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Structuralism Notes

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Structuralism Structuralism is the claim that mathematics is the study, not of particular objects, but of certain kinds of structure.

Motivation Many structuralists point to two considerations, both expressed by Paul Benacerraf (in his 'Mathematical Truth' and 'What Numbers Could Not Be' respectively):

1. The difficulty in including (a platonist reading of) mathematics within a plausible epistemology. Benacerraf takes this to be causal, but more weakly we could see this as just a demand a plausible epistemology for mathematics coheres with an epistemology for other subjects.

2. Multiple identification problems. We can also see the structuralist as making two observations:

1. That the (second-order) accounts of the real numbers, and of the natural numbers, are both categorical: all their models are isomorphic. In a certain sense, then, it doesn't matter which progression we choose for the natural numbers. (This realisation goes back at least as far as Dedekind's 'Was sind und was sollen die Zahlen?', where he proves a version of what we would now call the categoricity theorem.)

2. That from a mathematical point of view certain 'further' questions are irrelevant, particularly with respect to determinate reference or extra-structural properties. This is brought out both by Geoffrey Hellman and by Paul Benacerraf. Benacerraf argues that the philosopher's asking 'yes but what is an element of this [mathematical]
structure?', is mistaken: that he misses the point "of what arithmetic, at least, is all about." Hellman draws attention to similar "bad questions" which "seem entirely irrelevant to mathematical inquiry...utterly alien to the discipline", such as 'how do you know to which o-sequence you are referring when you speak of the natural numbers?' or 'how do you know the reals are densely ordered?' Identification problems: 'What Numbers Could Not Be' Benacerraf presents what has come to be known as the 'identification problem' for any attempted identification of particular objects with the natural numbers. His argument starts from the observation that we have an infinite number of ways of identifying the natural numbers with certain sets, most popularly the von Neumann and Zermelo approaches: 1 = {o}
2 = {{o}}
...

1 = {o}
2 = {o, {o}}
...

The question the identification problem asks is 'which column provides a correct analysis of, say, the number 2?' Clearly both accounts cannot be correct, at the risk of straightforward settheoretic contradiction: 2 = {{o}} = {o, {o}} is absurd. Hence, Benacerraf argues: 1

1. At most one of the accounts can be correct.

2. If one account is correct, then there ought to be some reason to favour that account over the other.

3. There is no such reason. Indeed, the differing accounts, when provided with the natural definitions of addition and multiplication, yield elementarily equivalent structures: they agree on the set of true sentences.

4. Therefore, no identification of numbers with sets cannot be correct. In other words: numbers are not sets. As Leon Horsten expresses the problem, "on the one hand, there appear to be no reasons why one account is superior to the other. On the other, the accounts cannot both be correct". Benacerraf then generalises his point beyond just set-theoretic identifications. The crux of the argument is the claim that choosing any progression in particular to identify with the natural numbers would fall into the identification problem: we could ask the question 'why this progression, and not another?'. As Gabriel Uzquiano reconstructs the argument:

1. Because of the identification problem, no identification of natural numbers with objects that have non-structural properties could be correct.

2. All objects have non-structural properties.

3. Therefore, no identification of natural numbers with objects can be correct. In other words: numbers are not objects. The lesson Benacerraf draws is that what matters is not the individuality of the objects, but their overall structure: numbers cannot be (particular) objects because the essence of what makes them perform the role of numbers - their forming a progression - is purely structural. As Benacerraf puts it, " 'Objects' do not do the job singly; the whole system performs the job or nothing does...arithmetic is the science that elaborates the abstract structure that all progressions have merely in virtue of being progressions." Evaluating Benacerraf's argument
? Overly positivistic in demanding some evidence for favouring one account over another?
? Do pragmatic reasons suffice? --- not in the kind of way Benacerraf is looking for. Advantages of the structuralist approach: first glance Many structuralists argue that the structuralist approach has significant benefits, particularly in its ability to steer a kind of middle course between what Geoffrey Hellman calls the "Scylla and Charybdis" of platonism (with its problematic epistemology), and constructivism (with its problematic view of mathematical truth). In his 'Mathematics Without Numbers: Towards a Modal-Structural Interpretation', Hellman presents four "desiderata" for philosophies of mathematics, and argues that structuralism comes closest to fulfilling all four. For Hellman, a successful theory ought to

2 1. Maintain that the statements of mathematical discourse are truth-apt, and have determinate truth value independent of our minds.

