Mathematics and AI: The Good, the Bad and the Existential.
Deep questions. Fundamental breakthroughs. Allegations of theft and scandal. Fascinating times in the world of maths.
Well, we’ve certainly been asking some big questions in AI of late, haven’t we?
The whole “is there a 10% chance humanity may be destroyed?” thing, understandably, got a lot of headlines.
But I want to take a moment in this NerdNews to look at my old bailiwick of mathematics. A fascinating microcosm of the bigger debates that are being had in many industries and fields of endeavour around AI.
There are three lenses through which I will look at this.
The good, the bad, and the existential.
The good.
Recently maths proofs have been falling like dominoes.
Many puzzles that have sat around for decades, in some cases almost centuries, have wilted under the glare of a well-focused AI model.
And in a lot of cases, the AI wasn’t producing radical new mathematics that no human could ever have come up with.
It was more a case of a model being loaded up with all the knowledge that we had in a specific domain, pointed at a question and told: ‘do your best and simply don’t give up’.
The equivalent, in some ways, as if we’d locked in a room the smartest 30 mathematicians in the world in a particular field and told them that this is the only problem we want you to concentrate on for the next month.
Now, in the real world, there are too many problems and not enough mathematicians and not enough time to take that incredible focus of all human effort and point it at a a single question.
But AI platforms have managed to do this.
Many of the solutions have just taken current knowledge or at best involved a small step of “genius” on behalf of the AI. Something equivalent to what a top-shelf human mathematician could have done.
I recall the words of my friend and one of the world’s great mathematicians, Geordie Williamson, who said that when he looked through some of these chains of reasoning he could see the moment where the large language model thinks: “Hold it, I’m onto something here.”
And then it just applies incredible focus and gigantic effort at that small gap it sees in the Death Star that is this maths problem.
The good is a world where someone like Geordie can clock off on a Friday and say to a large language model:
“I’m thinking of the following three lines of inquiry for when I’m going to come back on Monday.”
And have the model say:
“Look, I’ve run 50 million simulations of each of them. The first two don’t really seem to lead to much. But gee, line of inquiry three, particularly in the case of polynomials of degree 5 or more, really seems quite fruitful. I’d suggest you start looking there.” — hypothetical AI maths model.
In the good, nothing is replacing the work that Geordie could do. No one’s taking him out of the game in this situation.
But think of a great mathematician who might spend 99% of their time being wrong and chasing things that will never go anywhere. Imagine if they had an AI-angel on their shoulder just whispering ‘warmer, colder, warmer, colder’, such that they were wrong only 90% of the time.
Effectively we’ve got ten Geordie Williamsons, ten Terence Taos, ten insert-name-of-great-mathematician-here.
What a world to live in.
The bad.
This scenarios has been brought into sharp relief in just the last few days by the OpenAI controversy around the solution of the famous Navier–Stokes problem. And it is mired at this stage in ‘he said, he said’.
But the essential claim is that a group of mathematicians who’d done a lot of work on a famous problem and uploaded that work into an OpenAI platform might then have had that work purloined by OpenAI. Allegedly they then threw millions of dollars and incredible amounts of compute at it to smash out a proof to the problem.
OpenAI denies that this is what happened. But the important point here is it well could in the future.
And that fundamentally threatens the nature of mathematics itself.
You see, today mathematics is fundamentally a field of collaboration.
With the exception of someone like Andrew Wiles, who famously locked himself away from the world to prove Fermat’s Last Theorem, most mathematicians, most of the time, will happily collaborate with other people and say:
“I’m working on this problem. I’ve got this far. I’m stuck here, but I think I’ll get somewhere.”
They’ll present a seminar, they’ll speak at a conference, they’ll lodge a preprint of the paper or of discoveries on the way to a greater discovery for the mathematical community to look at.
And most of them do that, I think, happy in the knowledge that if someone else hopped in and said, “Well, I think I can help you out here,” they could collaborate. I’d happily trade with mathematician Jones and come up with the Spencer–Jones theorem if it means we got there together.
I would not consider that he or she has stolen my work in doing that. I’d happily share the love.
But would I be as willing to share where I’m at on a problem with the general mathematical community if there was a chance that over the weekend an AI model could just come through, bust it open with millions of lines of code and say, “Well there you go … I’ve proven it”?
Possibly not.
If mathematics loses this spirit of collaboration, if work is effectively stolen off people to get to some grunty proof, is that a better world?
