Ongoing Lean formalization of the proof for Fermat's Last Theorem
github.com
github.com
And if you want to read why we need additional types of science organizations, see "A Vision of Metascience" (https://scienceplusplus.org/metascience/)
It's probably a narrow set of problems with the right set of constraints and scale for this to be a win.
(OTOH, within the community they're effectively trying to build a massive, modern Principia Mathematica, so maybe they would...)
> Whereas broader research is full of risk and requires funders be enormously patient and willing to fund crazy ideas that don’t make sense.
Yah. I'm not a researcher, but I keep ending up tangentially involved in research communities. I've seen university labs, loose research networks, loose consortia funding research centers, FFRDC, etc.
What I’ve noticed is that a lot of these consortia or networks struggle to deliver anything cohesive. There's too many stakeholders, limited bandwidth, and nobody quite empowered to say “we’re building this.”
In the cases where there’s a clearly scoped, tractable problem that’s bigger than what a single lab can handle, and a group of stakeholders agrees it’s worth a visionary push, something like an FRO might make a lot of sense.
Scientific research is often not immediately applicable, but can still be valuable. The number of people that can tell you if it's valuable are small, and as our scientific knowledge improves, the number of people who know what's going on shrinks and shrinks.
Separately, it's possible to spend many years researching something, and have very little to show for it. The scientists in that situation also want some kind of assurance that they will be able to pay their bills.
Between the high rate of failure, and conflicts of interest, and inscrutability of the research topics. It's very hard to efficiently fund science, and all the current ways of doing it are far from optimal. There is waste, there is grift, there is politics. Any improvement here is welcome, and decreasing the dollar cost per scientific discovery is more important than the research itself in any single field.
But part of me feels like if they are going to spend the massive effort to formalize Fermat's Last Theorem it would be better to use a language where quotient types aren't kind of a hack.
Lean introduces an extra axiom as a kind of cheat code to make quotients work. That makes it nicer from a softer dev perspective but IMO less nice from a mathematical perspective.
This rule is fine in itself, but the Lean developers were not sufficiently careful and allowed it to apply for quotients of propositions, where it interferes with the computation rules for proof irrelevance and ends up breaking subject reduction (SR is deeply linked to computation when you have dependent types!) [0]. It is not really a problem in practice though, since there is no point in quotienting a proposition.
[0] see the end of section 3.1 in https://github.com/digama0/lean-type-theory/releases/downloa...
For example, if I want to prove that a+b=b+a, I wouldn't care if I'm directly in peano arithmetic or just have a construction of the peano axioms in ZFC, as in both cases the proofs would be identical (some axioms in PA would be lemmas in ZFC).
If that's the case with quotients, I wonder why it's such a big deal for some.
However, Lean's SR is broken in other ways which do show up in practice: definitional proof irrelevance is incompatible with Lean's computation rule for Acc. That one is much harder to fix.
> Throughout this section, we use A≈B to mean that A and B are essentially equal, in the sense that B is a suitable approximation of A in some sense that we will formalize in a later section. The reader may feel free to assume A=B when verifying estimates, even though A=B is generally false.
Is that when this would be needed?
If this does end up being the case, that translation becomes easy, then essentially all theorem proving efforts should be conducted in the language that is the easiest to work in. You can translate into the "mathematically superior" languages later.
So all the libraries they build up will have these holes in them that make it harder to do things like treat two isomorphic objects as the same -- something mathematicians do implicitly on a daily basis.
You can probably get a long way in Lean with the soundness axiom. But what I don't know is what happens when you build up a decade of software libraries in a system that adds a lot of manual and cognitive overhead when you want to use them.
My gut instinct is that by cutting corners now, they're creating a form of technical debt that could possibly escalate quickly and force mathematicians to reformulate their tools in a nicer way.
This actually happens continuously throughout the history of math. Sometimes it leads to errors like the so-called Italian school of algebra. Sometimes it just leads to pauses while we go back and figure out what the objects we're working with actually are before we can make more progress.
