MATHEMATICAL ABSTRACTION AS UNSTABLE TRANSLATION BETWEEN CONCRETE PRESENTATIONS

Roi Wagner

ETH (Eidgenossische Technische Hochschule}, Zurichroy.wagner@gess.ethz.ch

ABSTRACTThis paper suggests that mathematical abstraction be conceived not as the construction of some mental or ideal object, but as a practice of translation between various symbolic systems. This view aims to capture the intangibility of abstraction by framing it as an imperfect and somewhat unstable translation between symbolic systems. The imperfection and instability of the translation means that it does not establish an invariant core, but rather relates aspects of various symbolic systems in a dynamic and possibly even inconsistent manner. I support my argument with examples from mathematics education and the history of mathematics.

Available views on mathematical abstraction

The term “abstraction”, as used by mathematicians, usually refers to formalization or generalization (Dreyfus 2014, 5). Axiomatic representations are typically considered as more abstract than models (e.g. Peano axioms as opposed to decimal place value representation, Hilbert’s axioms for Euclidean geometry as opposed to a Cartesian model), as do objects that involve more general or clustered components instead of more specific or singular objects (e.g. a general set as opposed to a set of numbers, functions in the Hilbert space L2 as opposed to functions defined pointwise). This view associates the term “abstract” with a kind of object (supervening, more general, collective) or a kind of representation (formal, axiomatic).

In this paper, however, I would like to think of abstraction as a mathematical practice and describe how abstract mathematical objects emerge and grow. I therefore turn from the prevalent usage of the term among mathematicians to the discourse of mathematics education scholars.

Within the latter discipline, one approach can be summarized as follows: “An abstraction process occurs when the subject focuses attention on specific properties of a given object and then considers

1

these properties in isolation from the original” (Harel and Tall 1991, 39). This echoes the longstanding Aristotelian tradition where abstraction means subtraction or selective attention (see Bäck 2014, 11-23). A line drawn with a ruler, for example, becomes an abstract straight Euclidean line when we subtract or ignore its width, its small curvature fluctuations, and the physical constraints that prevent its indefinite extension. A group of permutations would likewise become abstract, if the representation of its element as permutations is subtracted or ignored, leaving behind only the relations between its elements with respect to the group operation.

However, this selective or subtractive view does not tell the whole story. Learners and teachers sense that abstraction is not simply about taking things away, but involves some ampliative and creative component. In mathematics education research, this intuition finds support in Piaget’s notion of reflective abstraction. Building on Piaget’s work, scholars articulate abstraction as transforming actions, processes or relations into objects that can be manipulated (Thompson 1985; Kaput 1989; Sfard 1991; 2008 Ch. 6; an operationalization of this approach is available in Dreyfus et al. 2015). An example here is the evolving relation of two quantities varying together (e.g. time and the spatial position of a moving object), which transforms into a new unified object: the function that relates these two variable quantities. This approach to abstraction portrays mathematics as a hierarchical accumulation of reified relations.

Another approach that does not reduce abstraction to subtraction can be traced back to the formal context of AI. Abstraction is defined there as a “mapping between representations” (Giunchiglia and Walsh 1992, 329). Such mappings translate the ground representation of a problem to a new (simplified) representation, which preserves the properties most relevant for the solution and facilitates its derivation. This approach can be integrated with the two previous approaches: abstraction as translation between representations may restrict attention to the invariants emerging from the translation and also turn these invariants from relations into objects. A classical example is the translation between Dienes blocks (sticks that encode numbers by length and color) and the numeral representation of numbers. Capturing the relations preserved by the translation was supposed to help young students come up with an abstract notion of number (Schoenfeld 1986).

Some issues with these views

While the above approaches from mathematics education fit well the prevalent images of mathematics, they still pose substantial problems. One problem is their ontological vagueness. They demand that we distinguish processes/relations/actions from objects, but also

2

that we be able to “substantiate” the former into the latter. Moreover, in many examples, the former are often only implicit before abstraction turns them into explicit objects (e.g. Noss and Hoyles 1996, 124), which means that one has to account for the slippery category of “implicit being” without falling into the trap of defining what is implicit by means of its explicit subsequent counterpart. I’m concerned that the above distinctions are often more confusing than helpful; I would therefore like to suggest an approach to abstraction that does not require substantiation and implicit being.

Moreover, the above formulations of abstraction often endow some formal or axiomatic definition with primacy over the representations and practices from which it emerges. This is the case, for example, in Wilensky’s (1991) proposal of a translation account of abstraction. He explains that a mathematical abstraction (say, the definition of a ring), can become concrete “provided that we have multiple models of engagement with them and a sufficiently rich collection of models to represent them” and “the more connections we make between an object and other objects, the more concrete it becomes for us” (Wilensky 1991, 198-199). In this narrative, the abstract object is given formally, and now the learner or mathematician must come to terms with it by means of examples and translations.

The attitudes presented in the last two paragraphs make sense in the context of mathematics education, where a teacher introduces a new formalism and is required to relate it to objects that the student already knows, or where the teacher is aiming to guide the student from diffuse practices that relate various mathematical objects toward a new mathematical conceptual goal. However, in the context of exploratory mathematical practice, whether in advanced mathematical research or more elementary problem solving, the formalized or objectified mathematical endpoint is not available. A hindsight description of the abstraction process by means of a conceptual target may therefore distort our understanding of doing mathematics outside the context of preconceived learning goals.

Mathematics education research is far from oblivious to the issues detailed above. Gray and Tall (1994) claim that successful learners read the same mathematical notation as indicating a process and an object, and can switch between them to optimize performance – rather than simply replace the former by the latter. Noss and Hoyles (1996) too, when discussing multiple representations in mathematical learning, state that learning mathematics involves “constructing multi-faceted connections between activities and experiences that are 'in some way' similar. Abstraction becomes a problem of how to add new friends and relations, not to ascend to unattainable heights” (47). Moreover “abstracting can be seen as a way of layering meanings on each other, connecting between ways of knowing and seeing, rather than as a way of replacing one kind of meaning with

3

another” (122). Ainsworth (1999), in turn, proposes a long list of functions served by relating multiple mathematical representations to each other, all of which contribute to mathematical performance, but do not reduce to formalizing, extracting invariants or objectifying relations.

In this paper I will attempt to follow this logic all the way through mathematical abstraction: instead of the formation of new objects, I suggest to think of abstraction as a collection of different translations between available mathematical representations. The innovation is that I do away with the suggestion that such translations generate a set of invariants that serve as the goal or content of abstraction.

A new definition of mathematical abstraction

Re-defining philosophical concepts is often a barren exercise. A philosophical definition may be considered good if it captures the most important aspects of everyday use, maximizes consistency, and allows to clarify philosophical problems. But there’s no common measure of how to balance these requirements and no agreement on what the “most important” aspects of usage are. This leads to a proliferation of technical definitions that end up in scholastic thickets. My point is, therefore, more modest. I would like to offer a definition that highlights a practical aspect of abstraction that tends to be set aside. I am not pretending to provide here the true essence of abstraction (whatever that may mean), only an important aspect that current discourse plays down.

I suggest to consider mathematical abstraction as:

the practice of incomplete, underdetermined, intermittent and open-ended translations between systems of presentations.

I will now specify the meanings of the terms used in this definition.

First, I should specify my notion of “presentation”. I use this term in order to omit the “re” of “representations”. This “re” suggests that an object is already there, and the representation simply denotes it. Here, I would like to set the collection of symbolic presentations as primitive, rather than as secondary coding of something else. However, presentations are not ex-nihilo. They depend genealogically on other presentations and practices, and evolve from them.

Presentations may involve any kind of symbols: material, graphic, auditory, motor, etc. In terms of this paper, all these presentations fall under the category “concrete”. Abstraction emerges, according to my suggestion, when one translates between concrete systems of presentations. Whether one’s arithmetic presentations are sheep, pebbles or numerals, one is working with a concrete presentation.

4

Sawed-off wooden cones, sketches of planet trajectories and algebraic quadratic forms are all concrete symbolic presentations, which eventually inter-translate to yield the abstraction “conic sections”. One is working with something material and sensual (or at least a simulation of sensation, if one imagines those presentations) that can be manipulated. Presentations thus include both what we usually call a mathematical model as well as what we usually consider as the modeled phenomenon.

The term “system of presentation” implies a collection of symbols, tools and rules that guide the manipulation and application of signs. The collection of symbols, tools and rules need not be well defined, closed or rigorous. In a mathematical setting, which typically values consensus, the system must be rigid enough for practitioners to be able to sufficiently (but not necessarily perfectly!) agree on correct application of the rules. Such agreement does not come for free – it is the result of training, a supervening social structure, and the cognitive features of the system of presentation. In particular, a mathematical system of presentation cannot be so complicated that practitioners could not be trained (given the available training institutions and technology) to follow it consensually, at least approximately or with respect to some key features.

I follow here a Wittgensteinean perspective, adopting a version of what he calls “language game”. The rules of such a game need not be fully explicit and are definitely not tracks to infinity. Creative interpretations and misinterpretations are always possible, and it is not always possible to distinguish the two. This is especially the case where the habitual scope of application is exceeded: trying to extend the rules beyond the scope where practitioners’ performance is already established and more or less consensual may lead to disagreements. Moreover, it is sometimes unclear how to individuate presentation systems, as their boundaries might be rather porous.

