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I made a comment recently speculating that Terence Tao may be a paid promoter of AI. After some off-site discussion elsewhere, our resident shaman on whom the mods cast a long-duration silence posted my comment as a circus exhibit on Twitter to demonstrate a dangerous form of psychosis. Fortunately, OpenAI deemed this a worthy time to intervene and publish an ad featuring Terence Tao, who has taken time out of his busy schedule to assist this struggling non-profit with their promotional.
I want to dissect what I think is really going on.
For starters, every time I sit down to watch WoW on YouTube, I'm greeted with my favourite streamer telling me about the super fun game Raid Shadow Legends, which I should definitely download and play because it's super fun. Is the OpenAI-Terence Tao relationship like this? Not really. Terence Tao does appear to actually spend some time playing around with LLMs. Further, he's not exactly saying a bunch of empty marketing blather, either. In fact, probably to the annoyance of some readers here, I want to take a couple paragraphs for a technical aside, because there is actually subtlety here. I'll bound this in horizontal bars so non-technical readers can skip it:
The main talking point is that automated theorem proving is a perfect fit for LLMs precisely because it's not vulnerable to their main catastrophic failure mode: hallucination. The model can hallucinate whatever it wants, but the text still goes into the theorem prover, and if it's bullshit, well, the prover just rejects it and you query the LLM again. Do this in a loop, burn whatever unholy amount of compute you want, and if the loop stops, you've got yourself a proof! (Well, or a bug in the theorem prover. Or a "You've run out of tokens on your budget" error message. But I digress). This story is largely true. There's a giant asterisk of "Uh, so how much compute we talkin' about?", and the answer is "As much as you need or can afford, whichever comes first!" Which is, of course, the business model.
I will point out one additional technical nit-pick that annoys me because Terence Tao is working in Lean 4, which is a dependently-typed theorem prover. In classical mathematics, one is concerned solely with whether a theorem is true or false, and the structure of the proof is basically irrelevant as long as it's valid. Lean 4 is not based on this model. Rather, it's based on a more computationally-motivated model of mathematics pioneered by Brouwer in the early 1900s called "constructivism." In this world, the question isn't the boolean notion of "Is this theorem true or false?" but rather the related but distinct notion of "Which proof do you have?" To ground this in practical terms, consider the following example: I can prove that True|False and Yes|No are isomorphic, but I can do so in multiple ways: I can map True to Yes and False to No, or I can map True to No and False to Yes (and then show that there are respective inverses which preserve identity, obviously). It is in this sense that one can meaningfully say "Which proof of isomorphism?" when I say I have a proof of isomorphism. Perhaps this all sounds like technobabble, but to connect it to the preceding paragraph: you can immediately see how this does reveal some cracks into the narrative being sold there. It does actually matter which proof is produced, not merely in a social sense of "can any human understand this wall of text the LLM spit out", but in a technical, computationally-relevant sense. For pure mathematics, this distinction is often not considered important -- in fact, many classical mathematicians aren't even aware of the difference, and will be confused if you try to explain it to them and think this is all a bit silly. However, it's not a silly or minor distinction for the following reason: one of the motivations of this computational model for "theorem provers" (it's really a programming language + compiler, rebranded for mathematicians) like Lean is so that formal methods can be applied not just to classical mathematics, but to software in general. And as soon as you enter software formalisation, this distinction is no mere intellectual curiosity, but of paramount relevance. The classical-style logic in the preceding paragraph does not apply to constructivist logic used for software formalisation! I'm sure this distinction is not lost on Terence Tao. But that doesn't concern OpenAI. OpenAI is more concerned with whether the distinction will be lost on the MBAs listening to Terence Tao, and the answer is "absolutely."
Ok, no more technical details like that, I promise. Back to the social level:
So, I mentioned Raid Shadow Legends is a poor metaphor for the OpenAI-Tao relationship. Let me propose some better ones: Michael Phelps and Wheaties (with the added benefit that Terence Tao never smokes weed. See, this is why mathematicians are better than athletes), or better yet, attending Harvard University. This may seem like a strange juxtaposition, but I've done so intentionally, because the marketing is obvious in one but subtle in the other, but it's actually the same trick: the goal is to misattribute performance. With Wheaties, the goal is to sell the notion that Michael Phelps is a great swimmer because he has a healthy diet of stuff like Wheaties, and if you eat Wheaties, maybe you'll perform well, too! Of course, in reality, he was eating sugar-coated french toast and chocolate chip pancakes because he needed 10k Calories/day just to break even on energy, and the reason he's such an awesome swimmer is in large part genetics. Wheaties, or anything similar to it, has virtually no relevance to Michael Phelps at all. But what about Harvard?
Well, Harvard sells the image so well that most people outside this forum outright believe the illusion. The illusion is, of course, that attending Harvard makes you smart and likely to succeed, rather than Harvard accepting only people who were smart and likely to succeed in the first place and thus redirecting credit for these future achievements to Harvard. Mark Zuckerberg may see Harvard as a pointless waste of time, but the world sees Harvard as "The university that made Mark Zuckerberg happen!"
I like the Harvard analogy because this is surely the intent with Terence Tao. There's a high chance sooner or later Terence Tao will prove something cool "using" ChatGPT, and if he does, it would be really awesome if we could make it sound like the secret ingredient in the ChatGPT-Terence Tao alliance was ChatGPT, when obviously the actual secret ingredient is Terence Tao. The analogy I always use for this is stone soup, a European folktale where starving travelers dupe gullible townsfolk into helping them make soup from stones by requesting "extra" ingredients bit by bit until they've just made actual soup, thus astonishing the gullible townsfolk.
There are a lot of other things I could say on this, especially on the technical side, as there are a lot of clever tricks you can pull to make it look like a model is doing more than it really is, but I'll stop for now and conclude with this:
Just be cognizant that OpenAI, and all the other LLM vendors, do marketing. They have an enormous budget that dwarfs anything you have ever seen before. Remember the reality distortion field of Black Lives Matter? Or trans people? Imagine that, but like... two orders of magnitude larger. That is the level of persuasive pressure we're dealing with here.
Just take it all with a grain of salt.
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.
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).
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)?
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 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.
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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.
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