My math news feeds have been filled with two number theory stories this month: the Oxford workshop on the abc conjecture and Piper Harron’s thesis. Both are in number theory, but that’s where the similarities end. At the risk of reading too much into both stories, they seem to represent two points along a cultural shift that may be taking place in mathematics.
The abc conjecture
Number theory is famous for having lots of easy to state, hard to prove theorems and conjectures (twin primes and Collatz conjecture spring to mind). The abc conjecture is a little more complicated. On a basic level, if a, b, and c are integers and a+b=c, the conjecture relates the prime factors of a and b to the prime factors of c; in other words, it connects the additive properties of integers to their multiplicative properties. (Such a connection is certainly unexpected. There’s no obvious reason to think the prime factors of 11 and 3 would be related to the prime factors of 14.)
Here is the rigorous statement of the abc conjecture: for every ε>0, there are only finitely many triples of coprime positive integers a+b=c such that c>d^(1+ε), where d is the product of the distinct prime factors of abc. It's a bit of a mouthful, but if you plug in some numbers, you can get a feel for it. In the case of 11, 3, and 14, the prime factors of the three numbers are 2, 3, 7, and 11, so d=2×3×7×11=462. 462 is already much larger than 14, so the expression is satisfied for those numbers.
In 2012, Japanese mathematician Shinichi Mochizuki announced that he had proved the abc conjecture, and the mathematical community…scratched its head. When the proof of a big conjecture like abc is announced, it’s not unusual for many mathematicians in the research area to start reading the proof to make sure it’s correct and for a journal to speed the peer review process along so it can be published as soon as possible.
That didn’t happen for abc. Many tried, but Mochizuki’s proof is so long and so strange that very few mathematicians, even experts in number theory, have been able to make any headway in understanding his techniques. The “inter-universal Teichmüller theory” Mochizuki developed to tackle the question is well nigh impenetrable. (I study Teichmüller theory of the non-inter-universal variety, and I have no idea whether or how Mochizuki's theory is related, but that's a question for another day.) So far, number theorists have not come to a consensus on whether Mochizuki’s proof is true.
Earlier this month, Oxford University hosted a workshop on Mochizuki’s work with the aim of disseminating the main ideas quickly to number theorists in hopes that they will be able to verify the proof. It was not an overwhelming success. Stanford mathematician Brian Conrad attended the workshop and wrote a detailed post about it on mathbabe.org. The impression I got from his post, along with Felipe Voloch’s Google+ updates and articles in Nature News and Quanta, is that a few researchers made a small amount of progress on understanding some of the foundations of Mochizuki’s theory but that in general they are frustrated with how opaque the theory and explanations seem at this point.
The liberated mathematician
“Respected research math is dominated by men of a certain attitude.” So starts the prologue to The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: an Artist’s Rendering (pdf), mathematician Piper Harron’s thesis.
The entire prologue is fantastic, so instead of trying to describe it, I might as well quote it.
“Even allowing for individual variation, there is still a tendency towards an oppressive atmosphere, which is carefully maintained and even championed by those who find it conducive to success. As any good grad student would do, I tried to fit in, mathematically. I absorbed the atmosphere and took attitudes to heart. I was miserable, and on the verge of failure. The problem was not individuals, but a system of self-preservation that, from the outside, feels like a long string of betrayals, some big, some small, perpetrated by your only support system. When I physically removed myself from the situation, I did not know where I was or what to do. First thought: FREEDOM!!!! Second thought: but what about the others like me, who don’t do math the “right way” but could still greatly contribute to the community? I combined those two thoughts and started from zero on my thesis. People who, for instance, try to read a math paper and think, “Oh my goodness what on earth does any of this mean why can’t they just say what they mean????” rather than, “Ah, what lovely results!” (I can’t even pretend to know how “normal” mathematicians feel when they read math, but I know it’s not how I feel.) My thesis is, in many ways, not very serious, sometimes sarcastic, brutally honest, and very me. It is my art. It is myself. It is also as mathematically complete as I could honestly make it.
“I’m unwilling to pretend that all manner of ways of thinking are equally encouraged, or that there aren’t very real issues of lack of diversity. It is not my place to make the system comfortable with itself. This may be challenging for happy mathematicians to read through; my only hope is that the challenge is accepted.”
What follows is a math paper, filled with real number theory but written in an informal style, with clearly labeled sections for laypeople, mathematicians who want a general overview of the ideas, and people who want to see some of the gory details. She explains concepts using unicycles and groups of well-behaved children. There are comics about going into labor while writing a thesis. The content of the paper is not easy, but the presentation is entertaining and refreshing.
Harron, whose website is called The Liberated Mathematician, writes, “My view of mathematics is that it is an absolute mess which actively pushes out the sort of people who might make it better. I have no patience for genius pretenders. I want to empower the people.”
