Synthese (2008) 163:119–131DOI 10.1007/s11229-007-9169-6

Structuralism as a philosophy of mathematical practice

Jessica Carter

Received: 6 February 2006 / Accepted: 17 April 2007 / Published online: 24 May 2007© Springer Science+Business Media B.V. 2007

Abstract This paper compares the statement ‘Mathematics is the study of structure’ with theactual practice of mathematics. We present two examples from contemporary mathematicalpractice where the notion of structure plays different roles. In the first case a structure isdefined over a certain set. It is argued firstly that this set may not be regarded as a structureand secondly that what is important to mathematical practice is the relation that exists betweenthe structure and the set. In the second case, from algebraic topology, one point is that anobject can be a place in different structures. Which structure one chooses to place the objectin depends on what one wishes to do with it. Overall the paper argues that mathematicscertainly deals with structures, but that structures may not be all there is to mathematics.

Keywords Philosophy of mathematics · Structuralism · Mathematical practice

Structuralism is presently seen as one of the more promising philosophies of mathematics. Theclaim that mathematics is the study of structures also seems to be supported by mathematicalpractice. Indeed, mathematicians often mention structures when talking about their subject.In Eilenberg’s words: “Among the most conspicuous trends in modern mathematics is theupsurge of modern algebra. Almost every mathematical theory today has an algebraic facet.The structures with which modern algebra is concerned have been compared to the grin of theCheshire Cat in Alice in Wonderland, which remained visible after the cat itself faded away”(Eilenberg 1969, p. 152). What I shall do here is to compare the statement “Mathematics isthe study of structure” with the actual practice of mathematics. We will focus on differentnotions of structure in mathematics and question whether there can be given a uniform notionof structure that captures the uses of structure in mathematical practice. Two different cases in

I wish to thank Colin McLarty as well as the anonymous referees for helpful comments on earlier versions ofthis paper.

J. Carter (B)Department of Mathematics and Computer Science,University of Southern Denmark, Odense, Denmarke-mail: jessica@imada.sdu.dk

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mathematics will be presented where the notion of structure plays different roles. In addition,I will show that, sometimes, which structure a certain object is taken to be depends on whatone wishes to do with the object. I stress here that mathematical practice consist of activities.

Before proceeding, it should be noted that there are many different motives or reasons toadopt a structuralist philosophy of mathematics apart from trying to describe mathematicalpractice. Some of these are as follows. Firstly, structuralism could be regarded as showingwhat mathematics is really about, namely structures. This is an ontological claim and it liesbehind Shapiro’s version of structuralism as presented in his ‘Philosophy of Mathematics:Structure and Ontology’ (1997). Secondly, structuralism could be regarded as providing afoundation for mathematics in the sense that structures are what the contents of mathematicscan be reduced to in an attempt to secure it. A structure can here be understood as ‘anymodel satisfying a certain collection of axioms’.1 Thirdly, there is another sense in whichstructuralism can be regarded as foundational, namely as a way to organize or “structure”mathematics. This was done by Bourbaki.2 Finally, structuralism is the outcome of tryingto answer certain philosophical questions about mathematics. In ‘What numbers could notbe’ (1965), Benacerraf found that there could not be given an answer to the question ‘Whichobjects are the numbers?’. This led him to conclude that there are no properties that sufficeto single out the numbers as objects and what matters are the relations between the number;he refers to his position as that of a “formist”. This resulted in the elusiveness of the Fregeannotion of objects as individuals. The more structuralist motive is to explain the freestandingnature of mathematical objects, namely that many different “objects” can fill the place of amathematical structure. These motives could be related in all sorts of ways but it is not myintention to identify those relations here. Neither is it my intention to criticize these motives.Rather I will argue that structuralism does not reflect the actual practice of mathematics,insofar as it is claimed that mathematics only deals with structure.

It is important to note that there are a number of different articulations of structuralist viewson the contemporary scene, and that not all of them are affected by the criticism presentedhere. I choose not to present or discuss any version of structuralism in detail, but merely pointto certain aspects of different structuralist positions that are relevant to my criticism.3 First Ipoint to the fact that a number of contemporary structuralists have expressed that they intendtheir positions to be compatible with mathematical practice. Shapiro describes this require-ment in the following way: “As I see it, the goal of philosophy of mathematics is to interpretmathematics, and to articulate its place in the overall intellectual enterprise. One desideratumis to have an interpretation that takes as much as possible of what mathematicians say abouttheir subject as literally true, understood at or near face value” (Shapiro, forthcoming, italics inoriginal).4 Furthermore, I note that there are also structuralists who have questioned the viewthat all mathematical objects fit into a structuralist description of mathematics. In ‘Struc-turalism and metaphysics’ (Parsons 2004), Parsons writes about ‘quasi-concrete’ objects,i.e., “objects that have concrete instantiations” (p. 70).5 These objects are for example basicgeometrical objects. It is one of the points of this paper that even objects of more abstractmathematics may not fit into some structuralist views of mathematics.

