## Caroline Chen on the ABC Conjecture

May 10, 2013

I was recently interviewed by Caroline Chen, a graduate student at Columbia’s Journalism School, about the status of Mochizuki’s proof the the ABC Conjecture. I think she found me through my previous post on the subject.

Anyway, her article just came out, and I like it and wanted to share it, even though I don’t like the title (“The Paradox of the Proof”) because I don’t like the word *paradox* (when someone calls something a paradox, it means they are making an assumption that they don’t want to examine). But that’s just a pet peeve – the article is nice, and it features my buddies Moon and Jordan and my husband Johan.

Read the article here.

Categories: math

“You don’t get to say you’ve proved something if you haven’t explained it,” she says. “A proof is a social construct. If the community doesn’t understand it, you haven’t done your job.” Brilliant comment!

Yes, this is exactly the issue — and raises a “paradox” — that when the proof is mathematically correct, is can also be wrong if it is not accepted or ignored.

This is the flip side of the open-source issue: not all open-source is genuinely open-source, that is, not if anyone can understand it.

My sense is, if mathematical resources are available across the mathematical community, disparate groups could pick up pieces, and even whole graduate programs emerge to train a cadre of mathematicians that can devote their careers to catching up with this new body of work.

Another easy solution, a short-cut actually, would be to demonstrate the validity of the proof, perhaps using computers. But — here again — is the black-box problem with code.

Wow! What an excellent article!

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It would take the equivalent of the Manhatten Project to get to where this guy is — so the question is whether or not the mathematics community has the resources to make the attempt.

It could very well be that the math community lacks to the commitment and/or the resources — and this might mean that mathematics has gotten as far as it will go, at least in our culture. Take for example theology, which bloomed explosively in the early decades of the last century, but collapsed as culturally irrelevant during the 1960s, when participation rates in religion declined, especially in Europe. The last real contribution was the Death of God theology, and that pretty much burned all its bridges.

This argument is as old as Oswald Spengler, who characterized western culture as worn out, senile and at a dead end.

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Yes, thanks for the great article.

I don’t see why you object to the article’s title. I think it is bringing attention to a true paradox — not avoiding it but highlighting it.

The proof may be logically flawless but he hasn’t proven it in any meaningful sense, as you point out, YET.

Nonetheless, while I understand the math community’s frustration, even annoyance, Mochizuki has given something of value to the math community. I assume that if there is valuable content in it (which seems likely) it will slowly be digested. Maybe, it will gradually become a real proof or contribute to one.

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Thanks for the link. Good article with a lot of interesting color. Funny to see what count as huge numbers.

When it came out, I did not appreciate how much all the machinery was novel and not understandable to anyone but the author. The Wiles proof definitely did have a group well prepared to dig in, and he was very forthcoming. (Both are far from what I know.)

I have a lot of sympathy for Mochizuki. But some engagement would go a long way. I guess he is temperamentally not disposed towards that and sees only downside. It won’t hang unresolved forever, but looks like it’s going to drag on a long while this way.

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I’ve been thinking that peer-review is one way of having Quality Assurance in the mathematics litterature. I remember Fermat’s marginal note in his copy of Diophantus’s book: “I have found a marvelous proof [ of FLT] but the margins are to small to contain it.” Many experts, including Andrew Wiles, don’t believe that Fermat actually had a correct proof of FLT. So, if we don’t know someone too well, how can we know that they have a correct proof of something? And, how well do we know ourselves and our fallibilities?

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I basically think that the article is inaccurate and sensationalistic. http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html This is Mochizuki’s homepage. He has many plans to give lectures in Japan, even a conference in June 2013. This is his homeground. Much wiser than say Perelman, I say. Almost all of his work is now published in Japan. Japanese mathematics is probably on the top level. They have at least 3 fields medalists.

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I’m not a mathematician, but I do have a background in math (Masters degree) and an interest in certain areas of it. I had a career as an actuary, and now that I’m retired, tutor high school students in math.

As I was learning math in high school, college, and graduate school, I was always acutely aware of what I regarded as a disturbing fact: most people in the math world are not very good at explaining math. My career as a tutor has only confirmed this to me.

I have worked with many very mathematically bright students who were having trouble only because they relied on their teachers (as most high school students do), who were incompetent. The teachers simply didn’t have the capacity to explain mathematics. This capacity (which doesn’t seem to be valued much by professional research mathematicians) is an ability distinct from the ability to understand higher mathematics.

