explanations, there are two important
obstacles in adopting her position to support fictionalism. The first is that believing fictions can
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genuinely explain may be at odds with scientific realism. One challenge for the Bohr model
explanation is to assert that the actual, genuine explanation is found in quantum mechanics; any
‘explanation’ that the Bohr model seems to provide is illusory. Bokulich’s responds by stating:
I think it is a mistake to believe that there can only be a single legitimate
scientific explanation for a given phenomenon. A closer examination of
scientific practice reveals that, not only can there be more than one genuine
scientific explanation for a given phenomenon (what might be called the
explanatory pluralism thesis), but that some of these explanations may turn
out to be deeper than others. To say that one scientific explanation is not
as deep as another is not the same thing as saying that it is no explanation
at all. (Bokulich, 2011, p. 44)
Granting that the quantum mechanical explanation is deeper than the one provided by the Bohr
model, Bokulich is correct in asserting that this does not imply that the Bohr model is itself not a
genuine explanation. However, is this explanatory pluralism something that a scientific realist
would assent to? Realists are interested in finding explanations that they can utilize IBE on; they
are interested in finding explanations that are most likely to be true. It is hard to see how any
pluralist thesis can be reconciled with a standard scientific realist position. The scientific realist
could surely be pluralist in that he could grant that the Bohr model is a nice explanation in the
epistemological sense, but it is not a genuine explanation. This is common practice as often
epistemological explanations are easier to use. Scientists still make use of Newtonian mechanics
when analyzing many simple dynamic problems, while at the same time recognizing that the
Newtonian picture is not genuinely correct. However, a key difference is that for the scientific
realist, genuine explanations are those which we would be willing to run an IBE on – they are
ontological explanations. Bokulich’s fictions are not like these.
Bokulich expands on her idea of explanatory pluralism when tackling an objection raised by
Belot and Jansson (2010). The worry is that if we accept explanatory pluralism then we end up
admitting far too many fictions as genuinely explanatory; the bar for a genuine explanation is set
too low. Bokulich’s response is to put forward a way to distinguish explanatory from non-
explanatory fictions.
My answer begins with the observation that some fictions are
representations of real entities, processes, or structures in the world, while
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other fictions represent nothing at all. We can even recognize that some
fictions do a better job of representing certain features of the world than
other fictions. What I want to say in answer to the challenge, then, is that
only those fictions that are an adequate representation of the relevant
features of the world are admitted into the scientist’s explanatory store.
(Bokulich, 2012)
The key to being a genuinely explanatory fiction is whether or not the fictions are representing
something real and physical. This leads to the second problem of asserting that fictions can
genuinely explain. As we have shown, the mathematics in GMEs specifically does not represent
anything at all. Their explanatory force is not due to any representational nature at all. According
to Bokulich’s account of genuinely explanatory fictions, then the mathematics in GMEs cannot
be fictions at all as they do not represent. We cannot coherently appropriate her suggestion that
fictions can genuinely explain to support mathematical fictionalism accounting for GMEs.
I have argued that there is no present way to account for the existence of GMEs from a
fictionalist perspective. If this is right, then Leng is incorrect in claiming that fictionalism is at
least as good as realism in explaining the applications of mathematics to science, and hence we
cannot use IBE to infer fictionalism. This by no means shows that there is no way in principle to
reconcile fictionalism and the existence of GMEs. One possible move would be to accept, contra
Bokulich, that fictions that do not represent anything can genuinely explain. This belief makes
fictionalism and GMEs perfectly consistent; however, the obvious problem is caching out this
belief in a coherent and appealing way without running into the problem above where the bar for
scientific explanations is set trivially low. Moreover, I suggest that any such system of
understanding scientific practice would be so complex that the appeal to Okham’s razor to
support fictionalism no longer seems realistic. Thus, granting that GMEs exist, fictionalism is not
the best explanation for the mathematical explanations in science.
3 Unjustified Inference to the Best Mathematical
Explanation
As argued in chapters 3 and 4, I believe that there are GMEs, such as the electron spin example,
which are the best explanation that we have. I also believe in IBE as a legitimate inference for
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the scientific realist. Lastly, I believe that there is no in principle objection to using IBE to infer
the existence of mathematical entities, nor do rival positions such as fictionalism pose a threat to
this inference either. Yet, at the same time I do not believe that we are presently justified in using
IBE to infer mathematical realism. I will argue that this is not a contradictory set of beliefs, and
is in fact supported by the actual way in which IBE is employed by scientific realists, and
advanced by its most ardent supporters.
The reason why believing in both GME and IBE, while at the same time rejecting the inference
to mathematical realism, may appear contradictory is due to an imprecise and naïve
understanding of IBE. As we saw above in 5.1, the best of a bad lot argument poses a serious
problem for any naïve presentation of IBE. This criticism from van Fraassen has led to more
precise and restrictive understanding of the inference. I will argue that once we adopt this more
sophisticated understanding of IBE that is motivated from the best of a bad lot criticism, it is
clear that inferring mathematical realism is not licensed by IBE.
The best of a bad lot argument states that there is no way for us to know that the true explanation
is among those which we are considering unless we assume that we are in some sort of
privileged position from which to make this judgment. Psillos assumes exactly this when he
asserts that scientific realists possess a background knowledge privilege that circumvents the bad
lot argument. The background knowledge from our best and strongly confirmed scientific
theories both restricts the potential set of explanations that we could consider to be true, and also
provides techniques that help us select the best explanation. The key for this process is that the
explanations must be compatible with our scientific theories. IBE does not work in a total
vacuum; rather we make use of our background knowledge as a critical guide for the inference.
While Lipton does not explicitly commit to a privileged position to avoid the best of a bad lot
argument, he does place additional criteria on the best explanation in order to ensure that the true
explanation is likely to be within the set we are considering. For IBE to be warranted, the
explanation needs to be the loveliest potential explanation available. By potential, Lipton just
means that we cannot already assume that we have the actual explanation as this would beg the
question. Still, defining what qualifies as a potential explanation is difficult. “We have to
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produce a pool of potential explanations, from which we infer the best one.” (Lipton, 2004a, p.
58) If our criteria for potential explanations are too broad, such as empirical adequacy, then our
pool would be far too large and would include explanations that we would not even consider
taking seriously. Lipton suggests that we restrict our pool to “only the ‘live options’: the serious
candidates for an actual explanation.” (Lipton, 2004a, p. 59) Doing so would restrict the pool
nicely, but this leads directly to van Fraassen’s complaint that we require a privileged position in
order to determine what counts as a ‘live option’, which he alleges is unjustifiable. Lipton
recognizes that this suggestion assumes some sort of epistemic filter exists, and that this filter
can both restrict our pool of potential explanations and serve to guide us in selecting the best of
them. This sounds very much like Psillos’ background knowledge privilege.
Lipton also discusses the difference between likely and lovely explanations. Likely explanations
are the most probable to be true, whereas lovely explanations are the most explanatory, or
provide the most understanding. It is certainly the case that sometimes the likeliest and loveliest
explanations are one and the same, but in other situations they will point to different explanations
entirely. What makes an explanation likely to be true is relative to our current body of evidence.