An obvious example for systems of presentation is simple systems of number presentation (number words in natural languages, Roman numerals, etc.). They are associated with clear rules concerning counting and basic arithmetic, but their extensions to large or small numbers and new kinds of numbers (zero, negative, irrational) is underdetermined. They only allow calculations up to a certain size, beyond which new tools must be invented or errors/disagreements are practically unavoidable. Such systems typically do not decide such questions as the existence of largest and smallest numbers, and their arithmetical predictions may conflict with actual measurements that are subject to various practical vicissitudes. Early modern infinitesimal calculus is another example for a system of presentation: a powerful tool for calculation, but also a fertile ground for disagreements and inconsistencies (see below). Among systems of presentation, modern formal systems are an extreme case –

5

extreme in their ability to generate consensus about correct derivations. But such systems are rare in actual mathematical practice.1

It is obvious from the examples above that I’m thinking here of “mathematical practice” as the wide range of activities involved in many forms of doing mathematics – from practical arithmetic to university research. Note that several different practices and kinds of presentations are relevant for a given context of doing mathematics: even pure mathematics researchers use gestures, vague suggestive diagrams, rules of thumb, imprecise analogies, empirical methods, and many other practices that cannot be reduced to formal mathematics. More rigorous systems of presentation are a crucial aspect of contemporary research mathematics, but they do not even begin to exhaust it.

Let us proceed now from concrete systems of presentations to abstraction. According to the approach outlined so far, decimal arithmetic is perfectly concrete. This system only becomes abstract in relation to other systems of presentation: one-one correlations of discrete objects, putting together continuous magnitudes, dividing things equally, etc. It’s the imperfection of the translation (discrete objects sometimes break or merge, measurements of continuous magnitudes inevitably involve deviations, large quantities may be handled by decimal arithmetic but not by actual counting) that turns the inter-translation of concrete systems of presentation into number as abstraction.

In the same vein, complex arithmetic as a formal calculus is no more abstract than real arithmetic (or than chess or Lego, for that matter) – these are all concrete systems of presentation, where one follows rather strict rules. However, none of these systems exists in isolation. They relate to many mathematical and non-mathematical practices. It is with respect to this relation that complex numbers are more abstract than real numbers: they involve more translations that introduce new instabilities. On the one hand, accepting a number whose square is negative violates one of the traditional arithmetic maxims, so we might want to think about imaginary numbers as foreign to real numbers. On the other hands, imaginary numbers historically evolved in the context of solving real polynomial equations, where they sometimes yielded correct real solutions, so they are intimately related to real numbers. It is precisely the open-ended problem of translating and relating the new system of presentation to older systems by various different means – something at once objectionable and well-motivated – that makes complex 1 Formal systems are indeed difficult to handle beyond short, simple arguments or without the assistance of still rather exotic computerized formal proof verifiers (Avigad 2018). They are therefore not sufficient as a framework for real-world mathematical practice (Azzouni (2008) and Avigad (2019) attempt – in different ways – to bridge the gap between mathematical practice and formal systems).

6

numbers feel intangible, unreal, abstract. They are more abstract than real numbers in the sense that they introduce additional layers of translation.

My qualification of mathematical abstraction as “incomplete, underdetermined, intermittent and open-ended translations between presentations” is meant to capture the instability of abstraction by translation. “Incomplete” means here that translations are not always well defined. Some bits of one presentation may not translate well into another, even if a translation between the presentations is part of the practice. “Underdetermined” means that there may be more than one way to translate one presentation to another. “Intermittent” means that one does not always have to use the translation, even if it is available and well defined. “Open-ended” means that the translation may change and evolve historically or within a single solution to a given problem. This notion of translation is therefore not some sort of surrogate to the notion of an inference-preserving morphism between cognitive domains (Lakoff and Núñez 2000; critically analyzed in Wagner 2013) or formal mathematical structures.2

Analytic geometry can serve as an accessible example for the kind of translations I have in mind.3 As long as geometry was not fully arithmetized (that is, as long as a linear continuum was not conceived as a set of points corresponding to real numbers, but as an entity that cannot be exhausted by points, even though points may lie on it), the scope of translatability between the Cartesian plane and the Euclidean plane was contested. This was only resolved around the turn of the 20th century, by giving primacy to a discrete-arithmetic point of view (Epple 2003). However, alternative notions of continuum do survive in some non-standard mathematical systems and prevalent but informal intuitions. Moreover, even if we accept the translation, not every curve has an algebraic presentation (given some determined scope of algebra, such as polynomials, elementary functions or power series), and not every algebraic expression translates into a geometric entity (e.g. a formal power series that is divergent everywhere). The translation is therefore still incomplete.

The historical growth of algebra and geometry to include new entities that correspond to given entities of the other field is an instantiation of the open-endedness of the translation between these fields. The translation of geometry to algebra in analytic geometry is also clearly 2 I note that we must understand a translation “between systems of presentations” as including the somewhat degenerate case of an automorphic translation, that is a translation between different terms of a single system. In that case, we sometimes apply to a certain term rules that apply to another kind of term in the same system (we will see an example when we discuss infinitesimals below).3 Another example is braid theory, analyzed under the terms “translation” and “hybridization” of diagrammatic and algebraic practices by Michael Friedman (2018), as well as the related discussion of knot theory by De Toffoli and Giardino (2014).

7

underdetermined: it depends on a choice of coordinate systems and projections. As every teacher (and successful student) knows, the compulsion to algebraize every geometric problem must be checked. Often enough, holding on to a geometric point of view turns what would have been a hopeless algebraic mess into an easy and intuitive argument. Therefore, the translation of geometry into algebra should remain intermittent. If we give up on the assumption that a translation between systems of presentation imposes a stable invariant core, then we cannot point to any unified object that emerges from the translation. Such translation gives rise to various mathematical practices, but not to a stable material or mental entity, not even in the sense of a structure or equivalence class. I believe that this state of affairs corresponds quite well to the elusiveness that the term “abstract” conveys. This elusiveness is, I believe, characteristic of the “abstract notions” of mathematics such as number, infinitesimal or set.

If we follow this approach, then a “perfect” translation between systems of presentation (a translation where nothing is lost) would yield, again, something concrete. If one could do the exact (or, more realistically, almost) same thing in one system and in the other (maintaining informal heuristics, cognitive associations, intuitions, complexity, etc.), then the systems would be practically the same, and working with both would be just as (or, more realistically, almost as) concrete as working with either one. No abstraction would ensue.

Moreover, according to this approach, abandoning previous presentations in favor of a new one (say, word numbers or pebble arithmetic in favor of decimal notation) would be just the replacement of one concrete system of presentation by another. Abstraction emerges from finding ways to relate systems of presentations without reducing them to a unifying “common denominator” or abandoning old systems for new ones.

Examples from mathematics education and history

In order to make what I call “abstraction” more concrete, I will draw on some elementary examples from mathematics education research, before bringing some examples from the history of mathematics. The first couple of examples are quoted from the research of Luis Radford and his colleagues. His semiotic approach to mathematics highlights precisely what my definition of abstraction brings forth.

8

The first example (Radfrod et al. 2007, 2) is a classroom task, where students are given the figures inside Figure 1, and are asked to come up with the number of circles in each figure if the sequence is continued (e.g. diagram 10, diagram 100, and later on diagram n). The authors provide a fine-grained analysis of what the students did as they tried to solve the problem. The following quotations are telling:

Doug says: “So, we just add another thing, like that.” Exactly as he utters the word “another” he starts making a rhythmic sequence of six parallel gestures. The gestures play a twofold role. First, they highlight the last two circles diagonally disposed at the end of each figure of the sequence. Second, by being rhythmically repeated, they express the idea of something general, something that continues further and further, in space and in time. (507-508)

Jay touches the two horizontal rows into which Figure 4 can be divided. Turning to Figure 10, Mimi … matches her utterance with two gestures that keep certain specific aspects of Jay’s: having one gesture for each row, and their vertical shift. But whereas Jay’s gestures point materially to the rows of Figure 4, Mimi’s are made in the air …. Indeed, Figure 10 is not in the perceptual field of the students, so new mechanisms of semiotic objectification have to be displayed. This, we suggest, is the role of gestures here. … This point becomes even clearer when the students address the question of Figure 100. The gestures are again made in the air, and this time at a higher elevation from the desk. (520)These quotations show how the object discussed here is not quite the drawn diagram. Gestures, words and rhythms take it, so to say, “off the page”. The translation between the figural, verbal and gestural modes of presentation plays out in their coordination: the authors show how different modalities are synchronized in the actions of a single student and inter-subjectively between different students. This practice of abstraction does not aim at something on the page, in the words or in the gestures. It also does not establish a common mathematical core of all representations (gestures, for example, capture very little of the numerical information here). Rather, it lies in our ability to coordinate them.The next example (from Radford and Puig 2007) considers a different level of translation. Here the problem is the following:

9

Sophie has 17$ more than Justin. Sophie has twice more than Manuel. Manuel and Justin together have the same amount as Sophie.Question 1: Let x be the number of dollars that Justin has. Find an algebraic expression for the number of dollars that Sophie has and an algebraic expression for the number of dollars that Manuel has.Question 2: Write an equation for this problem.Question 3: Solve the equation and clearly explain your steps. (153)

The two most salient systems of presentation are the word problem and the algebraic symbolism. This example highlights what the translation does not preserve. The authors claim that:

The original signification [money] has been suspended and the letter x is now a mark. During the transformation of the equation, the main “personages” of the formal algebraic text are no longer Manuel, Justin and Sophie, but the operations. Not exactly the original ones, however, for it is not the initial verbal meaning of operations that has primacy in the simplification of the equation. It is … a perceptual meaning, i.e. one that attends to the shape of the symbolic expressions. An algebraic equation is in fact like a diagram… (156)

symbolic algebraic thinking requires the cognitive ability to switch between verbal and perceptual meanings and to become conscious that the latter is governed by the shape of expressions whose syntactic complexity may lead to multilayered perceptual meanings (like in the expression ( x+172 )). (160)

Note that quite a lot changes in the translation, in particular the meaning of terms and their associated cognitive processing (verbal-semantic vs. visual-syntactic). Moreover, many of the algebraic manipulations (multiplying both sides by 2, moving 17 from side to side) make no sense in terms of the original word problem – and yet the algebraic solution is re-translated to the terms of the word problem. The translation is therefore incomplete and intermittent. The various possible algebraic approaches show that it is also underdetermined.

