In Adams’s book The Hitchhiker’s Guide to the Galaxy, the Deep Thought supercomputer took 7.5m years to answer the ultimate question of life, the universe and everything and came up with 42. Then it asked for another 10m years to work out what the question was.
Adams’s joke has become an accidental satire on AI because Tao and his colleagues are now wading through OpenAI’s answers to work out how it reached them.
At 51, Tao could be forgiven for not wanting to tidy up after Silicon Valley, although he has worked with OpenAI and still has contacts there. The Australian, speaking via video from California where he is professor of mathematics at UCLA, is one of the youngest winners of the Fields Medal (the maths Nobel), and has a slew of other prizes honouring his contributions. He attempts to distil the field in his book, Six Maths Essentials: How to Model the World, which is published on Tuesday.
But perhaps the worst part for Tao is that he believes OpenAI didn’t even make a new model to solve these difficult problems. Conundrums that people have spent careers addressing were merely raw material for training its models.
AI companies are “racing towards the most powerful model”, Tao says, so they need to “train and benchmark and give the AI lots of tests to see whether what they’re doing is improving their capability”. The practical problem is that the newest AI models are beating every test.
“Maths problems are almost the only thing left for them to train on, so suddenly all the big prizes and goals in our profession are benchmarks for them,” Tao says. “And as a byproduct, they’re generating these terrible proofs. Great results, but they don’t care how elegant the solutions are, or who understands them, or what to do next.”
An updated Deep Thought would probably involve a share offering; Tao says that if OpenAI or its main rival Anthropic, which makes Claude, suspect the other is close to a breakthrough “they just go insane”.
OpenAI rejected the code red analogy and said it had not suspended signups nor spent as much as Tao suggested.
[OpenAI] really, really wanted to beat Anthropic. It’s insane’
[OpenAI] really, really wanted to beat Anthropic. It’s insane’
Terence Tao
The dump of mathematical “toxic waste”, as Tao calls it, began when rumours surfaced online that Anthropic researchers were close to solving one of the seven $1m Millennium Prize Problems set by the Clay Mathematics Institute.
“My contacts at OpenAI said this became code red priority number one,” Tao says. “They dumped tens of millions of dollars. They set up this 10,000 AI agent swarm to attack these problems.” OpenAI focused its agents on the problem of the Navier-Stokes equations, which seem to perfectly describe the motion of a fluid, yet might fail in some conditions.
“They even suspended signups, so that for a few days you could not sign up to their premium AI service because they were redirecting all the compute for their flagship services to this one problem,” Tao says. “They really, really wanted to beat Anthropic. It’s insane.”
On 8 September, OpenAI published a proof that seemed to show that Navier-Stokes did fail, although reviews suggest there may be some errors. The backlash from the maths community left OpenAI with a problem, Tao says. “They had been testing their models on these problems for quite a while, and they actually had accumulated a bunch of very badly polished solutions to these problems. They didn’t know what to do with them.
“[When] you generate toxic waste, you have no good options. You can sit on it forever, or you can just dump it out all at once, or you can dole it out in drabs, or you can pay people to process it.”
OpenAI said it had neither withheld nor rushed out results.
Anthropic is taking a different approach, by paying mathematicians to write up results to its “trove of questions”, he says.
That creates a different problem. For years maths was a field with equality of arms; practitioners needed only some chalk and a blackboard, unlike their scientific counterparts – think of Cern or the Hubble space telescope. Now doing cutting edge maths research means getting access to a super-AI model. “They get to pick who writes up the paper.”
In the aftermath of the “toxic waste” dump, Tao became a meme: “Put the fries in the bag, Terence”, an allusion to mathematicians’ impending careers at McDonald’s. He laughs when I mention it.
“You know, social media is going to social media,” he says. It’s understandable that people “who had a bad experience with mathematics or intellectualism in general [would experience some] schadenfreude.”
The memesters miss the point though. Tao agrees that maths culture has to change – he hopes it becomes more collaborative, less focused on giving credit for solving a problem and more on explaining it.
“Maths is half science, half humanities,” he says. “We care about the outcome but also the process, how we got there, what we learned on the way.” He still uses theorems discovered by Euclid and Pythagoras 2,500 years ago. The story of solving a problem, the history of the arguments and the elegance of the solution are essential to the beauty of mathematics. Otherwise, “you can go to the gym and get a robot to lift your weights, but what’s the point?”
Six Maths Essentials is a short, pacy story about the evolution of numbers, algebra, geometry, probability, analysis and dynamics. The invention of the infinity symbol by John Wallis in 1657, and the decision to use the letter i to refer to the impossible square root of -1 (an imaginary number) created tools that allowed mathematicians to do things that had been out of reach. Is there a tool he’d like someone to create?
There is. We can describe shapes well – their volume, their width, their curvature. “But one notion which is still very elusive is how complex is the shape. Intuitively, some shapes look very simple, and some look very complicated.” Maths lacks the language to describe in detail the complexity of, say, a tennis ball compared to a steel wool scourer.
Large language models have thrown this problem into focus because they represent words as coordinates in space. The corpus of English becomes a vast cloud of points, plotted in thousands of dimensions, an “insanely complicated shape that there’s no way you could look at”.
“It turns out that this weird object has a simpler shape than we realised,” Tao says. “Not simple to humans, but AI machine learning algorithms can detect the shape, and they can use it to predict, given a portion of a sentence, what the next word is, and the next word after that, next word after that, and suddenly they can start speaking in fluent English.”
English has a similar shape to French, but quite a different one to Finnish. “There’s all these recipes in there that we do not know, in part because mathematicians have not actually developed a theory of complexity of shapes that would explain this. If we had, maybe we would understand [why] there are some languages that AIs pick up better and faster than others.”
An OpenAI spokesperson said: “Our work in mathematics is part of a longstanding effort to accelerate scientific discovery. We’ve worked on maths for years because better reasoning can make AI a stronger research partner, helping scientists tackle harder problems in biology, medicine and engineering.
“We are already working with leading mathematicians to improve how we share results, and we’ve committed $5m to help the maths community examine, understand and build on discoveries made with AI.”