A maths proof that is only true in Japan | Hacker News Reader

A maths proof that is only true in Japan

newscientist.com

75 pointsmonksdream77 comments

When the likes of Peter Scholze (fields medal!) and other very high profile mathematicians find (serious) flaws in every posted manuscript about this... I mean, it's pretty clear to me what's going on. The proof just doesn't go through.

I think the intrigue is mainly that it's at such a high level that lay mathematicians (like me) have no hope of understanding the debate. It's a situation that lends itself to crazy speculation, because nothing you say about it can easily be challenged.

> When the likes of Peter Scholze (fields medal!) and other very high profile mathematicians find (serious) flaws in every posted manuscript about this... I mean, it's pretty clear to me what's going on. The proof just doesn't go through.

On the other hand, Ivan Fesenko (also a heavyweight; he is for example the PhD advisor of the Fields medalist Caucher Birkar) insists that Mochizuki's proof is correct.

* Here is a popular scientific article from 2016 where Ivan Fesenko presents his perspective on this topic: https://inference-review.com/article/fukugen

* A popular scientific article by David Michael Roberts (also a renowned mathematician) from 2019 about where he believes an important contentious point in the different viewpoints of Scholze/Stix vs Mochizuki lies: https://inference-review.com/article/a-crisis-of-identificat...

Fantastic links, thank you. When I say Scholze and friends disagree I mean they seem to have specific mathematical criticisms with mochizuki's school that have not been addressed publically, not just "structural opinions" (for lack of a better word). For instance, see Sawin's answer here: https://mathoverflow.net/questions/467696/global-character-o...

But that's fair, it's not exactly one-sided, but to my (completely inexpert) judgement the matter seems heavily weighted against mochizuki?

The most damning part to me is that Mochizuki dismissed Joshi's work and insulted it. That's a crazy response to someone trying to improve on his theory, and shows more of a religious belief that a mathematical conclusion.

there are so many cases of genius weirdos that it seems unfair to judge them using a commoner social framework.

moreover we only have very partial information about the whole thing.

we simply have no idea and can't honestly deduce nothing of value. this was a funny read but i'll forget it in maybe 10 minutes.

It's interesting how the debate is transferable to other topics. In theory maths should be able to be broken down to its basic components and be proven to be all true, or if something is false, then the whole thing collapses. But in practice things like this become so complex that it becomes a matter of conviction, influenced by things like ego.

Now imagine taking something like biology and vaccines. What happens if you rely on your experts and other rely on theirs, and they disagree?

Yeah, it's true, there is politics in mathematical truth, for better or worse. That is slowly changing with the adoption of proof assistants, I think. A lot of well-known names (like Tao and Conrad for instance) are starting to formalise large swathes of modern maths in Lean, for instance. Perhaps it will never get to a point where it is so easy that formal proof is required to publish a result, but who knows? It seems like a start.

>>In theory maths should be able to be broken down to its basic components and be proven to be all true, or if something is false, then the whole thing collapses.

>>But in practice things like this become so complex that it becomes a matter of conviction, influenced by things like ego.

Isn't this like doing a bunch of AND , OR operations?

How does ego become a factor here? Either an expression evaluates to true or false. There are only two outcomes, why is there a confusion here.

That's true, but in practice mathematicians rarely check a proof to that level of detail. In fact, they rarely write a proof at that level of detail. There just isn't enough time to do that for every result/review, so people take shortcuts. Most of the time it's fine because trained mathematicians take good shortcuts, but sometimes things slip through.

At some point you need to trust the scientific consensus, you can only go so far in checking the data (or math).

I have a basic understanding of physics, despite having a PhD. I am not saying this to fake modesty - this is a fact. Most of what is happening in physics is beyond me, not to mention maths (which I had at an advanced level).

Physics taught me to have a bullshit detector when I read articles about "soft" science (and let's admit that this is not a very difficult task), but anything that requires deep, hard knowledge I must just trust.

