criticisms of mathematical structuralism

Question

A popular, pretty modern trend in the philosophy of mathematics has been to treat mathematical objects as only possessing properties within the context of a mathematical structure. Does anyone know of any books/papers discussing criticisms of, or problems for, mathematical structuralism?

Answer 1

The Internet Encyclopedia of Philosophy article on Mathematical Structuralism, written by Stewart Shapiro, is a brief but nice introduction to the subject. For Shapiro, there is a through line from David Hilbert and Paul Benacerraf to the current theme of the role of structures in the philosophy of Mathematics.

Shapiro poses that Structuralism tends to come in two distinct forms. Do you think that Structures are abstract objects (like Universals), such that the statements of mathematics have a kind of "face value" realist reading? Or is talk of Structures ultimately eliminable, shorthand fictions or generalizations that make mathematics easier but aren't constitutive of the ontology of mathematical reality?

A natural question on the outside to ask here is "how in the world are these the same position?" Platonists and Aristotleans have been debating this since ancient Athens, and if Structuralism doesn't answer it, does it really have a claim to being a distinct philosophy?

There is a really nice paper by Kate Hodesdon that proposes that if what joins these two positions together is a thesis about the "incompleteness" or "multiple realizability" of mathematical objects (that individuals in number or set theory, for example, are only positions in structures that might be instantiated in many different circumstances) then their theory is too general to successfully capture anything informative about mathematics as such. If Ante Rem and Post Rem structuralists are in agreement about structures, the way in which they agree is about Philosophical methodology.

So, while structuralist ideas in the philosophy of mathematics have useful contributions, being a Structuralist does not in itself answer key questions about what mathematical structures are, how one comes into epistemic contact with them, and thus how people do in fact practice mathematics to a degree of confidence in the grounding of their field. Some further description of mathematical practice and theorycrafting is required - in particular, how do Proofs and Representation work in practice, and how does one come to learn to do them.

Answer 2

I haven't found a broad set(!) of answers to your question, but I did find this:

In contrast, category theory represents a branch of abstract algebra, as its origin reveals. Thus it is, by its very nature, non-assertoric in character; it lacks existence axioms conceived as truths about an intended universe. For example, the Eilenberg-Mac Lane axioms of category theory are not “basic truths simpliciter”, but “schematic” or “structural” in character. They function as implicit definitions of algebraic structures, similar to the way in which the axioms of group theory or ring theory are “defining conditions on types of structures”. This point is related to another argument against the autonomy of category theory that Hellmann calls the “problem of the ‘home address’: where do categories come from and where do they live?” (2003: 136). Given the “algebraic-structuralist perspective” underlying category theory and general topos theory, its axioms make no assertions that particular categories or topoi actually exist. Classical set theories, such as ZFC with its strong existence axioms, have to step in again in order to secure the existence of such objects.

The quote concerns a brand of structuralism that is category-theoretic in nature. I don't see the argument as posing an insuperable problem for this brand: why not just add in some existence claims about categories/structures? Perhaps it has to do with the essential "flavor" of such theorizing (the taste is inimical to such existence claims?). At any rate, again, this is a critique of a subset(!) of structuralist claims, not the entire philosophical standpoint overall.