The abstraction is therefore not the algebraic symbol or some relation extracted from the word problem. Symbolic variables are as concrete as the terms of the word problem – they are all there on paper, manipulated in regulated ways (like the words of natural language, but subject to different rules). Working strictly within the system of a

10

single algebraic presentation (which I doubt ever happens in elementary or advanced mathematics) is no more abstract or concrete than playing chess. Rather, our practice of inter-translating the word problem and algebraic presentations, sometimes operating with several presentations, sometimes only with one of them, without ever reducing it all to some material or mental object that we can grasp or individuate4 – this practice is what renders mathematical objects abstract. It gives rise to something that cannot be concretely captured beyond its various presentations.

The incompleteness and intermittent nature of the translation is also salient in Arcavi’s (1994) attempt to articulate the components of “symbol sense” that algebra students are supposed to learn based on a synthesis of the author’s classroom observations. First, Arcavi acknowledges that “Symbol sense includes the feeling for when to invoke symbols and also for when to abandon them” (26). In other words, the translation into symbols should be intermittent, because symbolizing is not always an effective strategy. Moreover “symbol sense … consists of the search for symbol meaning, whether it is essential to the solution of the problem, or merely adds insight” (27). Here Arcavi points out the open-endedness of translation. For example, in the last example from Radford and Puig, the students could try to extend the scope of translation, that is, to try to incorporate into the story algebraic moves such as multiplying by 2 or moving the 17 (“each dollar split into 2” or “one person gave the other 17 dollars”). This may end up being confusing and ad-hoc, but is still part of how the abstraction emerges in the translation between the word problem and the algebraic calculus.

Arcavi recognizes the open-ended aspect of translation in more general terms:

A catalogue for symbol sense is inevitably incomplete because students will have little sense for a representation (in our case symbols) in isolation, if they are not able to carry the meanings of those symbols flexibly over to other representations. And, conversely, moving back and forth among different representations will result in enhancing the understanding of each particular one. Symbol sense will grow and change by feeding on and interacting with other "senses", like number sense, visual thinking, function sense, and graphical sense. (32)

The learners’ process of symbolic abstraction is not reducible to constructing an invariant or an object. Rather, it is expressed by the ability to integrate one more system of presentation – the algebraic

4 We must not confuse the rulings of the teacher or the experienced mathematicians with being ruled over by an object. Yes, some translations will be rejected and some endorsed by authority figures, but the rejection and endorsement of behaviors need not be conceived of as creating a material or mental object.

11

calculus – into the many systems of presentation already available to them.

First historical case study: Zero

In this first case study, I will consider several aspects of “zeroness” in various systems of presentation. These aspects cannot be superposed on each other in a perfectly coherent way, but are (almost) all relevant to contemporary practice. The translation between them constitutes what I would call zero as abstraction.

I do not claim that there are direct lines of transmission connecting all these historical systems of presentation to today’s mathematical practices, only that systems that are similar enough still play a role today. I prefer to bring examples of historical systems of presentation rather than contemporary ones, because they help isolate different aspects of zero, trace their contingent combinations, and remind us that such aspects always exist in concrete systems, rather than as ideal components of mathematical thought.

Zero as placeholder

A major aspect of zero is its role as a placeholder in place value number systems. Early examples of these include the Mesopotamian sexagesimal system, the Chinese rod-numerals, Mayan hieroglyphics, Indian decimal numbers, as well as non-written systems, such as quipo and counting boards (surfaces divided into columns where counters are placed, each column representing a different place value). In all these systems, a string of “numerals” represents the sum of their values, each multiplied by its respective place-value (which are not necessarily powers of a fixed number). The numerals may be incisions, traces of ink, or even pebbles moved around on a counting board.

In many of these systems an absent place value was marked by an empty space (e.g. in Babylonian tables – see for example Plimpton 322 reproduced in Robson 2008, 111-112 – or in Chinese rod numerals, Martzloff 1997). This was particularly useful where there was a clear tabular setting (for example, on a counting board) and where orders of magnitudes were clear from the context, so there’s no problem with missing placeholders at the end of the number. In other cases, such as later Babylonian times (Høyrup 2002, 294) and in Indian decimal numbers, a symbol (or one of several symbols) filled the gap. Note that placeholders need not play other roles that we associate with zero. Placeholder symbols need not represent a value, a number or their absence in other mathematical circumstances (the later Babylonian system is an example). Quantitative nothingness and the placeholder may belong to different presentation systems. Even

12

where the same sign is used both for the number zero and for a placeholder, the operations applied to the placeholder need not be identical to those applied to the number zero. In particular, when one needs to subtract from a placeholder in the context of place-value subtraction, one does something different from the equivalent operation on zero as number.

Zero as reference point

Evidence from ancient Egyptian sources (Lumpkin 2002, 164) indicates two zero-related uses of the hieroglyph nfr (which means “good”, “perfect” or perhaps a shorthand for “depleted” – see Imhausen 2016, 20). First, the hieroglyph was used to indicate a perfect balance (income equal to expenses in accounts). Second, it was used to indicate a reference point in construction with respect to which distance was measured upwards or downwards.

Note that the latter use does not imply negative numbers. Having “4 below” or “4 above” does not mean that “4 below” is ever manipulated arithmetically in a manner similar to negative numbers. Indeed, the available Egyptian sources don’t discuss “adding 4 below to 3 above”, and if they did, given the interpretation of these values as absolute distances, this would more likely mean “a total distance of 7” rather than “1 below”. Therefore, this is not an instance of what we would call “the number line” or “signed numbers”. Instead, what we have here is a translation between an accounting context (income and expenses) and a geometric-architectural context, but neither system involves proper signed arithmetic. Note also that ancient Egypt had no place-value system and hence no placeholders.

Zero as counter

I suspect most people would think of 1 as the beginning of counting and of 0 as a mark of absence, but this does not reflect all documented practices. Historically, this is attested in Mayan calendars, where the 20 day months (as well as some other calendrical periods) had their days counted from 0 to 19. Note that the Maya used many symbols that we read today as zeros in various contexts. In some date formats the zero is represented by the placeholder mark, but in others it is represented by other signs that have to do with beginning or end (Closs 1986, ch. 11, especially 298). Given the limited sources, it is not clear whether we should consider all these zeros as parts of the same system of presentation.

Today, we count the first day of a month as 1, but set the clock at midnight to 0h00 rather than to 1h01. A similar ambiguity exists in definitions of the natural numbers, which sometimes begin with 1 and sometimes with 0 – especially in the context of the so called

13

“standard model” of natural numbers in set theory, which involves a translation between numbers and sets (the first natural number is identified with the empty set and usually also with 0). This shows us that we can live at peace with inconsistent systems of presentations, where zero sometimes marks absence and sometimes serves to count entities.

Zero as the value of nothing

Reference points, marks of balance and counters need not be fully integrated into systems of arithmetic operations. By that I mean that expressions such as “nothing plus 2”, or “add day zero of the month to…”, or “add the reference point to 2-above” may be meaningless in the relevant system of presentation. Even placeholders need not be subject to arithmetic operations if the place-value representations in not actually used for arithmetic operations (for example, where place-value representations are used in books, but calculations are done in another system of presentation).

Some zeros are, however, explicitly integrated into arithmetic systems, where they represent the value of nothing. A well documented early example is Brahmagupta’s rules for adding, subtracting, multiplying and dividing with signed numbers and zero (Katz 2007, 429). Brahmagupta’s rules for division do not match our own: zero divided by a number does not seem to be simply zero, but zero divided by zero is set as zero.

This is a good example of underdetermination: what we make of operations with zero depends on how these operations are related to other practices. Such operations need not have any “natural” meaning, so they may be interpreted based on a linguistic or philosophical understanding of nothingness, on operations with the placeholder zero, or on adherence to relational rules (e.g. addition and multiplication should be inverses of subtraction and division, multiplication should distribute over addition, etc.).

Note also that it is not clear how stable the Indian place value system was at the time of Brahmagupta (I mean here a system where places indicate the multiplication of numerals by certain values without these values being otherwise indicated, and where a placeholder is used for missing places). Such a system of numerals is not present in texts by Brahmagupta or his predecessors, (I side here with doubts cast on the early dating of portions of the Bhakshali manuscripts in Plofker et al. 2017), although they probably did exist in some forms alongside these texts (see Plofker 2009, 43-48). In general, zero as a value integrated into an arithmetical system of presentation need not strictly depend on zero as placeholder.

