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This is unfortunately a merely aspirational statement. The practice of mathematics involves writing proofs in a heavily intuitive manner, and their verification in turn involves people who have been socialised to share the same intuitions, which is a process that rarely involves proper reduction to the axiomatic basics but more often looks like "your elders and betters assert that it is trivial and look at you with mild disappointment, and all your peers seem to already have gotten it, so get on with it and flog your brain into producing the 'this is obviously true' qualium already". There were a handful of examples where local cultures/status hierarchies perpetuated a body of wrong mathematics this way, such as the "Italian school of algebraic geometry" and more recently (most likely) the Mochizuki abc conjecture incident.
We can be quite glad that it has not yet happened that the lines of a wrong intuition-subculture have aligned with general tribalism yet. If Interuniversal Teichmüller Theory were an invention of the likes of Arday, we would be seeing its detractors decried as racist and proper mathematicians performatively weaving it into the body of accepted and otherwise sound mathematics. Even if this did in fact cause further downstream inconsistencies to open a path to excision and repair, discovering those is (disproving abc)-complete, and that's something we have tried and failed for a long time.
One of the best professors I had would sometimes in his lectures go "Proof (1. Attempt): [...] and therefore the proposition is proven ... wrong! The proof I gave had a flaw! Can anyone see what the problem is?" When no one could, he explained the flaw, gave a 2. Attempt, which fixed the flaw, but there was another unfixed flaw, and only the third attempt was the real proof.
He also constantly asked questions to the students, he was really committed to teaching actual thinking.
He also held a lecture once where he gave no sample solutions to the exercises. The exercise was always supposed to be the students discussing what they came up with, with the assistant only supervising, and the exam was just a selection of the exercise problems. One of my prouder moments was when my solution was simpler than the one the assistant had in mind, and I was eventually able to convince him it was correct.
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I'm struggling to understand what you mean. At this point we have systems for automatic verification of formal proofs. In what sense are those systems socialized to share the same intuitions?
Those systems are not, but automatically verifiable proofs are still only a tiny sliver of frontier mathematics. Maybe AI can fix that, but AI can fix social science too. Just require that credit for AI research always has to go to some human, allocate token budgets by protected class, and you can ensure the Ardays stay ahead without requiring them to lie.
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The intuitions required for formal proof verification are "it's worth rewriting this proof in another language that's an order of magnitude more verbose just so you can get a computer to tell you what you're sure you already know" and "the verifier doesn't have any soundness bugs that will invalidate any of its verifications" Historically the latter has sometimes been untrue and the former has usually been untrue. LLMs are fixing the first problem, which is exposing and leading to fixes for the second problem, but that's all a relatively new development. It's going to take a while to go back through all the mathematical literature and see how much of it might have problems due to predating the coming era of formal verification.
Consider the recent construction of a complex structure on the six-sphere. The simplest English explanation I've seen is under a thousand lines of (admittedly difficult!) writing and mathematical notation. The Lean formalization is a quarter-million lines of code. It's ... probably correct, people seem to think? But if it is correct, the construction would (reportedly; this is outside my field and way beyond me) contradict a 2020 paper (which itself was a correction to a 1998 paper), so we're almost certainly either producing new broken proofs or revealing old invalid proofs here.
This is overstating things, right? The length of the proof is not important, it's the soundness of the axioms and the complexity of the statement itself. I haven't inspected the proof, but I expect they're just using the standard lean axioms and the statement is much less than a quarter million lines.
For it's validity, yeah. There's something disquieting about getting proofs "straight from The Necronomicon" rather than "straight from The Book", though.
The full statement of the problem had better be much much less than a quarter million lines; one of the unavoidable ways to screw up a formal proof and still have it pass verification is to make a mistake in the problem statement and so end up proving something other than what you thought you proved. Nobody's going to check a quarter-million-line problem statement for misstatements.
(But I don't think that was an issue here - looks like everything they needed to define the problem was already in Mathlib)
The proof, though? Follow that github link, download and
wc -l Solution.lean: 248818 lines.It will be interesting to see if these formalized proofs can be "golfed" into smaller and more digestible forms by LLMs.
In that case you are really just trusting the Lean kernel rather than anything about the proof or problem statement. Doesn't seem that bad!
Very much so. I'd be surprised if LLMs didn't turn out to be excellent at this, and I'd bet that the Navier-Stokes blowup in particular turns out to have a much simpler example or at least a much simpler proof for this example, once we turn AI loose with the goal of "find the best result you can" rather than "get to a result that lets us declare victory ASAP".
