Antonelli, Nature & Purpose of Numbers [21 pgs]

7/29/2019 Antonelli, Nature & Purpose of Numbers [21 pgs]

1/21

1

The Nature and Purpose of Numbers

G. Aldo Antonelli

Dept. of PhilosophyUniversity of California, Davis

Numbers are abstract entities introduced for the purpose of counting. The presentpaper is dedicated to the explication of this claim, and in particular it addresses thequestions of what makes these entities abstract, in what sense they areintroduced, and what we mean by counting. Along the way, we investigate thelogical status of arithmetic, the function of abstraction principles, and the respective

merits of various strategies for reducing arithmetical notions to those of a theorythat is viewed as more fundamental. The main emphasis is on the conceptual,foundational and philosophical issues, with the technical details fully developedelsewhere.1

Two main conceptual threads are at the basis of the present approach: a deflationaryconception of abstraction and a non-reductionistversion of logicism. Each isimplemented through a specific device, i.e., respectively, an extralogical operatorrepresenting numerical abstraction and a nonstandard (but still firstorder)cardinality quantifier. The result is an account of arithmetic characterizing numbersas obtained by abstraction from the equinumerosity relation and emphasizing their

cardinalproperties (as used in answering howmany? questions) over theirstructural ones.

Abstract entities are obtained through the application of an abstraction operator towhat Frege would have called a concept, and which we will also refer to as a(possibly complex)predicate as long as we take care to distinguish predicatesfrom predicate expressions. However, not every mapping of the concepts into theobjects represents an instance of abstraction. Abstraction operators aredistinguished from other such assignments in that they are assumed to mapconcepts into objects while respecting a given equivalence relation. Frege, forinstance, postulated an abstraction operator assigning objects of a particular kind

which he called extensions to concepts in such a way that equiextensionalconcepts (i.e., concepts under which the same objects fall) are assigned the sameobject. This particular postulation, as embodied in an abstraction principle knownas Freges Basic Law V, turned out to be inconsistent. Nonetheless, many otherabstraction principles are indeed consistent, and among them, most notably for our

1 See G. A. Antonelli, Numerical Abstraction via the Frege Quantifier, Notre Dame Journal ofFormal Logic, forthcoming.

To appear, Journal of Philosophy


2/21

2

purposes, is a principle ofnumerical abstraction, also known as Humes Principle. HPpostulates an operator Num assigning objects to concepts in such a way thatconcepts Pand Q are assigned the same object precisely when Pand Q areequinumerous, i.e., when just as many objects fall under Pas they fall under Q. TheobjectNum(P) assigned to Pcan then be regarded as the number ofP.

It is natural to think of the abstract objects delivered by such operators as having asomewhat mysterious nature, with such properties as nonspatiotemporal existenceand causal inefficacy, for instance. But that would be a mistake, for abstractionprinciples do not force upon us any particular view of the entities they introduce. Infact, all that the adoption of a particular abstraction principle commits us to, is theexistence ofsome mapping of the concepts into the objects satisfying certain furtherconditions related to a given equivalence relation. The abstraction principlesthemselves do not tell us anything about the ultimate nature of these objects and that is how it should be. The role of these abstraction principles is not to single aspecial class of objects (the abstract entities living perhaps in separate sphere of

reality), but rather to make sure that the firstorder domain of objects is largeenough to accommodate an adequate account of arithmetic. Now one mightentertain specific worries about the ontologically inflationary nature of abstraction the fact that, for instance, HP forces us to accept the existence of an infinitenumber of objects but such worries are quite distinct from those concerning thecausal inefficacy or the nonspatiotemporal existence of numbers.

According to the view of abstraction proposed here, there is nothing special aboutabstract entities, which can be drawn from whatever firstorder domain we take ourordinary quantifiers there is and for all to range over. In this sense abstractentities can be taken to be just ordinary objects recruited for the purpose of serving

as proxies for the equivalence classes of concepts generated by the givenequivalence relation. Abstraction principles give a lower bound on the cardinality ofthe domain of objects, relative to the size of the class of all concepts, taken modulo agiven equivalence relation. We characterize this view of abstraction as deflationaryin that the main role it ascribes to abstraction is to provide such a lower bound,while denying the objects delivered by abstraction any special status.

Abstraction principles are represented linguistically by the explicit introduction oftermforming operators such as Num(P) or, in the general case, (P). For anypossibly complex predicate expression P, the term (P) will denote an object in thefirstorder domain. Such an object is introduced by abstraction as long as theassignment of objects to predicates satisfies the constraints associated with thecorresponding equivalence relation. This of course does not preclude the possibilitythat the same object might also be denoted by otherterms a possibility that, aswe will see, points in the direction of a possible dissolution of the socalled Caesarproblem.

The deflationary account of abstraction goes handinhand with the view that


3/21

3

abstraction principles are properly regarded as extralogical principles that, as such,do not enjoy a logically or epistemologically privileged status. On the contrary, onmany accounts inspired by some form or other of logicism, the logical character ofarithmetical notions is made to depend on some kind of reduction of arithmeticaltruths to Humes Principle, which in turn is claimed to be constitutive of the notion

of number, and therefore somewhat close to an analytical principle.

