21 ms·

A thoughtful and measured post, as usual from Gowers. The final note is neat and worth pasting out here in full: > A good sign that LLMs have reached human lev

by n4r9 2mo ago

A thoughtful and measured post, as usual from Gowers. The final note is neat and worth pasting out here in full: > A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. They should also be methods that are difficult to stumble on by accident. It is hard to say precisely what would count as such a proof, but I think we’ll recognise it when we see it.

tcp_handshaker 2mo ago
>>A good sign that LLMs have reached human level for a much wider class >> of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. I must be taking crazy pills and the AGI surely will pass me by... But TODAY, middle August 2026...And in the context of testing and evaluating the capabilities of current SOTA models to implement an Agentic application for job search, here is some simple inhouse built evals I run today, since I don´t trust LLM vendors published benchmarks... Models tested: GPT-5.6 Sol in Extra High mode and Opus 4.8 Max. TASK REQUEST: Clear, not too long not too short prompt, for LLMs to go out and research freelance consulting gigs for one specific IT domain, and in one specific country in Europe, including maybe opportunities driven from temp agencies based in geographically close countries. RESULT: Models go out, fetch the data, and completely misunderstand the task...offering on first results, permanent roles instead of freelance, and based on the country where the agencies are, not in the one it was request for. Think for example IT jobs in Ireland, while freelance agency in London. ANALYSIS: No intelligence I can call it shown by models, adding cognitive effort for human in the loop to detect subtle factors, and therefore totally useless for agentic app...Best practices would be I guess to add agents on top of agents but although in the p95 of cases that will reduce the errors...for the remaining 5% that could have hallucinations or logic hallucinations like these ones, compounding on top of other logic hallucinations. I dont care about the theorems being proven. At the end we will found out what most mathematicians were doing, was just exploring the same combinatorial and abstraction patterns. And because of that I am sure LLMs will make mince meat of a lot of mathematical domains. But right now, what we call intelligence is not existing where it matters, and Ed Zitron is right its a parlour trick.
- m348e912 2mo ago
  >TASK REQUEST: Clear, not too long not too short prompt, for LLMs to go out and research freelance consulting gigs for one specific IT domain, and in one specific country in Europe, including maybe opportunities driven from temp agencies based in geographically close countries. As a human, not an LLM, I could interpret "including maybe opportunities driven from temp agencies based in geographically close countries" as meaning "including opportunities in nearby countries outside of Ireland" (that happen to be driven by temp agencies). Before writing off LLM as simply a "stochastic parrot" or a "parlour trick" remember it can't read your mind, not yet anyway.
  - tcp_handshaker 2mo ago
    I am describing the contents of the prompt that was not the prompt. The prompt was very clear to the model that some freelance opportunities in country A, the only one in consideration could be available via agencies in country B and C. And it was a clear prompt. So what happen is a prompt said for example, find freelance opportunities in Ireland but keep in mind some of these might be available via temp agencies in London. If you offer me not freelance but permanent roles, and not in Ireland in London...that is a logic failure. Its this type of complexity with the normal world, that these SOTA constructions so badly fail at, and so spectacularly fail at the margins... despite maxing all benchmarks...Parlour trick.
    - someguyiguess 2mo ago
      Based on the rest of your writing I’m going to assume that the prompt was the problem.
      - coldtea 2mo ago
        He was perfectly clear in both cases. If a human misunderstood this, they'd be a dumb human.
        fn-mote 2mo ago
antonvs 2mo ago
> with hindsight come to seem beautiful and natural. …because they’ve been internalized. I wonder if the author has ever questioned where his notions of “beauty” and “natural” come from.
- n4r9 2mo ago
  I'm sure that he has, and I think that his point is solid. Mathematical beauty often has something to do with the transition from "surprising" to "natural".
  - antonvs 2mo ago
    It seems surprising before you've learned and internalized it. It seems natural after that. I think this is a case of people not being very introspective about their beliefs about their work.
    - n4r9 2mo ago
      Are you claiming that anything you learn and internalise becomes natural? Because there are plenty of proofs (such as the classification of finite simple groups) that remain ugly and unnatural well after they're learnt. Gowers' claim is that there are particular areas of (or proofs in) mathematics that become especially natural and intuitive after internalising them. This seems uncontroversial to me, and closely linked to ideas of beauty.
      - antonvs 2mo ago
        I was referring to the Gowers quote you provided: > A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. That last part, "new and surprising but that with hindsight come to seem beautiful and natural," is an almost perfect description of how processing fluency (https://en.wikipedia.org/wiki/Processing_fluency https://en.wikipedia.org/wiki/Processing_fluency) or the mere-exposure effect (https://en.wikipedia.org/wiki/Mere-exposure_effect https://en.wikipedia.org/wiki/Mere-exposure_effect) works. There's plenty of other work along these lines. Gian-Carlo Rota wrote about this more philosophically in "The Phenomenology of Mathematical Beauty" (1997). He argued (among other things) that the same proof can go from opaque to beautiful purely through familiarity. My understanding is that no single person has "learned" finite simple group classification from top to bottom, so it's easy to see why it wouldn't fit this description - there's no-one in a position to say "with hindsight, it seems simple and natural." That makes it a poor example for your/Gowers' position.