Harron has a reluctant maybe blog on her website and has written a great post on mathbabe.org. In both places, she says things a lot of us are afraid to say about the attitudes we feel like we have to pick up in order to fit into the mathematical community. Based on the way her thesis and blog posts have gone through my Facebook and Twitter, I think I’m not alone in identifying with many of the feelings she describes. I certainly felt the same pressure to conform that she felt as a beginning graduate student. As she puts it: “Please tell me the rules I must abide by in order to make no waves!”
Why I am writing about both the abc conjecture and Piper Harron’s thesis
Shortly after Mochizuki claimed he had proved the abc conjecture in 2012, Cathy O’Neil wrote, “proof is a social construct: it does not constitute a proof if I’ve convinced only myself that something is true. It only constitutes a proof if I can readily convince my audience, i.e. other mathematicians, that something is true. Moreover, if I claim to have proved something, it is my responsibility to convince others I’ve done so; it’s not their responsibility to try to understand it (although it would be very nice of them to try).” Later in the same post she writes, “I’m not saying Mochizuki will never prove the ABC Conjecture. But he hasn’t yet, even if the stuff in his manuscript is correct. In order for it to be a proof, someone, preferably the entire community of experts who try, should understand it, and he should be the one explaining it.”
Mochizuki has not fulfilled his end of the social construct that is a proof, while Harron has. But there’s more to it than that. Mochizuki's work can seem deliberately obscure, and people who have read and say they understand it can seem dismissive of other people's questions (see Ivan Fesenko's comments in this Not Even Wrong post). In the case of the abc conjecure, it's easy to sympathize with the questioners. If the august number theorists who attended the conference can't understand inter-universal Teichmüller theory, the problem is with the explanation, not the questioner. They have already earned the respect of the community. Harron's work, on the other hand, goes well beyond what is expected of a proof.
In an ideal world, a mathematical proof is just a way to convince other mathematicians that something is true. But there’s more baggage than that. We write mathematics in a particular style, a style that often removes intuition, examples, and signposts. There is a set of assumptions about what everyone knows, and woe to you if you don’t know some of them. Of course, the assumptions are useful; we have to start somewhere. We can’t define addition and multiplication at the beginning of every paper, but we also don’t want to instill the fear of admitting our ignorance that many math students pick up.
Around the same time I read Harron’s thesis, I read a remembrance of William Thurston in the Notices of the American Mathematical Society (pdf), and a quote from his MathOverflow profile stood out to me:
“Mathematics is a process of staring hard enough with enough perseverance at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others. I enjoy questions that seem honest, even when they admit or reveal confusion, in preference to questions that appear designed to project sophistication.”
I’m glad to see that attitude from a respected, established mathematician, and I hope people take it to heart. It is hard for those of us who are less secure, especially those from underrepresented groups in mathematics, to embrace it openly. I admire Harron not only because she is trying to peel back the curtain and help people understand that they can understand math but because as a woman of color, she could pay a higher price for doing it. A quality that is seen as humble, honest, and admirable in Thurston could be seen as a sign of incompetence in someone from a different background. (As is so often the case, there’s an xkcd for that.)
This is why Harron’s work is all the more necessary. As she writes in her thesis, she wishes to relay to students that “1) you are not expected to understand every word as you read it, 2) you can successfully use math before you’ve successfully understood it, and 3) it has to be okay to be honest about your understanding.” I wish more young mathematicians learned these lessons and weren't afraid to reveal their ignorance.
Another thing I thought was interesting about these two stories is who reported and shared them. My math friends shared the heck out of Harron’s thesis, and I saw more about the abc conjecture from nonmathematicians who follow popular science. This wasn’t a binary thing; mathematicians shared articles about the abc conjecture, and nonmathematicians wrote about Harron's thesis, but for the most part, it was the other way around, at least in what I saw.
I’m not quite sure what conclusions to draw from the way the stories were shared, but it feels important. I think it says something about how mathematicians
present themselves allow others to present them. On some level, we want others to put us on a pedestal. We don’t really mind it when someone gets the message “oh, it's too complicated—you wouldn’t understand.”
On the other hand, the strong positive reaction to Harron’s work within mathematics shows that many in the mathematical community are hungry for something different. Even people for whom the status quo has worked (after all, they’re still there) recognize that mathematics loses when we build barriers for some groups of people or encourage people to adopt the “certain attitude” Harron writes about in her thesis.
Does the abc conjecture represent a dying old guard, and does Piper Harron represent the way of the future? I'd guess nothing that dramatic is going on. But mathematicians should think about how they want to explain mathematics and who they are inviting in or leaving out in the process.