1 A number of authors have interpreted Hilbert as articulating a view of this sort (see, for example, Shapiro1996 and Chihara 2004).2 See their paper ‘The architecture of mathematics’ Bourbaki (1950).3 For the most extensive overview of current structuralist positions refer to (Landry and Marquis 2005).4 Shapiro makes similar claims in (Shapiro 1997), pp. 30–33. In addition to Shapiro, Resnik and Chiharaexpress similar views. See for example Chihara (2004), pp. 1–4, and Resnik (1997), p. 201.5 The notion of quasi-concrete objects is also mentioned in an earlier paper, see Parsons (1990).

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Among contemporary structuralists one finds Shapiro (1997), Resnik (1997), Hellman(1989), Awodey (2004) and Landry and Marquis (2005). Being advocates of structuralism,they all agree that mathematics in some sense deals with structures. Whereas Shapiro claimsthat mathematics ‘is the study of structures’,6 for example, Landry and Marquis hold thatmathematics deals with structured systems.7 Note that it has also been claimed that a structu-ralist view stating that mathematics deals with models of a collection of sentences, ‘model-structuralism’, is “philosophically uncontentious” (Hale 1996, p. 124).8 As we shall see, thisis not the view taken by Landry and Marquis. Furthermore the notion of structure adhered todiffers among these positions. Shapiro presents a structuralist view where structures are inthe ontology. A view of this sort is denoted Ante rem structuralism. As there is an ontology ofstructures, Shapiro chooses to axiomatize the notion of structure, such that anything fulfilledby the axioms constitutes a structure. Structures are then described in the traditional senseas a collection of places with certain relations on them or as determined by a collection ofaxioms. Another central point of Shapiro’s position is that he regards the places of a structureas genuine objects.9 This entails that he, according to the Quinean dictum ‘no entity withoutidentity’, needs to introduce criteria of identity on these places. But, as he also holds thestructuralist view that the properties of a place are only determined relative to the structurein which it has a place, there can be no identity across structures.10

Opposed to the ante rem view is in re structuralism, where there are no structures in theontology. On this view a structure is merely a generalization of all (possible) systems thatexemplify it: “[t]he program of rephrasing mathematical statements as generalizations is amanifestation of structuralism, but it is one that does not countenance mathematical objects,or structures, for that matter, as bona fide objects. Talk of numbers is convenient shorthandfor talk about all systems that exemplify the structure” (Shapiro 1997, p. 85). Note that thisalso entails that an in re structuralist does not countenance the places of a structure as objects.

Some versions of in re structuralism consider a structure as what is obtained by abstractingfeatures of concrete systems (a bottom–up view). When taking such a view one needs to ensurethat there is a background ontology of systems to fill the places of all relevant structures.One possibility is to let this background consist of set-theory, but then structuralism seemsto lose its advantage over standard realism. Hellman’s 1989 solution is instead to talk aboutpossible systems as he lets the background theory consist of modal logic.11

A third option for an in re structuralist is to adopt category theory as a framework fora structuralist position. This was suggested by McLarty in the paper ‘Numbers can be justwhat they have to’ McLarty (1993).12 Recently also Awodey (2004) and Landry and Marquis(2005) have presented versions of such a view. A major point in this position concerns thedirection of abstraction when going between systems and their structure. According to the

6 See Shapiro (1997, p. 5).7 See Landry and Marquis (2005, p. 20).8 Hale refers to Dummett (1991).9 See Shapiro (1997, p. 83).10 Resnik’s structuralism is similar to Shapiro’s in many respects, but one important difference is that Resnikdoes not claim that there is an ontology of structures. He does hold that the places of a structure constitutegenuine objects. The only way that a structure can be regarded as an object is for it to be a place in some otherstructure.11 Hale (1996) argues that even Hellman needs to postulate a background ontology. He concludes that, ifthe aim of structuralism is to avoid the problems of Platonism, then structuralism fails as a philosophy ofmathematics.12 McLarty simply advocates interpreting mathematics in categorical set theory (ETCS) exactly the wayphilosophers today like to interpret it in ZF.

