Lean offers rigour.

For years, people have been pointing me to Thurston’s work “On proof and progress in mathematics”, a work which I think of as a brilliant exposition of the importance of non-formalisation in mathematics. But it was only this week that I actually read the article Thurston was responding to — Jaffe and Quinn’s “Theoretical mathematics”. This paper opened my eyes. They have a word for how I now think about mathematics! They explain that I am now doing something called rigorous mathematics. This phrase gives me the language to explain what I am trying to tell mathematicians.

What is rigorous mathematics? It means mathematics which is guaranteed to be correct. Undergraduate-level mathematics is clearly rigorous mathematics: it has been checked from the axioms by many many people. What else? Perhaps SGA is a good example. It has been around since the 1960s and has been read by many people. The work of Grothendieck and others written up in this seminar was used in Ribet’s work on level-lowering, which was in turn used in Wiles’ proof of Fermat’s Last Theorem. But not all of our human literature is rigorous mathematics. We can basically guarantee that one of the papers published this year in one of the top mathematics journals will, within the next few years, be partially retracted, and nobody is asking the questions about whose job it is then to go through all the papers which cited it and to meticulously check that none of the citations were to parts of the paper which were wrong. I think defining rigorous mathematics in these human terms, trying to assert which parts of the literature are “rigorous”, is a hard problem. Perhaps Bourbaki is rigorous. Patrick Massot found a slip or two in it when formalising parts of Bourbaki’s Topologie Générale in Lean, but nothing remotely serious. Perhaps SGA is rigorous too — after all, hasn’t Brian Conrad read it or something? But actually Bourbaki and SGA both use different foundations, and neither of them use ZFC set theory (Bourbaki rolled its own foundations and then apparently ignored them anyway and just did mathematics, and SGA needs universes in places, but, as Brian Conrad has carefully checked, not in the places used to prove Fermat’s Last Theorem).

I am not sure I know a good answer for exactly which parts of the mathematical corpus as represented by our human literature is “mathematically rigorous”. Maybe the way it works is that anything published in a journal over 20 years ago and which hasn’t been retracted yet is mathematically rigorous. But probably some recently graduated PhD students would also say their work was mathematically rigorous as well. Perhaps the definition most mathematicians work with is that rigorous mathematics is “the part of published mathematics which the elders believe to be correct”.

There is another question too, of less interest to most mathematicians but of some importance when trying to figure out what we mean by rigorous mathematics, and that is the problem of saying exactly what the axioms of mathematics are. Are we allowed universes or not? Whose job is it to tell us? Scholze’s course cites, but does not use, a paper by Barwick and Haine which uses universes everywhere. De Jong puts some effort into explaining how the stacks project can be made to work in ZFC, but honestly who ever reads those bits of the stacks project? Scholze talks about infinity categories in his course and no doubt he can solve all set-theoretic issues — after all he did take the trouble to write section 4 of etale cohomology of diamonds. Emily Riehl and Dominic Verity are writing a book on infinity categories . I bet that book will be rigorous. Wouldn’t it be interesting checking it was rigorous in Lean? I think that this is an important question. Lurie needs infinity categories and his work is having wider implications across mathematics. The Clausen and Scholze work also needs infinity categories and is claiming to prove Grothendieck duality in a completely new way. Is this work definitely rigorous? What about this work of Boxer, Calegari, Gee and Pilloni, announcing some major new results in the Langlands programme but subject to 100 or so missing pages of arguments which are not in the literature but which experts know shouldn’t present any major problems (see the warning in section 1.4 on page 13). I have now seen a citation of that paper in another paper, with no mention of the warning. Is this what rigorous mathematics looks like? I am not suggesting that the work is incorrect, incomplete or anything, and I am not even suggesting that the human race is incapable of making the results in this 285 page paper rigorous. But is it, as it stands, rigorous mathematics? I have my doubts. When David Hansen stands up in the London Number Theory Seminar and claims that a certain construction which takes in a representation of one group and spits out a representation of another one is canonical — and then refuses to be drawn on the definition of canonical — is this rigorous? And if it isn’t, whose job is it to make it rigorous? I am well aware that other people have different opinions on this matter.

I think that every one of the computer proof systems — Isabelle/HOL, UniMath, Coq, Lean, Mizar, MetaMath and all the others, all represent candidate definitions for “rigorous mathematics” which are far better defined than the human “definition”. Which system is best? Who knows. I think mathematicians should learn them all and make their own decisions. I think Lean has got some plus points over some of the others, but I know people who think differently.

Perhaps a more liberal definition would be that rigorous mathematics is the theorems which are proved in the union of these computer proof checkers. So for example we could say that the odd order theorem was rigorously proved from the axioms of maths, as was the four colour theorem, the prime number theorem, the Kepler conjecture and the Cap Set conjecture. These are examples of theorems which I think all mathematicians would agree were rigorously proved. But we will need a lot more machinery before we can rigorously prove Fermat’s Last Theorem in one of these computer systems. In my mind, Fermat’s Last Theorem is currently a theorem of theoretical mathematics in the sense of Jaffe and Quinn, although I have no doubt that one day it will be a theorem of rigorous mathematics as well, because we human mathematicians have written some very clear pseudocode explaining how to make it rigorous. As for the ABC conjecture, I genuinely don’t know if Mochizuki’s ideas can be made into rigorous mathematics, although it seems that in general mathematicians are not happy with the pseudocode which has been produced so far. I can see a very simple way for the Mochizuki-followers to sort this out though, and we would happily talk to them on the Lean chat.