Zeros as unknowns and infinitesimals

14

Here I will mention two systems of presentations that use the zero symbol for purposes that are not clearly associated with zero in today’s mathematics.

The first is the use of zero as the sign of an unknown. I know of this practice in just two medieval manuscripts, but their geographical span suggests that it existed elsewhere as well: the Indian Bakhshali manuscript (Hayashi 1995) and Ibn Ezra’s 12th century Hebrew Book of Number, written in Italy (Katz et al. 2016, 231-232). In both cases an unknown answer to problems is marked by the same symbol as that used for zero.

This is obviously not part of contemporary practice, but it is not entirely unrelated. When children are being prepared to transition from arithmetic to algebra, they are often given exercises of the form “[empty box] + 2 = 5”. The empty box may as well be an empty circle, a dot or anything else that holds the place. The decimal placeholder (which holds the place empty) and the unknown (which holds the place for the value to be calculated) are thus not entirely unrelated. This translation may also be motivated by relating zero to absence, and through absence to lack of knowledge or even unknowability, but this takes us beyond the scope of mathematical systems of presentations.

This translation between systems of presentations, however, is not quite productive, as hardly any of the rules or practices associated with zero as a placeholder have little in common with those of operating on unknowns. Another, somewhat more productive translation relates zeros to infinitesimals.

The most famous proponent of this practice is Euler (who discussed infinitesimals in many different ways, not necessarily consistent with each other, as shown in Ferraro 2012). The motivation is quite clear: if we treat infinitesimals as zeros, we can drop them in calculations such as ( x+dx )2−x2

dx=2 x+dx=2 x. However, we can’t always treat

infinitesimals as we treat zeros. For example, infinitesimal calculus did not accept such claims as dx2 =dx, and divisions of infinitesimals by each other often have determinate values. Their zero-like behavior, therefore, is problematic at best.

The reason for including these last examples is to show how good mathematicians translated zero signs between systems of presentations in ways that do not necessarily fit contemporary practice. The problems of these translations alone are not enough to justify their rejection – we saw above that some contemporary aspects of zero are also incompatible with each other. We can only

15

state that in the historically contingent balance between inter-translatability and stability of procedure, some translations caught on and other were given up. This testifies to the underdetermination, intermittence and open-endedness of translations between mathematical systems of presentation.

The abstract concept of zero

What does the above teach us about the abstract concept of zero? One could say that it has very little to do with zero as an abstract concept. The abstract concept of zero is arguably the additive neutral terms in groups, rings or fields, which is defined by formal axioms. But if this is the abstract zero, then in practical terms, to have abstracted zero is simply to master certain concrete formal systems or certain procedures common to several formal systems. If we follow this approach, one can have the abstract concept of zero without ever having had any content-full concept of zero, and this abstract concept will have little to do with most of the practices cited above.

I suggest that, in practical term, to have an abstract notion of zero means to be proficient with placeholders, points of reference or balance, counters that precede 1, and arithmetic with elements valued as nothing. It means to be able to partially, productively and creatively translate between those and other systems of presentation, such as the formal axioms of a field.

There is no stable common denominator to all these systems: the different zeros obey different calculation rules (as with placeholders and arithmetic values) and describe different, incompatible things (lack or emptiness, arbitrary reference points, counted objects). The different systems do not necessarily depend on each other or follow from each other. The same symbol may invoke some of these systems some of the time, and suppress them at others. The collection of systems and the ways in which they relate to each other is not fully determined and is open-ended. In short, I suggest that to abstract zero means to be proficient in this delicate economy of translations.

Second historical case study: infinitesimals

Our second case study is infinitesimals. The first obvious objection is that there simply are no infinitesimals, as these “entities” lead to contradictions. But regardless of whether one could in principle “rigorize” some notion of infinitesimal (indeed, one can, as in Robinson’s 1974 Nonstandard Analysis and in Kac and Cheung’s 2002 Quantum Calculus), mathematicians did use infinitesimals successfully and had a fair idea of what they were doing. Indeed, mathematicians used infinitesimals even though they were well aware of the risks of contradiction, and found ways to mitigate those

16

risks. Even after the Weierstrassian expulsion of infinitesimals, infinitesimals still survived in some ways in mathematics, as is witnessed, for example, in Luzin’s letter, where he recounts his discovery that his own teachers, who had been teaching him to reject infinitesimals, were actually still open to their existence (Demidov and Schenitzer 2000, 76-78).

Infinitesimals as geometrical lines

Our historical starting point is Cavalieri’s indivisibles (I follow the interpretation of Andersen 1985). We know that if we take pairs of lines sharing the same ratios and put them together (that is, all the left hand lines together and all the right hand lines together), then the ratio of the added lines will maintain the original ratios. Cavalieri applied this principle not to finite collections of lengths, but to the collection of all line segments formed by intersecting an area with a moving regula, that is, with all lines parallel to a given line (and, similarly, to collections of areas formed by intersecting a solid with parallel planes).

According to Andersen, Cavalieri worked hard to avoid identifying an area with the collection of these line segments or with their sum. Instead, he tried to present such collections as a distinct kind of magnitude (in the sense of Euclid’s Elements book V), and to show that from statements about such collections one could derive statements about the corresponding areas. This endeavor, however, was regarded as problematic by many contemporary critics and over-simplified or misunderstood by others.

What we have here is not a translation between systems of presentations, but rather stretching the scope of the Euclidean system of presentation (or its 17th century counterpart): applying a certain reasoning (that of magnitudes and their ratios) not to one of its old and trusted objects, but to a new kind of object (collections of lines). At first glance, this does not fall under the definition of abstraction that I suggest.5

However, the notion of “magnitude” is abstracted here in my sense: it can be translated into a new system of presentation – that of collections of lines. Note that as Cavalieri’s himself worried (Andersen 1985, 303-304), treating collection of lines as magnitudes may be problematic. Therefore, the new translation has to be used cautiously – it does not simply follow the established reasoning.

Moreover, Cavalieri does translate reasoning about line-collections to reasoning about areas, as his eventual goal is to find ratios between

5 Under other definitions, this could be considered an abstraction (turning the process of cutting an area with a moving regula into an object – the collection of these lines – would be considered as abstraction according to some of the definitions surveyed above).

17

areas (and between volumes). The emerging new practice with areas is thus more abstract in my sense, because claims about areas are now translated into the terms of the new system of presentation.

Infinitesimals as very small rectangles

Soon enough, Cavalieri’s refined notion of a collection of lines was simplified and rearticulated as identical with the area that it covers. To counter the contemporary mathematical-philosophical objection to the reduction of one kind of magnitude (area) to another (lines), some mathematicians (e.g. Wallis, see Stedall 2008, 67-69) suggested thinking about these lines as infinitely small rectangles. Such rectangles actually have width, and are therefore two- rather than one-dimensional, but their width is so small that their sum equals the total area, rather than just approximate it (see Boyer 1959, Ch. 4).

Here we see another process of translation: the same entity is sometimes subjected to the rules that concern areas, and sometimes to those that concern lines. The result is a somewhat paradoxical practice of translation, which is, in our terms, abstract. The same translation occurs in the context of Leibniz’ infinitesimal diagrams. Leibniz alternately applies to the same diagram or sign two systems of rules: in the one it is a finite Euclidean diagram (e.g. his famous triangle), and in the other its dimension is reduced (e.g. to a point of tangency). Indeed, Grosholz (2007, 215-221) analyzes several of Leibniz’s examples in such terms as examples of what she calls “productive ambiguity”. When these disparate practices are superposed under the same name, sign or diagram, we get a new abstraction: the infinitesimal.

Infinitesimals as numbers

But already in the first generation of infinitesimals, the translation was not only between different kinds of geometric entities, but also between geometry an arithmetic. Wallis provides the prime example. When analyzing, for example, the ratio of volumes of a cylinder and a cone (with the same base and same height), he noted that the ratios between the areas of sections of the cone and cylinder that are parallel to the base are proportional to the square of the distance between the section and the tip of the cone. If we had just n+1 sections at regular intervals starting at the top, the ratios of the areas of the sections of the cone and cylinder would be ( 0n )

2

:1, ( 1n )2

:1, ( 2n )2

:1,

… until eventually reaching ( nn )2

:1, that is, 1 :1 at the base. These ratios would sum up to

18

(( 0n )2

+( 1n )2

+( 2n )2

+…1): (1+1+1+…+1 ), or

(02+12+22+…n2 ) : (n2+n2+n2+…+n2 )=13− 16n .

Now, taking an infinite series of sections at intervals 1∞ (this is indeed Wallis’s notation) so as to exhaust the cone and cylinder, that is, allowing the number of sections, n, to be infinite, the ratio of volumes turns out to be exactly 1 :3 (Stedall 2008, 89-95).6

We see here an additional level of translation – from geometry to arithmetic. The new line/rectangle infinitesimal now correlates with a new zero/nonzero number (recall also Euler’s infinitesimals as zeros mentioned above). Translating the infinitesimal from geometry into arithmetic constitutes an additional step of abstraction.

Getting around infinitesimals

There were several early attempts to avoid infinitesimals without giving up their mathematical fruit. Some were more geometric, and others more algebraic. My point here is to suggest that instead of seeing these attempts as successful rejections of infinitesimals, we should consider them as further levels of abstraction, as they translated infinitesimal analysis to the terms of other systems of presentation. Even if their authors wanted to get rid of infinitesimals, the impact on the mathematical community was to add new practices to the newly emergent method of infinitesimals.