These logicist or neologicist views are based, however, on an equivocation. Whenlogicism is properly understood in non-reductionistfashion, it is the notion ofcardinality, rather than that ofnumber, that appears logically privileged. Numbers,as we have seen, are objects, and matters of existence of objects fall outside thepurview of logic proper. On the other hand, from a logicist point of view, cardinalitycan be viewed as logically innocent. When taken at face value, this broad construalof logicism opens up the possibility of including cardinality as one of the basicbuilding blocks of a language suitable for the representation of arithmetic. Thenotion of cardinality that turns out to be at issue here, as we will see, is a

comparative notion, specifying a relation between concepts Pand Q that holds if andonly if there are no more objects falling under Pthan there are falling under Q. Thisnotion of comparative cardinality is linguistically represented through theintroduction of the Frege quantifier F, binding two formulas and, and expressingthe fact that there are at least as many objects satisfying as there are satisfying .

Numerical abstraction and the Frege quantifier are the two main devices that will beemployed to give a firstorder representation of arithmetic emphasizing the cardinalproperties of numbers.

It is worth noting here, before we get to some of the details of the project, that there

does not seem to be any obvious way to extend the present treatment to ordinalnotions, except trivially in finite domains, where ordinal and cardinal numberscoincide. Certainly it would seem that no such treatment could be developed usingthe Frege quantifier and the abstraction operator, which are aimed squarely atcardinal notions. But one might think that a similar treatment for ordinal notionscould be developed by introducing ordinal abstraction. But as we will see, such aprinciple gives rise to paradox. In this respect ordinal notions, while ordinarilyregarded as on a par with their cardinal counterparts (or perhaps even as primary,as in set theory where cardinals are defined as initial ordinals), appear instead to beintrinsically more complex than the latter, and indeed quite possibly beyond thereach of a firstorder treatment.

1. Numbers as abstracta

Arithmetic is the theory of the natural numbers 0, 1, 2, . . . with which we are allacquainted. Because of the basic character of the natural numbers as the foundationupon which mathematics and science rest, ever since Frege and Dedekind


4/21

4

philosophers have been concerned with the proper formalization of arithmetic.2This line of investigation has lead to a variety of approaches, including theDedekindPeano axioms that are nowadays standard, several settheoreticreductions, and finally a renewed interest in the Fregean project as championed bythe neologicist school of Hale and Wright.3

One possibility is to regard numbers as primitive objects that need no reduction toothers. A proponent of this view would then fully embrace the axioms of standardfirstorder arithmetic as extralogical characterizations of the fundamentalproperties of numbers. These axioms, first formulated by Dedekind4 and Peano,5identify the basic properties of the successor operation on the natural numbers (aswell as possibly the properties of addition and multiplication), and postulate aninduction schema expressing that any properties of natural numbers that hold ofzero and are preserved by the successor operation, hold of all natural numbers.Although Peano Arithmetic (PA), as the theory as come to be known, is sometimessupplemented by a secondorder induction principle, it is standardly expressed at

the firstorder level. The insistence on a firstorder axiomatization is motivated bythe desire to preserve certain properties of firstorder logic, such asaxiomatizability, compactness, LwenheimSkolem, etc., which fail at the secondorder level. 6 Let us refer to this view as the axiomatic approach.

A second approach was championed both by Frege (in his Grundlagen and, later theGrundgesetze7) and Whitehead and Russell,8 each one of whom provided an accountof the natural numbers as intimately related to classes of equinumerous concepts,i.e., as equivalence classes under the having the same cardinality relation. InWhiteheads and Russells (unramified) theory of types, numbers are concepts ofconcepts (that is, concepts of propositional functions) such that between any two of

2 Gottlob Frege, Begriffsschrift, eine der arithmetische nachgebildete Formel-sprache des reines Denkens, Halle, 1879, English transl. in J. van Heijenoort (ed.), From Fregeto Gdel. A source book in Mathematical Logic, Harvard University Press, Cambridge, MA,1967. See also G. Frege, Die Grundlagen der Arithmetik, eine logisch-mathematischeUntersuchung ber den Begriff der Zahl, Nebert, Breslau, 1884, English transl. by J.L. Austin,The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number,Blackwell, Oxford, 1950.3 R. Hale and C. Wright, The Reasons Proper Study. Essays toward a Neo-Fregean Philosophyof Mathematics, OxfordClarendon Press, 2001.4 R. Dedekind, Was sind und was sollen die Zahlen?, Brunswick, 1888.

5 G. Peano,Arithmetices Principia, nova methodo exposita, Bocca, Torino, 1889, Englishtransl. in van Heijenoort, op.cit.6 For an accessible, clear, and rigorous treatment of secondorder logic see H. EndertonSecondorder and higherorder logic, in E. Zalta (ed.), Stanford Encyclopedia ofPhilosophy, 2008, URL: http://plato.stanford.edu.7 G. Frege, Grundgesetze der Arithmetik, Hermann Pohle, Jena, 1903, English transl. by M.Furth, The Basic Laws of Arithmetic, University of California Press, 1967.8 A.N. Whitehead and B. Russell, Principia Mathematica, volume I, Cambridge UniversityPress, Cambridge, England, second edition, 1925.


5/21


6/21

6

center stage, at least on a par with the mathematical properties of sequences. It isthis emphasis on the applicability of arithmetic that is lacking in any account thatprivileges ordertheoretical properties of sequences over cardinal ones. In thisrespect, PA still needs to be supplemented by a separate account of how numbersare used in counting.

