Mochizuki’s Maze

mochizukiFrom Nature:

The mystery concerns an impenetrable but potentially groundbreaking proof — of a puzzle known as the abc conjecture — that appeared online three years ago. Whether the proof is valid is still not clear — a source of frustration for some of the leading specialists who gathered at the University of Oxford on 7–11 December to discuss the matter.

Others say that the workshop, in which the proof’s reclusive architect Shinichi Mochizuki made a rare, virtual appearance, has at least boosted prospects for a resolution.

The quest to understand Mochizuki’s proof dates back to August 2012, when he quietly posted four papers on his website in which he claimed to have solved the abc conjecture. The problem gets its name from expressions of the form a + b = c and connects the prime numbers that are factors of a and b with those that are factors of c. Its solution could potentially change the face of number theory, which deals with the fundamental properties of, and relationships between, whole numbers.

Here is the first part of his 500 page magnum opus in case anyone wants to take a crack at verification:

Trivial stuff, really.

I’d do it myself, but the guy who developed much of the underlying theory was a hardcore Nazi, and I for one am not going to continue the work of a Nazi. Sticking by ma principles, I am.

Anyhow, flippancy aside, this is a very good demonstration of Apollo’s Ascent theory in action. The discovery threshold for in pure mathematics today is very high. At least 160, probably 175.

Mochizuki is clearly very, very bright.

  • Attended Phillips Exeter Academy, regularly rated in the top 10 schools in the US (sometimes first).
  • Entered Princeton University at the age of 16.
  • Graduated 3 years later as a salutatorian.
  • Got his PhD at the age of 23 under the supervision of a Fields Medal winner.
  • Become professor a decade later.

So there are very few people in a position to even understand the stuff at the furthest frontiers.

“The decision about which topics to cover lacked some overall understanding of the proof,” says Jakob Stix, a number theorist at the Goethe University in Frankfurt, Germany. “Which is not really a complaint, because I sense that nobody really understands the proof.” …

Some, such as Felipe Voloch of the University of Texas at Austin, were more scathing. “The play showing today at the Hodge Theatre was a farce,” Voloch wrote online, referring to a theoretical construction that Mochizuki named a Hodge Theatre.

Attendees also restated familiar complaints about the proof itself. “The amount of language seems absurd,” said Artur Jackson of Purdue University in West Lafayette, Indiana, at the end of Thursday. And, Voloch told Nature: “I don’t know why he chose to make it so abstract.”

A follow-up workshop is expected to take place in Kyoto in July. Kedlaya plans to attend, unlike some of the disillusioned participants in the Oxford workshop. “The claim is an extremely important result,” he says, and the community deserves to know whether it is valid — even though the process will take several more years.

A couple further points.

Mochizuki is possibly the brightest mathmo on Earth, and he is Japanese (two other really big contemporary giants are the Chinese-American Terence Tao, and the Russian Jewish Perelman). Japan is the only major East Asian nation to have been developed for more than a generation. Clearly, pure IQ triumphs over any “q factor” let alone hazy Jaychickean notions of “clannishness” so far as intellectual accomplishment is concerned.

Second… We can envision some of the technologies that might conceivably play a big role in the coming century: Intelligence genes, geoengineering, SENS, autonomous robotics, self-assembling nanotech, superintelligence… It might be germane for serious futurists to start serious study of the sort of intelligence – loosely, in terms of standard deviations – that would be needed to actually discover them.


  1. [We can envision some of the technologies that might conceivably play a big role in the coming century: Intelligence genes, geoengineering, SENS, autonomous robotics, self-assembling nanotech, superintelligence… It might be germane for serious futurists to start serious study of the sort of intelligence – loosely, in terms of standard deviations – that would be needed to actually discover them.]

    Superintelligence was supposed to be a self-feeding, autonomous process which nobody could stop. If someone has to think to get it, I don’t believe betting on its happening would be wise. In the real world, peak future came and went in 1961 or so.

    [So there are very few people in a position to even understand the stuff at the furthest frontiers.]

    The trouble with Mochizuki is that he doesn’t seem to want to be understood very much, beyond the “understanding” of a few devoted Japanese cult followers. This is unusual behaviour for a mathematician. Possibly in the end he will be understood in spite of himself; far too early to say.