Science is very different from math. In math you can formally prove hypotheses. In science you cannot formally prove hypotheses, you can only reject hypotheses.

Yes, you are of course right.

My point is that even with a formal advanced education on an area (say -- physics), you are bound to weight the "consensual appraisal" of the authors of a claim to make up your mind.

With my PhD (and having left academia after it), I can easily determine that homeopathy is physical bullshit. It may very well work for different reasons (placebo effect), but when I read about people who try to explain that cosmic rays this or quantum energy that -- I know they are idiots.

And then came Jacques Benveniste who was published in Nature with his "water memory" hypothesis. Holy shit - physics has just changed! Why? Because it was in Nature.

Nature then realized this is a scam, retracted the paper, and physics is back on its (unfortunately) boring track (there is nothing that physicists hope more than a radical revolution in physics)

So we need to trust people who have the right titles, university affiliation, track record on TV and whatnot. In biology, despite having worked in some areas there, I have even less trust in myself.

Then comes the "-logy" soft science where it is not difficult to check that what they say makes sense or not just by looking at the math (which is simple, but can be very deceiving)

And then comes math itself. There can be a definitive answer, as you say, but the proof is completely beyond mere mortals. If you look at the Fermat's conjecture proof (a very simple conjecture, which also makes its beauty), it is completely unreadable for someone who is not in the subject. I did a lot of math and past the 10th line I am done.

I asked him more than 10 years ago if he would be interested in a formalisation of the proof, and he politely declined. I guess he was right to decline, my proposal would not have been viable then anyway.

Yeah, I was wondering how can debates like these exist nowadays when formal methods appear to my layman's eyes as the ultimate arbitrer of proof. Is that not how the math community looks at it?

When someone insists the only way to understand their <whatever> is to come in person and study under their direct tutelage, my scam/cult detector redlines.

https://www.reddit.com/r/math/comments/1bmp1vk/very_salty_mo...

Lot of gems in this thread. My favorite:

>>>After Mochizuki said that Scholze-Stix were “profoundly ignorant,” I’m starting to think that this phrase is a weird form of high praise from Mochizuki.

>>I feel like the most logical strategy for Mochizuki right now is to diss. Due to the currently prevalent (and not altogether unjustified) attitude towards Mochizuki and his "cult", any praise from him will condemn what he praises to oblivion, because anyone that he praises is guilty of being part of his "fan club" simply by association. In a way, this helps to give the perception that Joshi is "independent" and still worthy of being taken seriously, though Scholze has already been dismissive of Joshi's work from the beginning.

>Wow the implications of this perspective. Theatrical and operatic. If/when Joshi’s work is vindicated, Mochizuki comes out of the shadows and says “I’m sorry son I completely raked you through the coals so that you would gain sympathy and some credibility in the eyes of the wider mathematical community, so that eventually your ideas would be recognized and hence mine as well”. I would watch the fuck out of this movie.

    They've surrounded me. Cameras in every corner.
    Every move dissected in blogs, forums, peer-reviewed takedowns.
    "Cult leader". "Crank". "Outcast".
    Good. Let them watch.
    
    I'll solve equations with my right hand... and write names with my left.
    I'll take a potato chip... and eat it. [CRUNCH echoing like thunder]
    
    If I praise Joshi, he's tainted—marked as one of mine. Dismissed by association.
    But if I drag him... if I bury him in scorn... then they listen.
    Then they think, "Maybe he's different. Maybe he's not one of them.""
    I'll throw him under the bus... and save him!
    
    And the witness to my alibi... is the mathematical community itself.
    [A flicker of Scholze's blog. Stix's preprint. Joshi's strained silence.]
    They're all watching.
    
    They won't get it now.
    But when the theorems land... when every insult has aged into irony...
    
    ...they'll see it was all part of the proof.

I thought this would be something interesting, like the Sapir whorf hypothesis applied to mathematical reasoning - but no, it’s just the old classic professor/journal editor playing silly buggers in his power-tripping dotage scenario.

