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Discussion (36 Comments)Read Original on HackerNews
> A clear explanation can be found in Alex Kontorovich’s account of his own learning curve with formalized mathematics. In a nutshell: Mathlib, the dominant Lean library, is a human-curated formalization of an ever-growing fraction of existing human mathematics. It exposes clean APIs and abstractions, without which no autoformalization could take place. By contrast, Math Inc’s autoformalized proof of Viazovska’s results exposes no intelligible interface. Who in their right mind would merge a 200,000-line unaudited vibe-coded blob into the master branch of global human science?
https://davidbessis.substack.com/p/the-fall-of-the-theorem-e...
There are of course all the computer-assisted proofs (see 4 color theorem), as well as the partially-assisted ones (see Viazovska et al on packing problems in dimensions 8, 24). But even finding a solution numerically, then rigorously verifying its properties can leave a lingering sense of incompleteness, of a gap in understanding. I like this one quote by (allegedly) Wigner that illustrates it well:
"It is nice to know that the computer understands the problem, but I would like to understand the problem, too."
If you don’t understand the problem you can’t be sure that the computer does.
But I also definitely don't understand the problem if I can't get the computer to understand it, with tests.
In some sense I always considered programming to be more trustworthy than maths arguments without the certainty of a solver proof.
With all of these questions in the air, epistemology might be making a comeback.
A fundamental property of any formal proof is that it can be checked by a fairly stupid machine, automatically, because every step is a simple mechanical operation that names one of a handful of axioms and refers to a handful of earlier steps, the truth of which has already been established. So while coming up with a proof may require genius-level thinking, checking an existing fully fleshed out proof is simple -- just potentially very tedious because of the sheer number of steps.
That said, a typical human-written proof omits many steps considered "obvious" to a trained mathematician. Converting this to a formal proof involves interpreting what the original author "must have meant", which requires a lot of expertise and can go wrong -- or it may reveal that there is some inconsistency in the original claim itself.
The controversy around Mochizuki and the "abc Conjecture" proof is a contrary example.
The quality of the mathematics is a function of who has authored it?
Priests were interpreting the oracles (at least at a place like Delphi) according to the context of the people asking the questions aka participating in politics of that ancient times.
Subjectivity was a feature and I’m not sure that fits to mathematics though.
I wonder if mathematics as a science field moves more into engineering or if a different branch will emerge that is closer to that because to the point of the article, science is about understanding not just results.
This is a decidedly anti-enlightenment statement.
Would some one with tokens to burn mind checking that statement out and post back. Be sure to use long dashes too.
i.e. You have to be a good engineer to understand the well generated LLM code and a program
If mathematics is human understanding of logical consequences, understanding is the priority. But if AI proves something we can't understand but can utilize, that is a different sort of useful.
We are getting awfully close to "the answer of the universe is 42" and having it not be a joke...
I couldn’t build an internal combustion engine or even a plastic box, so maybe there’s nothing wrong with this approach.
> Some worry about the accessibility of AI tools. Traditionally, mathematicians have required little more than intuition, training, and a pen and paper to advance their field. If this slow, deliberative process is no longer valued by society, and particularly by research funders, then mathematics could become an elitist activity, only practiced by select organizations that can afford to work with proprietary AI models.
This can be true of anything LLMs do better than existing options. Because the best LLMs require enormous resources to develop, access to them can be very limited. Right now they are priced for market share. What happens when your small law firm attorney, or self-representation, goes up against a large firm with LLM expertise? Can the kid from the poor high school compete with the kid from the rich school with premium LLM access, in mathematics for example?
It would be great if someone could explain to me how AI improves this situation. Even if AI thinks it’s solved a problem, unless the proof is incredibly efficient and well explained, it will be difficult to verify the correctness. One hallucination in 300 steps of logic is enough to destroy the entire proof.
It's main utility is in the search step, not the verification step. The search is the bulk of the work and creativity. Separately, as the sibling commenter pointed out, it will likely get better at the verification step as well, with integrations of tools like Lean.
> One hallucination in 300 steps of logic is enough to destroy the entire proof.
The situation with human mathematicians is not much different. Eg, Wiles original proof of Fermat's Last Theorem contained errors found by reviewers, which he later repaired.
There's an interesting wrinkle though. The formal definitions and statements need to be correct, i.e. need to faithfully translate the human-readable (informal) definitions and statements. A few months ago, Kevin Buzzard created an auto-formalization challenge: https://gist.github.com/kbuzzard/778bc714030b3e974ab5f403878...
It's a handful of human-written lines, a skeleton that defines an "API surface". If you use AI to generate the rest of the code, the API surface (along with the Lean compiler) acts to guarantee that the whole generated solution (tens of thousands of lines) will be correct.
The discussion thread about the challenge, which was solved a couple weeks ago, is here: https://leanprover.zulipchat.com/#narrow/channel/583336-Auto...
Without the human-written or human-audited skeleton, you might get AI-generated slop Lean code that compiles, but contains nonsensical definitions/statements and thus vacuous proofs. Though from what I gather, recent models are much improved in their capability to generate correct statements and definitions and proofs, for both formal math and informal math.
A discussion thread about a different auto-formalization effort without human-written API surface, with some correct output and some slop: https://leanprover.zulipchat.com/#narrow/channel/583336-Auto...
You aren't having the AI check the proof, you interactively work on the same LEAN proof, handing off between the AI assistant and having LEAN check it and provide feedback for both of you when there's a mistake.
(edit: lol didn't realize the sibling comment below is essentially my comment)
It can definitely be a good research assistant though