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category-theoretical structuralist view one starts at the level of structures giving axioms ora description of a certain structure. Then one may exemplify this structure as a particularsystem that the mathematician is interested in. An approach going in this direction is denoted‘top–down’. In the opposite direction, ‘bottom–up’, one starts with a set of objects that arerelated in various ways and from this system one abstracts the structure. This correspond toShapiro’s and Resnik’s views. Landry and Marquis explain this distinction as follows: “theaxioms for a category provide the framework, or scaffolding, for what we can say aboutabstract kinds of structured systems independently of what those kinds are ‘made of’. Takingour top–down approach we begin with the notion of an abstract system; we do not seek toarrive at this notion by abstractly considering a kind of concrete system. In its most generalsense, then, an abstract system is considered in an Hilbertian, ‘algebraic’, sense as a schemafor our talk of the shared structure of an abstract kind of structured system: it allows us totalk about such abstract kinds of structured systems as instances of the same type without ourhaving to consider what these types are types of” (Landry and Marquis 2005, p. 33, italicsin original).

The structures that it is possible to formulate in categorical structuralism are those thatcan be described in category theory. This entails that the notion of structure becomes veryflexible. Furthermore, the theory also makes it possible to describe relations between differentstructures (or categories) via the notion of functors, and even relations between these vianatural transformations.

Shapiro’s version of structuralism as well as the category theoretical, structuralist versioninvolves the idea that certain axioms describe a structure. In the case of Shapiro, the structuresthat exist are described by his ‘structure-theory’, and in the case of Landry and Marquis acertain structure can be described by some axioms in a background theory for category theory.But there is an important difference with respect to their attitude towards the status of axioms.Both Shapiro and Landry and Marquis liken this difference to the difference of opinionsbetween Frege and Hilbert towards axioms. Landry and Marquis express the Hilbert/Fregedistinction as “viewing an axiom system as a framework, or scaffolding or schemata, andviewing axioms as truths, or assertions of some background theory” (ibid, p. 25). Landryand Marquis note that with respect to the places of structures, there is a difference betweenthe Fregean outlook and Shapiro’s structuralism, as Shapiro claims that the existence of aplace is relative to that of the structure in which it has a place. But with respect to Shapiro’sposition on structures they write “Shapiro retains the Frege-style demand for a backgroundtheory that fixes the meaning of the term ‘structure”’ (p. 25).

Before we turn to the examples from mathematics illustrating my claims, I note thatMac Lane 1996 has also pointed out different uses of the notion of structure in mathematics.Mac Lane’s examples include:

– Categorical structures where the theory of a structure only has one intended model, as infor example, N and R.

– Algebraic structures where the structure has more than one intended realization, as in agroup, topological space and metric space.

– “Structure theorems” stating that a certain kind of object can be split into simpler objectsof the same kind. The theorem of finite abelian groups provides an example of this.

Mac Lane concludes firstly that not all mathematical objects fit into structures. Secondlyhe claims that structuralism does not answer all interesting questions: “For example, there canbe quite different view of structure—as something arising in set theory and then formulatedin Bourbaki’s typical structures, or as something located in some ethereal category. Talkingof structures does not explain why certain structures (e.g., groups or topological spaces)

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play such a dominant role. Structure theorems do not really exemplify the important role ofcalculations or of geometric intuition in the study of particular structures” (p. 183).

One of the purposes of this paper is to describe some of the different roles structures playin mathematics. Mac Lane’s claim that ‘all mathematical objects do not fit as structures’ (p.183) depends, of course, on what one takes a structure to be. Mac Lane’s statement concernsthe notion of structure as used by Bourbaki. Previously I have argued (Carter 2005) thatnot all mathematical objects fit into a widely used definition of structures as ‘a collectionof places with certain relations defined on them’, whereas taking a structure to be describedby a collection of axioms seems to cover all mathematical objects. But, as we have seen,this view can have different interpretations, for example, depending on the status of axioms.Finally, Mac Lane writes “Many philosophers have followed the model set by Wittgenstein—discussing questions of putative philosophical importance, but with little or no attention tothe rich and varied actual aspects of mathematics” (p. 183). I certainly agree with the spirit ofthis remark, and in what follows I shall present examples from the practice of contemporarymathematics.

1 Different examples from mathematics

We consider examples illustrating the following uses of structure:

1. Structure over sets that is used to compute invariants of this set.2. A case where “structure” is extracted in order to change relations between objects.

1.1 Structure over a set

In contemporary mathematics the notion ‘structure over a set’ is used in many contexts.Generally one starts with a certain set. (Note that I use ‘set’ here in an informal way). Overthis set there is defined a structure and finally this structure is used to compute a so calledinvariant of the set. The following picture illustrates the general situation:

I nvariant

Structure

Set

���� R1

����R2

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A specific example from topology is a topological space, the structure is a vector bundleover this set (consisting of a finite vector space associated to each point in the topologicalspace) and the invariant could be the K0-group that is defined as a certain quotient of thevector bundle.13 Similarly, in algebraic geometry, the set is an algebraic variety,14 the structureconsist of a sheaf15 and it is again possible to compute the K0-group from the sheaf.16 Theidea behind this is to use the invariant to determine properties of the set. When computingthe invariant the aim is to obtain an object that is simpler, but still rich enough to provide thedesired information about the underlying set. I shall here emphasize that what are importantin this case are the relations that exist between the set, the structure and the invariant. It isbecause of these relations that it is possible to transfer information from the invariant to theset. The example I shall present to illustrate these ideas is somewhat simpler, but the generalidea is still the same.