Yeah, it's ironic how math is more or less the one "universal" truth, and we still long for somehow magically make it culturally dependent. I can definitely understand that temptation. Like the opposite one, e.g. the search for a "perfect" language (as in e.g. Umberto Eco's book). Both temptations are examples of a longing for an actual paradox or absurdity in the world.

Totally get your point, but math is still a human creation. The symbols, language, and frameworks we use are cultural, and disagreement over proofs like this one shows math depends on shared understanding, not just objective truth.

I definitely agree with that. But that is a pale cultural dependence compared to what one would wish for.

I’ve read about this a lot before. My gut tells me that if you’ve got a central genius with twelve adherents and no-one else, what you’ve got is a cult, not a proof. But also, it is frankly amazing to think that Galois’ original proof was very nearly lost. It wasn’t like he’d not tried to publish. He’d been laughed out by people like Cauchy saying it was nonsense.

Is that this controversy? https://hsm.stackexchange.com/questions/5341/did-cauchy-forg...

Interesting. I may be off here.

> The error concerned a part of the proof called Conjecture 3.12, seen as a vital part of Mochizuki’s efforts to solve the abc conjecture, which Scholze and Stix claimed suffered from an unjustified leap of logic. “We came to the conclusion that there is no proof,” wrote the pair, who didn’t respond to a request to comment for this article.

This is hard to understand. This element of the "proof" is named "Conjecture 3.12". Isn't that enough by itself to demonstrate that there is no proof? If there was a proof, Conjecture 3.12 would be a theorem, not a conjecture.

Well, I think it's not even called "Conjecture 3.12", but "Corollary 3.12", which is funnier :D

That's a pretty serious error for the journalist to make.

Is it even true in japan if only a subset of japanese mathematicians believe it?

> Another prize is also awarded annually to people who have made important progress in studying IUT – the first such prize, with an award of $100,000, went to Mochizuki and his colleagues.

Well that's hardly suspicious at all.

Reads like the Obama giving himself an award meme

Things have moved on since then, as artificial intelligence has started being used in formalisation, [...]

With how AI works fundamentally, wouldn't you still need to verify the results generated by AI? Doesn't seem like an applicable field for it, at least in its current state.

I asked a very similar question a couple of weeks ago here: https://news.ycombinator.com/item?id=44028051

The top answer helped me to understand.

> Presumably an AI would formalise the proof in a system such as Lean, then you only need to trust the kernel of that proof system.

I don't think trusting the Lean kernel is enough: you also need to trust that all of the Lean code is a valid translation of the informal proof. Given that the informal proof is already gigantic, and that there is no general mechanical way to verify if a formal statement corresponds 1:1 with an informal statement, it's far from trivial to trust that the Lean representation of the proof is the same thing as the original proof.

Now, if the proof works, presumably this problem goes away: Lean can show that based on this proof, the original statement holds. But if Lean says that this formal proof doesn't work, that doesn't tell you anything about the informal proof: the error may only be in the formalization.

Agreed; translating to Lean/Coq is more likely to prove the positive rather than the negative. It may still be useful in pinpointing where incorrect proofs go wrong.

They use LLMs to help write formal proofs (in languages like Coq) that are then checked by traditional programs; they're not using AI as the checker.

https://youtu.be/e049IoFBnLA

What is so special about IUT? They say the theory is "out of this world" but in what sense exactly? Did Mochizuki found a new interesting way to look at some ideas?

It's extremely complicated. The original document he wrote up was 500 pages of maths introducing effectively an entirely new theory. I studied number theory in uni, tried to read it, and understood barely anything of it.

Which obviously leads to the epistemological problem that the article points out. You had extremely good mathematicians like Scholze look at it and thought he found a flaw, then one guy from Arizona disagreeing that it is a fatal flaw and claiming to have fixed it, which Scholze doesn't agree with.

So what do you really make of it if only a handful of mathematicians can engage with it, and they can't even agree with each other. Probably the biggest value of IUT is that it puts to the test what even counts as a proof.