The first way around infinitesimals is the method of exhaustion, building on the proof of Euclid’s Elements 12.2. Instead of proving that the ratio between two areas is a :b by decomposing the area into infinitesimal slices, the method of exhaustion would use similar finite approximate decompositions to show that the ratio must be bigger than any given ration below a :b, and must also be smaller than any given ratio above a :b. The ratio thus would have to equal a :b. This was the most classical manner to “save” the results of infinitesimal analysis, but was also considered cumbersome and constraining in the contexts of discovery and understanding.

Another method was Newton’s first and last ratios. Instead of using infinitesimals as such, only their ratios were to be used, and instead of actual infinitesimals, one was supposed to relate vanishing finite quantities (that is, changing, rather than fixed magnitudes, which

6 I’m conflating here the language of Wallis’ De Sectionibus Conicis 1.1-1.3, where Wallis uses the infinity symbol arithmetically, and that of his Arithmetica Infinitorum 19-23, where his language is ambiguously shifting between a dynamic notion of infinitesimal and actual infinitesimal magnitude, which is not involved in explicit calculations. The point is, however, that infinitary reasoning was indeed translated between geometry and arithmetic.

19

grow unboundedly small, until they eventually vanish). If one thought about these dynamic ratios as magnitudes (as Newton did), and represented these dynamic magnitudes by line segments (as was the standard for all kind of magnitudes), then as the vanishing magnitudes vanished, these line segments hit an edge, or a limit. This edge marked the last ratio of the diminishing magnitudes – the ratio that they attain “not before they vanish, nor afterwards, but with which they vanish” (Ewald 1996, 58-60).

Note that this is not the modern notion of limit. Limit here has a rather geometric meaning: the end of a segment. This is precisely one of the aspects that Berkeley rejected in his famous critique: a line segment is indeed a magnitude with an edge, but thinking of ratios in this way was not within the scope of the classical conception of magnitudes (Ewald 1996, 60-92, especially par. 31).

There were also other interpretations of infinitesimals as variables, rather than constant “zero-ish” fixed magnitudes. One striking example is the “method of indeterminates”. Suppose one is looking for the slope of the tangent to the function y=x2 at a fixed point a. In order to find it, consider the chord connecting the values of the function at a and a+b. On the one hand, the slope of the chord equals the slope of the tangent plus a variable correction, which depends on b (let’s denote it T a+e (b)). On the other hand, it is easy to calculate that the slope of the chord is equal 2a+b. According to “method of indeterminates”, the equality between the former and latter is only possible if the constant part on the one hand (the slope of the tangent T a) equals the constant part on the other (2a). In order to set this idea apart from infinitesimalist ideas and the questions of arbitrarily small magnitudes, this is explained in analogy to equations of complex numbers: if a+bi=c+di, then the real parts, a and b, must be equal. Here the real part is analogous to the constant part, and the imaginary part is analogous to the variable part (Carnot 1832, Ch. 1, especially p. 35).

There are of course some other historical approaches, the most famous of which is Cauchy’s, who is considered as the father of the modern notion of limit – although I am much more convinced by Schubring’s interpretation (2005, Ch. 6), which sees Cauchy’s notion as a compromise between an infinitesimal and a variable. Another important functional approach is that of Lagrange, who, in his Théorie des Fonctions Analytiques, first followed Newton in rejecting infinitesimals in favor of their ratios (derivatives), and then defined derivatives as coefficients in power series expansions, rather than as limits.

The abstract concept of infinitesimal

20

None of the above methods was exactly translatable to the others and none was perfectly free of the danger of contradictions. Those maneuvers that were translatable to the method of exhaustion could be made consistent with Euclidean geometry, but infinitesimal calculus expanded into realms that went far beyond the Euclidean grasp, especially when power series expansions were involved, leading all the way to the analysis of divergent series.

The most convincing summary of this situation, indeed, one of my main inspirations for the current definition of abstraction, is the words of Carnot:

no method, should be used exclusively. Thus … amongst the methods spoken of above, it is necessary, with a view to the habitual employment of it, to choose that which effects its end in general by the shortest and easiest way, but not so as to reject any of the others, since they are fine speculations for the mind; and besides, there is not one amongst them, perhaps, which may not lead to some truth hitherto unknown, or procure in certain cases an unobserved result, or a solution more elegant than any other. (Carnot 1832, 121)

If the notion of infinitesimal managed to survive the Weierstrassian reformulation of analysis, remain an important mathematical intuition, and cross the threshold of modern formalization (without thereby becoming a standard mathematical feature), it is because of the complicated web of translations between different systems of presentation that provided it with a very broad safety net. This net kept infinitesimals around, without committing mathematicians to any of their particular presentations, or to some unattainable common denominator of all these different presentations. In this sense, the infinitesimal is one of the most abstract mathematical entities in existence – so abstract, that it has almost disappeared from view!

The Weierstrassian definition serves in many cases as an arbitrator: if some infinitesimal-inspired argument leads to a disagreement between mathematicians, the translation of the argument to Weierstrassian epsilon-delta terms can usually settle the argument. But this does not mean that the latter dominates or motivates the relevant mathematical reasoning either historically or today. In many “real life” cases, the argument would be presented in terms of a differential dx, which can be read in various different ways (cf. Thurston 1994, 163).

Third historical case study: variables

Variables are difficult to define, even in modern mathematics. From a formal-logical point of view, they are simply operands of quantifiers

21

and predicates. The set theoretical treatment considers variables as arbitrary members of a well defined set – even if in practice this set is often left ambiguous: a variable may start as real, then have certain values excluded from its range due to some arithmetical restrictions (such as division by zero), and finally end up as a complex number, vector or some other entity, due to some creative manipulation. But a variable can also be a formal variable, defined only in terms of the rules that govern its manipulation – but here too these rules may evolve as one goes along, rather than be set out once and for all (Wagner 2009; 2017, 100-111). Before all these late-mathematical inventions, a variable was also something that moves continuously through a span, like the time variable of classical mechanics. And in early school algebra, the variable is not even variable, but an unknown: a fixed value to be reconstructed from the conditions of a problem.

The review below will document some episodes in the history of algebraic terms. Again, I do not claim a historical-causal continuity between these episodes. They serve to show that different aspects of our practice with algebraic terms can and have been set apart. Abstraction, according to this narrative, is not the logical, set theoretic or formal codification of variable – the latter are just specific concrete systems of presentation. Abstraction lies in the fact that the variable signs are subject to practices that depend on different systems of presentation, without being reduced to any specific system.

Linear arithmetic and quadratic geometric proto-algebras

Our discussion of algebraic variables has two starting points. The first is Babylonian quadratic algebra. Typical problems in this setting relate the area, sides and diagonal of rectangles, and attempt to derive one from combinations of the others (e.g. given the area and sum of sides, find the sides). At first sight, the methods of solution appear to be arithmetic, but Høyrup (2002, ch. 2) convincingly showed that they recount geometric cut-and-paste manipulations.

The other starting point, which emerges in different forms in various places, is linear equations in one or more unknowns. Here the approach is arithmetical, and examples usually refer to real or imaginary commercial situations or to numbers without a specified referent. Examples include (among many others) the Chinese multi-variable “linear algebra” in the eighth chapter of the canonical Nine Chapters (Martzloff 1997, 249-258) and Aryabhata’s solution of linear equations in one unknown (Plofker 2009, 134). Specimens of this kind of linear algebras survive as distinct from other algebraic approaches at least as late as Fibonacci, who presents a variant of linear algebra under the title “regula recta” separately from “algebra et almuchabala” (Høyrup 2007, 105).

22

The Arabic synthesis

The next step of our story is al-Khwarizmi. Høyrup (1998, 77-78) and Oask and Alkhateeb (2005, 416-418) agree that his work joins together variants of the two kinds of proto-algebras mentioned above, but we do not have enough tracks to retrace the precise historical trajectory. What makes Al-Khwarizmi unique, however, is not the fact that the two traditions are presented together in a more or less unified manner – this was done earlier in various cultures. Al-Khwarizmi’s uniqueness stems from his translation of this combined algebra to the Euclidean language.7

Here is a step of abstraction. The two proto-algebras are not only inter-translated, but are also translated to a new system of presentation. Note that this translation is problematic: a number can be multiplied by itself an arbitrary number of times, but if this number is thought of as a line and multiplication as forming a figure by juxtaposing perpendicular lines, then one cannot go beyond three dimensions-multiplications. Moreover, Euclid’s lines never carry numerical lengths (although their lengths may have numerical ratios to each other), whereas algebraic terms can be numerically evaluated.

Omar Khayyam, who used geometry to solve cubic equations, stated that “the art of algebra … has as its goal to determine unknowns, either numerical or geometrical” (Wöpcke 1851, 1).8 Note that for Khayyam these unknowns are not detached from the two domains – they do not bracket out arithmetic or geometry or “reach up” to their common essence; but neither are they committed to either domain. Algebraic terms allow for a translation between arithmetic and geometry, while allowing these domains to remain distinct and not fully compatible. This process of partial translation is, in the terms of this paper, a step of abstraction.