“If we could distil down only the good bits of this issue it would be boon for mathematics. But that’s a colossally large ‘if’. ” — Dr Sean Gardiner, UNSW.
And the other part of the bad that is significant here is AI models are flimsy at acknowledging the shoulders of the giants upon which they stand.
Human mathematicians always give props to the people who helped them get there.
“In creating this proof, I just want to acknowledge the work of blah and blah whose lemmas I’ve used here and there,” etc.
But the spirit of crediting seems sadly lacking in some of these brute-force AI proofs. We often see humans later unravelling the chain of reasoning and noting the work of their colleagues that clearly played a crucial, unacknowledged, part.
In a word, bad.
The existential.
Just last week, Aussie maths legend Terry Tao, and 25 other Fields Medalists, authored an open letter that expressed concern about the world of AI proofs into which we might be moving.
If these proofs are presented not as human rationale in human language but as 13 million lines of verified computer code that most humans struggle to understand or would take years and years to read, have we benefited?
Yes, there’s a short-term sugar rush in an AI platform now saying: “Trust me, this is true.”
But if we don’t see the human struggle to get there at every step of the way, and the individual steps and understandings that are needed to create the greater whole, we’ve lost out on something.
I note that when Wiles dropped his proof of Fermat’s Last Theorem it ran 129 pages.
The AI verification ran to 13 million lines of code.
Now that 129-page proof of Fermat’s Last Theorem, at the time that it appeared, was impenetrable to the vast bulk of people in the mathematical world. There were literally a handful of mathematicians who were qualified to read it through and check if it held up. In fact a couple of them found a small hole. It took Wiles and a mate two years to patch up that hole, before, finally, a select few mathematicians signed off on his proof.
Thirty years later, the ideas and chain of reasoning behind that proof are taught in graduate courses and seminars around the world. Entire courses have been built around Fermat’s Last Theorem and the mathematics needed to understand Wiles’s proof.
Tens of thousands of mathematicians have learnt from the steps of what Wiles did and now have accumulated that as part of their own knowledge upon which they now build.
But if AI had just come out one morning and said, “Trust me, Fermat’s Last Theorem is true. Here’s 13 million lines of computer code if you want to read through it line by line,” would we have really benefited in total from that?
“The mass production at faster and faster pace of “true/false” statements could destroy fertile ground instead of breathing life into new ideas.” — Tao, et all, “A Severe Misalignment of AI in Mathematics”.
Let me draw an analogy.
If AI comes out tomorrow morning and says, “Here’s a drug that’s going to work on a certain sort of liver cancer,” and it does, that’s fantastic in the short term. Wonderful for sufferers.
But if we don’t understand why it works, if we weren’t part of the intellectual path to its creation, has our knowledge advanced about cancer, human physiology, or the science behind the drug?
If it hasn’t, does our knowledge atrophy as it becomes replaced by AI knowledge?
These issues are existential.
The pushback.
Now some people will argue: well, scientists, well, mathematicians, bad luck.
Call-centre workers are going to lose a lot of their jobs. Legal secretaries are going to lose a lot of their jobs.
Join the queue, bucko. Stop whinging.
But I’d respectfully suggest, and I do this without any arrogance or intellectual conceit, that it’s one thing for us to lose the ability to understand how to solve certain problems being asked in an airline’s call centre.
It’s one thing to lose the skill of people being able to type up legal drafts if AI can do it faster and more accurately.
It’s another thing altogether to lose the cutting edge of human knowledge that pushes us on to the next discovery.
It’s another thing to not understand why the very cutting edge of our knowledge is what it is.
It’s another thing altogether to dip our hat and just say to the AI: “Well, you’re smarter than us. We’ll take your word for it. Over to you from now on.”
We may eventually get to that point.
We may be creating machine intelligences so far beyond that of humanity that they’ll push back the boundaries of mathematics further and further and the smartest human minds will never comprehend why.
But surely for as long as we can actively be a part of that process, and as long as we can improve our own knowledge and pass it on to thousands of our colleagues, it benefits all of humanity for us to do so.
The sum of it.
A final word;
“It is chilling to hear some of my colleagues ponder, in the future will we just be verifiers for unreadably long, computer-generated proofs” — Dr Sean Gardiner.
Mathematics.
The good.
The bad.
The existential.
Further reading:
The Tao (and buddies) open letter.
NerdNews: AI ‘proves’ Fermat’s Last Theorem
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