  2. This kind of thing happens occasionally because some great mathematicians have poor social and language skills. It never lasts more than a few years. If Mochizuki can’t get any other mathematicians to make the effort, either he doesn’t really have a proof or he is an antisocial jerk, because the claimed result is absolutely, unquestionably important enough to be worth several years of effort to understand and explain. It would be the most important result in Number Theory in nearly a century.
    Yes, more important than the celebrated theorems of Faltings, Wiles, Tao, etc., only proving the Riemann Hypothesis would top it, and only proving the conjectures of Langlands or of Birch and Swinnerton-Dyer would be considered as comparable. Even coming up with the ABC conjecture as a hypothesis was a great insight.
    The layman’s version of the conjecture is “if A, B, and C all have different prime factors, and A+B=C, then the primes involved can’t have too many repeated factors.” The technical part is what “too many” means – it means that the product of the relevant primes without repetitions must be large relative to the product ABC (which includes all the repetitions). In other words, relationships like 2^3 + 1 = 3^2, or 5^3 + 3 = 2^7, are extremely rare.

  3. Pseudonymic Handle says

    Mochizuki built his own vast mathematics theory. For someone else trying to decipher it there are 2 big problems:
    1. Mochizuki doesn’t explain what he did, in a breach of math manners, so people have to try to figure a very obscure text by themselves
    2. this thing is so huge that those trying to find a proof may have to devote years of their lives to studying what in the end may be the ravings of a crackpot.

    Giving the grandiose names he gave to his concepts and his reluctance to help those seeking proof, I wouldn’t be surprised if the entire thing is a very elaborate lunacy. Actually, I’d be shocked if the entire thing is actually correct.

    About your final points:
    I don’t know what you mean by “Jaychickean notions of “clannishness”” but una hirundo non facit ver, so any human group can have one guy with very high IQ. hbdchick theories have more to do with behaviours and group averages rather than exceptional individuals.

    The idea of intelligence thresholds for solving certain problems is very important for the future of science, because science competes with far more profitable professions for high IQ people, so we may be unable to solve some problems because of societal misallocation of human resources. Dysgenic trends would also become a problem later in this century.

  4. Here’s the official statement. Suppose a+b=c and there are no common prime factors. Let d be the product of the distinct prime factors of abc (ignoring repeated powers). Then for any e > 1, there are only finitely many examples where c > d^e .

    Usually c will be less than d. If we let q be the “quality” of a triple a+b=c defined as the power you have to raise d to to get c, q is usually less than 1, though successively better examples of q are obtained from 1+8=9, 3+125=128, 1+4374=4375, and 2+6436341=6436343. That last one is the best example known, with q of log(6436343)/log(15042)=1.6299…

    There are infinitely many examples with q>1, but according to the conjecture there are only finitely many with q>1.1 or with q>1.01 or with q>1.001 etc.

    From the conjecture it follows immediately that “Fermat’s Last Theorem” (proved by Wiles) has only finitely many primitive (no common factor) counterexamples. Many other well-known and difficult results are also simple corollaries of the abc conjecture if you don’t care about finitely many exceptions.

  5. The Japanese are not noted for clarity of expression. Number theory can be interesting, but this looks more bother than it is worth.

    Also, don’t bank on there actually being a coming century, in any complete sense. Even half is looking dicey.

  6. Also, don’t bank on there actually being a coming century, in any complete sense. Even half is looking dicey.

    Apocalyptic attitudes have been present in basically every generation throughout history. This seems to be a fairly universal constant. There are always people who believe that the apocalypse is coming or is just around the corner.

  7. This sort of thing can only be appraised long after the fact though. He may just be a crank or pursuing a dead end. Lots of brilliant people end up as cranks or in dead ends.

  8. Are you sure this kind of pursuit is a good allocation of intelligent human resources? Me, I think smart people should rather search for cancer cures, or create new search engines, or run complicated businesses so that they don’t blow up. Even such unpopular activity as designing new financial instruments is of greater social usefulness than this.

  9. The increase in Japanese achievement in the sciences in this century is very interesting of course. I don’t think their mean IQ went up compared to Western countries’ mean IQs during the 100 years or so that IQ tests have existed – at least that’s not my impression.

    The Japanese are famous for fads. Maybe they’re experiencing a science-and-math fad right now? Or maybe their government increased funding of math and science late in the last century and we’re now seeing this bear fruit? That’s actually checkable.

  10. hazy Jaychickean notions


  11. So there are very few people in a position to even understand the stuff at the furthest frontiers.

    If the threshold for discovery is 175, what’s the threshold for merely understanding (initially) what’s been discovered? Do you have an estimate or guess?

    I see: 30 points down, other piece.

  12. Bao Jiankang says

    It is true that Mochizuki is not doing all the much to explain his work but most of the mathematicians who are trying to understand his work are not japanese cult followers. Here is an introduction to his work by Ivan Fesenko.

    Mochizuki is the real deal, he is an established mathematician who has done great work which is why we must take him seriously. However it does not negate the possibility he has gone off the deep end. Linus Pauling became a crackpot late in life. Hopefully this is not the case for Mochizuki.

  13. It’s already happening, the only people who can’t see it are the white supremacists.

    70% of civilian drones sold are from a Chinese company. Dgi

    Broad group from china.