1.2 Galois-theory: a simple example

Basically, given a polynomial f (t) = tn + an−1tn−1 + . . . a1t + a0, in Q[t], it is possible toassociate a group of permutations to it. Properties of this group will determine whether theequation f (t) = 0 is soluble by radicals. According to the Fundamental Theorem of Algebra,any polynomial will have a root over the complex numbers, but the question of whether agiven polynomial is soluble by radicals concerns whether these roots can be expressed onlyusing algebraic operations such as addition, multiplication and extraction of roots.

A simple example, illustrating the main idea:17

Let f (t) = t4 − 4t2 − 5. This polynomial can be factorized as

f (t) = (t2 + 1)(t2 − 5).

Roots are thus: α = i , β = −i , δ = √5 and γ = −√

5. These roots fulfil a number ofequations, like

α + β = 0 α2 + 1 = 0 δ + γ = 0 αδ − βγ = 0.

Some permutations of the roots do not change the truth value of these equations. Forexample, one may interchange α and β and interchange δ and γ . The set of permutationsthat do not change truth value are:18

R = (αβ) S = (δγ ) T = (αβ)(δγ ) and I.

13 Marquis (1997) gives a nice overview of how K -theory is used to determine properties of a topologicalspace. See, for example, pp. 263–264.14 Given a collection of polynomials over a field, an algebraic variety is defined as the intersection of all theroots of these polynomials.15 A sheaf consists of a triple (X, φ, F). X is an algebraic variety, F is the sheaf that associates to each pointof the algebraic variety an algebraic structure, for example an abelian group. φ is a continuous map fromF to X .16 Just to show how general this is in modern mathematics, I bring yet another example: In the beginning ofthe 1970s, Penrose found two tiles that could be used to tile the whole plane. It is possible to describe suchtilings by a space X . This space can only be given a trivial topology. But it is possible to define a C∗-algebraover it (the “structure”), and from this one computes the ordered group, (K0(X), K +

0 (X)) (the invariant) thatcan be used to obtain information about the space X of Penrose tilings.17 The following description is largely inspired by the exposition in (Stewart 2004).18 Note that the expression (αβ) means the permutation that takes α to β and β to α.

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These four permutations form a group, the Galois group. It is the structure of this groupwhich provides the information that f = 0 is soluble by radicals.

In general, one considers extensions of the roots to the field Q and associates a groupof field-automorphisms. The question of whether the polynomial is soluble by radicals thenbecomes a question of whether this group is soluble.19

Generally what goes on in this type of case is that one starts out with a certain set that onewishes to obtain information about. Then some structure (vector bundle, sheaf or C∗-algebra)is constructed over this set and finally from this structure one constructs an invariant. Thisinvariant could be a group, i.e., a structure of some kind. In the example of Galois theory,the polynomial over Q can be regarded as the “set”, the set of equations fulfilled by the rootsas the structure and the Galois group is the invariant that is constructed in order to obtaininformation about the polynomial. What I wish to stress is that when one constructs certainstructures over some set in order to obtain information about this set it is possible to doso because of the relations that exist between the set and the structure. Thus a claim thatmathematics is the study of structures should be modified to take into account the relationsthat can be defined between structures. Furthermore, it can be questioned whether the set onestarts out with is a structure. For example, is a polynomial a structure?20 Equally, it could beasked whether a topological space or an algebraic variety is a structure. This depends on thecontext. One way of talking about a topological space is that it consists of a set that has thestructure of a topological space. But a topologist would simply start out with the topologicalstructure. An algebraic variety can be thought of as given by a set of relations, namely thepolynomial equations that define it. Often sets have different kinds of structure. For example,an algebraic variety also comes with a topological structure.

1.3 Extracting structure

With this example, I intend to illustrate that structures are not always the focus in mathematicalpractice. In this case it will not even be clear what the structures are. With a little effort,however, we will see, after I have presented the case, that it is possible to identify variousstructures throughout the case. But it will be possible to do this in several ways. It will becomeapparent that which structure one should regard an object as being a place in, depends whatone intends to do with the object. The result that I shall discuss concerns a sequence, aso-called Mayer–Vietoris sequence. The particular sequence was formulated by Ranicki andYamasaki (1995). The problem with this sequence is that it is not exact, but the result thatI will present is that when placing this sequence in a certain category it does become exact.

