[mbid]

I didn't mean to critize all axiomatization but the concrete axiomatizations arrived at by PL research. In my opinion, one should start with a thorough understanding of the problem domain, and then try to come up with a syntax that allows expression of your intuition as faithfully as possible. Form follows function. Instead, I feel like PL research is often too focused on the concrete syntax considered, thus function follows form. Case in point: Building models for the syntax at hand is often an afterthought, done only to prove consistency. I'd argue that one should start with as precise an understanding of the intended models as possible, and then come with up with an axiomatization (thus, syntax) for the intended semantics.

For example, the notion of elementary topos has been invented because its creators wanted to capture the way Grothendieck toposes kind of behave like constructive sets. This I find very useful, and also the axiomatizations of elementary toposes. On the other hand, Martin-Löf type theory didn't have a formal semantics at first, then an erroneous one, and finally ~20 years later a kind of acceptable one. And its category of models is... not really interesting. Except for categories of assemblies, I don't know of a single model of ML type theory that's not also an elementary topos. And the interesting thing about assemblies is that they can be turned into objects of the associated topos... so yeah.

[joel_ms]

And yet MLTT led to Coq, Agda, Idris and Lean, while your ”PL practice” approach sounds like it would lead to, well, Go.