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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.
The main thing about constructivists is that they do not generally accept proofs where you you assume the negation of what you are want to prove and arrive at a contradiction (Reductio ad absurdum -- though it seems that some cases remain valid in intuitionist/constructivist logic).
As a layman, my gut feeling is that the constructivists would probably accept Euclid's proof of an infinite magnitude of primes, because it gives you a rather concrete algorithm for finding a prime which is not in your list of all primes, but not Hilbert's proof of his Nullstellensatz, because it does not give you a way to determine the algebraic relationship.
WP on constructivism
Moving on further
I am not sure I follow. Suppose you have a program and a claim about that program ("this Turing machine will halt for any input", "This C code will never invoke undefined behavior", etc). In my mind, a proof which is just half a gigabyte of gibberish without rhyme or reason would still be acceptable for such practical problems. If someone formally verified that a given version of the Linux kernel did not suffer from a given class of exploits, I would not complain about them using proof by negation.
By contrast, open problems in mathematics are not things where our main interest is in knowing the answer. Few people are interested in P==NP because they think there is a practical algorithm for solving SAT to be discovered. If an ASI told us what the answer is ("P is equal to NP, but the polynomial has coefficients and exponents so large that your mathematics can't even express them using all the protons in the universe"), that would be of little value. People are interested in these big open questions because their answer sometimes lead to the development of new and exciting branches of mathematics.
The textbook example of a problem which looked promising in that regard until it was proven would be the four color theorem. "So it can just be proven by brute-forcing 1834 configurations with a computer? Seems it was not a nice problem, after all."
With most of our existing software, the problem is even identifying what formal properties we would want, or constructing it so that has the desired properties (and ideally we can easily proof them).
The Hilbert thing is more a sign of the times. Back then, there was a lot of resistance and emotional slapfighting over this, because in ye olden days, people were very concerned about which postulates were True (TM), and viewed any attempt to discuss this as an attack on Truth.
Today, we worry less about this absolute notion of Truth and more about models: is Euclid's parallel postulate True or False? Well, there's actually nothing to fight over: you just get different geometries, but they're all meaningful and useful! So rather than saying "The parallel postulate is true, therefore XYZ", you can just say "When the parallel postulate is true, XYZ holds." Even questioning logical primitives that seem "very true" like "can you appeal to a theorem more than once in the same proof?" turns out to have surprisingly useful implications! For example, if you're modeling a cookie, it makes a lot more sense to say you can only eat it once, rather than you can eat it as many times as you want. It's not wrong; it's just a different thought model. A thought model that, incidentally, is the foundation of a very popular programming language.
One can say "Ok, but what if the postulates imply a contradiction? Surely that's bad, right?" Well... yes, but actually even here there's a lot of subtlety. Not for mathematicians, but for everyone else: see, the trick to defining logical systems that dodge Russell's Paradox is to have a cumulative hierarchy of universes, which is what a lot of dependently-typed proof systems use. But it turns out this is an enormous pain in the ass to work with for writing normal software, to the point that literally nobody does it, and instead we settle for simpler systems that are much more ergonomic to write in yet still, in practice, give a pretty strong (but not rock-solid!) guarantee that you haven't contradicted yourself.
Anyway, constructivism is a more grounded model compared to classical logic, in the sense that it actually computes results, and you can do the legacy thing by just saying "When the law of the excluded middle holds, XYZ."
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