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.
* 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.
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.
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?
>>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.
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.
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.
>>>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’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.
> 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.
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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.
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...
But that's fair, it's not exactly one-sided, but to my (completely inexpert) judgement the matter seems heavily weighted against mochizuki?
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.
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.
Now imagine taking something like biology and vaccines. What happens if you rely on your experts and other rely on theirs, and they disagree?
>>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.
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.
Deleted Comment
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.