When I was in graduate school, I was exposed to the “teaching” of approximately 20 professors, all of whom were obviously competent in math. Yet, only two of them also had this ability to explain math coherently. This was entirely consistent with my experience in college and high school. Essentially, any math I’ve ever learned I learned on my own. Classes were usually a waste of time. I often had the experience of coming to understand a concept and asking myself “why didn’t the teacher simply explain this”.

The idea that even high-powered number theorists can’t understand Mochizuki’s “proof” of the abc conjecture is absurd. He obviously has no capacity at all for explaining math, even though he may be brilliant enough to actually “see” why the conjecture must be true.

But, if he is unable to explain it to even the best minds in the field, his work is (in a sense) worthless. Yes, I suppose it is better to have the work, so that perhaps in 50 years mathematicians will finally untangle it. But, wouldn’t it be much better if Mochizuki simply had put some effort into making his work coherent? And, shouldn’t the mathematical community let him know that, in no uncertain terms?

The fact that the best minds in the field don’t even want to invest the time to (maybe) get to the bottom of the work is quite damning. And, I think it’s about time the mathematical community made a statement to this effect. It should be made clear to Mochizuki that he needs to “go back to the drawing board” and come up with a coherent explanation of his ideas, even if he has to hire an “interpreter” to help him do it.

In other words, he should be told the same thing I feel like telling all the incompetent teachers my students have encountered: you need to put some serious time into learning how to communicate what you know.

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Phil: I have to ask, did you read the article? This is hopefully a rhetorical question, so let me highlight the parts that I feel give your argument a fairly good beatdown:

Most importantly, he certainly has the capacity to explain mathematics. As de Jong points out, he has written several other papers solving serious questions that are well-regarded.

Furthermore, “the fact that the best minds in the field don’t even want to invest the time to (maybe) get to the bottom of the work” is actually quite commonplace. The article states this quite plainly. We could argue that this is not a good thing; fine, it’s probably not. But it is by no means “damning” in any sense I can imagine. What is damaging to the proof is his sparse commentary in the intervening year.

As a small concluding note, you have to look at this from Mochizuki’s perspective. All of his work was public. He knew that people were paying attention to it. As far as he knew, he was racing against a team of big-name number theorists to arrive at the correct interpretation of his ideas. So from his perspective: no, it would not “be better if [he] had [made] his work coherent” and that time caused him to be beaten to the punch.

Once it’s published, he has the rest of his life to make it simpler. He could be doing that now; if he’s planning on speaking about it in the near future, he should be doing that now.

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E. Stucky: I appreciate your comments, and find them quite interesting and thought provoking. But, I do have a few problems with some of them (aside from your gentle “put down” via your question “did you read the article”?).

You say,

“Furthermore, ‘the fact that the best minds in the field don’t even want to invest the time to (maybe) get to the bottom of the work’ is actually quite commonplace. The article states this quite plainly.”

Where is it “quite plainly” stated in the article that this situation is “quite commonplace”? I read this article quite differently (and I *did* read it – perhaps we read different articles?).

Of course, I’m certain it *is* commonplace for research mathematicians to sometimes forego plowing through a difficult piece of math for one reason or another. But, I don’t think the article is talking about that sort of thing. This is a highly unusual case, according to the article, in that a piece of work by a brilliant and respected mathematician has in effect been pronounced “gibberish” by the mathematical community. How often does this happen? And that’s precisely why mathematicians are are avoiding it. Not your “commonplace” reason.

The article has quotes from mathematicians working in the field. For example:

“Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” wrote Ellenberg on his blog.

“It’s very, very weird,” says Columbia University professor Johan de Jong, who works in a related field of mathematics.”

And, the article says:

“Soon, frustration turned into anger. Few professors were willing to directly critique a fellow mathematician, but almost every person I interviewed was quick to point out that Mochizuki was not following community standards”.

I find in hard to believe that the above statements indicate a “commonplace” situation. Indeed, the fact that this situation is so unusual is precisely why the article I read was written. And, I saw nothing that plainly stated or even implied otherwise. Please show me where you saw this in the article you read.

Rather, the article I read is talking about 512 pages of mathematics claiming to have proved a *very significant* result, that virtually every mathematician in (or out) of the field would certainly *want* to understand. And yet, even the best minds in the field, according to the article, seem to believe it is incomprehensible.