This evidence comes from our scientific theories, and how well the explanation fits with these
theories. So, when selecting the best explanation, should we choose the likeliest or the loveliest
explanation from our potential pool? Here, Lipton seems to differ from Psillos. Psillos looks to
the fit with background knowledge as his guide, which seems to point to the likeliest
explanation; however, Lipton argues that the loveliest explanations are those we should consider
to be the best. Lipton is quick to point out that loveliness and likeliness cannot be perfectly
separated, and in reality any defensible version of IBE would need to incorporate elements of
both in the selection process. (Lipton, 2004a, p. 61) In this light, Lipton is not as far from Psillos
as it first appeared. Still, Lipton emphasizes that explanations which are lovely, which provide
the greatest understanding, should be considered the best.
My strategy for showing that using IBE to infer mathematical realism is unwarranted is
straightforward. I will argue that all of our examples of mathematical explanation do not satisfy
the requirements for IBE advanced by Psillos and Lipton. I will not take sides on whose account
of IBE is better. If I can demonstrate that by the standards of the two most vocal defenders of
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scientific realism and IBE that inferring mathematical realism is unjustified, then that should be
sufficient to undermine the EIA.
Consider our optimization explanations: the cicada and the honeycomb examples. In both cases,
a mathematical explanation is advanced to explain the optimal behaviour of cicadas and
honeybees. However, in both cases there is a competing explanation that does not cite any
mathematical facts. Instead, these make use of physical facts such as sunlight, soil temperature,
and units of time in the cicada example, and facts of approximately Euclidean space-time in the
honeycomb example. Now, taking up Psillos’ version of IBE, do the optimization explanations
which depend on mathematical facts fit with our background knowledge privilege? I will not
make a knock down case here, as this would require a detailed look at what our background
knowledge genuinely consists of, but I will suggest that the optimizations explanations do not fit
well. The reason being is that in biology, and in typical biological explanations, it is the
biological, ecological, and physical facts that build up the core of our knowledge. The cicada and
honeycomb explanations, while interesting, are critically unlike almost all other biological
explanations in that they use mathematics for its explanatory force. This is not to take anything
away from the explanation in terms of its contribution to our scientific knowledge, but the
important point is that it does not satisfy the basic condition that would allow us to consider it the
best to warrant an IBE. This is made even clearer when considering the nominalist explanations
of cicada life-cycles and honeycomb shapes. Even if you reject the claim that optimization
explanations do not fit with our background knowledge, it is hard to deny that the nominalist
explanations do fit, and that they fit better than the mathematical explanations. So, even if both
the mathematical and nominalist explanations are in our pool, the nominalist versions are the
more likely to be true based on their compatibility with our background knowledge. Given this, it
is clear that using IBE based off of the mathematical explanations of the cicada or honeycomb is
not entirely justified. These mathematical explanations either do not meet the requirements of
fitting with our background knowledge, or if we are generous and grant them that fit, then they
do not fit as well and hence are not as likely as their nominalistic counterparts.
The situation is similar when we consider Lipton’s system. Are the mathematical explanations to
be considered ‘live’ options for us? It seems that when comparing them to the nominalist
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explanations that the mathematical explanations are not good candidates for being live by the
exact same reasons as above. This does depend on exactly what epistemic filter that Lipton has
in mind, so I will not base too much on the above conclusion. Still, there may be some hope as
Lipton’s focus is on the loveliest explanations. It can be argued that the main virtue of the cicada
explanation is that it genuinely enhances our understanding in a way that the nominalist version
does not, and thus while it may be true that the mathematical explanations are not more likely
than the nominalistic explanations, they are the more lovely and hence should be inferred as true
by IBE. While this sounds reasonable, it is, I feel, incorrect. While the mathematical explanation
invoking prime numbers does seem to answer some important questions about the life-cycle of
the cicada, it comes with an even larger question that remains unanswered. The question that is
blocking any actual understanding is how properties of abstract, noncausal, mathematical entities
influence, matter to, or make a difference to actual cicadas and honeybees. The nominalist
explanations, on the other hand, are lovely in that they explain the behaviour of cicada and
honeybees without introducing new unanswered questions. It is physical properties and physical
entities that relate to cicadas and honeycombs and are influencing or making a difference to their
behaviour. When comparing the two types of explanations, it is the nominalistic explanations
that lead to a greater understanding as there is no additional mystery as to how they work. Thus,
by Lipton’s criteria as well, it would be unwarranted to select the mathematical cicada or
honeycomb explanation to be the best explanation for an IBE as they are not the loveliest
potential explanation.
The Kirkwood gaps example does not fare much better when applying a more sophisticated
version of IBE. On the surface, it seems that the mathematical explanation does fit better with
our background knowledge as astrophysics is much more mathematical than biology. However,
even though mathematics may be ubiquitous in astrophysics, this is different than saying that
explanatory mathematical factors are standard in astrophysical explanations. When considering a
competing explanation that uses physical facts such as gravitation and mass as the key
explanatory factors, such an explanation certainly fits better. The same challenges that the
optimization explanations faced with fitting our background knowledge, or the live options for
Lipton, present themselves here. Moreover, while an eigenvalue analysis may explain the
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Kirkwood gaps in the sense of prediction, it is hard to say how eigenvalues contribute to our
understanding of the presence of the gaps. As above, the issue is in establishing some sort of link
between mathematical properties such as eigenvalues, and physical entities such as asteroids.
This issue is made even clearer when considering the nominalistic explanations that simply cite
how physical factors such as gravitation, planetary mass, etc., are the ones that influence the
asteroids. Without some sort of account as to how the mathematics is making a difference to the
asteroids, the mathematical explanation of the Kirkwood gaps explanation is not lovelier than its
nominalistic counterpart, and hence IBE would be unwarranted.
The criticisms of the cicada, honeycomb, and Kirkwood gaps explanations here mirror those
raised in chapter 3. One key factor that is making IBE unwarranted for these explanations is that
they may not be genuine mathematical explanations in virtue of the fact that there exists
nominalistic explanations of the same phenomena, which is precisely what was argued before as
a major downfall of these examples. Instead, we should consider the GME of electron spin. This
explanation is different as the mathematics is genuinely explanatory, and there is no nominalistic
explanation available to us. If IBE is able to infer mathematical realism at all, it would be most
likely to succeed when considering our best and uncontroversial example of a GME.
When considering how well the electron spin example fits with our background knowledge, it is
immediately apparent that the fit is much better as quantum mechanics is almost entirely
mathematical. Recall that the same claim was made for the fit between the Kirkwood gaps
explanation and astrophysics, but the problem was that explanations in astrophysics still almost
exclusively cite physical difference-makers. This problem does not present itself in quantum
mechanics. Many explanations in quantum mechanics point to mathematical difference-makers,
and lots of the mathematical formalism that is indispensable to doing any quantum mechanics
does not have any sort of physical interpretation at all. This certainly bodes well for using IBE to
infer mathematical realism; however, there is still a critical way in which a GME does not fit
well at all with our background knowledge. Some explanations in quantum mechanics have led
to our commitment to novel entities, such as quarks, through the use of IBE. But there is no case
yet where an explanation has led us to commit to an abstract, noncausal entity. Simply put,
mathematical entities just do not fit with the background knowledge of our best scientific
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theories no matter how inherently mathematical our theories are. This is because our best
scientific theories have only utilized IBE to infer physical entities. I do not mean this to be an in
principle objection to using IBE to infer mathematical entities. There is nothing about the
inference model that fundamentally restricts the possibility of such an inference. The issues
comes when we try to justify such a move using only the background knowledge from science.
Our background knowledge from our best scientific theories does not fit nicely with GMEs. It is
our background knowledge that is restricting our ability to infer mathematical realism.
Turning towards Lipton’s criterion of loveliness, the electron spin explanation fares no better.