A sequence C0f−→ C1

g−→ C2 is exact at C1 if the image of f is equal to the kernel of g,that is, the set of members of C1 that is in the range of the function f should equal the set ofmembers of C1 that is mapped to zero via the function g.

The sequence that we consider is built up from so-called controlled geometric modules.These were originally introduced to prove Whitehead’s conjecture (Whitehead 1939) thatsimple homotopy equivalence is a topological invariant.21 A Mayer–Vietoris sequence bearsthe name of Mayer and Vietoris, two early 20th century mathematicians. Mayer, inspired

19 A group G is soluble if there exists a finite sequence of subgroups Gi , 1 = G0 ⊆ G1 ⊆ . . . Gn = G suchthat i) Gi is a normal subgroup of Gi+1 for each i ∈ {0, . . . n − 1} and ii) each quotient group Gi+1/Gi isabelian.20 Considering a more modern approach, one would consider the ring Q[t] and to each member of this ringassociate its Galois group in which case it could be claimed that the ring of polynomials is a structure.21 Two complexes are simple homotopy equivalent if by a process of formal deformations (expansions andcontractions) they can be transformed into the same complex.

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by Noether, defined the notion of a homology group of a complex and Vietoris found that amap between complexes induces maps between their corresponding homology groups. Thismakes it possible to define an exact sequence that relates the homology groups of subsets of aspace.22 It is useful that this sequence is exact. It can then be used to compute the homologygroup of a set where the homology groups of parts of this set are known. The Mayer–Vietorissequence over controlled geometric modules formulated by Ranicki and Yamasaki was usedto prove Whitehead’s conjecture. This sequence, however, is not exact. Also in the beginningof the 1990s, Munkholm found a way to make the Mayer–Vietoris sequence exact by placingit in a certain category. The necessary details will be presented in the following sections.

1.4 Geometric modules and control

Let E be a topological space. A geometric module defined on E is a pair (S, σ ), where S isa set and σ : S → E .

Maps between geometric modules are defined in terms of so-called Moore paths. A Moorepath on E is defined as the pair (w, lw), where w: [0, lw] → E is a continuous function.The set of Moore paths on E is denoted M(E). Composition of two paths is defined byconcatenation of the paths (and is only possible if the end point of the first path coincideswith the starting point of the second).

A morphism f : (S, σ ) → (T, τ ) between two geometric modules consists of finite linearcombinations of Moore paths starting in σ(s) for s ∈ S and ending in τ(t) for t ∈ T 23 as inthe picture below:

fst =∑

w∈M(E)

zst (w) · w

S

• s

T

• t

E

•σ(s)

•τ(t)

ω

�� �� σ ����τ

To introduce control means essentially that one introduces a notion of size on sets andmaps making sure they are finite. To make this possible, one introduces a metric space to thesetting and a map from the topological space E to this metric space. Let X be a metric space.A continuous map pX : E → X is then said to be a control map. If pX : E → X and (S, σ )

is a geometric module σ : S → E , then (S, σ ) is said to be a controlled geometric module.

22 See for example, Eilenberg and Steenrod Foundations of Algebraic Topology, 1952.23 They are thus functions f : S × T → Z[M(E)] from S × T to the free module generated by Moore pathson E .

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Defining the size of a morphism f : (S, σ ) → (T, τ ) is done by introducing size on theMoore paths that it is composed of. A path in f has radius ε if the image of the path iscontained in the intersection of the closed ε-ball with center in the starting point of the pathand the closed ε-ball with center in the ending point of the path:24

Eω �� �� pX

X

����� ε pX (ω) �����ε

If all paths in f have size at most ε, then the morphism f is said to have size ε. Notethat when composing an ε morphism with a δ morphism one obtains an ε + δ morphism.The main problem with control then becomes to keep track of the size of maps as well as inwhich set they are placed. This gives rise to the notion of a stabilization map that will becomeessential when forcing the Mayer–Vietoris sequence to be exact. Let Y be a subset of themetric space X , then one denotes by Y δ the closed δ-neighbourhood of Y . A stabilizationmap can then be seen as a map sδ≤ε that enlarges the size of some set Y :

sδ≤ε : Y δ → Y ε .

