The Significance of Conway’s Construction of Number for Structuralism in the Philosophy of Mathematics

The Significance of Conway’s Construction of Number for Structuralism in the Philosophy of Mathematics

Allen Porter

Abstract

In this paper I argue for the significance of John Horton Conway’s construction of arithmetic (in his On Numbers and Games) for structuralism in the philosophy of mathematics. First I explicate the relevant details of structuralism, then I present the basic features of Conway’s method for constructing the “surreal number” system, i.e. for unfolding the entirety of numeric structure as such. I argue that the significance of the latter is twofold: first, it provides an easy way for the structuralist to rebut one of the most common criticisms of structuralism, namely the objection from the indeterminacy of inter-numeric identities; second, it provides a way to synthesize the two main variants of structuralism, namely in re and ante rem structuralism, because it can serve as a uniquely structuralist background ontology for in re structuralism. This allows the in re structuralist to have a fully or thoroughly structuralist theory, like the ante rem structuralist, without having to reify the various specific structures that the ante rem realist does.

Keywords: John Horton Conway ⋅ Philosophy of Mathematics ⋅ Structuralism ⋅ Arithmetic ⋅ Surreal Numbers ⋅ Stewart Shapiro

Allen Porter Rice University, Houston, TX, USA email: [email protected]

mailto:[email protected]

Allen Porter | Spring 2015 !2

0. Introduction

In this paper, I argue for the significance of John Horton Conway’s construction of arithmetic, in

his On Numbers and Games (henceforth ONAG), for the philosophy of mathematics—and in particular, for

that philosophy of mathematics known as structuralism.

First I explicate the relevant details of structuralism, then I present the relevant basic features of

Conway’s construction, and then I argue for the significance of Conway’s construction for structuralism,

which I argue is at least twofold. First, it gives the structuralist an easy way of directly refuting one of the

most common criticisms of structuralism (the problem of inter-numeric identities); second, it provides a

way to compellingly synthesize two of the three main approaches to structuralism, namely the ante rem and

in re approaches.

1. Structuralism

Structuralism is a philosophy of mathematics primarily associated with figures like Paul

Benacerraf, Geoffrey Hellman, Michael Resnik, and Stewart Shaprio (TM 257). As stated by Shapiro, its

“slogan” is that “mathematics is the science of structure” (PMSO 5).

The two most characteristic features of structuralism are the relationality of structure and the

reduction of mathematical objects to structural positions. Unlike, for example, the traditional platonist—

who “holds that the subject-matter of a given branch of mathematics, like arithmetic or real analysis, is a

collection of of objects that have some sort of ontological independence” (TM 257)—the structuralist

“rejects any sort of ontological independence among the natural numbers”, holding instead that the very

“essence of a natural number is its relations to other natural numbers” (TM 258).


For example , a structuralist might say that there is nothing more to the number 2 except its being 1

the second position, or the position between 1 and 3, in the natural series—and thus all of the properties

of 2 derive from its position and the relations that position has to other positions in the structure, such as

being the successor of the position of 1 and the predecessor of that of 3 (TM 258). A good historical

example of the contrast here is the difference between the approaches of Frege and Dedekind to the

construction of arithmetic and the definition of number.

Frege’s view that numbers are ontologically independent objects is explicit, and it is a

commitment that fundamentally orients his approach in The Foundations of Arithmetic. It is because he

believes that numbers must be “self-subsistent objects that can be recognized as the same again” (FA 68) 2

that he takes the “recognition-statement, which in the case of numbers is called an identity” (FA 116)—

i.e., identity-statements of the form “x is the same as y” or “x = y”— as the privileged context in which the

sense of the concept of number may be fixed. This, in turn, is the reason that Frege’s conception of

number in general turns out to be that of cardinal number in particular (cf. FPM 48). Frege’s fundamental

idea was to define number in terms of sameness of number or “equinumerosity” , and then to define 3

equinumerosity in terms of one-one correspondence, which—despite the name—Frege was able to define

through exclusively logical means (for Frege was of course a logicist, and his ultimate aim was to reduce

mathematics to logic).

