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PaperclipPerfector  6d ago (text post)   19374 thread views

Culture War Roundup for the week of May 25, 2026

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I do not understand where you were trying to go with your digression about constructivism. Is this just supposed to be FUD, as in you hope non-technical readers will look at it and walk away with an understanding like "what Terry produces with his AI is not a real proof"? Because the reality is rather the opposite: any proof in a constructive system like Lean is a proof in a non-constructive system like ZFC, and certainly it is especially a proof in the unspecified system that is mathematical practice outside of a handful of particularly rigorous subfields.

I don’t think we’re referring to the same constructivism. I’m referring to this: https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics)

as in you hope non-technical readers will look at it and walk away with an understanding like "what Terry produces with his AI is not a real proof"?

Come on, that is not an honest reading of what I said.

In constructive systems, one proof—in the Curry-Howard sense—is not interchangeable with another. For example, you can sort in O(n^2) (or worse), which would be a perfectly valid thing to do in classical mathematics, but a big oopsie in formally-verified software. This is doubly-so in cases like cryptography, where almost none of the "incorrectness" of compromised crypto is incorrect at all in the classical mathematical sense. Further, as someone who has a fair bit of experience playing around with these systems in the past, theorem provers often induce you to write code in a way that is easy to make formal statements about, rather than in a way that actually runs well enough that anyone would want to use it.

My point is the software verification story—which is surely of high importance to businesses—is far more nuanced than "throw the clanker at Lean and let it churn". I think the bucket of cold water I’m tossing is completely justified (even as someone who does indeed buy the baseline premise—that LLMs will be a big boon to working with theorem progress).

Your general argument is correct, because proofs won't scale without efficient tactics and smart sub-lemmas, which can't be algorithmically verified.

But Cunningham's Law obligates me to point out that proofs are interchangeable, it's called proof irrelevance. A value is basically only considered a proof if it's type's type is Prop, and once the proof is verified (i.e. value is type-checked), Lean can forget it and only remember that the theorem is proven (i.e. type is inhabited).

Well, if we're going to be pedantic, proofs of kind Prop are irrelevant. I haven't followed this stuff in a while, but I'm pretty sure this opens a big can of worms, especially wrt erasure. Google's AI assures me this is all totally resolved and there's zero overhead, then links me to this thread which is not that old and smells of all the swamp I remember from back when I followed this stuff.

Are they selling these things as proof of applicability to software verification? That seems silly, for many reasons. The domains share some language, but diverge a lot too.

The role of constructivity is a bit mysterious to me here. The headline results (at least the unit distance problem) were indeed constructive. On the other hand you can verify classical logic perfectly well by assuming, for example, a double negation oracle. Your programs won't run, of course, but they'll type check fine.

Are they selling these things as proof of applicability to software verification?

I mean, that was certainly my impression. Maybe I've self-selected into circles where that's the sort of thing people naturally care about, and incorrectly assumed this is how everyone is thinking.

Sure, if all you care about is Erdos problems, I guess much of what I'm saying is moot. But this whole "Mythos is finding all these vulnerabilities, the whole meta has shifted!" narrative sure leads me to the next part of the story of "Ok, so how do we actually build stuff correctly in the presence of tools like Mythos?" Because the mythos story, like the classical theorem prover story, is also basically invulnerable to the primary problem of LLMs, hallucination. If the model hallucinates a vulnerability that doesn't work, just throw it away and try again! (Or, in the case of low-human capital, spam the issue tracker with bogus vulnerability reports. But I digress).

I mean, that was certainly my impression.

I think you made up a thing that people are doing and then wrote a post about how it's dumb that they are doing that.

If the model hallucinates a vulnerability that doesn't work, just throw it away and try again!

I mean, yes?

I still can't make sense of what you are trying to say then. What is the sense in which you claim proofs to be "interchangeable" in classical systems, why do you believe this to be a desirable property, and how do you contend with the fact that constructive proofs are classical proofs?

There may be some HoTT-like sense in which two proofs in a given system are not equivalent (because you introduced some equivalence relation on proofs that does not relate them). If you have a sound embedding from proofs in this system to proofs in another system, and the other system comes with its own equivalence relation on proofs, the images of your two proofs under the embedding might still be equivalent in the other system. Is this not just what would happen here, if you assert that all classical proofs of a given sentence are equivalent (under an equivalence relation you picked) but not all constructive ones are (under an equivalence relation you picked)?

This is doubly-so in cases like cryptography, where almost none of the "incorrectness" of compromised crypto is incorrect at all in the classical mathematical sense.