Settheoretic reductions fare somewhat better than PA when it comes to theiremployment in the assignment of cardinal numbers, if anything because they areembedded into a richer theory allowing for many sorts of maps between sets ofdifferent kinds. The details here depend on the particular settheoretic reductionbeing adopted, but set theory does provide the beginnings of an account of thecardinal properties of the natural numbers. But again, this account is not based onan explicit consideration of such properties. Moreover, settheoretic reductionssuffer from a problem first pointed out by Benacerraf,10 who argues that there is noprivileged way to select one particular settheoretic reduction over any other one,and that in fact, in the end, any sequence will do. Given that all settheoretic

reductions (of which there are indeed infinitely many) provide for the sameintrinsic ordeertheoretical properties of the natural numbers, how is one to assessthe relative merits of, and decide between, say, Zermelo numerals and von Neumanncardinals? Benacerraf s answer is that there is no principled reason to choosebetween them, and that since the natural numbers cant be both Zermelo numeralsand von Neumann ordinals, numbers cant be sets at all.

This leaves us with a last option, the FregeRussell account of the natural numbers.The great advantage of this approach is that the intrinsic mathematical properties ofthe natural numbers are derived from their cardinal properties, rather than theother way around. Whereas for both the axiomatic approach and settheoretic

reductions cardinal properties require a separate account, according to the FregeRussell conception such properties are central to the account of the naturalnumbers. The essential lines of such an approach, therefore, appear to be intuitivelywell motivated and mathematically elegant. Unfortunately, the mathematicalimplementations are rife with problems: Freges own attempt in Grundgesetze wasnotoriously inconsistent, and Whiteheads and Russells imposition of a typediscipline, while blocking the inconsistency afflicting Freges theory, led toreduplications and restrictions that hardly do justice to actual mathematicalpractice.

Lately, Hale and Wright (op.cit.) have championed a somewhat different, neo

logicist approach, which addresses directly the idea that numbers are related to

10 P. Benacerraf, What numbers could not be, Philosophical Review, 74:4773,1965, reprinted in P. Benacerraf and H. Putnam (eds), Philosophy of Mathematics. SelectedReadings, Cambridge University Press, 1983, second edition, pp. 27294; and Recantationor any old sequence would do after all, Philosophia Mathematica, 1996 4(2), pp. 184189.


7/21

7

equivalence classes under the equinumerosity relation, dispensing with the wholeapparatus of concept extensions.11 They introduce a theory of numbers based onwhat we have been referring to as the Num operator and postulating that such anoperator is to satisfy Humes Principle. Since as it was already known to Frege theaxioms of PA can be derived, within secondorder logic, from Humes Principle, the

resulting neologicist system is thus adequate for the representation of secondorder arithmetic (and in fact equiconsistent with it).

There are two main issues with such an approach. The first is both philosophicaland conceptual: Hale and Wright rely on the logical character of Humes Principle inorder to characterize their project as continuous with Freges original logicist views.However, the extent to which Humes Principle enjoys a logically or evenepistemologically privileged status is debatable, for instance in the light of the factthat there are models of arithmetic where Humes Principle fails.12

But even discounting these worries, a main technical obstacle remains, namely the

fact that the neologicist program is wholly carried out within the framework ofsecondorder logic, characterized by the alreadymentioned failure of those metatheoretic properties that make firstorder logic so attractive. Nonetheless, the FregeRussell approach appears to be conceptually superior in its characterization ofnumbers as abstracta of the equinumerosity relation, in the way it derives basicmathematical properties from cardinal ones (and the concomitant emphasis on theapplicability of arithmetic), and in its intuitive motivation.

2. Deflationary Abstraction

In their most general form, abstraction principles such as HP govern the assignmentof objects to concepts according to a given equivalence relation: 13f(P) =f(Q) Rf(P, Q).

A principle of this form asserts that the objectfassigns to the conceptPis the sameas the object it assigns to the conceptQ if and only ifPand Q are appropriatelyrelated to each other by the equivalence relation Rf.

Because abstraction in some sense lives in conceptual space as evidenced bythe connection of abstraction to concept formation, e.g., in children or in science

11 Antonelli and May, op. cit., instead, go in the opposite direction, developing an explicitly

extensionalist program dispensing with the logical character of arithmetical principles.12 A counterexample to the righttoleft direction of Humes Principle can be found in G.Boolos, On the proof of Freges theorem, in Adam Morton and Stephen Stitch (eds),Benacerraf and his Critics, pages 14359, Blackwell, 1996; a counterexample to the lefttoright direction usually considered the less questionable one can be found in Antonelliand May, op.cit.13 See G. Rosen, Abstract objects, in E. N. Zalta (ed.), Stanford Encyclopedia of Philosophy,2006, URL http://plato.stanford.edu/entries/abstractobjects/, (Spring2006 Edition).


8/21

8

(whereby a concept is abstracted from a variety of instances) abstractionprinciples have often been thought to enjoy a particularly privileged status. Butclearly not all such principles are acceptable. For instance, one could not have aprinciple of the formf(P) =f(Q) P= Q, for then we would have at least as manyobjects as there are concepts over a given domain, contradicting Cantors theorem

(independently of whether the identity P= Q is taken extensionally or intensionally).There are two possible strategies, a combination of which can be used to obviatesuch a situation: the equivalence relation appearing on the right of the equivalencecan be made coarser, allowing possibly a great many distinct concepts to beassigned the same object; or the functionfon the left can be made to apply only to asubset of all the concepts (the definable ones, for instance).