1.5 Mayer–Vietoris defined over geometric modules

Now one defines the so called projective class, K̃0(X, ε), and the Whitehead group,W h(X, Y, ε) (See Ranicki and Yamasaki 1995 for details). It is possible to show that thesebecome abelian groups. (Direct sum forms a natural operation on the objects in these sets.)Defining maps between these enables the construction of the Mayer–Vietoris sequence fromcontrolled geometric modules. If X = X− ∪ X+, Y = X− ∩ X+, and W is a set containingsome (large) closure of Y :

W h(Y, ε) → W h(X−, ε) ⊕ W h(X+, ε) → W h(X, ε)

→ K̃0(W, ε′) → K̃0(X− ∪ W, ε′) ⊕ K̃0(X+ ∪ W, ε′)

It turns out that the controlled Mayer–Vietoris is not exact. For two consecutive maps, di

and di+1, we do have that di+1 ◦ di = 0 such that we have that im(di )⊆ ker(di+1). But theother inclusion does not hold unless we compose with stabilization maps, i.e., allowing thatthe sets we work in get bigger. Thus we have:s◦ ker(di+1) ⊆ s′◦im(di ), for certain stabilization maps, s, s′.

24 The condition is pX (w[0, lw]) ⊆ B(pX (w(0)), ε) ∩ B(pX (w(lw)), ε).

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1.6 How to make the controlled Mayer–Vietoris exact

In the previous section we noticed that what went wrong in attempting to make the sequenceexact is that we had to compose the image and kernel of certain maps with stabilization maps.The solution must then be to place the sequence in a category where the stabilization mapsdisappear.

The first step in doing this is to find an appropriate category25 in which to place the groupsK̃0(Y ε, ε) and the Whitehead group W h(Y ε, ε). Here we use the fact that they are abeliangroups and that they depend on the number ε. This means that one can consider the groupsas “maps” from the set of real numbers R+ to the set of abelian groups or, more precisely,using category theory language, they become functors26 from the category of real numbersto the category of abelian groups. The category consisting of such functors as objects is thecategory AbR+ , where:

– To each ε ∈ R+ corresponds an abelian group.– For each δ ≤ ε that is a morphism in R+ there corresponds a homomorphism between

abelian groups.

From this functor category one now constructs the so-called category of left fractions.This is constructed as follows: Given a category C and a set of morphisms S in morph(C)subject to certain conditions, then it is possible to construct a category, S−1C, the categoryof left fractions, consisting of the same objects as C but where the morphisms in S becomeisomorphisms.

A simple example will illustrate how this is obtained: Consider the category N that hasone object: * and where morphisms are given by morph(∗, ∗) = N ∪ {0}. Composition oftwo morphisms is defined by addition:

∗ n−→∗ m−→ = ∗ n+m−→ ∗The identity morphism 1∗ is thus zero.

Let S contain all morphisms from N . When forcing the morphisms (positive naturalnumbers) to become isomorphisms, i.e., adding inverses, the category of fractions turns outto be the integers (S−1)N = Z.

When constructing the category of fractions from the functor-category, we let the set ofmorphisms S consist of the stabilization maps. Then one shows that ker(di+1) ⊆ im(di ) holdsin the category of fractions when s◦ ker(di+1) ⊆ s′◦ im(di ) holds for certain stabilizationmaps, s, s′, in AbR+ . Furthermore, the inclusions im(di )⊆ ker(di+1) still hold in the categoryof left fractions. Thus in the category of fractions the Mayer–Vietoris sequence becomes exact.

1.7 Placing the Mayer–Vietoris sequence in the language of structuralism

We started by defining geometric modules. It is not obvious how to conceive of these asstructures. One could perhaps regard them as structures of the type consisting of places andrelations on these. This would entail that the elements of the set S are considered as the places

25 A category C consists of a certain collection of objects Ob(C) and for any two objects b, c ∈ Ob(C),there is a collection, morph(b, c), of morphisms between b and c. This collection may be empty, but theidentity morphism 1b must be contained in morph(b, b). Furthermore, if there are morphisms morph(b, c)and morph(a, b), then their composition must be in morph(a, c).26 Given two categories, C and D, then there can be defined a map between these, a so-called functor,F : C → D. A functor sends objects of C to objects of D and morphisms in C to morphisms in D subject tocertain conditions.

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of the structure and the relation that is given on this structure is the map σ that to each s ∈ Sassociates a point in the topological space E . Another way to consider a geometric moduleis to regard the set as a structure (actually it does not have any structure in the way that amathematician would use the word) and the map σ as providing a correspondence betweenthe two structures S and the topological space E . When maps are defined between geometricmodules one can consider the “global structure”, the category of geometric modules GM.Here the geometric modules are just the objects or the places of this structure. In this globalstructure one then defines the projective class and the Whitehead group being able to expressthe Mayer–Vietoris sequence. It is also possible in this category to show that the projectiveclass and the Whitehead group become abelian groups.

What happens next is that the information that the projective group and the Whiteheadgroup are abelian groups is extracted in order to place these in a new global structure,the functor category, AbR+ . Note that the “objects” or places in this category consist offunctors which are mappings from the category of real numbers to the category of abeliangroups. Finally, from this structure one constructs the category of left fractions, (S−1)AbR+ ,consisting of the same places but where the morphisms are changed.

