As one of those that do not like the “sets at the bottom” approach I just want to highlight why. For me, mathematics built on sets have leaky abstractions. Say I want natural numbers, I need to choose a concrete implementation in set theory e.g. Von Neumann, but there are multiple choices. For all good definitions, so get Peano arithmetic and can work with, but the question “Is 1 and member of 3” depends on your chosen implementation. Even though it is a weird question, it is valid and not isomorphic under implementations. That is problematic, since it is hidden in how we do mathematics mostly. Secondly, it is hard to formalize, and I think mathematics desperately needs to be formalized. Finally, I do not mind sets, they are great, and a very useful tool, I just do not like they as the foundation. I firmly believe we should teach type theoretic or categorical foundations in mathematics and be less dependent on sets.

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librexpr 2 days ago | parent [-]

> Say I want natural numbers, I need to choose a concrete implementation in set theory e.g. Von Neumann, but there are multiple choices.

You don't need to choose a concrete implementation. If you don't want to choose a construction, you can just say something like "let (N, 0, +, *) be a structure satisfying the peano axioms" and work from there.

> For all good definitions, so get Peano arithmetic and can work with, but the question “Is 1 and member of 3” depends on your chosen implementation. Even though it is a weird question, it is valid and not isomorphic under implementations. That is problematic, since it is hidden in how we do mathematics mostly.

Why is that problematic? The constructions are isomorphic under the sentences that actually matter. This kind of statement is usually called a "junk theorem", and they are a thing in type theory too, see for example this quote from a faq by Kevin Buzzard about why Lean defines division by zero to be zero:

> The idiomatic way to do it is to allow garbage inputs like negative numbers into your square root function, and return garbage outputs. It is in the theorems where one puts the non-negativity hypotheses.

https://xenaproject.wordpress.com/2020/07/05/division-by-zer...

> Secondly, it is hard to formalize, and I think mathematics desperately needs to be formalized.

Is that actually true? At the very least writing out the axioms and derivation rules is easier for set theory, since it's simpler than type theory. And there has been plenty of computer-verified mathematics done in Metamath/set.mm and Isabelle/ZF, even though less has been done than in type theory. Currently the automated tools are better for type theory, but it seems likely to me that that has more to do with how much effort has been put into type theory than any major inherent advantages of it.

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More generally, types in type theory are also constructed! The real numbers in Lean don't come from the platonic realm of forms, they are constructed as equivalence classes of cauchy sequences. And the construction involves a lot of type-theoretic machinery which I'd usually rather ignore when working with reals, much like I'd usually rather ignore the set-theoretic construction of the real numbers. And the great thing is that I can ignore them, in either foundation!

So I just don't really buy these common criticisms of set theory, which to me seem like double standards.

	▲	Atiscant a day ago \| parent [-]
		Sure you can work around it most of the time, but some times you cant. The whole point is that isomorphic is not equality in set theory, and sometimes proofs does not transfer along isomorphism because they refer to implementations. I agree that it is much preferable to work with abstract structure, but that not always what happens in practice. The natural number example is contrived but easy to see. My point of view is also that I do not like the Lean approach. It would actually like no junk theorems to exist in my theory. I am much more partial to the univalent approach and in particular univalent implementation that compute e.g. cubical. Regarding how easy it is to formalize, you are right. Lots of good work happens with set theory based type theory. My point was also that set theoretic foundations themselves are very hard to formalize, e.g ZFC + logic is very difficult to work from. A pure type theoretical foundations is much easier to get of the ground from. To prove that plus commutes directly from ZFC is a nightmare.