4 ms·

Next up the abc-conjecture! Claimed to be proven in 2012, and to 400+ page paper is online. But I don't many accept the proof.

nicklecompte 3y ago
There is a major difference though: Holmes's proof was broadly comprehensible, just extremely complicated and easy to get lost in the details. In particular Holmes really tries to make the reader understand, with a fairly gentle/apologetic introduction. But Mochizuki's "proof" is completely impenetrable even to experts on, like, page 3, and makes no effort to explain what is happening. Another key difference is that New Foundations is a niche field, so there simply was not a huge amount of human effort spent reading Holmes's work. That's not the case with Mochizuki's proof. There's a big difference between "a small number of mathematicians didn't understand the proof but suspect it's correct" and "a large number of mathematicians didn't understand the proof and concluded it was incorrect." And, most of all: Holmes's formulation of twisted type theory made the proof a natural candidate for dependently-typed formal verification. Mochizuki's proof is not type-theoretic and does not seem like a great candidate for the calculus of constructions - maybe it is! I actually have no idea. I suspect Mochizuki is the only person in the world who can answer that. But it's critical that Holmes did so much background work to simplify his proof and make this Lean program possible. Mochizuki should do the same. AFAICT, both in terms of the "sociology of mathematics" and the amount of work required to even attempt a Lean formalization, trying to verify Mochizuki's proof is a waste of time.
- Ericson2314 3y ago
  > And, most of all: Holmes's formulation of twisted type theory made the proof a natural candidate for dependently-typed formal verification. I am not so sure about this, actually. "twisted type theory" looks like type theory more in the original Bertrand Russel sense than in the modern (Martin Loef, Jean-Yves Girard, etc.) sense.
- alreadydone 3y ago
  I think TTT stands for "tangled type theory".
  - randallholmes 3y ago
    It does, but I rather like "twisted type theory" :-)
  - nicklecompte 3y ago
    Oops! Good catch. I'm not an expert - I read "tangled type theory" once and "TTT" about 75 times....
- ducttapecrown 3y ago
  Kirti Joshi from the IAS has made very serious attempts to patch up Mochizuki's work by developing new math and by making the exposition good. If you're an expert in arithmetic geometry presumably he's readable, if you're just interested, I found it fun to skim.
  - alreadydone 3y ago
    Latest update here: https://www.math.columbia.edu/~woit/wordpress/?p=13895 https://www.math.columbia.edu/~woit/wordpress/?p=13895 Unfortunately it seems Mochizuki and Scholze reached consensus that a part of Joshi's argument cannot possibly work.
  - Davidzheng 3y ago
    No comments on the correctness of his work (plenty of discussions online e.g on MO) but why do you say he's from IAS?
    - ducttapecrown 3y ago
      I write about his work on this website to bring notice to his work, because I don't believe Scholze and I'm not going to read Mochizuki (If I did, it would be his original actual contribution to algebraic geometry (and I don't know why I don't read Scholze but its scary mmkay)). So I mentioned that Joshi was from the IAS for the same reason, to convince my readers to read his work. Further edit: I guess he's at Arizona right now, but he was at IAS at some point....
nextaccountic 3y ago
There's a proposed proof of abc conjecture that is supposedly more readable. Here is a discussion thread about it: https://www.reddit.com/r/math/comments/1bhiz0s/construction_of_arithmetic_teichmuller_spaces_iv/ https://www.reddit.com/r/math/comments/1bhiz0s/construction_... It would be nice if this one were formalized in Lean (or Coq or HOL) though.
barfbagginus 3y ago
I seriously doubt mochi's going to be any help for the lean formalization effort. But a lean-assistive llm trained on mochi's work? Ahhh! What an intriguing possibility!
- Chinjut 3y ago
  I do not imagine LLMs will be of any use here.
  - barfbagginus 3y ago
    Then it's time to update your LLM reading! https://leandojo.org/ https://leandojo.org/ https://github.com/lean-dojo/LeanCopilot https://github.com/lean-dojo/LeanCopilot https://arxiv.org/abs/2404.07382 https://arxiv.org/abs/2404.07382
    - Davidzheng 3y ago
      These llm's are usually only proving some trivial lemmas right now. Hopefully it changes soon but...
      - barfbagginus 3y ago
        Just bolting llms onto proof search improves the state of the art. So if you want to improve the state of the art of proof search, bolt some llms onto proof search, and enjoy! As far as making LLMs understand mochi math... I'm going to go out on a limb and say it will probably take less time for us to build an AI that understands mochi than it would take to understand mochi ourselves.
        dimask 3y ago
        I would bet for this use case LLMs to be worse than random walk, as they would probably disregard certain possibilities in crucial steps, unless we are talking about easier stuff.
toneman 3y ago
I have studied IUT since 2012, and it is indeed, totally baroque. However, Motizuki Theory, is totally rebased and admits results of much interest. I will write more on this matter if my claim is of mutual interest. IRT, the topic and digression here, LLM and LEAN4 are of not much use for IUT. IUT Theory is much easier understood by a hacker than by a math guy, eventually tho, monitors and tv's did kinda become the same thing but there are, some minor differences.