It kind of introduces a fun thought experiment, of a super high-level, complex equivalent of the Monty Hall Problem (which is so counterintuitive that even very intelligent and mathematically literate people will outright refuse to accept the established truth). How would we ever establish truth on something so monstrously complicated that only ~10-100 people in the world could possibly understand and at the same time so divisive that there cannot be a strong consensus?

I'm not a mathematician (but I've seen exact sequences and commutative diagrams) and to me the stuff out of his IUT papers[0] looks borderline LLM-generated. I can only imagine what the LaTeX source looks like.

[0]: https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%2...

Holy moly, they weren't joking about being out of this world.

It reminds me of Lambda calculus

You can express `a + b` or `a * b` in their regular algebraic notation or you can express them as a lambda expressions

ADD = λab.(a S)n

MUL = λxyz.x(yz)

Manipulating these expressions instead of algebra, you can suddenly compute things such as "+ * +" (Plus times plus). That will yield you another expression for sure, but we don't even know what that means.

So maybe an analogy would be, it's like you developed a field where, from that mess, you could derive important insights and even turn them back into proofs

And there's debate on whether all invariants truly are maintained throughout the entire process

> Plus times plus […] but we don't even know what that means

Yes, we do. https://youtu.be/RcVA8Nj6HEo?t=1017

The video doesn't say what you think it says

Yeah, in Japan, it's well known that Pi is divisible.

https://kyoko--np-net.translate.goog/2005020901.html?_x_tr_s...

Is this the Japanese equivalent of The Onion?

Well he sounds like an arsehole either way...

He's, at least, weird.

This has been going on for 13 years. The difficulty on understanding the Inter-universal Teichmüller theory (valid or not) is it's all based on his Inter-universal geometry framework that only he and a handful of his students understand. So the work can't be peer-reviewed.

He has been offered to travel to work with other high level mathematicians to lecture them about his framework so other people can understand it but he has refused. He rarely travels (if at all) and he's very private, and doesn't even have lunch with his colleagues.

And I would speculate he sometimes disappear of the public eye, as he even has a section on his personal web site to notify he's alive [0].

I haven't asked friends in a couple years, but in math research centers the feelings were 'meh'. That there were probably some interesting things there, but it was going to be impossible to take something out of it unless something changes with Mochizuki or his students.

  0: https://www.kurims.kyoto-u.ac.jp/~motizuki/anpi-kakunin-jouhou.html

This title is extremely clickbaity. If a few people in Japan believe something that nobody else believes, that thing is not "only true in Japan", nobody knows if it's true anywhere.

> If a few people in Japan believe something that nobody else believes, that thing is not "only true in Japan", nobody knows if it's true anywhere.

Well, not quite. Sometimes believing in things makes them true.

https://en.wikipedia.org/wiki/Culture-bound_syndrome

>https://en.m.wikipedia.org/wiki/Ataque_de_nervios

Spaniard here, this is nonsense, I'm pretty sure everyone in the world experienced a nervous breakdown/light panic attack.

Spaniard here as well. What we call "ataque de nervios" (a nervous breakdown) in Spain may not match the usage of that term in Puerto Rico, as it is discussed in the Wikipedia article.

Just because Puerto Ricans and Spaniards speak dialects of the same language doesn't make our culture all that similar, as you surely know. I would even say there is little in common culturally.

Ah, TIL. Maybe centuries ago all of the breakdowns/dissociations were under the same mental basket...

We have the expression "attack of nerves" in English too, but I wouldn't expect it to represent anything like a nervous breakdown. It's not a strong term. Generally it would be provided as e.g. the reason you changed your mind about something.

I really do wish "extremely clickbaity" just meant something like this. Clearly our experiences with clickbait must be very different.

