Hacker NewsEvery mathematician has only a few tricks (2020)
https://mathoverflow.net/questions/363119/every-mathematician-has-only-a-few-tricks
#HackerNews #mathematics #tricks #mathematician #mathoverflow #problem-solving #2020
Terence TaoA third update on my previous posts https://mathstodon.xyz/@tao/115306424727150237 https://mathstodon.xyz/@tao/115316787727719049 https://mathstodon.xyz/@tao/115325228243131134 on the #MathOverflow problem https://mathoverflow.net/questions/501066/is-the-least-common-multiple-sequence-textlcm1-2-dots-n-a-subset-of-t . Out of curiosity, I asked an AI deep research tool for the known literature on this problem: https://g.co/gemini/share/0d41fcadfbaf . As is often the case with these frontier models, the results were mixed. On the one hand, the tool correctly identified a relevant paper of Alaoglu and Erdos from 1944 https://users.renyi.hu/~p_erdos/1944-03.pdf , where they claimed that the authors knew that the conjecture was false, but did not provide a proof. The AI then hallucinated by asserting that n=17 was a counterexample, which I knew to be false. Based on this latter failure, I did not initially give much credence to the initial statement; but later, after reading the Alaoglu-Erdos paper carefully, I did find (on pages 466-467) that they did indeed make the observation that "it would be easy to see" that the conjecture could only be true finitely often, without providing any hint of a proof. (1/4)
Terence Tao (@[email protected])
I was able to use an extended conversation with an AI https://chatgpt.com/share/68ded9b1-37dc-800e-b04c-97095c70eb29 to help answer a MathOverflow question https://mathoverflow.net/questions/501066/is-the-least-common-multiple-sequence-textlcm1-2-dots-n-a-subset-of-t/501125#501125 . I had already conducted a theoretical analysis suggesting that the answer to this question was negative, but needed some numerical parameters verifying certain inequalities in order to conclusively build a counterexample. Initially I sought to ask AI to supply Python code to search for a counterexample that I could run and adjust myself, but found that the run time was infeasible and the initial choice of parameters would have made the search doomed to failure anyway. I then switched strategies and instead engaged in a step by step conversation with the AI where it would perform heuristic calculations to locate feasible choices of parameters. Eventually, the AI was able to produce parameters which I could then verify separately (admittedly using Python code supplied by the same AI, but this was a simple 29-line program that I could visually inspect to do what was asked, and also provided numerical values in line with previous heuristic predictions). Here, the AI tool use was a significant time saver - doing the same task unassisted would likely have required multiple hours of manual code and debugging (the AI was able to use the provided context to spot several mathematical mistakes in my requests, and fix them before generating code). Indeed I would have been very unlikely to even attempt this numerical search without AI assistance (and would have sought a theoretical asymptotic analysis instead).
A second update on my previous posts https://mathstodon.xyz/@tao/115306424727150237 https://mathstodon.xyz/@tao/115316787727719049 on the #MathOverflow problem https://mathoverflow.net/questions/501066/is-the-least-common-multiple-sequence-textlcm1-2-dots-n-a-subset-of-t that had a first initial counterexample located with #AI assistance, and then the minimal counterexample obtained through a crowdsourced effort. As has happened with several previous projects, the crowdsourced effort has acquired a life of its own. In particular, the task of classifying *all* the counterexamples - which I had previously dismissed in my previous post as "somewhat tedious to perform", has in fact been completed, with the problem alternating between true and false for n between the first counterexample 71 and the last example 172, before settling down to a unending string of counterexamples. We now have some very efficient ways to either prove or disprove that a given number is highly abundant; some of the asymptotic analysis initially required some high powered number theory hypotheses such as the generalized Riemann hypothesis (GRH), but as our arguments became more efficient, this became unnecessary. We even have some recent contributions from the #LeanLang community, formalizing some of the counterexamples in Lean, almost in real-time! (1/3)
Terence Tao (@[email protected])
I was able to use an extended conversation with an AI https://chatgpt.com/share/68ded9b1-37dc-800e-b04c-97095c70eb29 to help answer a MathOverflow question https://mathoverflow.net/questions/501066/is-the-least-common-multiple-sequence-textlcm1-2-dots-n-a-subset-of-t/501125#501125 . I had already conducted a theoretical analysis suggesting that the answer to this question was negative, but needed some numerical parameters verifying certain inequalities in order to conclusively build a counterexample. Initially I sought to ask AI to supply Python code to search for a counterexample that I could run and adjust myself, but found that the run time was infeasible and the initial choice of parameters would have made the search doomed to failure anyway. I then switched strategies and instead engaged in a step by step conversation with the AI where it would perform heuristic calculations to locate feasible choices of parameters. Eventually, the AI was able to produce parameters which I could then verify separately (admittedly using Python code supplied by the same AI, but this was a simple 29-line program that I could visually inspect to do what was asked, and also provided numerical values in line with previous heuristic predictions). Here, the AI tool use was a significant time saver - doing the same task unassisted would likely have required multiple hours of manual code and debugging (the AI was able to use the provided context to spot several mathematical mistakes in my requests, and fix them before generating code). Indeed I would have been very unlikely to even attempt this numerical search without AI assistance (and would have sought a theoretical asymptotic analysis instead).