Take all this with a grain of salt: I haven't worked with Lean so I don't know how much this crops up in practice, and I don't know how large Lean libraries are at this point. This is all gut feeling.
But my sense is that what you really want is to get the foundations right, then build abstraction layers on those foundations that are nicer to use. Lean tries to build a "good enough" foundation and historically the gap between what we know is correct and what is seen to be "good enough" tends to show itself sooner or later in math. If you are just working in natural language then you can just forget it was a problem as soon as a fix is found. If you're working in software, though, you'll likely need to do a major rewrite or refactoring.
It really doesn't. I've been using Lean and Mathlib for about five years now, and Fermat's Last Theorem is definitely not going to depend on the reduction properties of quotient types in the large scale.
Mathematical reasoning in Lean is almost universally done with rewriting, not reduction. People have found reduction based proofs (colloquially "heavy rfls") to be difficult to maintain. It exposes internal details of definitions. It's better to use the "public API" for mathematical definitions to be sure things can be refactored.
Really, quotients almost should never use the actual `Quot` type unless you have no better choice. In mathematics we like working with objects via universal properties ("public API"). A quotient type is any type that satisfies the universal property of a quotient. All `Quot` does is guarantee that quotients exist with reasonable computation properties, if we ever need them, and if we need those computation properties — which in the kind of math that goes into FLT we often don't. We don't even need `Quot` for Lean to have quotient types, since the classic construction of a set of equivalence classes works. (Though to prove that this construction is correct surely uses functional extensionality, which is proved using `Quot` in some way, but that's an implementation detail of `funext`.)
https://github.com/ImperialCollegeLondon/FLT/blob/main/GENER...
Research level mathematics like this is as hard as it gets, and this proof is famously difficult: uses many branches of advanced mathematics, required thousands of pages of proofs, years of work.
It's somewhat like taking rough pseudo code (the informal proof, a mixture of maths and English) and translating that into a bullet-proof production app (the formal proof, in Lean), where you're going to have to specify every step precisely traced back to axioms, handle all the edge causes, fix incorrect assumptions, and fill in the missing parts that were assumed to be straightforward but might not be.
A major part is you also have to formalise all the proofs your informal proof relied on so everything is traced back to the initial axioms e.g. you can't just cite Pythagorus theorem, you have to formalise that too.
So it's an order of magnitude more difficult to write a formal proof compared to an informal one, and even when you have the informal proof it can teams many years of effort.
But I thought it was a widely celebrated result.
That's part of the motivation to formalise it. When a proof gets really long and complex, relies on lots of other complex proofs, and there's barely any single expert who has enough knowledge to understand all the branches of maths it covers, there's more chance there's a mistake.
There's a few examples here of errors/gaps/flaws found while formalizing proofs that were patched up:
https://mathoverflow.net/questions/291158/proofs-shown-to-be...
My understanding is it's common to find a few problems that then need to be patched or worked around. It's a little like wanting to know if a huge codebase is bug-free or not: you might find some bugs if you formalized the code, but you can probably fix the bugs during the process because it's generally correct. There can be cases where it's not fixable though.
And then, when I raised concerns in Zulip about Lean's metaprogramming facilities being used to trick the pipeline into accepting false proofs, he said the opposite. He even emphasized that the formalization efforts are not for checking proof correctness, but for cataloguing truths we believe in.
This kind of equivocation turned me away from that community, to be honest. That was an extremely frustrating experience.
I imagine one bias is because formal verification is such a huge effort, you're only going to do it for really interesting and impactful proofs, which means the proofs that get formal verified will already have been reviewed and critiqued a lot, and will be less likely to have glaring critical errors.
It takes thorough review by advanced mathematicians to verify correctness.
This is not unlike a code review.
Most people vastly underestimate how complex and esoteric modern research mathematics are.
The systems we deal with in software are massive compared with your typical mathematical framework though. But FLT is probably on similar scope.
That's actually where LLMs are already quite good at.