nicklecompte 3y ago

There is a major difference though: Holmes's proof was broadly comprehensible, just extremely complicated and easy to get lost in the details. In particular Holmes really tries to make the reader understand, with a fairly gentle/apologetic introduction. But Mochizuki's "proof" is completely impenetrable even to experts on, like, page 3, and makes no effort to explain what is happening. Another key difference is that New Foundations is a niche field, so there simply was not a huge amount of human effort spent reading Holmes's work. That's not the case with Mochizuki's proof. There's a big difference between "a small number of mathematicians didn't understand the proof but suspect it's correct" and "a large number of mathematicians didn't understand the proof and concluded it was incorrect." And, most of all: Holmes's formulation of twisted type theory made the proof a natural candidate for dependently-typed formal verification. Mochizuki's proof is not type-theoretic and does not seem like a great candidate for the calculus of constructions - maybe it is! I actually have no idea. I suspect Mochizuki is the only person in the world who can answer that. But it's critical that Holmes did so much background work to simplify his proof and make this Lean program possible. Mochizuki should do the same. AFAICT, both in terms of the "sociology of mathematics" and the amount of work required to even attempt a Lean formalization, trying to verify Mochizuki's proof is a waste of time.

Ericson2314 3y ago
> And, most of all: Holmes's formulation of twisted type theory made the proof a natural candidate for dependently-typed formal verification. I am not so sure about this, actually. "twisted type theory" looks like type theory more in the original Bertrand Russel sense than in the modern (Martin Loef, Jean-Yves Girard, etc.) sense.
alreadydone 3y ago
I think TTT stands for "tangled type theory".
- randallholmes 3y ago
  It does, but I rather like "twisted type theory" :-)
- nicklecompte 3y ago
  Oops! Good catch. I'm not an expert - I read "tangled type theory" once and "TTT" about 75 times....
ducttapecrown 3y ago
Kirti Joshi from the IAS has made very serious attempts to patch up Mochizuki's work by developing new math and by making the exposition good. If you're an expert in arithmetic geometry presumably he's readable, if you're just interested, I found it fun to skim.
- alreadydone 3y ago
  Latest update here: https://www.math.columbia.edu/~woit/wordpress/?p=13895 https://www.math.columbia.edu/~woit/wordpress/?p=13895 Unfortunately it seems Mochizuki and Scholze reached consensus that a part of Joshi's argument cannot possibly work.
- Davidzheng 3y ago
  No comments on the correctness of his work (plenty of discussions online e.g on MO) but why do you say he's from IAS?
  - ducttapecrown 3y ago
    I write about his work on this website to bring notice to his work, because I don't believe Scholze and I'm not going to read Mochizuki (If I did, it would be his original actual contribution to algebraic geometry (and I don't know why I don't read Scholze but its scary mmkay)). So I mentioned that Joshi was from the IAS for the same reason, to convince my readers to read his work. Further edit: I guess he's at Arizona right now, but he was at IAS at some point....

nextaccountic 3y ago

There's a proposed proof of abc conjecture that is supposedly more readable. Here is a discussion thread about it: https://www.reddit.com/r/math/comments/1bhiz0s/construction_of_arithmetic_teichmuller_spaces_iv/ https://www.reddit.com/r/math/comments/1bhiz0s/construction_... It would be nice if this one were formalized in Lean (or Coq or HOL) though.

barfbagginus 3y ago

I seriously doubt mochi's going to be any help for the lean formalization effort. But a lean-assistive llm trained on mochi's work? Ahhh! What an intriguing possibility!

Chinjut 3y ago
I do not imagine LLMs will be of any use here.
- barfbagginus 3y ago
  Then it's time to update your LLM reading! https://leandojo.org/ https://leandojo.org/ https://github.com/lean-dojo/LeanCopilot https://github.com/lean-dojo/LeanCopilot https://arxiv.org/abs/2404.07382 https://arxiv.org/abs/2404.07382
  - Davidzheng 3y ago
    These llm's are usually only proving some trivial lemmas right now. Hopefully it changes soon but...
    - barfbagginus 3y ago
      Just bolting llms onto proof search improves the state of the art. So if you want to improve the state of the art of proof search, bolt some llms onto proof search, and enjoy! As far as making LLMs understand mochi math... I'm going to go out on a limb and say it will probably take less time for us to build an AI that understands mochi than it would take to understand mochi ourselves.
      - dimask 3y ago
        I would bet for this use case LLMs to be worse than random walk, as they would probably disregard certain possibilities in crucial steps, unless we are talking about easier stuff.

toneman 3y ago

I have studied IUT since 2012, and it is indeed, totally baroque. However, Motizuki Theory, is totally rebased and admits results of much interest. I will write more on this matter if my claim is of mutual interest. IRT, the topic and digression here, LLM and LEAN4 are of not much use for IUT. IUT Theory is much easier understood by a hacker than by a math guy, eventually tho, monitors and tv's did kinda become the same thing but there are, some minor differences.