Dedekind is often regarded as a “direct forerunner” (PMSO 14) or even “the founder” (Reck 369)

of structuralism in mathematics. He disagreed with Frege over the question of “whether is is possible, not

merely to characterize the abstract structure of the system of natural numbers, but to identify the natural

This example is intentionally stated simplistically for purposes of illustration. The natural number 2 of course has 1

more inter-numeric relations than just those of being the successor of 1 and the predecessor of 3 (for example, it is the only natural number besides 1 that is a divisor of every even natural number). Nevertheless, the structuralist view would be that it is no more than all these relations, most or perhaps all of which could be derived from its position in the sense of succession or betweenness.

To be clear: this is not merely an uncritical assumption on Frege’s part; it is a result of his survey of other 2

conceptions of number and the inadequacies he finds in them.

“Equinumerous” is a frequent translation of Frege’s Gleichzahlig, a word he coined (FA 79). 3


numbers solely in terms of that structure. Unlike Frege’s, Dedekind’s natural numbers have no properties

other than their positions in the ordering determined by their generating operation, and those derivable

from them” (FPM 51). One result of Dedekind’s approach was that his conception of number is not

limited to cardinal numbers, but includes for example ordinals as well (PMSO 172). As we will see in the

next section, Dedekind’s method, which is indeed appropriately termed “structuralist”, is also a “direct

forerunner” of the method used by Conway.

There are three primary versions of structuralism, often designated as ante rem structuralism, in re

structuralism, and modal structuralism. In this paper I will only be concerned with the first two, and only

with some basic features of them. In fact, all that I need can be explained simply in terms of what Shapiro

calls the “places-are-objects” and “places-are-offices” perspectives on (our way of talking about) structural

positions: “Sometimes the places of a structure are treated in the context of one or more systems that

exemplify the structure” (TM 268)—for example, the office of the US President is a position in a

governmental structure, but we might say “the President was a senator eight years ago”, using the office to

refer to its occupant in the current “system” exemplifying that structure (here, administration). This is the

places-are-offices perspective. But we might also say “the President is the commander-in-chief of the US

military”, referring to a relation to another office in the structure that is entertained by this office itself, and

is thus exemplified by any occupant of the office in virtue of occupation; this is the places-are-objects

perspective.

The ante rem structuralist is a realist about structures, and so takes reference to structural positions

(the second position in the number structure) at face value. The in re eliminative structuralist, on the other

hand, “paraphrases places-are-objects statements in terms of the places-are-offices perspective”; this “plan

depends on being able to generalize over all systems exemplifying the structure in question. […] When

paraphrased like this, seemingly bold ontological claims lose their teeth. For example, the sentence ‘3

exists’ comes to ‘every natural number system has an object in its 3-place’” (TM 270-1).


2. Conway

In his introduction (in Chapter 0, “All Numbers Great and Small”) to ONAG, John Conway

compares the methods used by Dedekind and Cantor to construct arithmetic, and he says that he will

“show that these two methods are part of a simpler and more general one by which we can construct the

very large Class No of ‘Surreal Numbers’” (ONAG 4).

It is “very large” indeed: as Philip Ehrlich has argued in a number of works, “whereas the real

number system should be regarded as constituting an arithmetic continuum modulo the Archimedean

axiom , No may be regarded as a sort of absolute arithmetic continuum (modulo NBG)” (Ehrlich A 236). It is, 4

modulo NBG, the largest possible ordered field, and every ordered field can be embedded as a subfield

within it; as Ehrlich says, “No bears much the same relation to ordered fields that the system ℝ of real

numbers bears to Archimedean ordered fields” (Ehrlich B 6).

The adjective “surreal” in its mathematical sense was coined in 1974 by Donald Knuth to

describe Conway’s system; prior to that Conway had merely referred to “numbers”, and in ONAG as well

he “usually omit[s] the adjective ‘surreal’” (ONAG 4). In what follows, I will also omit the adjective

“surreal”, and simply refer to the numbers of Conway’s construction as “numbers” (in contexts when

contrast to other conceptions or constructions of number is needed, I will refer to them as “Conway

numbers”). Likewise, I will refer to the structure of his system as simply being the “structure of number”

or as “numeric structure” as such, without adjectival qualification.