Algebraic exponents as place-values

But algebraic terms did not only represent numbers, they also worked, in a sense, like numerals in a place-value system. One can think of each algebraic power (unknowns, squares, cubes,...) as a distinct place-value (units, tens, hundreds) marked by a numerical position (0,1,2…, in the case of modern exponents). Then one can translate algorithms for adding, multiplying, dividing and taking square roots from place-value number systems to combinations of different powers of unknowns (or, in contemporary terms: 7 Høyrup (2007, 103-104) notes correctly that al-Khwarizmi’s presentation is only superficially Euclidean, and has a strong Babylonian flavor, but some later Arabic authors were more faithful to the Euclidean standards.8 Khayyam’s is one of several attempts that preceded the well know Cartesian framework (see Wagner 2010a; Bos 2012 chs. 6-13).

23

polynomials). Indeed, Al-Samawal, one of the Arabic mathematicians who developed these methods, presented such “polynomials” as tabulated sequences of their coefficients, almost like a number is represented by place-valued numerals. Al-Samawal himself explicitly brought up the analogy to the Indian place-value system (Rashed and Ahmad 1972, Arabic page 45).

What is abstracted here is not the algebraic term, but its power. The number of times a term is multiplied and a position in a place-value system are brought into translation. This translation is rather imperfect – for example, when adding polynomials, one never “carries the one” as one does in number addition. Later on, when fractional exponents will appear, they will exceed the scope of the place-value translation. Nevertheless, this tentative, partial translation, means that algebraic powers are no longer simply the number of times a term is multiplied by itself; they become subject to practices carried over from a different system.

This translation also affects the content or value of the algebraic terms. Al-Samawal’s techniques allow him to derive such identities as (in anachronistic notation):

20x2+30 x6 x2+12

=103

+ 5x− 203 x2

−… (Rashed and Ahmad 1972, Arabic pages 50-55, French page 26).

The question that Al-Samawal does not raise is the meaning of this equality. If we are to think of it as a successive approximation of a numerical equality (as in the case of decimal number division with an infinite “tail”), then the procedure should be restricted only to some values of x, where the series converges. This limit of translation between exponents and place-value positions is left open for future mathematicians to observe and reflect on. But the very fact of this translation already makes the variable abstract in the sense of conflating various systems of presentation without being reducible to any single system, suggesting a path toward formal variables.

Economic and algebraic signs with underdetermined and unreal values

Another system of presentation that contributed to the abstraction of algebraic entities is the merchant arithmetic and bookkeeping of late medieval/Renaissance Italian abbacists (teachers of mathematics to children of merchants). It is often assumed that mathematical signs have stable and precise meaning. Abbacist mathematical practice is a great example for how false this assumption is (I follow here Wagner 2017, ch. 2; further references can be found there). There are many passages in the abbacist literature that testify to that. Here are two of my favorites:

24

The fiorino, to this money there is no fixed value, because sometimes it goes up by a few scudi and sometimes it goes down in price. … There is of course an imaginary value that’s called golden scudo, which is always stable, so that the fiorino is worth 20 golden scudi and the golden scudo 12 golden denari (Arrighi 1974, 34).

If 4 were the half of 12, what would be the 1/3 of 15? (della Francesca 1970, 48)

The first quotation testifies to the volatility of economic values, and to the existence of “imaginary” denominations, which do not correspond to any actual coin, but are part of economic accounting. The second appears more mysterious, but the mystery can be resolved by looking at the context: rule of three questions of the form “if 4 of this were worth half of 12 of that…”. My point is that these formulations show that under some circumstances, numbers needn’t represent their face values, and a “4” could actually mean “6”.

This economic practice was inter-translated with an algebraic system that had its origin in the work of North African and Iberian scholars and practitioners. This translation was eventually expressed in new practices with algebraic entities, leading to the introduction of algebraic terms that represented no specific values or values that did not exist. Two interesting examples present themselves in the work of Rafael Bombelli (both are taken from Wagner 2010b), a 16th

century engineer who wrote an important work on algebra following in the footsteps of Tartaglia and Cardano.

First, Bombelli uses algebraic terms to represent not only unknowns, but also arbitrary values. This occurs, for instance, when Bombelli expands the binomial ( x+2 )5 (in a somewhat different notation, but using the specific symbol that Bombelli used to represent unknowns; Bombelli 1966, 61). His goal was to figure out what we today call “binomial coefficients”. Here, there are no conditions that determine x, and no unknown value is sought.

Second, an algebraic term is used and operated on even though it represents a value that does not exist. Specifically, Bombelli writes (using what we would call today a change of variable): “having found the value of the [solution of x3+84=36 x] (if one could), if we’d add 3, the sum would be the [solution of x3+165=9 x2+9 x]” (Bombelli 1966, 262). The strange thing is that one can’t solve these equations (with a positive number, which is what a solution means in this context). Bombelli felt comfortable working with algebraic terms that represent hypothetical values that do not in fact exist. Given the attitude expressed in these examples – working with algebraic terms with no specific value or with values that do not exist – it is much less surprising that Bombelli dared to solve equations by means of roots

25

of negative numbers, even though he considered these roots “sophistic”.

The close connections between abbacist mathematical practice and the Renaissance algebra (to which Bombelli is a witness) imply that the indeterminacy and imaginary status of some abbacist economic values and economic symbols is related to the new algebraic practices, where algebraic signs carry indeterminate or even non-existent values (a more substantial elaboration of the argument is available in Wagner 2010b). This translation of practices is yet another step in the abstraction of algebraic terms. The abstraction here is not some formalization of algebraic practices – Bombelli’s conception of algebra seems to be less formal than, for instance, that of Al-Samawal. The abstraction here arises from the translation between algebraic and an economic systems of presentation.

Quantities and algebra

The last step in this (very partial) story draws on the notion of “quantity” reconstructed in Ferraro (2004). As the system of numbers grew and evolved in the 17th and 18th centuries, building, among other things, on the processes described in the previous paragraph, the notion of “general quantity” expanded far beyond the classically measurable geometric and numerical magnitudes to include “fictitious numbers”: irrationals, negative numbers, imaginary numbers and infinitesimals.

According to Ferraro, “general quantity” was not a formal-axiomatic notion, but a notion that carried over practices from measurable magnitudes to new kinds of quantity. The translation from the established system of numbers (integers and rational fractions) to “fictitious numbers” was based on circumstantial grounding of the latter in the former by means of analogous application of contextually appropriate manipulations. The rules governing irrational square roots, for example, were grounded in those that apply to rational square roots and to rational approximations of irrational roots; the rules that govern negative numbers depended on those that were applied to debts and to subtracted numbers (e.g. the fact that (8−2 )× (8−2 )=8×8−2×8−8×2+2×2 motivated the rule for multiplying negative numbers); and the rules governing infinitesimals were carried over from the treatment of gradually vanishing quantities, zeros, and relations between geometrical objects of different dimensions (such as surfaces and lines). These rules were applied in a context sensitive manner, not necessarily universal and consistent, rather than as universal formal rules – an intermittent, underdetermined translation.

26

This, in a nutshell, is how Ferraro explains the “wild” manipulations typical of Euler’s work regarding power series and divergent series. More specifically, regarding variables, Ferraro writes (2004, 36):

the calculus referred to indeterminate quantities [i.e. variables], subject to possible variations, whether increases or decreases, rather than to specific determinations of quantities or determinate quantities. A quantity [i.e. variable] could assume different values or determinations, although a quantity was not reduced to the enumeration of these values …. Indeed a quantity possessed its own properties, which might be false for certain of its determinations. Thus, given any property P of quantity, there might exist exceptional values at which the property fails and a theorem involving certain quantities x, y, …, [may be] valid and rigorous [only] as long as the variables x, y,…, remained indeterminate.

Variables were thus not reducible to values or geometric entities. The practices of dealing with variables were translated from systems of presentation involving the above, but were not always consistent with those the interpretation of the variable as a numerical or geometrical magnitude. This does not establish a theory of formal variables in the modern sense, only an abstraction in the sense of this paper.

By the end of the 18th century, another system of presentation emerges: the system that we would call today “operator calculus”, where operations such as derivation, summation and shift of power series were subject to practices carried over from algebra. Arbogast, a major proponent of this development, writes “This method consists of detaching from the function … the signs of operations that affect this function, and treating this expression … as if the signs of operations that are involved in it were quantities” (Arbogast 1800, viii-ix). With this translation, both operations and variables gain a new level of abstraction – the former gains a quantitative shade, and the latter an operatorial aftertaste.

And so on and so forth…

Much more can be added to this story: A translation between mathematical presentations and grammar gave rise to logical variables, and the tension between quantitative and operational interpretations led to the (never perfectly articulated) split between set-theoretic and formal variables. And then, applying the formal approach to set theoretic variables one finds oneself confronted with category theory… But what makes variables one of mathematics most abstract entities is not some strict highbrow formalization, but the fact that variables inter-translate so many different systems of presentation, many of which are still alive and kicking in mathematical practice.

27

Concluding remarks concerning rigor, consensus and abstraction

“But how can such a messy theory of abstraction explain something so rigorous and consensual as mathematics”? To answer this question, we should understand where rigor and consensus actually appear in mathematics.

First, I do believe that modern mathematics is one of the most consensual forms of human knowledge. But its exceptional consensus is not about what mathematical statements mean, how important they are, how useful or beautiful they are, and not even about whether they hold or not. The exceptional consensus concerns only whether a given argument proves a given statement. And even there we must qualify, because different mathematicians and different circumstances call for different kinds of proof. It would be more exact to say that if two mathematicians disagree on whether a given argument proves a given statement, then given enough patience and time, they are more likely to reach agreement than in almost any other domain of knowledge.