Describing things in this way shows that it is possible to identify various structures throu-ghout the example, suggesting that the claim ‘Mathematics is the science of structures’ makessense. This description, however, suppresses certain facts about mathematical practice. First,I noted that the mathematical entities can be regarded as structures in different ways. Thisholds for the geometric modules as well as the projective group and the Whitehead group;both could be regarded as being members of the category of geometric modules or the cate-gory AbR+ . Secondly, structures or objects may be created from properties of already existingobjects and placed in a new structure.

2 Discussion

This paper began by asking whether mathematics is the study of structures. So far I haveshown that part of the activity of mathematics can be described by claiming that it studiesstructures in order to determine properties of certain sets (or entities) via relations that canbe defined between them. In mathematics one also talks about sets having a certain structureor, that a set can be given a certain structure. The difference between the last two cases is thatin the first it is possible to show that the set with the given relations can be shown to fulfil therelevant axioms of the structure (as in the example in this paper, where K̃0 can be shown tobe an abelian group). In the last case, one has first to define certain relations on the set andthen one can show that the set together with these relations fulfils the relevant axioms. In allthree cases there is something more in the setting than the structures. In the first case there isboth the underlying set and the relation between this set and the structure. In the second andthird cases there is the set that ‘has a certain structure’. In mathematical practice one does notstress that this set has any structure. That is, mathematics sometimes studies sets, either viathe structures they have, or the structures they can be given or structures that can be definedover them.

In the example concerning the Mayer–Vietoris sequence, certain objects are part of dif-ferent structures. We have the projective class and the Whitehead group that consist of (equi-valence classes of) n-dimensional chain complexes. But they are also abelian groups. Whichstructure one considers these as part of depends on what one wishes to do with them.

This study also shows another role that structure plays in mathematics. I have noted thatat some point one extracts the information that K̃0(X, ε) and W h(X, ε) are abelian groups

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in order to place them in a new category. When doing this one chooses to disregard whatthe objects were before, i.e., one focuses only on their structural properties. Finally, whathappens is that the objects K̃0(X, ε) and W h(X, ε) are placed in the category of left fractionsin order to change their properties. Thus, sometimes mathematicians choose only to considersome structural properties of the entities they study.

The fact that objects or ‘places’ are moved between structures seems to go against thedictum that ‘places have no distinguishing features except those determined by the structurein which they have a place’ which is taken as implying the claim that ‘places from differentstructures can not be identical’.27 Firstly, we have seen that the properties of places or objectscan be determined by different structures that they are part of. Secondly, the properties ofan object in a given structure can be used to consider the object as part of another structure.Finally, when the category of left fractions has been constructed, we move the objects to adifferent structure. In this new structure, the objects are the same, but the relations are changed.However, one needs to be careful when using a word like ‘moving’ in mathematics. It does notnecessarily mean the same in mathematics as intended when talking about physical objects.One difference is that, in mathematics, one may talk as if one object is part of two differentstructures at the same time. We stress that it is the same object that is part of different structures,because when proving that this object has certain properties in the category of left fractions,one uses conditional statements. These statements claim that ‘any object having certainproperties in the first category will have certain other properties in the category of fractions’.

I return to the claim that the category-theory structuralist position will not be affectedby the problems posed here.28 First their claim that mathematics is the study of structuredsystems rather than structures fit the descriptions made above. Secondly, as mentioned earlier,they can naturally account for relations that are defined between various sets or structuresvia the notion of a functor that has a natural place in category theory. Note, however, thatalthough the result in the second case does depend on a category theoretic setting, our firstexample does not mention categories. One way of expressing some of the points made inthe case studies could be that ‘which structure a set has, or which structure an object istaken to be a place in, depends on the context’. The context depends on what one wishesto do with the set and the object, respectively. This point is fundamental to the categorytheoretical perspective: “This [top–down] approach is best characterized by an adherence toa category-theoretic ‘context–principle’ according to which one never asks for the meaningof a mathematical concept in isolation from, but always in the context of, a category” (Landryand Marquis 2005, p. 7).

Finally, I need to substantiate my claim that mathematical practice consists of activities.In the first case one constructs a structure over a set and then from this structure an invariantover the set. Then one studies this invariant in order to determine certain properties of the set.In the category theory case, first the Projective class and the Whitehead group are constructedfrom geometric modules and maps between them. They are then placed in another category(the functor category). From this category one constructs the category of left fractions.Words like these indicate the activities performed in mathematics. A realist-inclined opponentmight claim that it is merely our use of language that fails to reflect that these objects existindependently of our constructions and that we do not move objects around. I do not wishto discuss ontological claims here, but merely to point out that when discussing knowledge

27 Shapiro writes: “places from different structures are distinct. This fits at least with one of my repeatedremarks/slogans: ‘mathematical objects are tied to the structures that constitute them”’ (Shapiro, forthcoming).28 Although the category theoretical structuralist position can account for the examples presented here, Shapiro(2005) suggests that it has problems with respect to meta-mathematics.