    They are the ones who invented e cigarette and hover boards.

    The list goes on and on. The East asian creativity theory does not seem to be holding up. It seems that pure g and luck are the only things that determine innovation.

  14. Fesenko isn’t Japanese, sure, but the great majority of M’s followers are, and none of them has convinced outsiders that they truly understand what they claim to.

  15. I wanted to comment on your “Introduction to Apollo’s Ascent” post, but I see the comments there are closed… so I will do it here, and please feel free to move this post if you like.

    This is very interesting and impressive. I haven’t read all the way through yet, but a few thoughts come to mind:
    1) Would it be fair to say that a higher GDP, at least to some degree, is a result of a higher percentage of human activity being expressed in terms of money? If so, would it not make sense that countries which more greatly emphasize non-monetary metrics of human activity (such as the former communist countries, Bhutan which prefers to measure its population’s happiness instead of GDP, etc.) would have depressed GDPs? (It is no secret that a large portion of the US’s GDP consists of money which is not used for anything except to make more money, with nothing “real” being involved in the chain.)

    2) Common sense suggests the GDP for countries that were invaded by others (by army or by coup) would be lower, just as the GDP of countries who invaded and plundered others, or found unexpected resource windfalls, would be higher… A potential example:
    Most of the Soviet-allied countries suffered a collapse of trade routes and were deliberately de-industrialized by new owners, perhaps their current bad results are due to that, rather than to former central planning.

    (I am unconvinced that central planning consistently causes lower GDP… To prove this, one would need to compare not just USA/USSR (the typical comparison), but the whole economic system that those countries were part of. In the USA’s case, this involved trade with (or exploitation of) much poorer countries. In other words, it is possible that the USA’s higher GDP/capita was simply the result of capitalism’s greater wealth inequality, with the “low GDP” being off-shored to foreign countries, and thereby made invisible in the statistics. Since I have never seen anybody attempt a study of this sort, the question seems to be unresolved…)

  16. The most deplorable one says

    two other really big contemporary giants are the Chinese-American Terence Tao,

    As far as I am aware, Terence Tao was born in Adelaide, Australia, of Hong Kong Chinese parents.

    Also, that Japanese guy has a suspiciously European looking face (except for the black hair.)

  17. The Japanese have played a very prominent role in mathematics for the last 100 years. They had 3 Fields Medalists – Kodaira, Hironaka, and Mori – in the 20th century and before the time of the Fields Medals they had guys like Takagi and Oka.

  18. No question about the importance of the ABC conjecture if some version of it is true. A valid proof of some version of it would be one of the biggest breakthroughs ever in mathematics.

  19. I have heard rumours that Mochizuki mother is an American Jew, so he might only be half goyim.

  20. Academic Gossip says

    re: Mochizuki’s origins,

    I was in the same place as Mochizuki for a couple of years when he was a postdoc, and talked to him here and there. He is fairly tall and robust-looking by Asian standards, even in comparison to those of his generation raised in the USA, and does look plausibly part-European. The photograph in this blog post understates how Western his features look, and many people who interacted with him might well have wondered whether his ancestry is 100 percent Asian or not. It is not a question many people would have asked him, and he seems fairly private. From web searches, his mother is named Anne.

    Culturally, Mochizuki fit the same profile as many other immigrants in North American academia. He was brought to the States as a kid, attended very good US institutions, and speaks the unaccented and colloquial American English of an educated native, but beneath the surface might have also been anywhere on the assimilation spectrum between “totally Americanized” and “not quite comfortable here”. I saw him speaking Japanese, no idea how fluently, with visiting scholars from Japan, so he apparently retained the language before the move to Kyoto.

  21. Having looked at Mochizuki’s manuscript, I get the strong impression, as a professional mathematician, that he’s probably got a real proof. What he has done, if correct, is more difficult than what Perlman did, because Perlman’s work was much more closely connected to previously known work. It’s also clear that Mochizuki can communicate in English perfectly and knows how to organize mathematical work.

    Therefore, the difficulty is entirely due to the obscurity, abstractness, and novelty of the mathematics itself. I predict that within 2 years there will either by something close to a consensus that his proof is good, or a clearly defined gap in the proof (as was found in Wiles’s original proof).

  22. No Serre, Deligne, Tate? The half-Jewish Grothendieck is still alive. And Peter Scholze is already the most promising mathematician of his generation.
    Perlman and Tao are both closer to problem solvers than theory-builders. If you talk to those in the community, you’ll see that construction of new mathematical machinery is really considered the pinnacle of mathematical achievement. Aside – Kiran Kedlaya, a multiple IMO gold medalist/ Putnam fellow, is a southern indian brahmin.

  23. Perelman. Why have two of you got his name wrong? It’s as annoying as “Feynmann”.