The property of spin itself does increase our understanding and is a lovely explanation. In
addition, no one doubts the predictive power of the mathematical formalism of spin, but this is a
far cry from claiming that we are any closer to understanding what spin is. We saw above that
the primary reason for our lack of understanding is due to the absence of an account as to how
mathematical entities and properties influence physical entities and properties. This same
problem presents itself here in an even more bizarre way. Quantum mechanics as a whole is
famous for being incredibly accurate in its predictions, and notoriously difficult, if not
impossible, to understand in its present form. As the ever quotable Feynman remarked, “I think I
can safely say that nobody understands quantum mechanics.” (Feynman, 1965, p. 129) This cleft
between predictive power and understanding has been made famous by the Einstein-Podolsky-
Rosen and Schrödinger’s cat thought experiments. The core idea is that while mathematical
explanations in quantum mechanics, such as the mathematical formulation of spin and psi
function of superposition, are incredibly accurate predictors, they do not enhance our
understanding of the physical phenomena; in short, they are not lovely explanations. Even if
GMEs from quantum mechanics are among our pool of potential explanations, they are not
lovely enough to infer their (approximate) truth.
I admit that the above arguments have not been rigorously developed. I do hope that I have done
enough to motivate the claim that simply being a GME is not enough to satisfy the conditions of
IBE. The crux of my position is that there is a difference between being the best explanation that
we have, and being the best explanation with respect to running an IBE. GMEs satisfy the
former, but at present they do not satisfy the latter. While I have used the formulation of IBE
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advanced by two of its most ardent defenders, a brief look at the history of science also supports
the idea that there are times when an explanation is the best that we have, is incredibly useful in
terms of predictive power, but at the same time the scientific community has rejected any IBE
that would lead to new ontological commitment. In chapter 2 we saw such an example with
Maddy’s story of the discovery of the atom. Explanations that made use of the atom were
arguably indispensable, but still scientists rejected the existence of the atom. It was not until
Perrin’s 1905 experiments on Brownian motion that ultimately led to an almost wholesale
acceptance of the atom by the scientific community.
Another example of this surrounds J. J. Thomson’s discovery of the electron. Based on his
experiments on cathode rays, Thomson advanced the hypothesis that tiny ‘corpuscles’ with
negative charge are constituents of the atom.51 These electrons nicely explained experimental
results from Thomson, Lenard, and others in 1897; however, the scientific community still did
not accept the electron as a real entity. It took years of further experimenting to increase the
understanding of the electron before being accepted as a real entity. (Baigrie, 2007, p. 127) This
situation was summed up by Thomson almost 40 years later. “At first there were very few who
believed in the existence of these bodies smaller than atoms. I was even told long afterwards by a
distinguished physicist who had been present at my lecture at the Royal Institution that he
thought I had been 'pulling their legs.'” (Thomson, 1936, p. 36) The acceptance of the electron is
similar to the atom example, but a key difference is that in the case of the atom a crucial
experiment was critical in convincing the scientific community that atoms exist. For the electron,
increased understanding came more gradually. In both cases though, there was a time where the
best explanation of physical phenomena was still not good enough to be considered the best
explanation for an IBE.
My belief is that we are in the exact same position with GMEs as the scientific community was
with respect to the atom and electron prior to their ultimate acceptance. Some GMEs may be the
51 Thompson also claimed that these negatively charged corpuscles are the only constituents of the atom, which was
proven false.
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best explanation we have, and they may be excellent predictive tools that explain physical
phenomena, but we do not have neither a crucial experiment nor the developed understanding
that makes using IBE legitimate. What makes my position different from other critics of IBE for
mathematical realism is that my rejection of using IBE in this way is not an in principle rejection.
I do not believe that we can never use IBE to infer mathematical realism, but rather that we are
presently unjustified. This is supported by those who advance sophisticated versions of IBE, and
also by the historical record.
The only remaining question is how can we move forward such that an inference to
mathematical realism will be justified? This is difficult to answer, and I will merely suggest two
possible ways forward. One route that scientific realists typically take is to look for additional
confirmation. This was the case for the atom. If the atom actually existed, then Einstein predicted
we would be able to witness Brownian motion. Psillos also recognizes this with his example of
the neutrino.
When the neutrino’s energy is taken into account, there is no need to abandon
the principle of conservation of energy during β-decay. This degree of
confidence in the existence of the neutrino will depend on many factors. But,
sure enough accepting its existence will guide looking for further experimental
and theoretical confirmation. Pretty much like the mouse-case, the presence of
neutrino in a β-decay implies certain further predictions of neutrino-related
phenomena. (Psillos 1999, p.212)
So, if we take GMEs seriously, then we should be looking for further predictions of
mathematical-related phenomena. Herein lies the problem with this route of justification. It is
unclear what, if any, predictions are entailed by the existence of mathematical entities. The main
reason for this stems from their abstract, noncausal nature. The only hope for this way forward is
to gain an increased understanding of how noncausal explanations work in general. This is
important for not just mathematical explanations in science, but also any other type of noncausal
explanation such as geometrical or structural. While somewhat farfetched, it is possible that an
increased understanding of how these explanations work could yield interesting ways to
manufacture novel predictions and to test for them.
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A more promising route towards justification would be to show how mathematical objects truly
enhance our understanding of the physical world. For example, an increased understanding of
how the mathematical formulation of quantum mechanics is such a successful predictor, and how
it relates to the physical world, could lead to a critical mass of mathematical explanations and
understanding that would warrant an IBE. While this suggestion is highly speculative and
somewhat vague, the basic idea is that it more closely follows the path of the acceptance of the
electron where acceptance came gradually through increased understanding. Some inroads were
made in section 4.4 when we analyzed some of the possible contributions and ways that
mathematics can be a difference-maker in scientific explanations. What we need to be focusing
more on is how mathematics makes a difference to the physical world. The danger with this route
is that it may be impossible to gain this understanding without first presupposing some sort of
relationship between mathematics and the physical world. Undoubtedly, such a presupposition
would be charged as totally unsupported by our best scientific theories by the nominalist. This
would clearly make an inference to mathematical realism question begging as mathematical
objects or truth would already be taken for granted.
I have argued that when we consider a sophisticated version of IBE, inferring mathematical
realism based on GMEs is presently unjustified. I then presented two potential ways forward that
would allow for IBE to work on GMEs, but at the same time it should be clear that both these
ways are challenging and come with their own set of problems. By no means is this meant to be a
condemnation of either IBE or mathematical realism. Rather, the point of this analysis is just
meant to illustrate that if we take IBE on its own as our sole inferential tool for ontological
matters, then mathematical realism is not yet attainable. This leads to one final way in which we
can make use of IBE to infer mathematical realism legitimately: we can supplement IBE so that
it does not stand on its own. I will briefly consider two ways in which we can add to our
inferential tools, one below, and one in chapter 6.
4 The Not-So-Enhanced Indispensability Argument
We are now in a position to identify precisely what the problem is with the first premise, (EP1),
of the Enhanced Indispensability Argument (EIA). (EP1) states that “we ought to rationally
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believe in the existence of any entity that plays an indispensable explanatory role in our best
scientific theories.” This is meant to be an appeal to IBE. However, what is clear now is that this
conception of IBE is naïve. It does not reflect the understanding of IBE advanced in the
literature, nor does it accurately portray the actual way in which scientific realists have
historically utilized the inference. The version of IBE advanced in the EIA is not accurate, and
thus the argument is unsound.