However, I do so for a principled reason, which is that I take Conway’s system to constitute a

unifying framework for the concept of number as such. Indeed, perhaps the foremost merit of Conway’s

approach is that, on it, “there is just one kind of number” (ONAG 27)—and, again, this kind of number

The “Archimedean axiom asserts: for all a, b ∈ A, if 0 < a < b, there is a positive integer n such that na > b. […] An 4

ordered field is Archimedean if and only if it contains neither infinite nor nonzero infinitesimal elements” (Ehrlich A 272).


includes every other kind of number (from the more familiar kinds like the rational, real, and complex

numbers, to more exotic kinds such as the superreals and the hyperreals).

The method through which Conway generates or unfolds the entire structure of number is truly

elegant in its simplicity. It is essentially the method by which Dedekind “(and before him the author—

thought to be Eudoxus—of the fifth book of Euclid) constructed the real numbers from the rationals”,

which was “to divide the rationals into two sets L and R in such a way that no number of L was greater

than any number of R, and use this ‘section’ to define a new number {L | R} in the case that neither L

nor R had an external point” (ONAG 3).

As Conway notes, this method “produces a logically sound collection of real numbers”, but “has

been criticized on several counts”—of which “[p]erhaps the most important is that the rationals are

supposed to have been already constructed in some other way, and yet are ‘reconstructed’ as certain real

numbers” (ONAG 3-4). Conway’s approach, of course, remedies this defect, in that it constructs not just

the rationals but all numbers by the same method. Conway’s articulation of his own method of

construction is given as follows (ONAG 4):

If L, R are any two sets of numbers, and no member of L is ⩾ any member of R, then there is a number {L |R}. All numbers are constructed in this way.

This definition of the rule for construction allows Conway to preempt the aforementioned

criticism of Dedekind’s account by making essentially the same move that von Neumann made as a

response to the same sorts of criticisms when they were leveled at Cantor’s approach, which latter yielded

the ordinal numbers (including the transfinite ordinals) by identifying them with their order-types.

Incidentally, the definition of order-type proceeds along Fregean lines, in that it is defined in terms

of sameness of order-type: two well-ordered sets have the same order-type if they are “order isomorphic”,

i.e., if they have the same cardinality and order. In other words, whereas two sets have the same

cardinality if there is a function that maps a one-one correspondence between their members

(disregarding their order in the set), they have the same order-type if there is a function that maps a one-

one correspondence between their members and preserves the order of those members. So, for example,


the set of positive integers and the set of positive even integers have the same order-type, because they can

be placed in an order-preserving one-one correspondence (2 in the latter set would correspond to 1 in the

former, each occupying the first position in their respective ordered series—and 4 to 2 as second, 6 to 3 as

third, etc.). But notice that the rationals do not have the same order-type as the integers, even though they

have the same cardinality, because the requisite order-preserving one-one correspondence does not exist . 5

Obviously, order-type tells us more about the structure of a given set than cardinality alone.

Cantor’s method was to suppose the positive integers given, and then to observe “that their order-

type ω was a new (and infinite) number greater of all of them”, that “the order-type of {1, 2, 3,… ω} is a

still greater number ω + 1, and so on, and on, and on” (ONAG 4). Von Neumann remedied the flaw of

having to presuppose the positive integers as given or already constructed by making the Fregean

observation that “it is natural to start at 0 rather than 1”, and then by identifying each ordinal with the set,

rather than order type, of its predecessors—so that, “for von Neumann, 0 is the empty set, 1 the set {0}, 2

the set {0, 1}” and so on (ONAG 4).

Likewise, when we consider Conway’s rule for construction, it is natural to ask how construction

can “possibly ‘get off the ground’” in the first place—since it appears to proceed by constructing new

numbers on the basis of given (i.e., already constructed) numbers, and “since initially there will be no

earlier constructed numbers” (ONAG 7). Conway thus makes the same move as von Neumann: he notes

that “even before we have any numbers, we have a certain set of numbers, namely the empty set ∅! So the

earliest constructed number can only be {L |R} with both L = R = ∅, or in the simplified notation, the

number { | }. This number we call 0” (ONAG 7).