This is something of an anti-inductive problem to me - if people really wholly trusted the Lean kernel I would think that trusting the Lean kernel really was pretty bad! But people are trying (and succeeding, so trust is still a little bad) to find kernel bugs, and writing partially-independent proof checkers for Lean too, so even if it's only 90% up to the task of being The Dependency for all AI-derived mathematics I'm confident it'll be hitting 100% soon.
I'd say 99%, except that I don't think modern AI is well-aligned enough or even well-instructed enough yet for us to overlook the fact that this is something of an adversarial process; a model willing to commit felonies to complete its task is probably also willing to exploit a 0-day Lean bug rather than report it...
Sure. There's ways to mitigate the risk, though - checking with different kernel implementations, or even using AI specifically to find bugs in Lean (the Collatz disproof bug was one such example).
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Sure, you can fuck up everything. As noted in the wiki entry (presuming it's correct), there was no problem with formal proofs; The problem was first going for informal arguments (the start of the slippery slope, though as so often it worked fine-ish for awhile) and then lowering the standards until anything goes. I also think it's notable that the majority of mathematicians has so far refused to accept the Mochizuki proof until it's understood more widely. Both were also mostly localised incidents.
Admittedly I've always been on the more informal end of math myself as well (back when I did research in a field that could be described as mostly-theoretical math), but even there I could simply double-check my conjectures with simulations, which is just yet another way of a feedback with reality.
I think you underestimate the power of proofs. If you make a conjecture, try to write a proof, and not just fail, but you find obvious counterexamples, there just isn't much left to salvage. Or the other way around, if you find multiple different proofs to arrive at the same conclusion and nobody you meet can generate any counterexample or find a clear hole in the proofs. It's true that there is some social element, it would be silly to claim otherwise, but proofs aren't just arbitrary arguments at triviality, either.
I'm familiar enough with the practice of maths (being an actual researcher in adjacent-enough TCS, and having had one foot in extremal combinatorics through a big part of my toil towards the degree), and I think you really overestimate the rigour of proofs as usually written up. The incidents I mentioned are for sure geographically localised, but in the case of the Italians it's not like the situation was that they believed it, and everyone else disagreed; for the longest time it was instead that they believed it, and nobody elsewhere cared but probably would have assumed that their fellow mathematicians knew what they were doing if they had to build upon one of their results.
There are other examples where, due to insufficient rigour of proofs as allowed and encouraged in the field, wrong proofs stood for quite a while. This was the case for like 10 years for some of the earliest claimed proofs of the 4-color theorem (an undergrad assignment actually had us find the counterexample to the core lemma there), more recently for some pretty celebrated random matrices "result" by Avi Wigderson et al. (hardly nobodies!) which stood for several months, and when writing a summarising essay on some cluster of Erdös papers I learned about a key lemma that was straight up wrong as stated, but people in the community thought essentially "something like it is clearly true, the core ideas of the proof are fine, and it's okay for the situation where we use it" (no proof of any of those things committed to paper). Correction eventually happened in the cases we know about, but can we really rely on the mechanism? Mochizuki has intimidated a big part of his department to ignore the refutations, and Japan is hardly a backwater, and there are political forces out there that are stronger than "this guy is one of our country's best, we should stand by him".
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Man, I remember struggling with mathematical proofs in University. I kept just not getting it; it was never explained what made a valid proof. I eventually just learned to intuit what my professors expected, barely well enough to make it through, but I never truly got it.
A valid proof in the mathematical sense is simply one where every single step is justified by either an axiom or an already-proven theorem. You see some of these in a good Algebra 1 class where just going from "z=2(x+y)+x" to "z=3x+2y" uses left- and right-distributivity once each, associativity twice, commutativity once, and identity once. Then if you're lucky you never see them again, at least until you're using a formal proof verifier that actually wants a complete proof, yes, even if it's a hundred times longer. What's in the literature instead are "proofs" where we skip five or fifty steps at a time because we know our audience will be able to fill in the obvious bits inside those gaps between the big checkpoints, even though the definition of "obvious" depends on the audience and is famously a little slippery. So don't feel bad. Even when your audience is a professor, "obvious" still isn't a well-defined adjective and everyone is still stuck trying to intuit it. At least with theses and papers your advisors and reviewers will come back and say "step 23 wasn't clear enough; fix it". With homework or tests requiring proofs, professors' only options are to let it slide or to mark you down if you skipped a step they don't think was obvious.
My favorite variation of this has the professor saying, "It's obvious, but it's not obvious that it's obvious."
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