There is a pseudocode app that depends on a bunch of pseudocode libraries. They want to translate that pseudocode app into a real runnable app. They can do that, and it's a good amount of work, but reasonable. The problem is to get the app to run they also need to translate the hundreds or thousands of pseudocode libraries into actual libraries. Everything from OS APIs, networking libs, rendering libs, language standard libs all need ro be converted from specs and pseudocode to real code to actually run the app. And that's a ton of work.
It's, as far as I know, quite hard to teach an LLM things it doesn't know.
It's a bit frustrating because, before Copilot, new users would come with proofs and you could spend some time helping them write better proofs and they'd learn things and gain skills, but now it's not clear that this is time well spent on my part. Copilot is not going to learn from my feedback.
A "two-page document" (perhaps 200-300 lines of code) would typically check in a matter of seconds to, at most, a few minutes. If it took hours, it would mean the file contains a pathological case that a developer would be expected to identify and fix. It is absolutely not a normal or "not unusual" occurrence.
1. Load a complex data structure, e.g. from JSON. (I do not consider typing out the JSON to be part of the human's job.)
2. Process the data, using the magic of functional programming. (Performance? What's that? Performance is not a priority.)
3. Add some helpful lemmas.
4. Prove the data processing stage was correct… but proving this in general is haaaard, so just make automation that's capable of proving it for most cases… eventually.
5. Great! It takes 3 minutes to run on my tiny examples. Now run it on the real data.
And that's where hours come from!
> Lean's metaprogramming capabilities enable users to extend the language with domain-specific notations and new proof automation techniques.
In particular, note that a key lemma of crystalline cohomology rests on a mistake. Experts think that it is fixable by virtue that results have depended on it for a long time and no issue was found, but it is not fixed.
I admit that my original comment was inaccurate, as it seems to suggest that the gap still exists.
Plus, Kevin Buzzard is a world expert with some ideas for how to better organize the proof. In general, formalization leads to new understanding about mathematics.
Something people outside of mathematics don't tend to appreciate is that mathematicians are usually thinking deeply about what we already know, and that work reveals new structures and connections that clarify existing knowledge. The new understanding reveals new gaps in understanding, which are filled in, and the process continues. It's not just about collecting verifiably true things.
Even if somehow the OpenAI algorithm could apply here, we'd get less value out of this whole formalization exercise than to have researchers methodically go through our best understanding of our best proof of FLT again.
(gemini llm answer to google query: constructive math contradiction)
https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem
"3. Proof by Contradiction: The core of the formal proof involves assuming ¬Fermat_Last_Theorem and deriving a contradiction. This contradiction usually arises from building a mathematical structure (like an elliptic curve) based on the assumed solution and then demonstrating that this structure must possess contradictory properties, violating established theorems. 4. Formalizing Contradiction: The contradiction is formalized in Lean by deriving two conflicting statements, often denoted as Q and ¬Q, within the context of the assumed ¬Fermat_Last_Theorem. Since Lean adheres to classical logic, the existence of these conflicting statements implies that the initial assumption (¬Fermat_Last_Theorem) must be false."
(gemini llm answer to google query: Lean formalization of fermat's last theorem "proof by contradiction")
https://www.quora.com/In-math-are-there-any-proofs-that-can-...
https://math.andrej.com/2010/03/29/proof-of-negation-and-pro...
also
"It’s fine to use a proof by contradiction to show something doesn’t exist. When the assumption that it does exist leads to a contradiction, then that shows it can’t exist.
It’s not so fine to use a proof by contradiction to show something does exist. Here’s the situation. The assumption that it does not exist leads to a contradiction. What can you conclude from that? You would like say “therefore it exists”. But you haven’t got any idea what it is. You may know it’s out there somewhere, but you have no idea how to find it. It would be better to have a proof that tells you what it is.
That’s a difference between what’s called “classical logic” and “intuitionistic logic”. In classical logic, proof by contradiction is perfectly accepted as a method of deductive logic. In intuitionistic logic, proof by contradiction is accepted to show something doesn’t exist, but is not accepted to show something does exist."
David Joyce, https://www.quora.com/In-math-are-there-any-proofs-that-can-...