Structuralism as a Philosophy of Mathematics

Mathematical structuralism was developed to explain the ontology of mathematical objects, and is often contrasted with Platonism, which is the position that numbers are real things like you or I.

The point is that if you are interested in the structure of finite sets, then there are structure-preserving isomorphisms between the vertices of a triangle and the set {A, B, C} so that (for example) permutations of the set and reflections/rotations of the triangle are coincident.

But if you're interested in the structure of plane geometry then there is no such structure-preserving isomorphism because any map into a finite set will "delete" information about side length and angles.

The structuralist idea is that interesting mathematics can be "most easily" found by considering structure-preserving isomorphisms and not the structures themselves. In particular dealing with the structures directly can obscure the mathematics you are trying to discover, e.g. dealing with a full Euclidean group when the dihedral group is all the problem requires.

Well, given that structuralism as a program didn't exist until basically the 19th century, you can look at the entire history of mathematics to see what mathematics without structuralism looks like. It's sort of hard to argue that the isomorphism between symmetries and permutations is "obvious" when group theory wasn't invented until the 18th century, and symmetry groups weren't formalized until the 19th century. Your comment is basically this:

One fish says to another fish: "The water's nice today." and swims off, the other fish says "What's water?". Your entire mathematical world view is so permeated with the language of structuralism that you can't see it any more.

Representation theory should do it:

https://en.wikipedia.org/wiki/Representation_theory_of_finit...

Lawvere theories should be fine as well:

"The rough idea is to define an algebraic theory as a category with finite products and possessing a “generic algebra” (e.g., a generic group), and then define a model of that theory (e.g., a group) as a product-preserving functor out of that category."

https://ncatlab.org/nlab/show/Lawvere+theory#the_theory_of_g...

There are two kinds of mathematicians, or one can say, two cultures of mathematics: (a) problem solving, proving conjectures, etc (b) theories. Alexander Grothendieck belongs to (b). Paul Erdos to (a).

You can read Gower's paper on "The two cultures of Mathematics" at: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

Categories are the main and most developed tool building around relations that are the building blocks of structure.

The quote says why you should care about it. Without categoricity the axioms of a theory don't define a specific structure.

The fact that Peano Arithmetic isn't categorical, and that if you want to nail it down you need either bigger theories (within which you can of course nail down a single model) or move to theories which don't have proof theories like second order arithmetic, suggests all might not be as easy as it looks here.

Being an ex-proof theorist, I'm a bit dubious of not having one.

I think you are onto something. Study mathematical objects and relations among these objects, without bringing in Categoricity. That's what Structuralism in Philosophy of Sciences is: "In the philosophy of science, structuralism[α] (also known as scientific structuralism or as the structuralistic theory-concept) asserts that all aspects of reality are best understood in terms of empirical scientific constructs of entities and their relations, rather than in terms of concrete entities in themselves" [1]

[1] https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_s...

Or start with the morphisms, then look for the idempotents, so further up the structures fall out naturally?

(this program may have an advantage in that it motivates passing from the continuous to the discrete?)

If you want to see an organization of mathematical ideas, I'd recommend digging into https://us.metamath.org/. Great intro to formal systems, though many more layers of definitions than the OSI model.

Though, there may be canonical structures, any universal structure is illusive if not non-existent.

In a sense computer and electrical engineers/scientists did largely map out the base layers, over ~150 years from the 19th century through the mid 20th. I think equipment (broadly speaking) is foundational for mathematics. The "stack", as far as I interpret it, is something like

1. Being

2. Communication

3. Equipment -- a device you can put marks on and read off of

4. Discipline -- ability to reliably and skillfully manipulate the device

5. Submission

The stage is set at this point for some "elementary mathematics" -- think back to elementary school.

6. Symbolism -- the equipment is not just equipment. Mathematical relevance springs here.

7. Geometry -- from vision we see area and edges, objects of perception, they are interpreted as mathematically relevant and hence symbolized.

8. Algebra -- manipulating our equipment with discipline, an equivalence is perceived between different sequences of operations.

9. Proposition -- conviction the relevant facts of geometry and algebra can be formulated clearly in declarations of the sort "if ... then ..., and ... (... and etc)".

Higher level mathematics

10. Refinement 11. Proof

12. Application 13. Theory

14. Computer science, engineering, and design

...

Yes, typically the base layer is set theory (with ZFC being the "standard" formulation).

https://en.wikipedia.org/wiki/Foundations_of_mathematics

The computer and computer science is, in a sense, the result of the somewhat failed attempt to put mathematics on a solid logical grounds, in a way that resembles a stack like OSI. Gödel's incompleteness theorems and Turing undecidability results showed that it's not that easy. Briefly, the proof for the halting problem was the Turing machine.