You will have to elaborate on this statement. What do you take to be an instance of compromised crypto that is not "incorrect in the classical mathematical sense"?

Well, in the classical mathematical sense, crypto doesn’t work at all: just factor the composite number. The entire premise relies on the relatively ill-specified (by mathematical standards) notion of relative computational cost disparity.

What is the sense in which you claim proofs to be "interchangeable" in classical systems

What I mean is when I sit down to write something in a theorem prover, I speak in terms of Peano nats and inductive lists. When I write software anyone would ever want to actually use, I use machine ints and arrays. There is definitely a sense in which I’m doing the same thing in both cases, but nailing this down precisely is… well, non-trivial. (aka, a royal pain in the ass). Like, the discrepancy here is show-stoppingly problematic.

EDIT: basically, the computation is relevant, and more specifically the speed of computation is also relevant, both in terms of practical usability and outright security in the case of crypto. In the classical world, there is no notion of computational relevance at all -- in fact, you outright end up with these counterintuitive weirdo theorems like Zorn's Lemma. The counterintuitiveness depends entirely on this conspicuous lack of computational awareness in the model: as soon as you put that ingredient back in, you're in the construcivist world, and silly results like Zorn's Lemma don't hold anymore unless you postulate them. My contention is the class of problems for which one can say "I don't care about which proof, just that one exists" is relatively narrow and of little practical relevance (in the sense that one can get an incomputable proof, which is, in a fundamental technical sense, useless), while the class of problems about which one does care which proof, in the Curry-Howard sense, is large and of enormous practical consequence. I want to argue that enthusiasm over the former should not translate to enthusiasm about the latter.

I think he means it in the sense that "a nice man on the phone told me he was from BitPanda and asked me to read my password to him so he could check if it was secure" or "SBF stole all my money" can't be detected by enforcing code correctness. All of the badness has happened outside the code, everything inside the code is a perfectly valid transaction.

But this general argument (ChatGPT can claim something in English, then formally prove something completely different) still applies if the proofs are classical.

Who is saying anything about AI assisted software verification via Lean? I've literally never heard of anyone suggesting that we should do this.

Terrence Tao literally talks about this in his Lex Fridman interview:

https://youtube.com/watch?v=hh4cjZOddQA

No, he's talking about theorem proving, not formal software verification.

Yeah, I'm not sure what his point about "the proof isn't canonical" is all about. A proof is a proof, and the underlying statement is just as valid regardless. The fact that there's no canonical isomorphism between a finaite dimensional vector space and its dual doesn't mean the two spaces aren't isomorphic or any such isomorphism is less "real".

Part of the misunderstanding is the word "proof" has different flavour in classical mathematics vs theorem prover math, which is why the word is rarely used in this world except when appealing to legacy mathematicians. The native vocabulary is "3 is a value of type Nat" or "Refl is a value of type 0 + 0 = 0". A legacy mathematician would use the word "proof" only to describe the latter, and think it odd to use the word for the former. But internal to the formal logical framework, these are same notion, and so they should have the same word.

The native vocabulary is "3 is a value of type Nat" or "Refl is a value of type 0 + 0 = 0". A legacy mathematician would use the word "proof" only to describe the latter, and think it odd to use the word for the former.

Neither of these would be considered a "proof" by a mathematician. Those are just statements of fact. A mathematical proof is a logical argument of deduction that shows that some statement must be true if some set of premises are true. Statements of fact are used in proofs, but a single statement of fact wouldn't actually constitute a proof.

I think you've never spent any time with a formal theorem prover lol. You're conflating the definition of + with the proof Refl : 0 + 0 = 0.

Mathematicians can say whatever they want, but the story here is about mechanically-verified formal verification, and this is how formal verification works, whether legacy mathematicians think this pedantically or not.

Mathematicians can say whatever they want

A legacy mathematician would use the word "proof" only to describe [("Refl is a value of type 0 + 0 = 0")]

(emphasis added)

So it seems like you're walking back the claim you made about what a (legacy) mathematician would say. Again, a legacy mathematician or any mathematician wouldn't describe that as a "proof."

An appeal to a definition is a proof ("By the definition of XYZ, we conclude..."). A definition itself is not a proof ("Let xyz be defined as ...").

The confusion is coming from the fact that Refl : 0 + 0 = 0 is the syntax of an appeal to the definition of plus. It is not the syntax of defining the + operation.

Look, I'm not going to run around in circles all day about this. I'm pointing out that mathematicians are more sloppy in their vocabulary than the theorem prover, and that there are subtle differences over how terminology is used. I do not care to bikeshed endlessly over this. It's obviously true to anyone with any experience in these fields.

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