Humes Principle implements the first of these strategies, by taking the equivalenceof the right to be equinumerosity. As mentioned, neologicists such as Hale andWright claim for HP a status not unlike that of a logical or analytical truth, andconsider HP somehow constitutive of the notion of number. However, several

objections have been raised against such a privileged status, beginning with the socalled Bad Company objection, i.e., the fact that there are principles very much likeHP that turn out to be inconsistent. One such example of course, is Freges own BasicLaw V. Another example replaces equinumerosity with orderisomorphism:according to such a principle, given relational predicates (x,y) and (x,y), one saysthat the type() of equals the type () of if and only if (x,y) and (x,y) areorderisomorphic. Attractive as this treatment of ordinal notions might appear, it isinconsistent, for it gives us the BuraliForti paradox.14

Against this, neologicists can (and did) rebut that consistency itself ought to betaken as the hallmark of acceptability for abstraction principles. But both R. Heck15

and A. Weir16 have criticized this line of argument, pointing out that there areindividually consistent but pairwise incompatible abstraction principles: if bothprinciples in such a pair are acceptable, hence at least to an extent analytic, thenboth need to be regarded as true, which is of course impossible (this is Weirs socalled Embarrassment of Riches objection). There is a vast literature attempting tofind a general and wellmotivated demarcation between acceptable abstractionprinciples and unacceptable ones.17 But in the end there are reasons to believe thatany such attempt to find a completely satisfactory demarcation might turn out to befutile.

The issue of the privileged status of abstraction principles can be approached also

14 See J. Burgess, Fixing Frege, Princeton University Press, 2005, pp. 16470 for an excellenttreatment of these issues.15 R. G. Heck, On the consistency of secondorder contextual definitions, Nos, 26:49194,1992.16 A. Weir, Neofregeanism: An embarrassment of riches, Notre Dame Journal of FormalLogic, 44:1348, 2003.17 See . Linnebo (ed.), The Bad Company Objection, special issue ofSynthse, 170 (3).


9/21

9

from a different angle. There is a long tradition, going back to the work ofAlfred Tarski, according to which logical notions are those that are invariantunderpermutations of the domain of objects, ostensibly because logical notions arecompletely general and do not have any specific subject matter (this proposal hassince come to be known as the TarskiSher thesis18). Tarski first introduced this idea

in its full generality in a posthumously published 1966 lecture,19

explicitly inspiredby Kleins Erlangen program.

Let us call a predicate Plogically invariantif and only if for any permutation andobjecta, (a) falls under Pif and only ifa does. This idea can be generalized tonotions of arbitrary type, including connectives, quantifiers, and other higherorderobjects. While Tarskis proposal is nowadays considered too liberal (in that itcountenances as logical notions that appear to be properly mathematical, ratherthan logical, in character), there is widespread consensus that permutationinvariance provides at least a necessary (although likely not sufficient) condition forlogicality of predicates and quantifiers.20 Somewhat surprisingly, the invariance

criterion was not applied to the question of the status of abstraction principles untilrelatively recently.

Kit Fine first considered criteria of logical invariance for abstraction principles intheir full generality.21 Among the several possible ways in which such criteria can beformulated, we follow the reconstruction given by John Burgess,22 and call anabstraction principle of the form

f(X) =f(Y) Rf(X, Y)invariantif and only if for any permutation , it holds thatRf(X, [X]), where [X] isthe pointwise image ofXunder . Clearly this implies the weaker criterionaccording to which Rf(X, Y) holds precisely when Rf([X], [Y]) also holds.

Since permutations preserve the cardinality of a set, it follows immediately that HPis logically invariant, in the sense that ifR is the equinumerosity relation betweenconcepts (viewed as sets) then for any such conceptX, we have R(X, [X]). But weshould not make too much of this. First of all, it is not at all clear that there is aprivileged way to express invariance for abstraction principles. But evenconsidering just this version of invariance, there is another principle, very close toHP, that is invariant in this sense, but inconsistent: this is the principle mentioned inthe bad company objections to HP assigning abstracts to binary relations R and Sin such a way thatR and Sare assigned the same object precisely when there is an

18 After the proposal was endorsed and further developed by G. Sher, The Bounds of Logic,MIT Press, Cambridge, MA 1991.19 A. Tarski, What are logical notions?, History and Philosophy of Logic, 7: 14354, 1986(posthumously edited. by J. Corcoran).20 See D. Bonnay, Logicality and Invariance, The Bulletin of Symbolic Logic, 14(1) 2008, pp.2968.21 K. Fine, The Limits of Abstraction, Oxford University Press, 2002.22 J. Burgess, Fixing Frege, op. cit.


10/21

10

order isomorphism between them. The principle is invariant because a relation Rand its image Sunder a permutation are orderisomorphic (with itself providingthe isomorphism), but as already mentioned it gives rise to the BuraliForti paradox.

The approach of this paper aims to sidestep the issue of the privileged status of

abstraction principles completely. Abstraction principles, when properlyunderstood, just provide an assignment of representatives to the equivalenceclasses induced by equivalence relations. Nothing more is said about theserepresentatives, other than the assignment must respect the equivalence relation.Accordingly, these principles are best viewed as extra-logicaldevices, whose mainfunction is to provide inflationary thrust on the firstorder domain D of objects byimposing a lower bound on the cardinality ofD, relative to the size of the space ofconcepts over D.23

Some abstraction principles are good and some bad, and among the bad ones weshould certainly count those that are inconsistent. Good abstraction principles can

be put to use in a variety of philosophical contexts, and they are capable ofaccomplishing a variety of tasks. But by classifying them as good no principledgood/bad demarcation is implied, but only the statement that they have turned outto be useful in some context or other.

