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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 don't really understand your overall point. To whatever extent Phelps' success was said to be thanks to wheaties, it's obvious the same can't be said for Tao for the simple reason that wheaties are from 1924 (predating Phelps) and Tao won the fields medal in 2006 (predating ChatGPT). It's obviously a logical contradiction.
The other thing is that we are at the point where LLMs are solving open problems with minimal or no involvement from humans:
Automated theorem proving across open Erdos problems. Each solved problem cost a few hundred dollars, which is hardly an unbelievably large number.
A problem relating to sumset combinatorics. Gowers is also a Fields medalist and seems pretty impressed with ChatGPT here. Reading the post, it does not seem that it takes a Fields medalist to do the prompting here: "the era where you could enjoy the thrill of having your name forever associated with a particular theorem or definition may well be close to its end." This also doesn't seem to have cost an inordinate amount of money, although there isn't a figure given.
A disproof of a conjecture related to the planar unit distance problem. I'll grant that it's not explicitly clear what the human prompting looked like, but there's no mathematician whose name is attached to this so I can't imagine it was something that only a select few could have done.
This is not an exhaustive list.
In other words, you don't have to be Tao to find new results with these tools. I am sure your response will be that nobody cares about these results, but unless you were predicting beforehand that we would get to this stage but no farther, it's hard to take this seriously.
Sure but that still doesn’t eliminate the problem exactly. It’s somewhat reminiscent of when Fermat’s Last Theorem was proven by Andrew Wiles. The early stage formulated results he came up with he remained confident about, until it took other mathematicians joining in to show him where he was off track. You don’t have to be a Tao to “find” these results per se, but how do you validate them?
It's frequently the case that it's easier to check a result for correctness than it is to generate the result in the first place. This is especially the case if the problem can be formalized in Lean.
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