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.
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0: https://www.kurims.kyoto-u.ac.jp/~motizuki/anpi-kakunin-jouhou.htmlPeople 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.
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.
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.
The concepts such as true, false, equal, greater than - all refer to human experience with counting things or perception of existence etc.
Well, not quite. Sometimes believing in things makes them true.
Spaniard here, this is nonsense, I'm pretty sure everyone in the world experienced a nervous breakdown/light panic attack.
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.
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.
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.
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.
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 that's hardly suspicious at all.
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?
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.
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.
[0]: https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%2...
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
Yes, we do. https://youtu.be/RcVA8Nj6HEo?t=1017
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.https://kyoko--np-net.translate.goog/2005020901.html?_x_tr_s...
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