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acquisition, it certainly seems as if we have to perform these constructions in order to obtainknowledge in mathematics.29

To conclude, it is certainly true that mathematics deals with structures. But I doubt thatthere is a single description of structure that can be used to describe mathematical practice, asstructures are used for different purposes. Furthermore, returning to Eilenberg’s quote fromthe beginning, we suggest that structure is not all there is to mathematics: there is also theCheshire Cat that fades away when we choose only to consider the smile.

References

Atiyah, M. F. (1974). How research is carried out. Bullet in of IMA, 10, 232–234.Awodey, S. (2004). An answer to Hellman’s question: ‘Does Category theory provide a framework for mathe-

matical structuralism?’ Philosophia Mathematica, 12, 54–64.Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74, 47–73.Bourbaki, N. (1950). The architecture of mathematics. American Mathematical Monthly, 57, 221–232.Carter J. (2005). Individuation of objects - A problem for structuralism? Synthese, 143, 291–307.Chihara, C. (2004). A structural account of mathematics. New York: Oxford University Press.Dummett, M. (1991). Frege: Philosophy of mathematics. London: Duckworth.Eilenberg, S. (1969). The algebraization of mathematics. In G. Boehm (Ed.), The mathematical sciences

(pp. 153–160). Cambridge, MA: MIT Press.Eilenberg, S., & Steenrod, N. (1952). Foundations of algebraic topology. Princeton: Princeton University

Press.Hale, B. (1996). Structuralism’s unpaid epistemological debts. Philosophica Mathematica, 4, 124–147.Hellman, G. (1989). Mathematics without numbers. New York: Oxford University Press.Landry, E., & Marquis, J.-P. (2005). Categories in context: Historical, foundational, and philosophical. Philo-

sophia Mathematica, 13, 1–43.Mac Lane, S. (1996). Structure in mathematics. Philosophia Mathematica, 4, 174–183.Mancosu, P. (2001). Mathematical explanation: Problems and prospects. Topoi, 20, 97–117.Marquis, J.-P. (1997). Abstract mathematical tools and machines for mathematics. Philosophia Mathematica,

5, 250–272.McLarty, C. (1993). Numbers can be just what they have to. Nous, 27, 487–498.Parsons, C. (1990). The structuralist view of mathemtical objects. In W. D. Hart (Ed.), Philosophy of mathe-

matics (pp. 272–309). Oxford: Oxford University Press.Parsons, C. (2004). Structuralism and metaphysics. The Philosophical Quarterly, 54, 56–77.Ranicki, A., & Yamasaki, M. (1995). Controlled K-theory. Topology and Its Applications, 61, 1–59.Resnik, M. (1997). Mathematics as a science of patterns, New York: Oxford University Press.Shapiro, S. (1996). Space, number and structure: A tale of two debates. Philosophia Mathematica, 3, 148–173.Shapiro, S. (1997). Philosophy of mathematics. structure and ontology, Oxford: Oxford University Press.Shapiro, S. (2005). Categories, structures, and the Frege-Hilbert controversy: The status of meta-mathematics.

Philosophia Mathamatica, 13, 61–77.Shapiro, S. (forthcoming) Structure and identity, to appear in F. Mac Bride & C. Wright (Eds), Identity and

modality: New essays in metaphysics. Oxford University Press.Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34, 135–151.Stewart, I. (2004). Galois theory. Boca Raton: Chapman & Hall (CRS).Whitehead, J. H. C. (1939). Simplicial spaces, nuclei and m-groups. Proceedings of London Mathematical

Society, 45, 243–327.

29 It could be objected that if knowledge follows from any justification of the results, and that justificationcould be obtained without performing such activities, then my claim that knowledge requires activities fail. Ican not rule out that there could in fact be given proofs of the results presented here, for example, some sortof a formal proof in a formal system that do not at all contain activities. But a number of philosophers andmathematicians have claimed that proofs play different roles in mathematics (see for example, Atiyah (1974),Steiner (1978) and more recently Mancosu (2001)). Some proofs merely provide ‘justification’ whereas otherproofs are ‘explanatory’ in the sense that they explain why a result holds or provide some insight about theinvolved objects. In any case, mathematicians seem to prefer proofs that are not merely justificatory, andas this paper is concerned with mathematical practice, I will take this as an indication that there is more tomathematical knowledge than justification.

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