theHigherGeometerLess than 24 hours for users to self-nominate for the #MathOverflow moderator elections https://mathoverflow.net/election the button/text link to click is at the bottom of the page.
Community moderation election on #MathOverflow is happening! Currently in nomination phase: https://meta.mathoverflow.net/q/6199/4177 We are looking to fill two positions.
An interesting (unscientific) experiment on #MathOverflow from a few months ago, where a user gave 15 different MO problems for o1 to answer, with the aim of verifying and then rewriting the answer into a presentable form if the AI generated answer was correct. The outcome was: one question answered correctly, verified, and rewritten; one question given a useful lead, which led the experimenter to find a more direct answer; one possibly correct answer that the experimenter was not able to verify; and the remainder described as "a ton of time consuming chaos", in which the experimenter spent much time trying to verify a hallucinated response before giving up. https://meta.mathoverflow.net/questions/6114/capabilities-and-limits-of-ai-on-mathoverflow This success rate largely tracks with my own experience with these tools. At present this workflow remains less efficient than traditional pen-and-paper approaches; but with some improvement in the success rate, and (more importantly) an improved ability to detect (and then reject) hallucinated responses, I could see one soon reaching a point where a non-trivial fraction of the easier problems in MO could be resolved by a semi-automated method.
I found the discussion for possible AI disclosure policies for MO in the post to also be interesting.
In the spirit of old questions, here's another one without an answer: can we get a "holomorphic model" for a K(Z,2)?
A recent answer by Simon Henry about coproducts of C*-algebras in a certain category reminded me of this old question on pushouts in the same category, which is still unanswered:
An anecdote that I shared about rolling around on the floor back in 2000 to solve a math problem, both in my #Masterclass at https://www.masterclass.com/classes/terence-tao-teaches-mathematical-thinking/chapters/transforming-problems , and on #MathOverflow at https://mathoverflow.net/a/38882/766 , as well as the #NewYorkTimes https://www.nytimes.com/2015/07/26/magazine/the-singular-mind-of-terry-tao.html , has for some reason recently gone viral on various social media. Just for the record, I wanted to add some mathematical background behind the story, which eventually led to my paper https://arxiv.org/abs/math/0010068 . At the time, I was trying to construct solutions to an equation known as the wave maps equation on the sphere: the solution was like a solution to the wave equation, except being forced to take values in a sphere rather than in a vector space.
I was trying to solve the equation iteratively, breaking up the solution to a low frequency base solution and a high frequency correction. As a first approximation, the low frequency base could also be assumed to stay on the sphere and solve the wave maps equation, so the main problem was to work out what the high frequency correction was doing.
Because the high frequency correction also had to keep the solution on the sphere, one could assume as a first approximation that the high frequency correction was tangent to the low frequency base. So, at any given point in space and time, the low frequency base solution was located on some point on the sphere, and the high frequency correction basically lived on the tangent plane to the sphere at that point. But because the base solution evolved (slowly) in space and time, this tangent space kept rotating around the sphere.
(1/3)
Given the interest in these posts, I thought I would share some other minor experiments I had also made with my preview of the model. In 2010 i was looking for the correct terminology for a “multiplicative integral”, but was unable to find it with the search engines of that time. So I asked the question on #MathOverflow instead and obtained satisfactory answers from human experts: https://mathoverflow.net/questions/32705/what-is-the-standard-notation-for-a-multiplicative-integral
I posed the identical question to my version of #o1 and it returned a perfect answer: https://chatgpt.com/share/66e7153c-b7b8-800e-bf7a-1689147ed21e . Admittedly, the above MathOverflow post could conceivably have been included in the training data of the model, so this may not necessarily be an accurate evaluation of its semantic search capabilities (in contrast with the first example I shared, which I had mentioned once previously on Mastodon but without fully revealing the answer). Nevertheless it demonstrates that this tool is on par with question and answer sites with respect to high quality answers for at least some semantic search queries. (1/2)