2. Extend to a reasonable account of mathematical knowledge, which is capable of being integrated with the rest of human knowledge.

3. Account for (if not uphold) the prima facie a priori status of mathematics.

4. Provide a reasonable explanation of how mathematics applies to the real world. The structuralist argues that by turning to structural considerations he can have his cake and eat it: that he can hold on to the benefits of realism (1, 3, 4) while avoiding the problems the nature of platonic objects brings (2). However, to be successful, he needs to flesh out his view of mathematical structures, and his treatment of mathematical objects. Applications to the physical world Charles Parsons defends the structuralist on his ability to account for 'external' relations. As Parsons expresses it, the objection runs that the application of arithmetic requires relations of numbers that are not even part of the world of mathematical objects. If counting certain objects is just placing those objects in a 1-1 correspondence with the numbers 1, 2, ..., n, how can the eliminative structuralist interpret these relations?
Parsons' reply is that the relations between elements of a progression will only count as relations of numbers if they are invariant under choices of "realisation" of the structure. His argument runs as follows:

1. Suppose that and are simply infinite systems. By categoricity there is an isomorphism h mapping N onto M.

2. Now suppose that for n in N there is an 1-1 correspondence f between some Fs and
{m: m ie N and m [?] n under S}.

3. Then we may set g(x) = h[f(x)], so g is a 1-1 correspondence between the Fs and {m: m ie M and m [?] h(n) under R}. Hence, Parsons concludes, if one concludes that there are n Fs on the basis of f, then using M one would conclude on the basis of g that there are h(n) Fs, which is right. Thus we can establish the kind of external relations we need. Indeed, Parsons extends this in general. Suppose we have structures S and T with domains S' and T' respectively, and an isomorphism h of S onto T. Then for a relation R of elements of S to some set U we can define R' such that xR'y IFF [?] z [?] S [ x=h ( z )[?]zRy] . Then for any x in S', and y in U, xRy holds IFF h(x)R'y and R' can 'do the work' of R.

Varieties of structuralism The different varieties of structuralism differ primarily in how they flesh out the notion of structure. At a general level of ontology, we can follow Stewart Shapiro's distinction of how an object might be taken to exist:

1. Ante rem ('before the thing'): accept a realist, platonist ontology in which the object in question exists independently. 3

2. In re ('in the thing'): accept a moderately realist, Aristotelian position according to which the object exists in so far as it is exemplified in the concrete.

3. Post res ('after the thing'): accept an anti-realist position, paralleling nominalism. Deny the existence of the object and attempts to parse away talk of the object through some translation scheme. Accordingly, we can divide up structuralist views along the same lines. We neglect the in re approach because it is highly problematic (for example, in its contingency on the concrete) and so we get the following:

1. According to the ante rem structuralist [abstract structuralist, non-eliminative structuralist], surface form is logical form. Mathematical structures are entities in their own right, and places in a structure are then construed as bona fide objects, capable of being denoted by a singular term. However, these elements have no nonstructural properties: as Bob Hale puts it, they are " 'bare positions' in the structure, having no properties save those which derive from, or consist in, they bearing structurally relevant relations to one another".

2. According to the post res structuralist [pure structuralist, eliminative structuralist], apparent talk of structures is a mere facon de parler. As Bob Hale puts it, "any given mathematical theory is purely structural in that the information it conveys is completely general". Mathematical 'objects' are not genuine singular terms: they are purely schematic, and capable of being eliminated by an appropriate translation scheme. Hence, surface form does not reflect underlying logical form. Post res/ eliminative/ pure structuralism The pure structuralist approach is the one closest to the conclusion that Benacerraf draws from his identification problem, and seems (on one reading) to come out of Dedekind's considerations. According to this approach, talk of mathematical objects is just a way of expressing the relevant generalisations over all systems: a sentence S in the language of arithmetic is to be interpreted as through a paraphrase of the form 'if X were any progression, S would hold in X'. Advantages of the pure structuralist approach The pure structuralist seems to have three advantages:

1. He appears to be able to provide a successful epistemology for mathematics: to know the truth of statements in a theory is to know what follows from the description of the theory. The theorems of mathematics are thus maintained as objective, truth-apt, and mind-independent.