Their agreement may be of the form “this argument and statement can be reformulated so as to satisfy both of us that the former proves the latter” (but they may still disagree whether the reformulated argument is more-or-less the same as the original argument or should be considered an altogether different argument!). The agreement may also be of the form “this argument has a problem that we can’t fix, and which prevents it from being a proof of the statement”. I do not claim that one of these kinds of agreement is always reached – only that it is more likely to be reached in mathematics than in other domains, provided enough patience and time are invested.

Two (somewhat related) prerequisites have a role in this kind of consensus. First, one needs to separate the question of endorsing certain rules from the question of whether a certain argument obeys these rules (this assumes, obviously, that people agree explicitly or implicitly on what the rules are). Indeed, many mathematicians disagree on which frameworks, axiom systems and mathematical structures are relevant, “real” in some sense, or are worth studying; but they would still be highly likely to agree (in the above sense) whether a given argument proves a given statement within a given framework. The limit of this tolerance is reached with frameworks that are considered as not well articulated or suspect of inconsistencies. With regard to such doubts mathematicians are indeed somewhat more likely to be at long-term odds, and the suspect frameworks may be marginalized by the mainstream community.

28

Second, in order for a consensus to emerge, the many informal, intuitive, rule-of-thumb like or analogy based methods that mathematicians actually use should be translatable to specific, highly coded systems or presentations, which serve as arbitrators in disagreements. This does not mean that consensus depends on actually translating proofs to formal languages in the Hilbertian sense. It means that building on the array of informal, intuitive, rule-of-thumb like or analogy based methods, mathematicians narrow down disagreements, isolate problematic portions of proofs, and if still necessary, translate those portions to one of the “more rigorous” systems of presentation that lie between “everyday mathematical language” and the almost unattainable ideal of a Hilbertian formal system. Such translations are clearly not the only way, perhaps not even a very common way of settling disputes, but they have a power of arbitration when other, less formal forms of reasoning, fail to settle disputes.9

Now, the actual occurrence of these two prerequisites is very rare in the history of mathematics. For most of the history of mathematics, the separation between endorsing rules and just following them was not clear, so the first prerequisite did not hold .Think, for example, of Heaviside: did the objections to his methods target his system of rules or his failure to follow them? It is very hard to tell, because, like many of his predecessors and followers, Heaviside never tried to introduce an explicit system of rules, or, for that matter, to distinguish the mathematical part of his argument from the physical part (Hunt 1991; Lützen 1979).

Second, usually there was no “arbitrator system of presentation” that was successful enough in producing consensus and could absorb a large enough portion of mathematics. One historical exception obviously comes to mind: the Euclidean core of classical geometry, which includes Euclidean proportion and number theory (I follow here the depiction of Netz 1999, 56-57, 309-310). There, the combination of rigorous formulaic language and diagrams allowed a very high degree of consensus. Debates over the axioms and methods of geometry were indeed prevalent in antiquity, but did not prevent agreement on what holds within classical geometry. However, as

9 The current state of the alleged proof of the ABC conjecture by Mochizuki is an interesting example. As far as I understand, the people who doubt the proof (specifically, Peter Scholze and Jakob Stix) have narrowed their objection down to a specific lemma, and provided an analogy between Mochizuki’s argument and a fallacious inference. The people who endorse the proof claim that the analogy is wrong, but have not yet provided a reformulation of the proof that shows why this analogy fails. This is either a case of not enough time and patience investment, or perhaps one of those relatively rare cases where even despite the investment of time and patience, consensus is simply beyond reach. Most mathematicians, it seems, tend to fault the defenders of the proof, and would suspend their belief at least until those defenders reformulate the proof to refute the critics’ specific objection.

29

new, more complicated mathematical objects were introduced, even in antiquity, dissensus readily emerged (Netz 2004, 60-63, 180-181).

In many other, more practically oriented mathematical cultures, the question of justification was not the central question, and therefore the above debate (agreement concerning proofs) was not a major issue, and consensus about proposed solutions to given problems was of a more practical sort. In yet other cultures, like that of early modern European calculus, justification was a major issue, but the Greek consensus could not be mimicked – at least not until Hilbert’s approach became dominant.

If we think of abstraction as a general process of mathematical object formation and of consensus as a special contingency that depends on specific mathematical practices, then abstraction obviously does not have to account for consensus. Consensus should be accounted for in terms of culture-specific epistemological ideals, specific systems of presentation, and specific practices and institutions – not in terms of general mathematical phenomena such as abstraction, which were there even when mathematics was much less consensual.

To emphasize how far my definition of abstraction shies away from mathematical rigor, one needs to observe that the main underlying notion, that of systems of presentation, is highly problematic. As I insinuated in my Wittgensteinian definition of this notion, systems of presentation are not well defined or properly individuated. In fact, more than anything else, they are actor-categories, and their individuation is relative to practical, educational and institutional articulations.

Since the notion of system of representation is not well individuated, an analysis of abstraction along the lines of this paper is not foundational. It simply has no proper starting point, on which one can build up a hierarchical Lego-castle. My analyses of abstraction always start “from the middle”, and always depend on a contingent articulation of systems of presentation. Anything that is presented as a concrete system of presentation may be the result of a genealogy of abstractions.

Since this definition of abstraction does not assume anything very coherent as that between which one translates, it can only describe relative, qualitative degrees of concreteness and abstraction. Indeed, looking at some systems of presentation, one can argue whether they form a single system (the concrete pole), or an inter-translated collection of systems (the abstract pole). This ambiguity, I believe, need not be resolved. Instead of deciding whether something is strictly concrete or abstract, one should consider one mathematical system more abstract than another if it adds to the former more

30

systems of presentation and translations (this is of course only a partial ordering).

Hegel and several philosophers after him (e.g. Deleuze) interpreted infinitesimals as entities that exist in relation to each other without having any determinate existence outside this relation (Somers-Hall 2010). In other words, one can determine the value of dydx , but not dy or dx themselves. In this interpretation, infinitesimals are a kind of relation without relata. I would say that the present theory of abstraction portrays mathematics as a “translation without translata”. Mathematical practices of translations take place between concrete systems of presentation, but these are not rigorously articulated formalisms from which invariants are to be extracted. Objects emerging from mathematical translations do not exist except in translation, that is, abstractly.

ReferencesAinsworth, Shaaron. 1999. “The Functions of Multiple Representations.”

Computers and Education 33 (2): 131–52. https://doi.org/10.1016/S0360-1315(99)00029-9.

Andersen, Kirsti. 1985. “Cavalieri’s Method of Indivisibles.” Archive for History of Exact Sciences 31 (4): 291–367. https://doi.org/10.1007/BF00348519.

Arbogast, L. F. A. 1800. Du calcul des dérivations. Levrault. Strassbourg.Arcavi, Abraham. 1994. “Symbol Sense: Informal Sense-Making in Formal

Mathematics.” For the Learning of Mathematics 14 (3): 24–35.Arrighi, Gino, ed. 1974. Tractato d’abbacho Dal codice acq. e doni 154

(sec. XV) della Biblioteca Medicea Laurenziana di Firenze. Pisa: Domus Galilaeana.

Avigad, Jeremy. 2018. “The Mechanization of Mathematics.” Notices of the AMS 65 (6): 681–90. http://dx.doi.org/10.1090/noti1688.

Avigad, Jeremy. 2019. “Reliability of Mathematical Inference.” Preprint. http://philsci-archive.pitt.edu/16283/.

Azzouni, Jody. 2008. “The Compulsion to Believe: Logical Inference and Normativity.” ProtoSociology 25: 69–88.

Bäck, Allan. 2014. Aristotle’s Theory of Abstraction. Cham: Springer.Bos, Henk J. M. 2012. Redefining Geometrical Exactness: Descartes’

Transformation of the Early Modern Concept of Construction. New York: Springer.

Boyer, Carl B. 1959. The History of the Calculus and Its Conceptual Development. New York, NY: Dover. http://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment.

Closs, Michael P. 1986. Native American Mathematics. Austin: University of Texas Press.

De Toffoli, Silvia, and Valeria Giardino. 2014. “Forms and Roles of Diagrams in Knot Theory.” Erkenntnis 79 (4): 829–42. https://doi.org/10.1007/s10670-013-9568-7.

31

Della Francesca, Piero. 1970. Trattato d’abaco. Dal codice ashburnhamiano 280 (359*-291*) della Biblioteca medicea laurenziana di Firenze. Edited by Gino Arrighi. Pisa: Domus Galilaeana.

Demidov, S. S., and A. Shenitzer. 2000. “Two Letters by N. N. Luzin to M. Ya. Vygodskii.” The American Mathematical Monthly 107 (1): 64–82. https://doi.org/10.2307/2589382.

Dreyfus, Tommy. 2014. “Abstraction in Mathematics Education.” In Encyclopedia of Mathematics Education, edited by Stephen Lerman, 5–8. Dordrecht: Springer. https://doi.org/10.1007/978-94-007-4978-8_2.

Dreyfus, Tommy, Rina Hershkowitz, and Baruch Schwarz. 2015. “The Nested Epistemic Actions Model for Abstraction in Context: Theory as Methodological Tool and Methodological Tool as Theory.” In Approaches to Qualitative Research in Mathematics Education: Examples of Methodology and Methods, edited by Angelika Bikner-Ahsbahs, Christine Knipping, and Norma Presmeg, 185–217. Advances in Mathematics Education. Dordrecht: Springer. https://doi.org/10.1007/978-94-017-9181-6_8.