There are two possible ways forward to rescue the EIA. The first is to redefine the argument so
that it reflects a more sophisticated form of IBE. (EP1) would need to be changed to stipulate, for
example, that the explanation needs to fit nicely with our background knowledge, or that it needs
to be the loveliest explanation. While this would fix the first premise to be a more faithful
representation of IBE, it actually renders the EIA invalid. As shown above, our present stock of
GMEs do not satisfy the criteria such that making use of IBE is legitimate. To circumvent this,
the second premise, (EP2), which states that GMEs exist, would also need to be changed to
assert that there exist GMEs which do fit our background knowledge, or are the loveliest
explanations available. It is no longer enough to just produce a GME, and this means that the
status of (EP2) is again cast into doubt. What we actually need is a GME that satisfies the criteria
for being the best explanation with respect to IBE. With these changes the debate would shift
away from the indispensable explanatory role of mathematics to other complicated issues such as
the relationship between prediction and understanding. This is certainly a reasonable way to
proceed for defenders of the EIA, but it raises such challenging issues that the argument is no
longer the nice and easy path to mathematical realism that it is meant to be. Suddenly, the EIA
depends on spelling out the relationship between mathematical and physical objects such that we
can make sense of how mathematical explanations lead to understanding. This is a tall order and
it is unclear if it is even possible to achieve without prior metaphysical assumptions.
The second option for the EIA is to note that all the objections that have been raised here seem to
hinge on the assumption that there are real and important differences between mathematical and
physical entities. For example, the inability to generate further mathematical-related predictions
was due to the assumed abstract and noncausal nature of mathematical entities, which is, of
course, different from the concrete and causal nature of physical entities. This assumed
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difference seems fine, but actually part of this assumption is that this is a difference in kind rather
than in degree. Similarly, the bad fit between GMEs and our background knowledge is because
of the difference between standard physical explanations and abstract, noncasual, mathematical
explanations. Again, this assumes that the explanations are significantly different in kind, and not
just different along the same continuum of scientific explanation. Rejecting these assumptions
would be a radical move, but it would open up the possibility that we could make further
predictions and test for the existence of mathematical objects, and for GMEs to
unproblematically fit with our background knowledge. The reason why is that there is no longer
a cleft between the mathematical entities, predictions, and explanations in our best scientific
theories and the physical entities, predictions and explanations. Surely they are different in some
important ways, but fundamentally they are all of the same kind. The obvious problem with this
second path is that abandoning the assumption that mathematical entities are different in kind
from physical entities would be unappealing to the bulk of scientific realists and mathematicians
alike. This would make it a nonstarter and not a good way in which to craft an overarching
argument for realism.
Another problem with treating mathematical entities and explanations exactly like physical ones
is that such a move seems ad hoc in nature. There would need to be significant additional reasons
to abandon the belief that they are different beyond simply rescuing the EIA. Fortunately, there
is one set of beliefs that would provide independent reasons for maintaining the equality of
mathematical and physical entities and explanations: Quine’s thesis of confirmational holism
which Quine believed faithfully reflected the actual practice of science. Confirmational holism
has two important features. The first is that theories are confirmed in a holistic manner. We do
not separate entities within our theories by their properties; what matters is simply what role
these entities play. Secondly, there is no real difference in kind between the unobservable entities
utilized in our best scientific theories. Terms such as abstract versus concrete merely indicate a
difference in degree. If the EIA adopts confirmational holism, then even when using a
sophisticated version of IBE we will be able to infer mathematical realism. No longer is there
any problem with our GMEs fitting with our background knowledge. Our best scientific theories
are to be treated holistically, so any scientific explanation, be it mathematical or otherwise, is just
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another part of science itself. The differences that we had previously identified assumed that we
could separate GMEs from regular scientific explanations, but this goes against the spirit of
conformational holism. Similarly, there is no difference in using IBE to infer abstract
mathematical entities compared to concrete unobservable physical entities, as these entities are
not different in kind. They are all unobservable posits in our theories, and hence there is no
significant difference at all in the inference either.
Confirmational holism also helps us escape the challenge that our GMEs are not lovely
explanations – that they do not contribute to our understanding. It is a mistake to look at specific
explanations in isolation. We need to consider how the scientific theories that our GMEs are a
part of contribute to our understanding of the world. When thought of this way, then it is clear
that GMEs are lovely explanations. Even the electron spin example which is certainly an
excellent predictor contributes to our understanding as the concept of spin itself is critical in
understanding observed behaviour of things like electrons. Before we were able to isolate the
mathematical formalism from the rest of the theory and allege that we did not understand how
this aspect of the explanation functioned. When we adopt confirmational holism, this criticism is
no longer possible. This all points to confirmational holism being a perfectly reasonable and
effective way at ensuring that using IBE to infer mathematical realism is entirely justified.
I feel that fully embracing confirmational holism is the only reasonable way forward for the EIA.
However, at the same time it also spells the end for the argument itself. Unlike the two above
suggestions, adopting confirmational holism is not merely altering the premises of the EIA.
Rather, it requires supplementing the argument with an entirely new inferential tool. By adopting
confirmational holism the EIA simply collapses to the Quinean indispensability argument. The
EIA differentiates and enhances itself from the original Quinean version by pointing to IBE as
the inferential process that leads to realism. The catch is that, as I have argued, IBE alone cannot
presently deliver the desired conclusion. Confirmational holism is needed to circumvent the
problem that any sophisticated expression of IBE does not justify inferring mathematical realism.
If confirmational holism is adopted, then in actual fact the inferential process leading to realism
is not IBE at all, but rather a product of our holistic beliefs. The differences between the EIA and
the QIA vanish entirely. In no way then is the EIA actually enhanced at all. It points to the exact
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same inferential process, depends on the exact same claim that mathematics is explanatorily
indispensable52, and is thus susceptible to the exact same criticisms that we examined in chapter
2. The Enhanced Indispensability Argument collapses entirely to the Quinean indispensability
argument, and ultimately the appeals to inference to the best explanation have done nothing to
strengthen indispensability arguments for mathematical realism.
Perhaps there is some other inferential tool that we can add to the EIA to rescue it other than
confirmational holism. Although I will not make the case for it here, I believe that confirmational
holism is necessary for any indispensability argument that aims to lead us to mathematical
realism.53 IBE forces us to examine the precise role of the entities in our explanation. Moreover,
any justified use of IBE also requires us to examine how well our supposed explanations fit with
what we already know, and how they lead to a greater understanding of the world. The trouble
with GMEs is that this type of examination will always lead to trouble due to the differences
between mathematical and physical entities. This difference is not only due to the abstract,
noncausal nature of mathematics, but also stems from our lack of understanding of the
relationship, if any, between mathematics and the physical world. These problems cannot be
avoided when we look at GMEs and mathematical entities in isolation. Thus, the only way to get
around these issues is to not look in isolation at all, and to not consider mathematical entities
truly different from physical. This is exactly what confirmational holism allows us to do, and is
why the EIA needs it to work.
What does this mean for mathematical realism? The failure of the EIA notwithstanding, by
confirmational holism alone and the positive results from chapters 3 and 4, it seems that we
ought to be mathematical realists. The pressing question then is whether or not we should believe
that confirmation is applied in a holistic manner. In chapter 2 I argued that the criticisms of
52 Quine never pointed to being explanatorily indispensable in his writings. As mentioned in chapter 2, Colyvan
shows that it is simple to alter Quine’s argument to take explanation into account.
