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Philosophy Notes Philosophy of Mathematics Notes

Structuralism Notes

Updated Structuralism Notes

Philosophy of Mathematics Notes

Philosophy of Mathematics

Approximately 53 pages

Notes made for the Philosophy of Mathematics paper at the University of Oxford.

Each set of notes brings together in detail all the major areas needed to write a first-class essay on that topic. Key arguments and positions from both primary and secondary sources are summarised clearly: perfect as a basis for an exam essay or as a primer on the subject. Also includes a summary of key positions, including strengths and weaknesses, within the philosophy of mathematics.

The notes draw on key re...

The following is a more accessible plain text extract of the PDF sample above, taken from our Philosophy of Mathematics Notes. Due to the challenges of extracting text from PDFs, it will have odd formatting:

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 ω-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 = {ø} 1 = {ø}

2 = {{ø}} 2 = {ø, {ø}}

… …

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 set-theoretic contradiction: 2 = {{ø}} = {ø, {ø}} is absurd. Hence, Benacerraf argues:

  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

  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....

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