by n4r9 2mo ago

tcp_handshaker 2mo ago

>>A good sign that LLMs have reached human level for a much wider class >> of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. I must be taking crazy pills and the AGI surely will pass me by... But TODAY, middle August 2026...And in the context of testing and evaluating the capabilities of current SOTA models to implement an Agentic application for job search, here is some simple inhouse built evals I run today, since I don´t trust LLM vendors published benchmarks... Models tested: GPT-5.6 Sol in Extra High mode and Opus 4.8 Max. TASK REQUEST: Clear, not too long not too short prompt, for LLMs to go out and research freelance consulting gigs for one specific IT domain, and in one specific country in Europe, including maybe opportunities driven from temp agencies based in geographically close countries. RESULT: Models go out, fetch the data, and completely misunderstand the task...offering on first results, permanent roles instead of freelance, and based on the country where the agencies are, not in the one it was request for. Think for example IT jobs in Ireland, while freelance agency in London. ANALYSIS: No intelligence I can call it shown by models, adding cognitive effort for human in the loop to detect subtle factors, and therefore totally useless for agentic app...Best practices would be I guess to add agents on top of agents but although in the p95 of cases that will reduce the errors...for the remaining 5% that could have hallucinations or logic hallucinations like these ones, compounding on top of other logic hallucinations. I dont care about the theorems being proven. At the end we will found out what most mathematicians were doing, was just exploring the same combinatorial and abstraction patterns. And because of that I am sure LLMs will make mince meat of a lot of mathematical domains. But right now, what we call intelligence is not existing where it matters, and Ed Zitron is right its a parlour trick.

m348e912 2mo ago
>TASK REQUEST: Clear, not too long not too short prompt, for LLMs to go out and research freelance consulting gigs for one specific IT domain, and in one specific country in Europe, including maybe opportunities driven from temp agencies based in geographically close countries. As a human, not an LLM, I could interpret "including maybe opportunities driven from temp agencies based in geographically close countries" as meaning "including opportunities in nearby countries outside of Ireland" (that happen to be driven by temp agencies). Before writing off LLM as simply a "stochastic parrot" or a "parlour trick" remember it can't read your mind, not yet anyway.
- tcp_handshaker 2mo ago
  I am describing the contents of the prompt that was not the prompt. The prompt was very clear to the model that some freelance opportunities in country A, the only one in consideration could be available via agencies in country B and C. And it was a clear prompt. So what happen is a prompt said for example, find freelance opportunities in Ireland but keep in mind some of these might be available via temp agencies in London. If you offer me not freelance but permanent roles, and not in Ireland in London...that is a logic failure. Its this type of complexity with the normal world, that these SOTA constructions so badly fail at, and so spectacularly fail at the margins... despite maxing all benchmarks...Parlour trick.
  - someguyiguess 2mo ago
    Based on the rest of your writing I’m going to assume that the prompt was the problem.
    - coldtea 2mo ago
      He was perfectly clear in both cases. If a human misunderstood this, they'd be a dumb human.
      - fn-mote 2mo ago

antonvs 2mo ago

> with hindsight come to seem beautiful and natural. …because they’ve been internalized. I wonder if the author has ever questioned where his notions of “beauty” and “natural” come from.

n4r9 2mo ago
I'm sure that he has, and I think that his point is solid. Mathematical beauty often has something to do with the transition from "surprising" to "natural".
- antonvs 2mo ago
  It seems surprising before you've learned and internalized it. It seems natural after that. I think this is a case of people not being very introspective about their beliefs about their work.
  - n4r9 2mo ago
    Are you claiming that anything you learn and internalise becomes natural? Because there are plenty of proofs (such as the classification of finite simple groups) that remain ugly and unnatural well after they're learnt. Gowers' claim is that there are particular areas of (or proofs in) mathematics that become especially natural and intuitive after internalising them. This seems uncontroversial to me, and closely linked to ideas of beauty.
    - antonvs 2mo ago
      I was referring to the Gowers quote you provided: > A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. That last part, "new and surprising but that with hindsight come to seem beautiful and natural," is an almost perfect description of how processing fluency (https://en.wikipedia.org/wiki/Processing_fluency https://en.wikipedia.org/wiki/Processing_fluency) or the mere-exposure effect (https://en.wikipedia.org/wiki/Mere-exposure_effect https://en.wikipedia.org/wiki/Mere-exposure_effect) works. There's plenty of other work along these lines. Gian-Carlo Rota wrote about this more philosophically in "The Phenomenology of Mathematical Beauty" (1997). He argued (among other things) that the same proof can go from opaque to beautiful purely through familiarity. My understanding is that no single person has "learned" finite simple group classification from top to bottom, so it's easy to see why it wouldn't fit this description - there's no-one in a position to say "with hindsight, it seems simple and natural." That makes it a poor example for your/Gowers' position.