53 Bangu (2012) argues at length that confirmational holism is central to any indispensability argument. He claims
that the best an indispensability argument can do is to establish a conditional conclusion: if one is a scientific realist
and a Quinean holist, then one should also be a mathematical realist. The challenge, which he admits is unaddressed,
is why we should ever want to be a Quinean holist in the first place.
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confirmational holism are powerful enough to reject it, and I stand by that analysis here. Looking
at it more broadly, we can see now the trouble with indispensability arguments that depend on
confirmational holism such as the QIA and EIA. All the nominalists needs to do to avoid the
conclusion of mathematical realism is to reject confirmational holism as an inferential tool in our
best scientific theories. The tradeoff for the nominalist is minimal. Adopting holism leads to
unwanted ontological commitments, whereas rejecting it they are able to maintain the sparse
ontology they desire while at the same time the nominalist arguably does not lose any significant
understanding of the practice of science. The upshot to this is that now philosophers of science
can look to examine and understand the explanatory role of mathematics in science free from the
trouble of worrying about ontological commitment. The downside is that this may spell the end
of the recent revival in interest of indispensability arguments.
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Chapter 6 Conclusion
In a brief survey on mathematical explanation, Mancosu suggests three ‘philosophical pay-offs’
that could result from an increased understanding of mathematical explanations of physical facts.
First, in the direction of a better understanding of the applicability of
mathematics to the world. Indeed, understanding the ‘unreasonable
effectiveness’ of mathematics in discovering and accounting for the laws
of the physical world (Wigner, Steiner) can only be resolved if we
understand how mathematics helps in scientific explanation. Second, the
study of mathematical explanations of scientific facts will serve as a test
for theories of scientific explanation, in particular those which assume that
explanation is causal explanation… Third, philosophical benefits might
also emerge in the metaphysical arena by improved exploration of various
forms of the indispensability argument. Whether any such argument is
going to be successful remains to be seen but the discussion will yield
philosophical benefits in forcing for instance the nominalist to take a stand
on how he can account for the explanatoriness of mathematics in the
empirical sciences. (Mancosu, 2008, p. 138)
The principle aim of this dissertation was to analyze and assess the Enhanced Indispensability
Argument (EIA) for mathematical realism. Ultimately the EIA was found to be lacking as it
relied on a naïve understanding of inference to the best explanation (IBE). Regardless, as
Mancosu suggests, several important philosophical pay-offs were arrived at along the way. The
first important result was to generate a set of criteria for a genuine mathematical explanation
(GME) that nominalists and realists alike could agree to. This step, which is part of Mancosu’s
third pay-off, has been overlooked in the literature and is a major factor in why there has been no
agreement whatsoever on the status of GMEs. The key to success for this generation was in
refining and understanding the ways in which nominalists reject supposed examples of GME via
the indexing argument. With the criteria in hand, presenting an example of GME turned out to be
a much easier task as it immediately became clear what sort of features in the explanation were
needed in order to circumvent the nominalists’ arguments. The electron spin example, which
explains the splitting of a beam of electrons passing through a Stern-Gerlach apparatus, was thus
shown to satisfy all the criteria of a GME.
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From there we were able to address a second overlooked area in the literature; we looked
towards accounts of scientific explanation to corroborate the claim that GMEs exist. Strevens’
kairetic account was an ideal account to make use of due to its commitment to a two-level
approach to explanation that could identify difference-makers in any domain. In addition to
tackling our stock of mathematical explanations, this process was interesting for two other
reasons. First, it presented an interesting stress test for the kairetic account. Strevens’ belief that
the kairetic criterion can be applied to other domains, such as noncausal or mathematical, is
appealing but up till now untested. GMEs represent a serious challenge for any account of
scientific explanation, and attempting to appropriate the kairetic account in order to analyze
GMEs without a doubt pushes the limits of Strevens’ account of scientific explanation. Secondly,
by corroborating GMEs as a legitimate form of scientific explanation, this paves the way forward
for supporters of noncausal explanation to both put pressure on exclusively causal accounts of
scientific explanation. All this work speaks to Mancosu’s second philosophical pay-off; GMEs
serve as a challenging and arguably necessary test for accounts of scientific explanations.
Once the electron spin example was shown to be a GME we were able to free ourselves from the
burden of proof and focus on the more interesting question of how it is that mathematics
explains. We briefly surveyed some of the promising suggestions for how to make sense of the
explanatory role that mathematics plays in scientific explanations. Though this work was quite
preliminary, it represents progress in Mancosu’s first philosophical pay-off of an increased
understanding of the applicability of mathematics through understanding mathematical
explanation. Moving forward, this area of research looks to have the greatest impact in terms of
our understanding of mathematics in science.
In my opinion, one of the greatest impediments to understanding applicability via mathematical
explanation in science comes from the staunch opposition by nominalists due to their fear of
ontological commitment. This was a key motivating factor in my methodological approach of
treating mathematical explanation and metaphysical conclusions as independently as possible.
To that end, taking a precise look at IBE was essential as it showed that the nominalist has
nothing to fear from arguments such as the EIA. This analysis has two immediate benefits. The
first is that no longer is the idea of mathematical entities playing a genuine explanatory role in
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scientific explanations equivalent to mathematical realism. This separation means that further
research can continue on the applicability of mathematics in explanation free from metaphysical
biases. Secondly, understanding how IBE works made clear that our present indispensability
arguments critically depend on confirmational holism to succeed. This speaks directly to
Mancosu’s third pay-off of an increased understanding of indispensability arguments and what
they truly need to succeed.
1 Moving Forward
I have just suggested that a promising area for future research is to continue the work started in
section 4.5 where we looked at the difference-making role of mathematics in GMEs. I will close
by considering two other areas for further development. Throughout this dissertation I have
remained purposefully silent on two important overarching themes: naturalism and mathematical
realism. I will now briefly address these topics.
Quine’s naturalism plays an incredibly important role in the Quinean indispensability argument
(QIA) and for all indispensability arguments that stem from it such as the EIA. Naturalism serves
as the backdrop for Quine’s motivations for being a mathematical realist, the inferential
framework that leads to the realist conclusion, and also as a first line of defense against many
potential objections to the QIA and mathematical realism. Beyond this foundational role,
naturalism also specifies the arena in which the debate for mathematical realism takes place.
Science is the sole arbiter of our ontology. The actual practice of mathematics is simply not
taken into account as Quinean naturalism explicitly rules out looking anywhere else for
guidance. But is such a restriction necessary? Are we cutting ourselves off from potentially
fruitful avenues of exploration? For the naturalist it seems reasonable to want to rule out certain
other domains, such as religion, voodoo, or astrology. These things are often found to be in
conflict with the practice of science, be it in their conclusions or in their methodologies.
However, mathematics is different in that its practice is acceptable and in fact is indispensable to
the practice of science, whereas the other domains are certainly not. In short, mathematics seems
special.