Could you please briefly explain the indispensibility argument? (this being the first time I've run across the term; although if it's some philosophical thing I'm afraid I'm only interested in it as far as I'd be in the theological creation of the universe*)

My view of the relation between maths and physics is set out in https://news.ycombinator.com/item?id=39220159 , and because we go from physics to maths (f) and then back from maths to physics (f^-1), we can conjugate with any g,g^-1 pair, meaning that on the maths side we might as well mod out by isomorphics (only care about the partial order of equivalence classes of the preorder of structures)

(Similarly, we should find that we only care about the equivalence class [singular] of isomorphic realities [plural]?)

---

* this is not to say that I'm not interested; see https://news.ycombinator.com/item?id=39886966

Assuming a God who creates Creations, do we wind up with a best possible (principal) Creation? all possible (perhaps trivially so) Creations? if there is a set of Creations created, are they directed? Theologically, what happens if we have an infinite number of finite Creations, each an appropriate approximation, and we pass to their supremum?

Numbers have properties beyond their order, right, e.g. factorization.

Maybe the field you are looking for is Number theory? Or maybe algebra and group theory?

But I don't quite understand your sentence:

> [...] I feel a lot of detail is lost when we only think about the structure of an object as its total count, and not its composition/order.

Maths is concerned a lot with how numbers can be constructed or composed from other numbers, ir other mathematical structures.

Still, there is a difference between the abstract notion of 4 and objects where we assign some measurable quantity (e g. counting similar objects, let's say turtles, and saying in total it's "4 turtles").

Numbers are not concerned with counting alone, but that's the easiest way to construct numbers.

In some sense, basically every field of math makes this kind of distinction. e.g. it's acknowledged that for a vector space V, the tensor products V⊗(V⊗V) and (V⊗V)⊗V are not equal, but they are equivalent in the sense of e.g. a linear isomorphism, or maybe in the sense of an algebra isomorphism or something, and mathematicians do have machinery to track in what sense they consider two non-equal things to be equivalent, and they take the time to prove that operations they want to define on equivalent but non-equal things give equivalent results. Category theory is partly concerned with this kind of machinery.

So they do care about this, but then invented ideas like equivalence relations and quotients so that they can not have to constantly care about it.

With programming languages or a proof assistant like lean, by default non-equal things are not the same, and you need to do some bookkeeping to carry around proofs that things are equivalent in the way you need them to be. The tricky thing is actually making it ergonomic to be able to think of (1+1)+1 and 1+(1+1) as "the same" without constantly needing to think about it. It's hard to do math in most programming languages partly because they lack the machinery necessary to have contextual notions of equivalence (but also because they lack dependent types).

In some fields, that extra bookkeeping is really important. E.g. in geometry you might be able to choose coordinate systems at each point of some space, but perhaps not in a way that is globally defined and consistent with the types of equivalence that you need, so you need to track everything through local coordinate changes as you move around.

Or a simpler example is a vector space and its dual: they're isomorphic, so in some sense equivalent, but not canonically so, so you need to track the equivalence you're using, and maps on one are "mirrored" when applied to the other. More generally, any finite dimensional vector space is characterized (in the sense of equivalence up to linear isomorphism) by a single number (its dimension), so in some sense it "is" that number, but we also think of them as each being different, depending on context. So f.d. vector spaces can in some sense be thought of as "numbers that remember which vector space they are".

Well, there's the fundamental theorem of arithmetic, that there is exactly one composition of prime factors for every natural number (excluding ordering). https://wikipedia.org/wiki/Fundamental_theorem_of_arithmetic

That’s how integers are defined in set theory.

The distinction between "first order" and "second order" is in the first instance a distinction at the level of formal languages. A second order language has more complicated syntax and semantics: it allows variables in predicate position, which (in the standard semantics) take values from the entire powerset of the underlying domain.

This makes second order languages, including the language of arithmetic, much more expressive: they can distinguish models that first order languages can't. Those infinitely many non-isomorphic models of arithmetic expressed in a first order language can be distinguished, and excluded, as models of arithmetic expressed in a second order language. That's why second order arithmetic is categorical: all of its models are isomorphic.

Yes, a model of second order arithmetic contains a model of first order arithmetic, but within the second order language, you can say "which model it is" (up to isomorphism). It's only if you restrict yourself to a first order language that you can no longer say anything which will be true in that model, but false in any non-isomorphic one.

see Pierre Bordintello, "La Démonstration" (1984):

> Mathematics entails, and it entails the entailer.

The basic problem with computation is that identity among the inputs is not necessarily identity among the outputs, which explains why we often bring in bigger guns than Boolean logic.