2. On the other hand, he avoids what Bob Hale calls "ontological inflation". Through his translation scheme, the pure structuralist can hope to avoid all commitment to abstracta, or reduce the commitment to an unproblematic, limited set of abstracta.

3. The kinds of 'bad questions' highlighted by Benacerraf and Hellman are blocked: there simply is no further question about whether or not the real reals are densely ordered. In studying the reals, we are studying a dense order: we may study other kinds of structure, which are not densely ordered, but these are not the reals. 4

Non-modal eliminative structuralism According to the non-modal approach as described by Charles Parsons - the approach possibly closest to Dedekind's own - we construe a statement in the language of arithmetic with a set N, a distinguished element 0, and a mapping S such that induction holds by A(N, 0, S). Let us follow Dedekind (and Parsons) in calling a simply infinite system a structure
with N, 0, S as above, and denoting it O(N,0, S). Then a statement about the natural numbers becomes elliptical for 'for any N, 0, S, if O(N,0, S), then A(N, 0, S)'. Hence, by categoricity, this claim is true if A(N, 0, S) holds in any simply infinite system. There is an immediate problem with this approach, however: what if there simply aren't any simply infinite systems in the relevant sense? Then the antecedent becomes vacuously true and hence any arithmetical statement comes out true under the translation. So, on pain of accepting the possibility that arithmetical claims such as '2=1' as true, some kind of background ontology is required. It faced with this difficulty that many authors have turned to a modal approach. Set-theoretic structuralism Can a non-modal approach be made to work? One version of this approach, advocated by
Oystein Linnebo, would be to accept an ontology of sets for all of mathematics. Then every mathematical theory apart from set theory is understood in eliminative terms, with set theory requiring special justification: perhaps in the vein of Penelope Maddy's 'set-theoretic perception', or by building on George Boolos' 'stage theory' for ZFC. Geoffrey Hellman rejects the set-theoretic approach for two reasons:

1. Pending a structuralist view of set theory, it relies on the assertion of the set theoretic axioms - and all the ontological commitments that come with this.

2. It makes number theory dependent on set theory, and thereby saddles number theory with all the set-theoretic problems of 'Cantor's universe'. For Hellman, number theory ought to be able to "stand on its own two feet". According to Hellman, moving to a weaker set theory with fewer ontological commitments is not a solution. It would be too weak: it would restrict the generality of structuralism to only those structures expressible within the weaker framework. But then we would lose the study of arithmetic as about arbitrary structures of the right type. Arithmetic would instead become the study of arbitrary structures recognisable in this framework. Certainly a commitment to the axiom of infinity in the foundations of mathematics would appear to be problematic, in that it would forego the advantages of the eliminative reading's unproblematic ontology. We would be left in a position akin to Russell's: trying to justify why there ought to be a brute stipulation for an infinity of objects. It is worth noting that the axiom of infinity is one which Boolos struggles to provide an intuitive basis for within his stage theory, and therefore may be seen as lacking justification in a way that, say, Extensionality does not.