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The disrespect of mathematical practice was highlighted in chapter 2 but played no role in our
analysis of the EIA. However, it does stand to reason that if one finds this disrespect to be
offensive, then any sort of indispensability argument that depends on the naturalistic perspective
that science is the sole arbiter of our beliefs is simply a nonstarter. Maddy is one such person
who finds the lack of respect towards mathematical practice inherent in Quinean naturalism to be
objectionable; so much so that she proposes a mathematical naturalism to operate alongside her
scientific naturalism.54
What I propose here is a mathematical naturalism that extends the same
respect to mathematical practice that the Quinean naturalist extends to
scientific practice. It is, after all, those methods – the actual methods of
mathematics – not the Quinean replacements, that have led to the
remarkable successes of modern mathematics. Where Quine holds that
science is ‘not answerable to any supra-scientific tribunal, and not in need
of any justification beyond observation and the hypothetico-deductive
method’ (Quine, 1981a, p. 72), the mathematical naturalist adds that
mathematics is not answerable to any extra-mathematical tribunal and not
in need of any justification beyond proof and axiomatic method. Where
Quine takes science to be independent of first philosophy, my naturalist
takes mathematics to be independent of both first philosophy and natural
science (including the naturalized philosophy that is continuous with
science) – in short, from any external standard. (Maddy, 1997, p. 184)
Since mathematical practice answers to no ‘extra-mathematical’ tribunal, then issues such as
deciding on the continuum hypothesis or new axioms of set theory are made solely within the
set-theoretic community, which Maddy takes to be representative of the mathematical
community as a whole. This also means that the question of whether or not mathematical objects
exist is also only answerable within mathematics. Indispensability arguments have no say in
issues of ontology regarding mathematics. This, however, ends up being problematic. Science is
54 Maddy’s scientific naturalism is similar to Quine’s in that she believes that science is not answerable to any
external or ‘supra-scientific’ tribunal. However, her position differs in three important ways. First, we have already
seen that Maddy rejects confirmational holism. Second, she argues that her naturalism is a much more subtle
position compared to Quine’s. For example, Maddy considers her naturalism more of an approach, and that belief in
scientific results does not stem from “some general meta-thesis about the reliability of science... but the detailed
scientific evidence specific to each individual case.” (Maddy, 2002, p. 61) The last point of departure between
Maddy and Quine is in their treatment of mathematics. For more details see Maddy (2007)
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clear in telling us that many physical objects, such as tables and chairs, and even unobservables
like protons and quarks, exist. Science also tells us many properties of these objects – they exist
objectively and are located in space-time. But mathematics tells us almost nothing about
mathematical objects. At best, mathematics only tells us that a small number of specific
mathematical objects exist as stated in mathematical axioms. Yet mathematics tells us nothing
about what this existence entails; whether or not they are spatio-temporal, objective, etc. Maddy
notes that so little is specified that realist and anti-realist positions, such as fictionalism and
formalism, are all perfectly compatible with mathematical practice. (Maddy, 1997, p. 192) She
also claims that from a methodological perspective, questions about the existence of
mathematical objects have historically proven to be essentially useless for the development and
expansion of mathematics. In this light, questions regarding existence and ontology should be
dismissed as they turn out to be irrelevant to the development of successful mathematics, and to
regular mathematical practice.
In a more recent work, Maddy (2007) is careful to distinguish between the methodological and
the metaphysical question. Her original argument was that issues of existence and ontology play
no role in the advancement and practice of mathematics. However, Maddy does recognize that
there are still interesting and important metaphysical questions that remain, such as ‘does
mathematics have a subject matter?’, or ‘are mathematical claims true or false?’. Interestingly,
Maddy admits that the answers to these questions will not stem from mathematical practice, but
rather will be inspired by traditional philosophy of mathematics. She considers three possible
positions that purport to answers the metaphysical questions and assesses how well each fits into
her overall naturalistic approach to mathematics and science.
The first position Maddy calls Robust Realism. Robust Realism is actually one of any of a
collection of realist positions such as ontological platonism, epistemological platonism,
structuralism, etc. By my definition, a robust realist accepts all three of the realist theses and can
adopt further theses as he sees fit. Maddy is quick to reject Robust Realism as a good answer as
it conflicts with mathematical practice which is sacrosanct according to her mathematical
naturalism. Her evidence for this is the case of the continuum hypothesis and whether or not we
should accept V = L as a set theoretic axiom. Recall that the set theoretic community wishes to
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reject V = L as doing so would maximize the potentialility of set theory. What would the robust
realist say? Maddy alleges that the robust realist would need to appeal to the fact that V = L must
be objectively false, as there is an objective world of sets that we are trying to discover. But this
is not the type of reasoning that actually goes on in mathematical practice. Set theorists dislike
V = L on grounds of its limiting nature. The problem is, “granting that selecting for
maximization generates theories we like, what reason do we have to think it likely to generate
theories that are true?” (Maddy, 2007, p. 366) The robust realist is adding metaphysical
assertions on top of the methodological concerns of the mathematician. In essence, they are
adding some ‘extra-mathematical’ interpretation of mathematics that is just not supported by
practice, and thus should be rejected by the mathematical naturalist.
Thin Realism is the second position that Maddy considers. The thin realist takes the practice of
set theory at face value and accepts that the axioms and theorems of set theory, and hence the
sets generated by the axioms, are true and exist because set theory tells us so. But what about
questions regarding independence, abstractness, etc.? Set theory and mathematical practice say
nothing about this. Answers to these sorts of questions are typically found within science. The
thin realist looks towards science and finds that scientific practice does not ascribe spatio-
temporal or causal properties to things like sets (nor do they deny them either). Sets, then, “have
the properties ascribed to them by set theory and lack the properties set theory and natural
science ignore as irrelevant. There is nothing more to be said about them.” (Maddy, 2007, p.
369) Maddy contends that Thin Realism has the advantage over Robust Realism in that it does
not need to inject anything over and above mathematical practice, such as an appeal to objective
truth or reality. What we know about mathematical objects just comes from the practice of
mathematics and science. Issues such as whether or not the continuum hypothesis is true or false
will be strictly determined by whether or not one day we have well motivated mathematical
reasons for adopting one or the other. It has nothing to do with objective truth value of the
continuum hypothesis. Of course, Thin Realism will fail at giving satisfactory answers to many
of the metaphysical questions that we are interested in. It cannot tell us anything insightful into
the properties of mathematical objects, their nature, etc., beyond simply pointing to what set
theory tells us. This, as we know, amounts to basically nothing.
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Finally, Maddy considers Arealism. Arealism is just like anti-realism but they differ in their
reasons for adopting their positions. Maddy aims to distinguish between a priori arguments
against realism that are typically associated with anti-realism, and reasons that are motivated
from within a naturalistic framework and free from prior prejudice. Ultimately their beliefs are
the same, but it is how they arrive at these beliefs that are different. The Arealist does not take
the axioms and theorems of set theory to be true, and thus maintains that sets and all other
mathematical entities do not exist. Is this compatible with mathematical practice? Haven’t we
already acknowledged that certain axioms of mathematics assert that some mathematical objects
exist? The Arealist needs to somehow explain how they can take back the existential claim of
these axioms, and also provide an account of how it is that something that is false can be so
useful. Fortunately we have seen several options already such as fictionalism and the weaseling
argument. Maddy provides another:
set theoretic claims, including existence claims, generated by its most
effective methods should be adopted as appropriate means towards
theories that serve our goals, but natural science is the final arbiter of truth
and existence, and it confirms neither the truth of mathematics nor the
existence of sets. (Maddy, 2007, p. 384)
Natural science is where we should look, and thus we do not need to believe in the truth of the
axioms of set theory.
The final task is to compare the respective positions and see which fares best. Robust Realism is
already off the table as it conflicts with mathematical practice. But when adjudicating between
Thin Realism and Arealism, Maddy comes to a surprising conclusion.