I's only extremely clickbaity in Japan

I feel like the implication was fairly clear here (Or maybe not "extremely" clickbaity as far as headlines are concerned)

Really? I clicked through and was wondering what weird and interesting mathematical twist I'd read about, and it turned out to be "well only a few people believe it, and they happen to live in Japan".

Yeah, exactly, me too. I was expecting something about Japan using 50 and 60 Hz at the same time, something about counting in units of 万 instead of 1000s, some WWII story, heck even some Fukushima one, but instead it was just something akin the Kirisuto no Hata (the belief that Jesus ended up in Japan by some).

Agreed - it wasted my time.

I agree - it should have been "A math proof only accepted by Japanese mathematicians"

Not even that, "only accepted by some mathematicians". The fact that they're all in Japan is entirely incidental.

When I saw the headline, I really hoped the article would be about some neuro-linguistic phenomenon that made native speakers of Japanese uniquely able to understand some mathematical proof.

Same! I think we may have more success with Lojban. Speaking of, are there any children whose native language is Lojban? That would definitely make for some interesting "case studies". I certainly want my kid to be bilingual, English and Lojban. :)

I was also disappointed.

Well, I guess it isn't really so easy when you're big in Japan.

Finally, a proof that’s less ‘universal truth’ and more ‘regional dialect’. Next up: the Schrödinger’s theorem—proven and unproven, depending on your timezone.

There's a method of death that's only true in South Korea too: https://en.wikipedia.org/wiki/Fan_death

What can be proven depends on what is allowed be a part of mathematics and logic. Zero, negative numbers, imaginary numbers and a lot other stuff had go through the acceptance first before they can be used in proofs. A lot of foundational concepts in logic, reality, causality, boolean exclusivity, spatial locality - had to be rewritten due to advances in quantum physics etc.

We went through this over a 100 years ago, math now sits on very solid axioms (look up ZFC), they're not questioning that.

> math now sits on very solid axioms (look up ZFC), they're not questioning that.

People question C all the time. That might be the most prominent ideological difference in mathematical philosophy.

Does it matter? Of course not, but people question it anyway.

Logicians and philosophers of mathematics have also questioned ZF set theory and "set theory" more generally.

For example the axiom of infinity (by finitists), the power set axiom and first-order theories in general (the downward Löwenheim-Skolem theorem implies that the infinity and power set axioms can't guarantee the existence on an uncountable power set), the fact that ZF doesn't allow a set of everything, and in particular no proper set complements, the fact that the axiom of regularity seems to be useless, etc.

Of course most ordinary mathematicians don't care about all that, because they don't care about ZF(C) or set theory or the foundation of mathematics in general. They rather care about problems in their specific field, like algebraic topology or whatnot.

200 years passed by between Newton and Einstein. 100 years is tiny in the evolution of thought and is no basis for shutting the questions down.

I wasn't trying to make an appeal to authority due 100 years having passed, that was tangential to my point that almost all modern math now sits on formalized axioms, which it did not do before the foundational crisis in math was resolved (about 100 years ago).

Comparing the axioms of math to relativity in physics is just nonsensical. Math is independent of observation, if a proof is formally correct now, it will always be correct under that chosen axiomatic system. Sure, we can play with different axioms (as others commented, it's common to drop the axiom of choice), but that doesn't invalidate the previous work at all.

Math is not independent of observation. Math sits on logic which itself sprouts from human experience with the world around them. Math and logic are not alien pure forms isolated from this world. There is not even single concept of logic that is not fully tied to the human experience and perception (of the world).

The concepts such as true, false, equal, greater than - all refer to human experience with counting things or perception of existence etc.

While there used to be resistance to coming up with new formal systems that played with loosening certain restrictions in long-used systems, I think this has not been true for a long time. If you want to come up with a new set of axioms of arithmetic today in which pi = 3, and you can actually come up with a set of meaningful axioms and prove some interesting property of this formal system, I don't think it would be that hard to get mathematicians to accept it and occasionally use it.

In taxicab geometry, π=4 https://en.wikipedia.org/wiki/Taxicab_geometry

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