Introduction to Homotopy Type Theory
I realize it&#x27;s Christmas Eve, but this post tempts my inner curmudgeon.<p>I do not understand why homotopy type theory posts are so popular on this website. My view is that all the &quot;philosophical&quot; arguments in favor of it (vs. the standard set theory foundations) misunderstand the issues at play. Further, the &quot;practical&quot; arguments in terms of facilitating formalization are not so compelling given the HoTT people haven&#x27;t actually (as far as I know) formalized much mathematics - whereas (seemingly) less ideological communities like users of Lean have made great progress.<p>To expand on the comment about the philosophical arguments: take for example the abstract of this article. It states:<p>&gt; It is common in mathematical practice to consider equivalent objects to be the same, for example, to identify isomorphic groups. In set theory it is not possible to make this common practice formal. For example, there are as many distinct trivial groups in set theory as there are distinct singleton sets. Type theory, on the other hand, takes a more structural approach to the foundations of mathematics that accommodates the univalence axiom. This, however, requires us to rethink what it means for two objects to be equal.<p>It is sometimes quite useful in practice to recognize that two isomorphic objects are not literally the same. So I am skeptical of any approach that wants to blur those distinctions.<p>Also, more to the point: ZFC does everything we need a foundation to do extremely well, except serve as a basis for practical formalization of proofs.