If this is a fair description of the state of debate between the Thin Realist
and the Arealist, then it’s hard to see that there is any fact of the matter
here about which we can be right or wrong... [T]he decision between Thin
Realism and Arealism appears to hinge on matters of convenience, taste,
and preference in the bestowing of these honorific terms (true, exists,
science, knowledge). (Maddy, 2007, p. 389)
Thin Realism and Arealism are perfectly compatible with mathematical and scientific practice.
They are equally supported by our mathematical naturalism. In fact, the actual differences
between Thin Realism and Arealism are quite trivial. The thin realist observes that science says
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nothing about the causal or spatio-temporal nature of mathematical objects, so he excludes these
properties from mathematical objects. All he allows for is existence due to the axioms of set
theory. The Arealist also notes that science says nothing about the existence of mathematical
objects either, so she believes they do not exist. Asides from existence, they agree on every other
aspect in that there is simply nothing more to say about mathematical objects at all. What
choosing between the two positions boils down to is whether or not we think science should be
the sole arbiter of existence, in which case we are Arealists, or whether we think that
mathematics can also determine existence, in which case we are Thin Realists. The difference
between the two positions is minimal, and furthermore, Maddy claims that there is no objective
way to judge between them.
Although I am quite sympathetic towards Maddy’s goal of treating mathematics with greater
respect within a naturalistic framework, I find her analysis quite unsatisfactory. Maddy rules out
Robust Realism as it appeals to the objective truth or falsity of V = L as justification instead of
purely methodological concerns. However, this need not be the case. It is perfectly reasonable
for a robust realist in the face of an independent question to look towards mathematical practice
for guidance. This makes sense as it is this very practice that has been refined, improved, and so
successful over so many years in discovering objective mathematical truths. The robust realist
need not appeal to objective notions of truth prior to there being good mathematical reasons for
adopting or rejecting V = L. In fact, Gödel, a Robust Realist if there ever was one, weighed in on
V = L and ultimately sided against the axiom. His reasons seem to be the exact types of reasons
that Maddy says that we should use, and that Robust Realists do not. “[The] axiom [of
constructibility] states a minimum property. Note that only a maximum property would seem to
harmonize with the concept of set.”55 (Gödel, 1983a, pp. 478–479) Regardless, Maddy does not
consider other realist positions, such as plenitudinous platonism56, that would be happy to accept
55 Maddy (1988) suggests that it is possible that Gödel’s position here was actually motivated by his disbelief in the
continuum hypothesis, and that this disbelief stems entirely from an unacceptable appeal to mathematical intuition
and truth. Whether or not this is actually the case is debateable.
56 Plenitudinous platonism claims that all logically consistent mathematical theories are true and refer to real
mathematical objects. See Balaguer (1998) for details.
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multiple independent sets of axioms for set theory, so long as each set is consistent, or, the
situation could resemble Euclidian and non-Euclidian axioms of geometry where all are equally
valid. I see no reason to conclude that all Robust Realist positions conflict with mathematical
practice.
Another issue is that Thin Realism and Arealism seem to both violate the mathematical
naturalism that Maddy seeks to protect. The limited realism that Thin Realists adopt adds
properties of mathematical objects that are informed by science. If science suddenly says that
mathematical objects are causal, then the Thin Realist would have to agree. This suspiciously
resembles an ‘extra-mathematical’ tribunal for mathematical objects. Granted, it is not so
grievous in that such input from science regarding the properties of mathematical objects would
never contradict what mathematical practice says, as mathematical practice says next to nothing.
Still, given Maddy's efforts to treat mathematics as an independently respected practice, it is
peculiar that we look externally to what science says. The conclusion of Arealism is also
suspicious. The Arealist believes, like Quine, that science is the sole arbiter of existence. But the
Arealist notes that nothing in science implies the existence of mathematical objects (although it
does not imply non-existence either).57 Again, if science changes its tune in the future by
explicitly stating that mathematical objects exist, then the Arealist would be forced to change
their position. But isn’t this the very type of situation that we were trying to avoid in the first
place? The point of mathematical naturalism is that mathematics is understood through
mathematical practice alone. Appealing to science for discovering properties of mathematical
objects, or answering truth or existence claims is simply unnaturalistic.
Maddy wants to have her cake and eat it too. As a committed scientific naturalist, she
acknowledges that questions regarding existence and ontology are legitimate questions. At the
same time she is committed to the importance of respecting mathematical practice as much as we
respect scientific practice. In her analysis of mathematical practice she discovers that
57 Recall that Maddy already rejects the QIA as she rejects confirmational holism so we cannot say that
indispensability points to the existence of mathematical objects.
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mathematics says next to nothing regarding the existence of mathematical objects. It would stand
to reason then that that is the end of that. There is simply nothing to say about the existence of
mathematical objects. Here, Maddy’s commitment to ontological questions betrays her. She
insists that the big metaphysical questions regarding mathematics are still reasonable. However,
seeing as we know that mathematical practice cannot help us, our only recourse is to look
towards scientific practice. Her conclusion is that two positions fit with both mathematical and
scientific practice: Thin Realism and Arealism. There is no fact of the matter about which one is
true as both are equally supported. So, what it all comes down to in the end are our personal
preferences.58 Do we prefer the existence claims of mathematics to hold, or do we prefer letting
science rule the day? This is a shocking admission as nothing could seem further from the
naturalistic position that Maddy so wishes to defend. Recall that Maddy’s choice of the term
Arealism was meant to highlight that arguments for Arealism are not a priori but based on
mathematical and scientific practice. But in the end, what leads one to Arealism ends up being a
preference that has nothing to do with practice, and everything to do with a priori feelings! The
Thin Realist fares no better as ultimately the justification for their beliefs also stem from a priori
beliefs. This appears to be incredibly unnaturalistic.
Given that mathematical practice does not conclusively decide questions regarding existence and
ontology of mathematical objects, Maddy is left in a difficult dilemma. On the one hand she can
accept that mathematical practice says nothing and that there is nothing more to say. Ontological
questions are essentially meaningless within mathematics. The downside to this is that she must
abandon her naturalistic commitment that all metaphysical questions are meaningful and
important. Alternatively, Maddy can affirm her commitment to metaphysical questions, but this
leads to an un-naturalistic conclusion where the answers all depend on a priori personal beliefs
and external, ‘extra-mathematical’ tribunals.
58 One could cite this as another example of underdetermination where both Thin Realism and Arealism are equally
supported so there is no need to choose between them. This certainly is compatible with Maddy's naturalism, but
does it pay enough respect to her commitment to ontological inquiry? I believe that it does not.
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There is a third option that evades the horns of the dilemma. This third option has the added
advantages of respecting mathematical practice and remaining true to our naturalistic beliefs that
questions of ontology matter. A problem with the QIA is that it looks only at science and not
towards mathematical practice for guidance in answering metaphysical questions regarding
mathematical objects. The problem with Maddy’s mathematical naturalism is that the pendulum
has swung too far to the other extreme. In looking solely at mathematical practice there is no way
to answer questions of existence. Her solution is to take a cursory look at science to provide
answers, which I have argued is no solution at all. Surely there is value in looking at both
scientific and mathematical practice together in order to help us make sense of our mathematical
knowledge. Taking lessons from both disciplines and appreciating the important role that
mathematics plays in science, and that science plays in mathematics, opens up many avenues of
exploration in order to understand the nature of mathematics. Separating science and
mathematics and arguing that one or the other is the sole arbiter is a legacy from Quine’s hard
line naturalism, but we need not be forced into it now. The third option then is to free ourselves
from the confines of the naturalistic tendency to treat mathematics and science separately, and
also from any talk of ‘sole-arbiters’ with regards to ontological issues.
