Comment by Atiscant

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.