We characterize this view of abstraction as deflationary, in that it denies abstractobjects any privileged status (the fact that the deflationary account emphasizes theinflationaryrole of abstraction principles should not it is hoped lead toconfusion). Any worries about the special ontological status of abstract objects, orthe special logical status of abstraction principles, to the extent that they have anycogency at all are no longer foisted upon us by a consideration of abstraction. For

from a logicalpoint of view, we need not assume anything logically (orepistemologically) privileged about such principles; and from an ontologicalpoint ofview, we need not assume that abstract objects make up a separate, privilegedontological realm.Anything at all even ordinary objects can play the role ofthese abstracta, as long as the choice respects the equivalence relation.

One of the recurring worries connected with the introduction of natural numbersvia Humes Principle is the socalled Caesar Problem, i.e., the fact that HP onlygives us enough information to settle the truthvalue of identities in which bothterms are abstracts, but says nothing about identities involving an abstract and aterm of a different kind. The worry then is that no account of the natural numbers

can be regarded as complete unless their identity conditions settle all suchquestions.

Such worries are misplaced. The answer to the question, What prevents thenumber of the planets to be equal to Julius Caesar? is: nothing. The number of the

23 Kit Fine, and others following him, use inflationary in connection with abstractionprinciples in a different sense.


11/21

11

planets will indeed be equal to Julius Caesar in some models, and distinct from it insome other models. Nothing much is to be made of this, for the correspondingabstraction principle, HP, is silent about it. To worry about this is not to understandthe proper nature of abstraction as imposing a lower bound on the size of thedomain of objects, rather than opening up a metaphysically separate domain of

abstract or objects.24

The existence of such a separate realm has nothing to dowith abstraction, for abstraction is perfectly compatible with there being only onedomain of discourse, populated by objects to which we should have no qualmsappealing for whatever philosophical, logical, or mathematical purpose we might bepursuing. So, in this sense, the present account is deflationary as regards the Caesarproblem as well.

It is worth noting that in his classic paper on the nature of numbers (and its sequel)Paul Benacerraf also reached a generally deflationary point of view. His conclusionthat any sequence will do after all deflates metaphysical worries about theultimate nature of numbers. But Benacerraf s account moves in a different direction

from the present one, focusing on the ordertheoretical rather than the cardinalproperties of the natural numbers. Being an sequence is an ordertheoreticalpropertypar excellence, and such properties are best viewed as supervenient uponcardinal ones or so we submit. Benacerraf s account, then, is incomplete, but notbecause it fails to provide a characterization of the ontological status of numbers;Benacerraf himself showed that no such strategy could succeed. Rather, the accountis incomplete because it does not address the role of cardinal properties inarithmetical applications such as counting.

Benacerrafs views have long been regarded as promoting a version ofstructuralismas regards arithmetic. His claim that any sequence sequence can play the role of

the natural numbers has been interpreted as implying that there is nothing more tothe natural numbers than their ordertheoretical properties. Be that as it may,Benacerraf certainly did argue against the claim that there is a privileged way toselect objects to be arranged in an sequence. A similar argument can then be putforward after we shift the focus from the ordertheoretical properties to the cardinalones. The lesson to be learned from Benacerraf s argument is this: once we have anaccount of natural numbers in terms of cardinal properties, it does not make anydifference which objects are chosen as representatives of the equinumerosityclasses, as long as we have enough of them to satisfy the inflationary thrust of thecorresponding abstraction principle.

24 It is then perfectly possible for abstracta to have additional nature beyond theproperties they inherit in virtue of HP. This issue has recently been taken up by B. Hale andC. Wright in Abstraction and Additional Nature (Philosophia Mathematica 16 (2), 2008, pp.182208) in response to M. Potter and P. Sullivan, What is Wrong with Abstraction?,Philosophia Mathematica 13 (3), 2005, pp. 18793. The line taken here is that abstracta areallowed to have such additional nature, a fact that presents no obstacle to theiremployment in mathematics and science.


12/21

12

3. Non-reductionist Logicism

Traditional logicism and neologicism rely on the special status of abstraction

principles such as HP to establish the logical character of arithmetical notions. Butonce we adopt the deflationary point of view and regard these principles as extralogical and not epistemically privileged, what is then left of logicism? What are theprospects of the ambitious program initiated by Frege and revived by the neologicist school?

Freges original program aims to combine two largely incompatible views: logicism,construed as the view that arithmetic is interpretable into (higherorder) logic; andextensionalism, construed as a theory of concept extensions qua abstract objects.Dummett25 referred to the latter as Freges platonism the view that there arelogical objects in the form of concept extensions and pointed out that the view is

not only independent from, but in fact in direct tension with Freges logicism, a factthat underpins the contradiction uncovered by Russell. And in fact, on the naturalview of logic, there are no logical objects: all that is needed or required for thecompletion of the logicist program is an interpretation of all mathematicalstatements (or at least the arithmetical ones) into a logical language.