5 Charles Parsons points out, however, that this kind of approach need not require the axiom of infinity. By taking the natural numbers to be the finite von Neumann ordinals, we could significantly reduce the ontological extravagance at issue. Doing so, however, would likely come at the cost of admitting impredicatively defined classes. Given the controversial nature of the vicious-circle principle, this is something we might well wish to avoid. If we are willing to admit impredicatively-defined classes, then, and to tie ourselves to a Maddy- or Boolos-style explanation of the ontology of set theory, we could deflate the ontological objection. The force of Hellman's rejection then turns on the commitments we want to saddle number theory with. His point has intuitive force: number theory ought to be independent in some sense from set theory. It is not clear that the discovery of a contradiction in the set-theoretic development of number theory would cause us to change our arithmetical practice, for example. However, this does not appear to be a knock-down argument. As such, we might conclude that the set-theoretic structuralist's approach is at least a possible one, pending these further considerations. Non-modal nominalist structuralism A different approach to the problem would be to get rid of any reference to abstract at all, to completely deflate the ontology. Under this view, we could prevent the vacuity problem by describing a domain of actual objects for the quantifiers to range over. However, this approach can be rejected. As Russell made clear, mathematics ought not be contingent on features of the physical world. In fact, the problem is even more acute here: the non-modal nominalist structuralist faces the prospect of arithmetic becoming inconsistent if there is no progression to satisfy the antecedent. But even if the physical world happens to instantiate a progression at this moment, there is no guarantee that it will in the future. So it is possible that arithmetic is consistent at the moment, but won't be in the future. Surely this goes too far. Even a Hartry Fielidan appeal to the nature of space and an uncountable number of spacetime points will not satisfy us on this score. Charles Parsons argues that this approach presupposes a hypotheses that is "stronger and more specific than needed". Any such approach makes mathematics contingent on the physical world, and so vulnerable to refutation. But this refutation will not affect mathematics: it seems strange to say that the discovery that spacetime is quantised (as, I believe, some string theorists believe) should undermine our belief in simple arithmetical claims. Modal eliminative structuralism If we wish to reject an ontology entirely, therefore, a modal approach - following Hilary Putnam's move in his defence of the claim that arithmetic could be reduced to second order logic - seems attractive. Geoffrey Hellman works out this position in some detail in his 'Mathematics Without Numbers: Towards a Modal-Structuralist Interpretation'. Hellman's approach has two steps. In the first, hypothetical component, a suitable elimination scheme is provided. So statements in the language of Peano arithmetic, for example, become elliptical for statements saying that 6

'if there were progressions satisfying PA's axioms, the statement would hold in them'. The second categorical component is to assert the right kinds of structures are logically possible. His approach is to take modal operators as primitive - with their use explicated by the system as a whole - and use them to reconstruct platonist discourse so as parse away all commitment to abstract objects. As Hellman puts it, he seeks "an alternative, non-literal interpretation of mathematical discourse which can be understood as realistic but in which ordinary quantification over abstract objects is eliminated entirely". For the hypothetical component, he parses the translation scheme as follows. A arithmetical claim A in the language of second order PA is given a modal mathematical translate Amsi 2

[?]P A - A

(where msi stands for 'modal structuralist interpretation') given by

X ( sf)

#[?]X[?]f

,

2 where [?]P A is the conjunction of the second-order Peano Axioms, the two-place relation

variable f replaces the constant s throughout the conditional (else the sentence contains a relation constant for the successor that is best understood as set-theoretically defined), and the relatavisation to the domain X is to ensure the translation is not schematic (so that the metatheory is not doing the work of expressing structuralism). To avoid the problem of vacuity, Hellman takes "as a basic thesis of modal mathematics" the 2

[?]P A

claim that the appropriate kind of progression exists, i.e. that

X (fs)

*[?]X [?]f

. For Hellman it

is "absolutely essential to affirm, categorically [this claim]". As he puts it, "[the claim]
affirms the coherence of the notion of an o-sequence, something that is generally taken for granted, but which nevertheless forms an indispensable 'working hypothesis' underlying mathematical practice". This "modal-existence postulate" is supposed to be a starting point for our ordinary reasoning 'about numbers'. Applications to the physical world are understood in the following way. Mathematics is applied by establishing an appropriate isomorphism between parts of mathematical structures and the structures representing the material situation. Such an isomorphism establishes the relevant structural equivalence between the mathematical and non-mathematical. However, as Otavio Bueno notes, it seems difficult for the structuralist to explain how we can know the relevant equivalence holds in a non-circular fashion: to simply claim its existence is to beg the question; not to would presumably require some appeal to physics, which themselves typically rely on mathematics. But the issue at stake is precisely to explain how physics and other sciences are so amenable to mathematical treatment.

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