### The Liquid Tensor Experiment

#### Posted by David Corfield

Peter Scholze has just published a challenge to the automated mathematical formalisation community in a post – Liquid tensor experiment – on Kevin Buzzard’s blog. Peter explains there the motivation for the theorem he would like to be formalised, and his reasons to want a formal computer-checked proof.

I see a couple of Café interests intertwined here: formalisation in dependent type theory, and the nature of (mathematical) space.

Regarding the former, there was an intense discussion recently on MathOverflow arising from a comment on the advantages of dependent type theory by Kevin, who for some time has been promoting the prover *Lean* as the best means to formalise the mathematics of typical mathematics departments.

In the comments to the main question, there is some discussion of why much more has been achieved in this regard by Lean compared with HoTT/Univalent Foundations approaches. We’ve heard plenty about this latter approach at the Café, in particular about its use to develop a kind of synthetic mathematics, due to HoTT being the internal language of $(\infty, 1)$-toposes. At the end of the references in the last link are some articles using modal HoTT, in particular relating to the *cohesive* modalities, as in Mike’s

*Brouwer’s fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28(6), (2018): pp. 856-941 (arXiv:1509.07584).

That brings me to the second issue – the nature of (mathematical) space. My question on the comparison of condensedness/pyknoticity and cohesion generated an interesting discussion through which I think it became clear that the former is pointing in a somewhat different direction to the latter.

One of the main motivations for the ‘solid’ and ‘liquid’ mathematics in Peter’s condensed mathematics program is to allow a better approach to functional analysis. Of course, Lawvere, having defined cohesion, also looked to found functional analysis in terms of something else, namely bornology, and Peter told us there that

condensed sets are bornological sets equipped with some extra structure related to “limit points”, where limits are understood in terms of ultrafilters.

I suppose one day there might be a synthetic treatment of condensedness, but the following from Peter’s post suggests an analytic approach is in order for the challenge, where he describes the proof to be formalised as:

…a very delicate argument fighting with estimates against homological algebra. There is a hierarchy of implicit constants that have to be chosen, and you have to do it in exactly the right order. In the end, one formulates Theorem 9.5 which can be proved by induction; it is a statement of the form $\forall \exists \forall \exists \forall \exists$, and there’s no messing around with the order of the quantifiers. It may well be the most logically involved statement I have ever proved. (On the other hand, if I want to claim that the theorem on liquid vector spaces makes it possible to black box functional analysis, hard estimates have to be

somewhere. Better be sure the estimates actually work…!)

## Re: The Liquid Tensor Experiment

Hi. Just to give you the goss about condensed v pyknoticity – right now we are sorely tempted to just ignore all the cardinal bounds and formalise condensed sets using the Barwick/Haine approach, based on the computer science maxim “build one to throw away”. Basically we’re going to assume a couple of universes exist and see what happens, on the basis that it will simply make our lives much easier, at least when it comes to actually getting started.