In recent versions of neologicism, Freges extensionalism has been replaced by atheory of numbers qua logical objects delivered by HP. But it is indeed possible,perhaps for the sake of conceptual purity, to pursue one part of Freges originalprogram independently of the other. For instance, one approach26 is to pursueextensionalism by providing a theory in which concept extensions are explicitly

governed by extralogical principles, and arithmetic recovered as a secondordertheory identifying numbers with particular concepts. But can logicism also bepursued for its own sake without relying on a theory of numbers as logical objects?

Arithmetical logicism27 is generally characterized as the view that arithmetic is, in asubstantial sense, logic. This view is usually taken to comprise the two distinctclaims that arithmetical notions are definable in terms of purely logical ones andthat under this interpretation arithmetical theorems can be proved from purelylogical principles. Ever since Frege, the view that arithmetic is logic is most oftenarticulated in a reductionistfashion by identifying some principle, claimed to enjoysome logically or epistemologically privileged status, to which (translations of)

25 M. Dummett, Frege. Philosophy of Mathematics, Harvard University Press, Cambridge,Mass., 1991, p. 301.26 Antonelli and May (op. cit.).27 By arithmetical logicism we mean logicism as applied to arithmetic. There is also a moreambitious version of logicism claiming that mathematics as a whole is also in the samesense interpretable as logic. Here we restrict our attention to the more limited version, towhich we refer as logicism simpliciter.


13/21

13

arithmetical truths turn out to be prooftheoretically reducible. The need for such aprinciple is clear. Even when arithmetical notions have been appropriatelytranslated in a purely logical language, one cannot expect translations ofarithmetical theorems to be provable using only the most general axioms and rulescharacterizing reasoning in terms of connectives and quantifiers. For one thing,

arithmetic implies the existence of great many objects, and pure logic alone cannotestablish existence claims. Hence the need for some intermediate principle carryingenough inflationary thrust to allow the derivation of such arithmetical truths, whileat the same time retaining its purely logical character. Such a principle wasidentified by Frege in Basic Law V, whose completely general nature does indeedlend some plausibility to the claim that it is a proper part of logic. But unfortunatelyBasic Law V is too inflationaryand therefore unsatisfiable. The option pursued bythe neologicists, instead, is Humes Principle. While HP is inflationary, it is alsoconsistent. It is arguable, however, that HP enjoys the full generality that madeclaiming logical status for Freges Basic Law V plausible. And indeed, as we haveseen, there are reasons to question the logical character of HP.

The reductionist implementation of logicism seems thus to fall short of the desiredgoal. But why go the reductionist route in the first place? Reductionism did indeedenjoy a certain currency in the philosophy of science, a role later codified in thework of Nagel,28 who championed intertheoretic reduction via bridge principlesplaying a role not too dissimilar from that of Basic Law V or Humes Principle inlogicist or neologicist theories. But even if reductionism could be defended in thecase of empirical science, in the case of arithmetic such a strategy is not the only, themost general, or indeed not even the most natural interpretation of logicism.

On the broadest interpretation of logicism, cardinality is alreadya logical notion,

and it does not need a definition in terms of a special kind of logical objects to makeit so. This is a point already made by Dummet,29 although for Dummett it applies tothe notion of cardinal number, rather than directly to the more basic notion ofcardinality. The two, as mentioned, are indeed distinct. Whereas cardinal numbersare objects introduced by abstraction, cardinality expresses a property of conceptsor, more generally (in the case of equinumerosity), a relation between concepts. Andwhile the case for the logical character of cardinal numbers is philosophicallysuspect, those same objections hardly apply to the general notion of cardinality.

Once we recognize that cardinality is also available to the logicist as a genuinelogical notionper se, independently of the status of any abstraction principles

involved, we come to a broader, and more natural construal of the logicist project, inwhich cardinality is employed as one of the basic, or perhaps the fundamentalbuilding block in designing a formal framework adequate for the representation ofarithmetical facts. As mentioned, the basic notion of cardinality involved is that of arelation Fbetween conceptsXand Ythat holds whenever there are no moreXs than

28 E. Nagel, The Structure of Science, Harcourt, Brace, and World, New York, 1961.29Op. cit., p. 224.


14/21


15/21

15

4. The modern view of quantifiers

According to the modern view, a firstorder quantifier over a domain D is a

collection of, or more generally a relation among, subsets ofD. This idea can betraced back to the work of Frege, and specifically 21 of his Grundgesetze derArithmetik, where Frege asks us to consider forms of the conceptual notationcorresponding to the modern formulas a(a2 = 4) and a(a > 0): these forms can beobtained from the general form a (a) by replacing the functionname placeholder( ) by names for the firstlevel functions 2 = 4 and > 0 (a function is at the firstlevel if it takes its arguments from the domain of objects). These two functions takenumbers as arguments and return the value true if those numbers are square rootsof 2 or (respectively) positive, andfalse otherwise. In other words, they are exactlywhat Frege refers to as concepts. It follows then that the general form of aquantifier, a (a), is that of a secondlevel concept, i.e., a function taking firstlevel

concepts as arguments and returning truth values. The modern view, made precisein the theory ofgeneralized quantifiers, 33 identifies such secondlevel concepts withcollections of subsets of the domain. So for instance:

The ordinary existential quantifier can be identified with the collection of allnonempty subsets ofD;

Dually the universal quantifier can be identified with the collection ofsubsets ofD that contains D itself as its only member: = {D};

The quantifier there exist exactly k, usually written !k can be identified withthe collection of all kmembered subsets ofD.