Now the system is off the ground, and 0 (or rather the set containing it) can be used for L and R,

yielding in the second step of construction the two numbers {0| } = 1 and { |0} = –1, the “earliest

constructed” numbers to the right and left, respectively, of 0. Intuitively, one might think that {0|0} = 0,

This is because the rationals are not well-ordered. To see the point clearly, pick some rational number to start the 5

series, e.g. 1/2, map it to the corresponding integer, i.e. 1, and now try to pick which rational will be next and thus mapped to 2: 1/4? But there are rationals between that and 1/2, and will be no matter what rational you pick.


but thanks to the total ordering that is guaranteed in the rule of construction (“and no member of L is ⩾

any member of R”), we know that {0|0} cannot in fact be a number. It is, however, a “game”, which is

what Conway calls “the more general notion” that results from dropping the requirement of total

ordering (ONAG 5). Conway makes great use of games in the “game theory” that is the concern of the

second half of ONAG.

At this point (passing over a wealth of details that exceed the scope of this paper), Conway is able

to unfold “all numbers great and small”, or rather the structure of all numbers, which can be graphically

represented as an absolutely extensively and absolutely densely branching “tree of number”: 6

There is one more feature of Conway’s approach that is relevant to my purposes, which is that

each Conway number has a unique position in the structure of number (as a node in the branching tree),

Ehrlich notes that these features of absolute density and absolute extensivity “collectively ensure that absolute 6

linear continua have no order-theoretic limitations that are definable in terms of standard set theory”, and also that they are devoid of “algebraic limitations”, i.e. have no holes that can be filled by “supplementing the field with solutions to polynomial equations with coefficients in the field—and in a previous work Ehrlich had proved that “⟨No, <⟩ is (up to isomorphism) the unique absolute linear continuum” and that “No is (up to isomorphism) the unique real-closed ordered field that is an absolute linear continuum” (Ehrlich B 7).


and can thus be given “a unique name” (ONAG 3). This unique name is known as the number’s Conway

normal form, by analogy with Cantor normal form for transfinite ordinals. It is given by the following

formula, where α and β are ordinals, yα is a (possibly empty) descending sequence of members of No, and

rα is a sequence of members of ℝ – {0} (Ehrlich A 254):

In terms of the Conway tree, this formula essentially tells us how, for a given number and starting from

the origin position (0), we are to move down the branching tree such that when we stop moving, we will be

at the position of (i.e., the position that is) the number in question.

3. Inter-numeric Identities

One of the most common objections to structuralism concerns its issues with questions of identity.

As Fraser MacBride puts it: “The fact that mathematical objects are identified in terms of their basic

relations leaves the mathematician unable to answer certain questions: whether mathematical objects of

one kind are identical to objects of a different kind, or what intrinsic features mathematical objects

possess” (MacBride 564).

The second question is of course not worrying at all to the structuralist, who simply, and as a

matter of fundamental principle, holds that mathematical objects do not possess intrinsic features. The

first question arises because structuralists hold that a given mathematical object, like the natural number

2, is fully determined by its being a position in the natural number structure, i.e. through its relations to

other positions in that structure. This means that determinate criteria for the identity of a given

mathematical object will be relative to the structure in which it is a position—which raises the question of

inter-structural identities.

A classic example has to do with the attempt to reduce number theory to set theory by identifying

numbers with corresponding sets in the set-theoretical hierarchy. As Benacerraf famously observed, there


are two ways such a reduction could go, for example with regard to the number 2: on the Zermelo

approach, 2 would be identified with the set {{∅}}, while on the von Neumann approach it would be

identified with the set {∅, {∅}}. So which is it—which set is the number 2, “really”?

Structuralists have various ways of dealing with this objection, none of which are satisfying to the

critics of structuralism. In general, they simply end up biting the bullet, averring that questions of identity

between or across structures “are either trivial and straightforward”—e.g., by saying that anything can

occupy the position of 2 in a given system that exemplifies the natural number structure, including the

Zermelo 3 and the von Neumann 3 and even Julius Caesar —“or else the questions do not have 7

determinate answers, and they do not need them” (TM 267). Shapiro even goes so far as to say that such

questions do not make sense: “One can form coherent and determinate statements about the identity of

two numbers: 1 = 1 and 1 ≠ 4. And one can inquire into the identity between numbers denoted by

different descriptions in the language of arithmetic. […] But it makes no sense to pursue the identity between a

place in the natural number structure and some other object” (TM 265-6).