It is incomprehensible to me why any mathematician in the field would forgo going through something as important as this paper (especially by a mathematician of Mochizuki’s stature) unless they have no choice. And according to the article, they have no choice because:

“This is not just gibberish to the average layman. It was gibberish to the math community as well.”

It is quite understandable why a mathematician would not want to spend precious time trying to understand what appears to be what the article describes as “gibberish”. And, I don’t believe this situation is “quite commonplace”. That is, it is not commonplace for the best minds in the field to avoid a paper because it is “gibberish”.

You point out that Mochizuki *does* have the ability to explain mathematics, because he has done so to the satisfaction of other research mathematicians in some of his other papers.

So, you seem to be saying, it’s okay for him to put out 512 pages of “gibberish”, because (from his perspective) he doesn’t want to be “beaten to the punch” by someone else or “a team of big-name number theorists”. Well, I don’t think it’s okay.

Also, if he *does* have the capacity to explain his work, but chose not to, as you seem to imply, well, that makes the situation even worse, in that he is then deliberately withholding a clear explanation from others who are seriously interested in understanding it.

You might argue here that he didn’t have *time* to explain it. But, he *did* have the time to write 512 pages of gibberish. Why not use a bit of that time to explain things coherently?

What if someone next month comes out with a crystal-clear proof of the conjecture? Then, Mochizuki can come forward and claim credit, and explain to other”s satisfaction that his “gibberish” actually says the same thing as the new proof.

And, indeed, it may. But, here’s the problem. It could actually be that while Mochizuki actually did *see* the proof of the conjecture, he did not have the ability at the time to *explain* it in words other than gibberish. But, after he sees an explanation, he now knows how to make his own proof comprehensible. So… who should get credit? The person who was able to “see” *and* explain it, or the person who was able only to *see* it and write up his insights in 512 pages of gibberish… and then explain the gibberish after someone else has shown him how to do it?

Another possibility: another person, X, manages to decipher Mochizuki’s gibberish and present a coherent explanation, embraced by the math community. Shouldn’t that be worth most of the credit? Probably not in Mochizuki’s opinion. But, certainly in mine. The credit could read: “solution by X, based on 512 pages of gibberish (as judged by the best minds in the field) by Mochizuki”.

I think one solution to some of this absurdity is for the mathematical community to declare somehow that any work deemed to be “gibberish” by enough people capable of judging it to be so, is *not* “credit worthy” and that the first person who makes it comprehensible (including the original author) will get the major credit for the proof.

Of course, there is no way to make this idea “rigorous”. But, according to the article, Mochizuki has clearly crossed the line. His work is gibberish, and therefore, until he explains it, he should receive little credit. And if someone else explains it satisfactorily, that person should get at least half the credit (maybe more) depending on how significantly Mochizuki’s ideas are used in the coherent explanation.

As Einstein once said “if you can’t explain it simply, you don’t understand it well enough”. In our context, I would say that if Mochizuki’s work is incomprehensible to the best math minds in the country, he doesn’t understand it well enough. He at least is not capable of explaining it. And, if he is (as you seem to believe) he doesn’t *want* to explain it. And that is “damning” either of him, or the “system”.

You say:

“We could argue that this is not a good thing; fine, it’s probably not. But it is by no means “damning” in any sense I can imagine”.

You seem to agree there is a problem here. I’m hoping that some of my remarks have enhanced your imagination to the extent that you see that my use of the word “damning” might be justified here. I actually think “absurd” is the more appropriate word.

One final point. You say:

“So from his perspective: no, it would not “be better if [he] had [made] his work coherent” and that time caused him to be beaten to the punch”.

Of course you can justify any absurd, ridiculous thing from the standpoint of the person who did it. Everyone has their “reasons” for their actions. But, what’s the point of doing that? The word for arguing that incoherent work is at least as good as coherent work is, I think, “ridiculous”.

I appreciate that this whole issue is extraordinarily complex. I recall a “squabble” that arose in the math world when Perelman published his work on the Poincare Conjecture (see the article in the August 28, 2006 edition of the New Yorker magazine). Some other mathematicians filled in the gaps and were trying to take credit for the definitive proof. I wouldn’t have had a problem with that if virtually the entire math community in that field had characterized Perelman’s work as “gibberish”. But that didn’t happen. It has happened to Mochizuki, so I think it’s imperative for him to come forth very soon with some coherent explanations of his work. That is, if he expects to receive significant credit for it.

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