What I propose is that the best way to understand mathematical practice and ontological issues is
to treat mathematics and science together. In essence, naturalism should not cover only scientific
practice, nor should there be a separate naturalism for mathematical practice. Instead, we need to
consider both disciplines using the exact same standards and methods. The immediate benefits of
this joint approach are obvious. No longer would there be tension between respecting
mathematical versus scientific practice as this brand of naturalism would treat them both
together. There would not be any of the issues that Maddy faces where her mathematical
naturalism says next to nothing about ontological matters and she is forced to look externally for
answers. Beyond this, focus will be clearly placed on the integration and interplay of
mathematics and specific scientific domains such as physics. Such integration goes beyond mere
applicability, but can speak towards methodological approaches and interesting topics such as
reduction. Of course there are major obstacles that would need to be overcome for this project to
get off the ground. First, how can we treat mathematics and science together when, according to
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the standard view, mathematics has always been developed entirely independently from the
needs and demands of science? Secondly, what should we do regarding the abstract unapplied
areas of mathematical research? These, and many more issues are daunting, but it is my belief
that the Quinean form of naturalism is a relic from the past, and that removing this cleft between
mathematics and science represents a fruitful way forward in understanding our best
mathematical-scientific practices.
My suggestion that naturalism should refer to both the practice of science and mathematics
together betrays my metaphysical beliefs. My commitment to mathematical realism does not
stem solely from the indispensability of mathematics in science, as it does for Quine, Colyvan,
Bangu, and other indispensabilists. At the same time, my belief in mathematical objects does not
come solely from the practice of mathematics, as it does for epistemological platonists such as
Gödel. Instead, my belief in mathematical realism comes from both mathematical and scientific
practice together. Because of this I find traditional platonist arguments as well as the modern
indispensability arguments unsatisfactory as they each are instructing individuals to look at a
singular domain for the answers to ontological questions in mathematics. I feel that this is
misguided, and looking towards both mathematics and mathematical applications in science is
reasonable and makes for a stronger argument.
Looking towards both scientific and mathematical practice has been suggested before, although
it seems to have been forgotten. The QIA is closely associated with Putnam as he did much to
advance and defend the argument. In fact, he supported the argument so much that many call it
the Quine-Putnam Indispensability Argument. I have purposefully avoided this label as it
obfuscates the differences between Putnam and Quine’s take on the argument. One difference
already discussed is in the type of conclusion generated by the argument. Quine is vague and
non-committal, whereas Putnam advocates a form of semantic realism. Another important
difference that is almost entirely overlooked is the naturalistic component. Putnam has no
problem drawing on inspiration from both mathematical and scientific practice. In fact, for him
the QIA only has force once you do.
In my view, there are two supports for realism in the philosophy of
mathematics: mathematical experience and physical experience... If there
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is no interpretation under which most of mathematics is true, if we are
really just writing down strings of symbols at random, or even by trial and
error, what are the chances that our theory would be consistent, let alone
mathematically fertile? (Putnam, 1979, p. 73)
When considering mathematical practice, Putnam says that we cannot help but notice the
apparent consistency and fertility of the discipline. This leads him to believe that there must be
some interpretation for which mathematics is true. This inference is definitely not rock solid.
Putnam is making a type of no-miracles argument; it would be a miracle if mathematics was not
true under any interpretation given how our experience of mathematics seems to be consistent
and is certainly fruitful. We do not want to explain this via a miracle, so the only recourse is to
believe that there must be an interpretation for which mathematics is true as it is the best
explanation of our experience. As discussed in chapter 5, this type of IBE leaves much to be
desired, but for now let us grant Putnam’s conclusion.
At this point, Putnam turns towards the applications of mathematics to try to determine which
interpretation makes mathematics true.
The interpretation under which mathematics is true has to square with the
application of mathematics outside of mathematics... I argued in detail that
mathematics and physics are integrated in such a way that it is not possible
to be a realist with respect to physical theory and a nominalist with respect
to mathematical theory... Mathematical experience says that mathematics
is true under some interpretation; physical experience says that that
interpretation is a realistic one. (Putnam, 1979b, p. 74)
It is scientific practice that leads Putnam to conclude that the realist interpretation of
mathematics is the correct one. The realist interpretation is what makes mathematics true and
fruitful. The route to this conclusion is through an indispensability argument. Given that we are
scientific realists, and that mathematics is indispensable to the practice of science, then we must
also take a realist attitude towards mathematics as to do otherwise would be ‘intellectually
dishonest’. For Putnam, an analysis of both mathematical and scientific practice work together to
lead to mathematical realism.
Putnam’s indispensability argument can be formulated as follows.
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(P-P1) Mathematics must be true under some interpretation.
(P-P2) We ought to have ontological commitment to all those entities
that are indispensable to our best scientific theories.
(P-P3) Mathematical entities are indispensable to our best scientific
theories.
Therefore:
(P-C) The correct interpretation of mathematics is the realist
interpretation.
The justification for (P-P1) is a no-miracles argument which I will not critique here. (P-P3) is a
standard indispensability premise, but could easily be altered to specifically point to mathematics
being explanatorily indispensable. What makes Putnam’s argument interesting and unique is his
second premise. (P-P2) is a softer, more open version of Quinean naturalism that removes the
clause that only those entities that are indispensable to science be considered. Removing this
restriction is what allows us to look towards mathematical practice as an aid for understanding
mathematical knowledge and mathematical entities, and this allows us to make use of his first
premise which is a conclusion drawn from mathematics alone. It is this move of rejecting the
idea that science (or mathematics) is the sole arbiter of ontological questions in mathematics that
I find immediately appealing about Putnam’s version of the argument.
Certainly there are problems with this argument. It still critically depends on confirmational
holism which I have consistently rejected, the conclusion of (P-P1) is based on a suspect
inference, and the relationship between true interpretations and realism needs to be made clear in
a non-question-begging way. Regardless, there are several important points that are worth
highlighting from this examination of Putnam. The first point is a historical one: it is a costly
mistake to lump Putnam too closely with Quine. Surely Putnam agreed with Quine in several
important ways, but as we have seen their differences are significant. Secondly, mathematical
and scientific practice together can contribute towards leading us to a realist conclusion. Our
naturalism need not be so overly restrictive as to bar potentially fruitful avenues of exploration.
This point is made even more salient since the focus has shifted towards mathematical
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explanations in science. Finally, arguments for mathematical realism should look to draw from
as many areas of mathematics as possible in order to make a broad and convincing case. Moving
forward, the goal would be to learn from the lessons of Putnam and create a stronger argument
for mathematical realism in line with the joint naturalism suggested above.
2 Final Thoughts
The recent surge in interest for mathematical explanations was most certainly sparked by
indispensability arguments for mathematical realism. I hope that I have shown that while
indispensability arguments are interesting, the worth of studying them goes far beyond looking at
a simple argument for mathematical realism. Understanding the role that mathematics plays in
scientific explanations is an exciting and fruitful area of research. This understanding helps us
make better sense of the practice of science, scientific explanation, and inference to the best
explanation regardless of metaphysical worries. Finally, looking at mathematical explanation
leads to at least three areas of further research that could significantly advance the field. Focus
can now be placed on understanding how it is that mathematics explains physical facts, the key
explanatory role of mathematics can motivate the rejection of exclusive forms of naturalism, and
lastly we can draw on all this to craft a superior argument for mathematical realism that makes
use of both mathematical and scientific practice together.
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