Epple, Moritz. 2003. “The End of the Science of Quantity: Foundations of Analysis 1860-1910.” In A History of Analysis, edited by Hans Niels Jahnke, 105–36. Providence, RI: American Mathematical Society.

Ewald, William, ed. 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford: Clarendon press.

Ferraro, Giovanni. 2004. “Differentials and Differential Coefficients in the Eulerian Foundations of the Calculus.” Historia Mathematica 31 (1): 34–61. https://doi.org/10.1016/S0315-0860(03)00030-2.

Ferraro, Giovanni. 2012. “Euler, Infinitesimals and Limits.” halshs-00657694v2.

Friedman, Michael. 2018. “Mathematical Formalization and Diagrammatic Reasoning: The Case Study of the Braid Group between 1925 and 1950.” British Journal for the History of Mathematics 0 (0): 1–17. https://doi.org/10.1080/17498430.2018.1533298.

Giunchiglia, Fausto, and Toby Walsh. 1992. “A Theory of Abstraction.” Artificial Intelligence 57 (2): 323–89. https://doi.org/10.1016/0004-3702(92)90021-O.

Gray, Eddie M., and David O. Tall. 1994. “Duality, Ambiguity, and Flexibility: A ‘Proceptual’ View of Simple Arithmetic.” Journal for Research in Mathematics Education 25 (2): 116–40. https://doi.org/10.2307/749505.

Harel, Guershon, and David Tall. 1991. “The General, the Abstract, and the Generic in Advanced Mathematics.” For the Learning of Mathematics 11 (1): 38–42.

Hayashi, Takao. 1995. The Bakhshālī Manuscript: An Ancient Indian Mathematical Treatise. Groningen: Egbert Forsten.

Høyrup, Jens. 1990. “Sub-Scientific Mathematics: Observations on a Pre-Modern Phenomenon.” History of Science 28 (1): 63–87. https://doi.org/10.1177/007327539002800102.

32

Høyrup, Jens. 2002. Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin. New York: Springer.

Høyrup, Jens. 2007. Jacopo Da Firenze’s Tractatus Algorismi and Early Italian Abbacus Culture. Basel: Birkhäuser. http://public.eblib.com/EBLPublic/PublicView.do?ptiID=337900.

Hunt, Bruce J. 1991. “Rigorous Discipline: Oliver Heaviside versus the Mathematicians.” In The Literary Structure of Scientific Argument: Historical Studies, edited by Peter Dear, 72–95. Philadelphia: University of Pennsylvania Press.

Imhausen, Annette. 2016. Mathematics in Ancient Egypt. Princeton: Princeton University Press.

Kac, Victor, and Pokman Cheung. 2002. Quantum Calculus. New York: Springer-Verlag. //www.springer.com/de/book/9780387953410.

Kaput, J. J. 1989. “Linking Representations in the Symbol Systems of Algebra.” In Research Issues in the Learning and Teaching of Algebra, edited by S. Wagner and C. Kieran, 167–94. Hillsdale: Lawrence Erlbaum. https://asu.pure.elsevier.com/en/publications/experience-problem-solving-and-learning-mathematics-consideration.

Katz, Victor J., ed. 2007. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton: Princeton University Press.

Katz, Victor J, Menso Folkerts, Barnabas Hughes, Roi Wagner, and J. L Berggren. 2016. Sourcebook in the Mathematics of Medieval Europe and North Africa. Princeton: Princeton University Press.

Lakoff, George, and Rafael E. Núñez. 2000. Where Mathematics Comes from: How the Embodied Mind Brings Mathematics Into Being. New York: Basic Books.

Lumpkin, Beatrice. 2002. “The Zero Concept in Ancient Egypt.” In Proceedings of the International Seminar and Colloquium on 1500 Years of Aryabhateeyam, 161–67. Kochi: kerala Sastra Sahitya Parishad.

Lützen, Jesper. 1979. “Heaviside’s Operational Calculus and the Attempts to Rigorise It.” Archive for History of Exact Sciences 21 (2): 161–200. https://doi.org/10.1007/BF00330405.

Martzloff, Jean-Claude. 1997. A History of Chinese Mathematics. Berlin: Springer.

Netz, Reviel. 1999. The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History. Cambridge: Cambridge University Press.

Netz, Reviel. 2004. The Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations. Cambridge: Cambridge University Press.

Noss, Richard, and Celia Hoyles. 1996. Windows on Mathematical Meanings: Learning Cultures and Computers. Mathematics Education Library. Dodrecht: Kluwer. //www.springer.com/de/book/9780792340737.

Oaks, Jeffrey A., and Haitham M. Alkhateeb. 2005. “Māl, Enunciations, and the Prehistory of Arabic Algebra.” Historia Mathematica 32 (4): 400–425. https://doi.org/10.1016/j.hm.2005.03.002.

33

Plofker, Kim. 2009. Mathematics in India. Princeton: Princeton University Press.

Plofker, Kim, Agathe Keller, Takao Hayashi, Clemency Montelle, and Dominik Wujastyk. 2017. “The Bakhshālī Manuscript: A Response to the Bodleian Library’s Radiocarbon Dating.” History of Science in South Asia 5 (1): 134–50. https://doi.org/10.18732/H2XT07.

Radford, Luis, Caroline Bardini, and Cristina Sabena. 2007. “Perceiving the General: The Multisemiotic Dimension of Students’ Algebraic Activity.” Journal for Research in Mathematics Education 38 (5): 507–30. https://doi.org/10.2307/30034963.

Radford, Luis, and Luis Puig. 2007. “Syntax and Meaning as Sensuous, Visual, Historical Forms of Algebraic Thinking.” Educational Studies in Mathematics 66 (2): 145–64. https://doi.org/10.1007/s10649-006-9024-6.

Rashed, Roshdi, and Salah Ahmad. 1972. Al-Bahir en Algèbre d’As-Samaw’al. Université de Damas.

Robinson, Abraham. 1974. Non-Standard Analysis. Amsterdam: North Holland.

Robson, Eleanor. 2008. Mathematics in Ancient Iraq: A Social History. Princeton; Oxford: Princeton University Press.

Schoenfeld, Alan H. n.d. “On Having and Using Geometric Knowledge.” In Conceptual and Procedural Knowledge: The Case of Mathematics, edited by James Hiebert, 225–64. New York: Routledge. Accessed December 20, 2018. https://www.taylorfrancis.com/books/e/9781136559761.

Schubring, Gert. 2005. Conflicts between Generalization, Rigor and Intuition. New York: Springer.

Sfard, Anna. 1991. “On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin.” Educational Studies in Mathematics 22 (1): 1–36. https://doi.org/10.1007/BF00302715.

Sfard, Anna. 2008. Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing. New York: Cambridge University Press.

Somers-Hall, Henry. 2010. “Hegel and Deleuze on the Metaphysical Interpretation of the Calculus.” Continental Philosophy Review 42 (4): 555–72. https://doi.org/10.1007/s11007-009-9120-2.

Stedall, Jacqueline A. 2008. Mathematics Emerging: A Sourcebook 1540-1900. Oxford: Oxford University Press.

Thompson, Patrick W. 1985. “Experience, Problem Solving, and Learning Mathematics: Considerations in Developing Mathematics Curricula.” In Learning and Teaching Mathematical Problem Solving: Multiple Research Perspectives, edited by Edward A. Silver, 189–243. Hillsdale: Lawrence Erlbaum. https://asu.pure.elsevier.com/en/publications/experience-problem-solving-and-learning-mathematics-consideration.

Thurston, William P. 1994. “On Proof and Progress in Mathematics.” Bulletin of the American Mathematical Society 30 (2): 161–77. https://doi.org/10.1090/S0273-0979-1994-00502-6.

34

Wagner, Roi. 2017. Making and Breaking Mathematical Sense: Histories and Philosophies of Mathematical Practice. Princeton: Princeton University Press.

Wagner, Roy. 2009. “Mathematical Variables as Indigenous Concepts.” International Studies in The Philosophy of Science 23 (1): 1–18. https://doi.org/10.1080/02698590902843351.

Wagner, Roy. 2010a. “The Geometry of the Unknown: Bombelli’s Algebra Linearia.” In Philosophical Aspects of Symbolic Reasoning in Early Modern Mathematics, edited by Albrecht Heeffer and Maarten Van Dyck, 229–69. London: College Publications.

Wagner, Roy. 2010b. “The Natures of Numbers in and around Bombelli’s L’algebra.” Archive for History of Exact Sciences 64 (5): 485–523. https://doi.org/10.1007/s00407-010-0062-1.

Wagner, Roy. 2013. “A Historically and Philosophically Informed Approach to Mathematical Metaphors.” International Studies in the Philosophy of Science 27 (2): 109–35. https://doi.org/10.1080/02698595.2013.813257.

Wilensky, Uri. 1991. “Abstract Meditations on the Concrete and Concrete Implications for Mathematics Education.” In Constructionism: Research Reports and Essays, 1985-1990, edited by Idit Harel and Seymour Papert, 193–203. Norwood: Ablex Publishing.

Wöpcke, Franz. 1851. L’algèbre d’Omar Alkhayyâmî. Paris: B. Duprat.

35