These examples apply to a single open formula (x) at a time: they are, as we willsay, unary. But in fact, some quantifiers are not only best viewed as applying to morethan one such formula, but they are also such that no other interpretation ispossible. One such example is the quantifier Most. The statementMost x((x),(x))represents most s ares and it is true when more s ares than s that arenots. It is well known thatMost cannot be represented by a formula of ordinaryfirstorder logic.34 In contrast, the quantifier Only, applying to formulas and andjust in case alls are s, can be expressed using the ordinary universal quantifierand Boolean connectives.

A further distinction concerns the dimension of a quantifiers arguments, as distinct

33 The theory originated with A. Mostowski, On a generalization of quantifiers,Fundamenta Mathematic, 44:1236, 1957; and R. Montague, English as a formallanguage, in R.H. Thomason (ed.), Formal Philosophy, Yale University Press, 1974, originallypublished 1969.

34 See Peters and Westersthl, op. cit.


16/21

16

from their number. For instance, a quantifier can simultaneously bind two variablesxandy(thus having dimension 2), as in the case of the quantifier Qxy(x,y) whichreturns value true if and only if expresses the universal binary relation over D.All the abovementioned quantifiers arefirst-order, a notion that can becharacterized precisely in semantic terms. A unary quantifier is firstorder if and

only if it represents a collection of subsets ofD (and similarly, a binary quantifier isfirstorder if and only if it expresses a relation between subsets ofD). According tothis definition, some quantifiers are firstorder even though, like Most, they are notdefinable by a firstorder formula.

The same is true of the Frege quantifier. The Frege quantifier represents a relationbetween subsets of the domain the relation that holds between Fand G whenthere are no more Fs than Gs. Hence, the quantifier relates Fand G precisely whenthere is an injective function mapping the Fs into the Gs. Thus it might appear thatthe Frege quantifier inherently appeals to a second-ordernotion. After all, existenceclaims for relations, functions, etc., are properly expressed at the second order. But

appearances are deceiving: the Frege quantifier is no more at the second order thanOnlyorMost, and just like them itexpresses, but does notassertthe existence of arelation between the concepts appearing as arguments. The distinction betweenexpressing and asserting the existence of higherorder entities is a crucial one, onethat properly demarcates the first from the secondorder realm.

The property ofpermutation invariance plays a crucial role also in the modernconception of quantifiers. Quantifiers such as andanswer the question Howmany? with no concern for the specific nature of the objects in question and aretherefore invariant under permutations that swap around objects of the domain.Whereas notions of invariance for abstraction principles can be formulated in at

least a few nonequivalent ways, it is easy to make precise such a notion in the caseof quantifiers:

If is a permutation ofD, then a binary firstorder quantifier Qispermutation invariantif and only iffor every subsetsA and B ofD, Q(A, B)holds precisely when Q([A], [B]) holds as well.

While the standard quantifiers of firstorder logic are permutationinvariant in theabove sense, many more quantifiers enjoy this property, most notably those dealingwith cardinalityconstraints (including the Hrtig and the Rescher quantifiers).Among the latter, of course, is the Frege quantifier F. Our first task is to explore theexpressive properties of the logical framework resulting from taking the Fregequantifier as the basic building block.

5. The language of the arithmetic

Formally, we consider a firstorder language LF with formulas built up from


17/21

17

(individual or predicate) constants by means of Boolean connectives and thequantifier F; specifically, F takes two arguments, so that if (x) and (x) areformulas, so is Fx((x), (x)).35 The language of the Frege quantifier can be given astandard interpretation by supplying a recursive truth definition la Tarski. Modelsfor LF look just like firstorder models, providing a nonempty domain D and

interpretations for nonlogical constants. The recursive clauses for the connectivesare as usual, and formulas of the form Fx((x), (x)) are satisfied in the model if andonly if there are no more objects in D that satisfy (x) than there are objectssatisfying (x).

The language thus defined is quite expressive. First observe that the standard firstorder quantifiers are expressible in LF: a universally quantified formula x(x) canbe represented by saying that the complement of is empty, i.e., that there are nomore objects satisfying (x) than there are satisfyingx x. Dually an existentiallyquantified formula x(x) can be represented by saying that there is no injection of into the empty set. But the language turns out to be much more expressive than

ordinary firstorder logic. For instance, while infinity cannot be characterized infirstorder logic using only the standard existential and universal quantifier, there isan axiom of infinity in the pure identity fragment ofLF (such an axiom states that theuniverse is Dedekindinfinite). The negation of such an axiom, then, is true in all andonly the finite domains, a fact that shows that, as a consequence, compactnessfails.36 And in a similar way, for any formula one can express the fact that the setof objects satisfying is Dedekindfinite.

There is, however, an alternative interpretation of the Frege quantifier, which weregard as equally attractive we refer to it as thegeneralinterpretation ofF onwhich the Frege quantifier is much less expressive. Recall thatsecond-orderquantifiers can be given, beside a standard interpretation, also a socalled generalinterpretation (first introduced by Henkin37). On such a general interpretationsecondorder quantifiers are taken to range not over the true powerset ofD, butover some previously given universe comprising some, but not necessarily all,subsets ofD. So while standard models for secondorder logic are indistinguishablefrom firstorder models, general models carry, beside a domain D, also a universe ofnplace relations over D (for each n).38 All this is well known.