In some cases this is less of a problem for structuralism than in others, but it seems to be a

particular problem in the case of arithmetic, i.e. with regard to inter-numeric identities. As Shapiro says,

“[m]athematicians sometimes find it convenient, and even compelling, to identify the positions of different

structures”, and to give an example of this he says that “it is surely wise to identify the positions in the

natural number structure with their counterparts in the integer, rational, real, and complex number

structures” (TM 267). The question is how a structuralist can account for this mathematical practice.

Resnik recognizes that the indeterminacy of inter-numeric identities is likely to be particularly

troubling to the non-structuralist, but unlike Shapiro, he does not personally feel the force of this

objection. He says that “[i]t has been suggested to me that while there may be no fact of the matter as to

whether the natural number 2 is identical to the Zermelo Two, surely the natural number 2, the positive

integer 2, the positive rational 2, the positive real 2, and the positive complex number 2 are all the

For example, if he were the second person in a line of people.7


same” (Resnik 214). His threefold response to this worry is essentially a denial of the “wisdom” that

Shapiro refers to: “I don’t think we should make much of this facet of mathematical practice”—because,

first, “it may just be a manner of speaking”; and second, “mathematicians do not speak with one voice on

the issue of whether these numbers are the same”; and third, mathematicians generally just “take the

numbers as given, and don’t even consider” such questions (Resnik 214-5).

Shapiro likewise has to find a way of defusing the ontological commitments apparently expressed

in this practice, and he offers two possible ways of doing this. The first is a “threefold theory of the

copula, distinguishing the ‘is’ of identity, the ‘is’ of predication and the ‘is’ of fiat” (MacBride 578). For

example: we can say that the natural number 2 “is” (identity) the second position in the natural number

structure; we can say that 2 “is” (predication) {{∅}} in the Zermelo system, i.e. that {{∅}} has the same

position in that system as 2 does in the natural number structure; and we can say that 2 “is” (fiat) {∅,

{∅}} in the von Neumann system. It is the last of these supposed types of copula use, identity by fiat, that

is used to explain the mathematical practice in question—and to critics of structuralism, it is also the most

“dubious” (MacBride 578). Shapiro’s alternative answer is to simply declare all cross-structural identities

false (TM 266).

Conway’s construction solves the problem of inter-numeric identities for the structuralist outright

because, as we have seen, it gives all numbers as unique positions of the same structure, which is ordered

by the single basic relation (⩾) and which is the structure of number as such. Thus a structuralist like

Resnik, for instance, no longer need accept that “there is no fact of the matter concerning whether the

natural number 2 is identical to or distinct from the real number 2 (since these numbers are introduced by

distinct theories)” (MacBride 570)—nor need the structuralist attempt to explain away the frequent and

intuitive identifications made, as Shapiro and Resnik both do.

Is the natural number 2 the same as the integer +2? In Conway arithmetic, the only signless

number is 0, so unless we are going to identify the natural 2 with –2, the answer would be yes. Is +2 then

the same as the rational 2, the real 2, etc.? Again the answer would be yes, and Conway provides


identifications of our “familiar” numbers with their counterparts in his system . But notice that, strictly 8

speaking, these questions do not really make sense within Conway arithmetic (except with regard to

external theories, translations, etc.), because in Conway arithmetic there is just the one type of number.

4. Synthesizing Ante Rem and In Re Structuralism

The primary difference between an in re or eliminative structuralism like Resnik’s and an ante rem

structuralism like Shapiro’s is one of ontology. Specifically, the in re approach “paraphrases places-are-

objects statements in terms of the places-are-offices perspective”, and thus requires a background

ontology of objects to fill the offices (TM 270). Since this “plan depends on being able to generalize over

all systems exemplifying the structure in question” (TM 270), the background ontology “must be quite

robust. The nature of the objects in the ontology does not matter, but there must be a lot of objects

there” (271). On the in re approach “the only relevant feature of the background ontology is its

size” (PMSO 88)—if the ontology was finite, for example, statements about the structure of even just the

natural numbers would be vacuous.

