We have just got some very strong evidence about the way in which LLM-based systems solve mathematical problems and this evidence supports what many have already suspected including myself.
Here's what I'm talking about. On 10 August Anthropic released an article [1] claiming that:
An unreleased research version of Claude has improved on a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. Drawing on extensive prior research by mathematicians over the past decades, it has increased this bound from 41.6% to 67.2%.
The same article describes the methodology followed by Anthropic's employee, Jarred Sumner, who prompted Claude, as follows:
Jarred Sumner, an Anthropic staff member (and non-mathematician), prompted Claude to “take a real stab” at the hypothesis itself, leaving the mathematical choices from there up to the model. Initially, Claude generated and tried 650 ideas, none of which worked. Jarred prompted Claude to try again, and it spent a day and a half coordinating about 60 Claude subagents, which this time went much deeper: between them, they ran 2,400 shell commands and wrote hundreds of Python scripts.1 The subagents ran thousands of numerical checks against known zeta zeros and refereed one another’s work. Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”).2 This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.
Jarred got Claude to throw stuff at the wall repeatedly (650 initial "ideas" plus unspecified more by "60 Claude subagents" ... running "2400 shell commands" and "hundreds of Python scripts") and then kept whatever happened to stick. In this case, by happy accident, what stuck was an improved bound of the zeroes of the zeta function etc.
This is how every single mathematical result reported by an AI company has ever been generated. They throw stuff at the wall and take whatever happens to stick.
This approach works. Not only it works, it is, in principle, a universal problem solver. "Millions of monkeys on typewriters" will eventually produce a proof of the Riemann hypothesis; or a disproof of it.
The key point being "eventually". Is this a way to do mathematics research? Can that replace mathematicians?
In AI, this method is well-known as the "generate-and-test" method. It is ancient, basal to AI if I may be so bold. It first appeared to my knowledge in the Logic Theorist, the proof-finding program that Simon and Newell presented in the 1956 Dartmouth convention that named "Artificial Intelligence", to such luminaries of AI and CS as John McCarthy (the real "godfather of AI" who named the field), Marvin Minsky, Claude Shannon and others.
We've had the ability to brute-force all of mathematics "eventually", given "enough" compute for nearing a century now. Why haven't we solved all of mathematics? Are LLMs really so special that they can out-brute force search every previous brute force searcher?
Well, you tell me, HN. I say: no.
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