My last post was, amongst other things, a challenge for the community behind every modern computer proof verification system to start formalising “proper mathematics”. This term has some problems, so how about the equally vague but perhaps less offensive “fashionable mathematics”. What mathematics is fashionable? Just take a look at the work of the recent Fields Medallists. That’s a pretty good way of telling.
But fortunately, unlike many other fashions, “fashionable mathematics” is not controlled by the whim of big companies or some cabal. Fashionable mathematics is mathematics which justifies itself by its ability to answer questions which have been previously deemed interesting or important. Peter Scholze’s definition of a perfectoid space opened a door. In the last ten years, perfectoid spaces have been used to prove new cases of the monodromy-weight conjecture, to prove the direct summand conjecture, to give a new proof of purity for flat cohomology, a strengthened version of the almost purity theorem and so on (and I didn’t even mention applications to the Langlands philosophy). These are results whose statements do not mention perfectoid spaces, and some have been open for decades. This is what makes this kind of mathematics fashionable — it is giving my community new insights.
Each formal proof verification system (Lean, Coq, Isabelle/HOL, UniMath, all of the others) has its own community, and it is surely in the interests of their members to see their communities grow. These systems are all claiming to do mathematics, as well as other things too (program verification, research into higher topos theory and higher type theory etc). But two things have become clear to me over the last two years or so: