### The nightmare

In September 30 the mathematician Vladimir Voevodsky passed away. Voevodsky, a Fields medalist, is a mathematician of whom I barely heard earlier, but after bumping into an obituary I was drawn to read about him and about his career. His story is remarkable in many ways. Voevodsky comes out as brilliant, intellectually honest giant, who bravely and honestly confronted the crisis that he observed “higher dimensional mathematics” was in.

Here are a few of links that I read and on which I base this post: an obituary by John Baez, with some links, including to  an account by himself of the origins of “univalent foundations”, and also this obituary on the IAS site.

Here I want to write about several aspects of Voevodsky’s story which struck me. Note, it is written from the point of view of a mathematician who has not studied his work at all. I surely am not qualified to give an account of his development of motivic cohomology and his solution of Milnor’s conjecture, achievements for which he won the Fields Medal, nor the development of Univalent Foundations or Homotopy Type Theory (though I am certainly determined to read the first chapters in the book on homotopy type theory whenever I find the time). What really drew my attention in what I read about Voevodsky is the human story of a mathematician and his struggle. It is a story that can be understood by “human-level-IQ mathematicians” – in fact by any person – and it raises some disturbing and disheartening issues. Beyond the human story, there is the story of mathematics – our fractal and fragile profession, which at times seems to be standing on firm ground, and at times seems to be hanging on thin air.

Here are some key parts the story, brutally retold. (The quoted texts below here taken from this account by Voevodsky. The personal information is from the Wikipedia page or the obituaries linked above.)

The existential nightmare. Voevodsky apparently did not finish his undergraduate studies at Moscow State University (wiki says that he “flunked”!). However, as a first year undergrad he started reading a manuscript of Grothendiek’s and since then tried to develop his own mathematical ideas. He met Michael Kapranov and together they published a paper “$\infty$-Groupoids as a Model for Homotopy Category”, where they “claimed to provide a rigorous mathematical formulation and a proof of Grothendieck’s idea…”. Based on this exceptional achievement (presumably), Kapranov arranged for Voevodsky to be accepted to Harvard graduate school (Voevodsky did not apply, and didn’t even know that this was being arranged!) where he worked under the supervision of David Kazhdan. He continued to do outstanding work, and went on to solve famous conjectures, get appointed to the Institute of Advanced Studies, and win the Fields Medal.

What a romantic story! But Voevodsky tells us what happened later:

In October 1998, Carlos Simpson submitted to the arXiv preprint server a paper called “Homotopy Types of Strict 3-groupoids.” It claimed to provide an argument that implied that the main result of the “∞-groupoids” paper, which Kapranov and I had published in 1989, cannot be true. However, Kapranov and I had considered a similar critique ourselves and had convinced each other that it did not apply. I was sure that we were right until the fall of 2013 (!!).

Voevodsky is telling us that his first paper, which boosted his stellar career, turned out to be flawed – the main result was not true! Moreover, he was not able (maybe it was an emotional block, maybe too much work) to settle the issue of who is right for 15 years!! The horror of this situation is unbearable. Or maybe it is not so horrifying – maybe at times he did not care any more, not enough to resolve it?

And another question comes to mind: what if he found his mistake, when he was writing the paper? What if he could not fix it (it was not fixable), and gave up on mathematics? So, should we, should he, be happy that he made this mistake? He also says that Kapranov and he considered this critique, but convinced themselves that it did not apply. Well, what if they still had doubts? Would ignoring these doubts have been the right thing to do? Was scratching the paper the right thing to do? But then maybe there would never have been an arrangement to have Voevodsky study at Harvard, maybe he would have not continued his mathematical pursuits.

Does it make any difference if a paper on $\infty$-groupoids is correct or not? If a result is proven in a paper, and nobody ever finds the mistake, is it as good as true? If a person got a job, or tenure, on the basis of wrong paper – should he be dismissed? If you write a paper, and find a big mistake, should you withhold the information until the situation gets clearer? After all, its not your fault that you were even more diligent than Voevodsky, and found your own mistake, is it?

The referee’s concerns.  But these are not the only mistakes coming up in this story. Voevodsky tells:

The field of motivic cohomology was considered at that time to be highly speculative and lacking firm foundation. The groundbreaking 1986 paper “Algebraic Cycles and Higher K-theory” by Spencer Bloch was soon after publication found by Andrei Suslin to contain a mistake in the proof of Lemma 1.1. The proof could not be fixed, and almost all of the claims of the paper were left unsubstantiated.

A new proof, which replaced one paragraph from the original paper by thirty pages of complex arguments, was not made public until 1993, and it took many more years for it to be accepted as correct. Interestingly, this new proof was based on an older result of Mark Spivakovsky, who, at about the same time, announced a proof of the resolution of singularities conjecture. Spivakovsky’s proof of resolution of singularities was believed to be correct for several years before being found to contain a mistake. The conjecture remains open.

The approach to motivic cohomology that I developed with Andrei Suslin and Eric Friedlander circumvented Bloch’s lemma by relying instead on my paper “Cohomological Theory of Presheaves with Transfers,” which was written when I was a Member at the Institute in 1992–93. In 1999–2000, again at the IAS, I was giving a series of lectures, and Pierre Deligne (Professor in the School of Mathematics) was taking notes and checking every step of my arguments. Only then did I discover that the proof of a key lemma in my paper contained a mistake and that the lemma, as stated, could not be salvaged. Fortunately, I was able to prove a weaker and more complicated lemma, which turned out to be sufficient for all applications. A corrected sequence of arguments was published in 2006.

What’s going on? So many flawed papers. Makes one wonder who were the charlatans who refereed these papers and accepted them for publication. Of course, I am kidding. It really makes one wonder: am I, as referee, accepting flawed paper after flawed paper? Doesn’t it happen to all of us that we review a paper, it is a hard and technical paper, and then there is this lemma, which we can convince ourselves is true, but is it really true? It would be really hard to get to the bottom of this, and the other parts of the paper seem fine, and it is Voevodsky, mind you, who is author… I don’t really have time to check each and every lemma in this paper! It’s not my job! Can we let just this lemma pass? In fact, maybe we should, we do not want to block the next Voevodsy?

The working mathematician’s toil. If there are these truly important papers out there, by the leaders of our field, that are flawed, some of them even dead wrong, then what is the meaning of all this? Maybe there are more wrong papers, and nobody ever noticed? Does it even matter? Should I quit my job and become a carpenter, build real thing? Voevodsky says:

But to do the work at the level of rigor and precision I felt was necessary would take an enormous amount of effort and would produce a text that would be very hard to read. And who would ensure that I did not forget something and did not make a mistake, if even the mistakes in much more simple arguments take years to uncover?

To me, the most inspiring part of Voevodsky’s story, is the way that he chose to handle the crisis that he observed mathematics is in. First of all, he honestly admitted that there is a problem, and he decided to confront it.

And it soon became clear that the only long-term solution was somehow to make it possible for me to use computers to verify my abstract, logical, and mathematical constructions.

But his pursuit for truth was much deeper than a superficial slogan “use computers”:

The primary challenge that needed to be addressed was that the foundations of mathematics were unprepared for the requirements of the task. Formulating mathematical reasoning in a language precise enough for a computer to follow meant using a foundational system of mathematics not as a standard of consistency to establish a few fundamental theorems, but as a tool that can be employed in ­everyday mathematical work.

And so he undertook the herculean task of developing new foundations for mathematics !! (of course, not alone). Could this enormous pressure, coming from within, from his own intellectual honestly, be what drove him to a breakdown? Probably, but not much information is given in the obituaries, since this stuff is very personal. Is it possible that it is not this pressure that led to his death at the very young age of 51?

So let us return to the paper “Cohomological Theory of Presheaves with Transfers”, a very important paper which was certainly studied in many seminars worldwide. And imagine the budding graduate student, who doesn’t understand this lemma.

“Can you explain this?” he asks, and everybody volunteers to explain: “think of it this way…” says the veteran grad student, “it’s like bla bla blab la” and waves his hands. “Well, I see why its morally right”, says the budding one, “but I don’t understand the proof…”. Some others try to help, while the graduate starts to regret having asked. The postdoc moves uneasily in her chair. “What a waste of time!” she thinks, to be explaining lemmas to graduates students, and encourages: “I also had problems understanding that one, it’s one of those things that you have to work out on your own”. The supervisor recalls vaguely that he too had to work to understand that lemma (in fact, it was when he refereed the paper!) and that one could fix it somehow….  “um, technically I am not sure that this is precise, but we don’t really need the full power of the lemma, um, one can fix it” he says “now how did that go?”. Everybody waits “You know what, I’ll have to check my notes, why don’t we assume the lemma now and proceed”. And the budding graduate student is left with the feeling that everyone here is a clown except himself, or alternatively that everyone here is genius except himself, and maybe it isn’t that difficult and perhaps all this is not for him…

The tired mathematician’s worry.

So at the end, Voevodsky and many other mathematicians have set off to develop new foundations for mathematics, which, among other things, might make it easier to use computers to check proofs. Is this a good development? Is it necessary? Will it really help?

(If they can check our proofs, maybe the computers can do research on their own? Maybe they can also read one another’s papers. Imagine a world where all research mathematician are actually computers: how different would that world be? )

Do I have to invest myself in learning these new foundations? Should I wait? Maybe my field requires a different foundations and a different computer system to check it – is it a good idea to pursue these ideas?

Maybe all that “univalent nonsense” is important only if you want to work on Grothendiek-style shenanigan. If you do honest mathematics that actually relates to reality, you’re probably on safe ground and have nothing to worry about. Should I be worried?

* * * * *

To be honest, I am not very worried. I am split between being two opposite opinions. On the one hand, I am somewhat angry and disappointed at Mathematical Culture for not putting enough emphasis on correctness and understanding. It is clear that different people have very different notions of “understanding” and “knowing”. On the other hand, I think that mistakes are part of life, and also part of science, and therefore can and should be permitted be part of mathematics. These things happen, and by a process of mutation and selection, we hopefully evolve. (In passing, a request: please post corrigenda to your papers/books/etc.; perhaps math will move on without the corrigendum, but at least you can help that budding graduate student survive grad school in one piece).

And although I am very curious about univalent foundations, I cannot learn it in any deep way without stopping everything I am doing, and this won’t happen (one of the reasons why it won’t happen is that I am very skeptical). The details of Voevodsy’s mathematics, I feel, are not the important part of the story. The heart of the story is the determination to follow truth according to one’s standards and convictions, which is relevant far beyond mathematics, and which everyone can follow within their limitations. And maybe in this story there is also is a warning, or a calling, that those who come too close to the light, might burn.