>>11671244>>11671287>Why is there a meaningful difference between saying that two things are equal and that two things are the same?I don't want to get into language games. Informally, you never properly capture what you talk about, but you always need an informal level to communicate. Sentences like

>Two sides of an equilateral triangle are the same>The groups of two elements "(0,1) and addition mod "2, as well as "(1,-1) and multiplication" are the same.Are two ways to start a discussion and distionction.

It would be easier to give you a satisfying answer if you pin your words down a bit, maybe even formalize things a bit. A lot can be said about equality and fields of math and philosphy are dedicated to it.

One must also take good care about mathematics "in-theory" statements, their meta-theoretical ramifications as well as those on the informal meta-level thereof.

Keywords are structural vs. material theories, extenionality and intentional equalities, substitution, isomorphism, explicit equivalence or apartness demands in constructive theory, etc., etc.

Here's some link relating to ways to approach the broader issue

https://ncatlab.org/nlab/show/structural+set+theoryhttps://en.wikipedia.org/wiki/Intuitionistic_type_theoryhttps://en.wikipedia.org/wiki/Confluence_(abstract_rewriting)https://en.wikipedia.org/wiki/Apartness_relation>Has anybody studied...Yes, the philosphy of 40's category captures this indirectly - two objects found to be isomorphic (like the (0,1) and (1,-1) example, as modeled in a category of sets, found to be groupy objects) are understood of having invertible arrows between them such that one can translate the objects between any diagram expressing one and the same thing.

Martin Löf Type theory formalizes this, even.

>can you do set theory with...Check out Bishop set theory for one view on it. Also afaik you get far even without extensionality in set theory, by considering equivalence classes.