The obvious choice is set theory: “Because, historically, one purpose of set theory was to provide

as many isomorphism types as possible, set theory is rich fodder for eliminative structuralism. A structure,

on this account, is an order type of sets, no more and no less” (PMSO 87). Now, as Shapiro notes—and

this is what allows the in re structuralist to resist reifying all sorts of abstract structures—it is “crucial” here

that “the background ontology is not understood in structuralist terms” (273). On the in re approach, all

that ultimately exists is sets, and set theory would thus be about sets (rather than about a structure the way

number theory is about the numeric structure) for fear of regress: “from a different point of view, set

theory can be thought of as the study of a particular structure U, but this would require another

background ontology to fill the places of U” (TM 273). Let me emphasize: on the in re approach, the

background ontology of non-reduced objects is required to ensure that the structures talked about are not

One important reason to do this is to show that Conway arithmetic preserves our “familiar” arithmetic truths, such 8

as that –8/4 * –2 = +4.


vacuous—because the structures in question (qua isomorphism types ranging over exemplifying systems)

do not themselves independently exist (or they would be ante rem universals).

I propose that in re structuralism take the Conway structure of number as its background ontology.

But have I not offered a structuralist interpretation of Conway numbers, such that they are “offices”

rather than “objects”, and thus will require a new background ontology in turn, so as to assure that they

are not vacuous? The answer is no, and here is why.

The crucial observation is that “the distinction between office and office-holder—and so the

distinction between position and object—is a relative one, at least in mathematics”, since what “is an

object from one perspective is a place-in-a-structure from another”, such that “the background ontology

for the places-are-offices perspective can even consist of the places of the very structure under discussion” (TM

269). For example, the even natural numbers (viewed as objects) exemplify the natural number structure.

The answer is then to assert that a single “object” is all that exists in our background ontology,

namely the (absolute linear) continuum. There is only one existent in the ontology, but it is structured and

moreover is a structure—and its positions are not “empty offices”, but are precisely the “nodes” or points

of symmetry-breaking in the Conway tree of number.

The continuum exhibits the “relativity” of the two perspectives on structural positions (offices and

objects) in a particularly pronounced sense, perhaps even the sense underlying all other instances, as we

can see by considering where Conway construction begins and what it produces in the end: it begins by

“cutting” the continuum (anywhere will do—we need only presuppose its existence, not, precisely, its

structure), and then unfolds the entire structure of the continuum from that first cut. As can be seen in the

tree of number, a given numeric position qua place-as-office is not empty, for despite the talk of “cuts” and

“offices”, the positions are just points where the symmetry of the continuum breaks—which is enough to

distinguish them structurally, which is all that is needed (they can then be viewed from the places-as-

objects perspective and play the object-side roles in the structure’s self-exemplification). From one

perspective, the first cut { | } is precisely a hole or gap we make in the continuum, which can be filled by


the number 0 (in the exemplifying system we are constructing) we construct to fill it—it is thus an office;

but from the other perspective, we know that what “fills” this gap is nothing other than the entire

continuum itself (0 as the original break of symmetry in the number line, from which the entire number

tree can be unfolded). And on this option for structuralism, there will certainly be no question of whether

the ontology is big enough!

In this way, in re structuralism can have a purely structuralist ontology like ante rem structuralism,

but it can still avoid having to reify the innumerable structures that ante rem structuralism does (for these

are just isomorphism types of exemplifying systems, guaranteed to be non-vacuous by the cardinality of

the background ontology, which is indeed the proper class of all numbers qua “joints” in the continuum).

Bibliography

Conway, John Horton. On Numbers and Games (second edition).

Dummett, Michael. Frege: Philosophy of Mathematics.

Ehrlich, Philip (A). “The Absolute Arithmetic Continuum and Its Peircean Counterpart”. In New Essays on Peirce’s Mathematical Philosophy. Ed. Matthew E. Moore. Chicago: Carus Publishing Company, 2010.

Ehrlich, Philip (B). “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”. The Bulletin of Symbolic Logic, Vol. 18, No. 1 (March 2012).

MacBride, Fraser. “Structuralism Reconsidered”. In The Oxford Handbook of Philosophy of Mathematics and Logic. Ed. Stewart Shapiro.

Reck, Erich H. “Dedekind’s Structuralism: An Interpretation and Partial Defense”. Synthese 137: 369-419 (2003).

Resnik, Michael. Mathematics as a Science of Patterns.

Shapiro, Stewart. Philosophy of Mathematics: Structure and Ontology.

Shapiro, Stewart. Thinking about Mathematics.

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The Significance of Conway’s Construction of Number for Structuralism in the Philosophy of Mathematics