35 In the most general presentation, we will allow the formulas (x) and (x) to contain

parameters, and the quantifier to bind one or more variables simultaneously.36 Since Hrtigs quantifier is interpretable in LF (by the SchrderBernstein theorem setsAand B have the same cardinality if and only if there are injections fromA to B and viceversa), any results about the expressiveness ofI as detailed, e.g., in Herre et al. (op. cit.),carry over to the Frege quantifier.37 L. Henkin, Completeness in the theory of types,Journal of Symbolic Logic, 15 (1950) pp.8191.38 In practice, such a universe of relations will satisfy some closure conditions it will be,e.g., closed under definability, thereby satisfying the secondorder comprehension axiom.


18/21

18

Perhaps more surprisingly,first-orderquantifiers can also be so interpreted (anapparently hitherto unnoticed fact). Consider for instance the ordinary existentialquantifier: as we have seen, in a classical firstorder language this quantifier rangesover the collection ofallnonempty subsets of the domain. But an alternative,general interpretation is possible as well, on which ranges over some collection

of nonempty subsets of the domain.39 The set of sentences valid on such aninterpretation of the quantifiers turns out to be well known, if unexpected: it is theset of validities of positive free logic.40

In a similar vein, we consider a less expressive interpretation of the firstorderquantifier F, which, just as in the case of, is specified by singling out a particularclass of models. By a general model for LF we understand a structure providing anonempty domain D and interpretations for the nonlogical constants, as well as acollection F of 1to1 functions between subsets ofD. On this account, a formulaFx((x), (x)) is satisfied in the model if and only if there is a functionf in F takingthe set of objects satisfying into the set of objects satisfying . In practice, we

want the collection F to satisfy also certain closure conditions, which ensure that thelanguage is still powerful enough for an adequate formalization of arithmetic.41 Onesuch closure condition, for instance, ensures that if there are no more Fs than Hsthen there are no more Fs and Gs than Hs.

Thus we have two equally attractive ways to specify a semantics for the language ofthe Frege quantifier. This language can be given either the standard interpretation,in which F ranges over all injections between subsets of the domain, or the generalinterpretation, in which less comprehensive collections of functions are allowed. Weregard the two interpretations as equally attractive.

One more ingredient is missing in order to specify completely the language, viz., theabstraction operator. We thus introduce an abstraction operator Num mappingformulas into terms: Num() picks out an object to be construed as the number of. (Strictly speaking, Num is a variablebinding operator so that it should properlybe written as Numx(x,y) whereyis a placeholder for possible parameters; inpractice, the bound variable is understood). It is clear what a model for such alanguage would look like: on thegeneralinterpretation, besides supplying a nonempty domain D and interpretations for the nonlogical constants, a model wouldalso supply both a collection F of injections between subsets ofD and a function

39 So the nonstandard existential quantifier can range over any collection of subsetsomitting the empty set; then dually, the nonstandard universal quantifier would range overany collection of subsets as long as that collection contains D itself.40 See G. A. Antonelli, Free quantification and logical invariance, in A. Paternoster, M.Andronico and A. Voltolini (eds), Il significato eluso. Saggi in onore di Diego Marconi,Rosenberg & Sellier, Torino, 2007 pp. 6173 (special issue ofRivista di estetica, vol. 33 (1)).

41 The reader is referred to G. A. Antonelli, Numerical Abstraction via the Frege Quantifier,cit., for the technical details.


19/21

19

taking subsets ofD into D: the former, of course, is used for the interpretation of theFrege quantifier, while the latter provides an interpretation for the abstractionoperator Num. On the standardinterpretation, on the other hand, there is no need tospecify the collection F (or, equivalently, F can be taken to be the collection ofallinjections between subsets ofD.)

6. Formalizing Arithmetic

We now have both main components of our approach to arithmetic: the Fregequantifier, embodying a nonreductionist take on logicism; and the Num operator,construed according to a deflationary view of abstraction. Special extralogicalaxiom schemas formulated in the language LF augmented with Num will be neededto govern the interaction between the cardinality quantifier and the abstraction

operator. We will give just a sketch of these axioms here, since the details are fullydeveloped elsewhere.42

These extralogical axiom schemas fall naturally into three main categories. The firstgroup of axioms contains definitionaland uniqueness principles, beginning first andforemost with Humes Principle. HP can be easily expressed by asserting that theidentity Num() = Num() holds if and only if there is a bijection between the sand the s, i.e., if and only if both Fx((x), (x)) and Fx((x), (x)) hold. In a similarvein one can define an ordering relation by means of a schema saying thatNum()Num() if and only ifFx((x), (x)) holds (where is taken as primitive). Next,one characterizes the natural numbers by introducing a primitive predicate N(x)

along with the axiom stating thatN(x) holds if and only ifxis the number of thepredicate natural number less thanx, i.e.,x= Num(N(y) &y


20/21


21/21

21

by providing a reduction to some other principle), we set out to explore itsconsequences by introducing cardinality, in the form of the Frege quantifier, as themain building block in the language of arithmetic.43

28 December 2009

43 I am grateful to Elaine Landry, Albert Visser, Ed Zalta and an anonymous referee forhelpful comments, suggestions, and criticism.

Date post:	03-Apr-2018
Category:	Documents
Upload:	gzalzalkovia
View:	218 times
Download:	2 times

Antonelli, Nature & Purpose of Numbers [21 pgs]

Documents