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Comment by moomin | Hacker News Reader

I’m not sure what a mathematical approach _without_ structuralism would look like. Like, if you consider the operations created by combining the rotations and flips of a triangle, and the set of permutations of the letters A,B and C, it’s pretty obvious they’re isomorphic and also obvious that they’re different. My question is: is there a mathematically useful way of expressing that difference?

Or to put it a different way, I’m not sure anything interesting is being said here.

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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.

I think sometimes obvious concepts like symmetries are hard to distill into the appropriate mathematical language. I’m willing to bet the isomorphism there is obvious to most folks, but the expression of its mathematical essence is not.

This is really an important point for mathematical philosophy. It is a single enterprise going back to ancient Babylon, China, India, Greece, Egypt, and before. The elementary meta-theory needs to be syntonic to mathematical activity and knowledge predating (or otherwise practiced without) formalism, structuralism, categoricity, platonism, etc.

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.

If you want, it's even possible to fully commit to the isomorphism point of view by saying that you'll represent the structures themselves by their identity isomorphisms, but then if you "delete" the little tag telling you which particular endo was the identity (would a physicist say "up to phase"?), you might discover other interesting things...

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

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.

I'm not real. I'm not sure about you.

Found the co-solipsist.

The working mathematician is a Platonist on weekdays, a formalist on weekends.

Reuben Hersh

A funny insight about Platonism (if it's not funny, treat this as a bad joke)

I think aka "∅" therefore I know I thought aka "{∅}" therefore I know I knew I have thought aka "{{∅}}"

and ...

boom! The entire Math system is imported. (BTW, limitations known as "computation theory" is also introduced)

So maybe we are actually mathematical being on a manifold named as "real world". To me it is more concise and profound than "philosophy". As we are real, so do all mathematical objects.

The problem is: do numbers exist? One group says, they don't. Another say, they do exist in the Platonic world, but this Platonic world is accessible from the world we inhabit in. Next time, look at the folks who looks for Platonic love:)

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...

Representation theory should do it:

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

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