AI är så smart att det är experterna som har räknat fel när modellerna rättas.
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I don't know of any individual who can solve any significant fraction of our final collection of problems without AI assistance. For instance, I can solve the problems I contributed myself and understand a few others. Most problems don't even make sense to me.
Our initial evaluation suggested that GPT-6-Astra's score was in the late eighties followed closely by Opus 5.5
But, as we started to investigate the mistakes that the models might have made, the models, especially Astra, insisted that they were right and that it was the expert-supplied answers that were erroneous. When we went back to the experts it turned out, in multiple instances, that the experts found some subtle errors buried deep inside their calculations. As they revised their answers, Astra's score started to inch up and it finally ended up with 98% (49/50).
The one problem it doesn't solve is not because it makes a "mistake". This is a trick-prompt, and requires the model to make an inspired guess, and then bend the rules a little. It can't do that yet.
We comment a little on the broader implications in our conclusions. Clearly, these agent are enormously powerful tools that will accelerate discovery in the short run. They also raise serious structural concerns about access, equity and pedagogy.
Bad en AI att översätta Sylvias föreläsning och transkribera den på engelska. Fetningen är min. 2023 skrattades det gott åt att AI skulle kunna matte. Om många år sa folk då, vissa sa tio år, tjugo år eller aldrig. Tre år senare är det klart. Kan vara bra att komma ihåg när vissa säger att den minsann aaaaldrig kommer kunna ha mänsklig kreativitet, intuition, resonemang, självmedvetande.
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Why artificial intelligence changes the question
When I was preparing this lecture, I asked myself what the theme of “scholarly flavours” made me think of.
It seemed to me that there was one subject I could not ignore. Today is October 9, and a great deal has happened between September and October that you may have heard about.
In particular, on October 6, just three days ago, we witnessed the forceful arrival of artificial intelligence in mathematics. There had already been spectacular announcements on September 9, followed by further extraordinary results from OpenAI on October 6.
The sheer quantity of these results was astonishing. There were no fewer than 400 results. To give you an idea of their significance, many of them would arguably merit something on the level of a Fields Medal.
This led me to a question that I had been considering for some time, not merely since this month: What is non-artificial mathematics, or, in other words, what is human mathematics?
Until recently, this did not seem like such an important question. Human mathematics? Of course mathematics is human. But suddenly, the question has become much more immediate and pressing.
What does it mean to do mathematics in a human way? What distinguishes our practice from that of a machine?
When I began writing my book, I wanted to start with a personal account and explain what mathematics meant to me. I wrote it over several years, in small pieces, gradually developing my ideas.
As I wrote, I watched AI advance. I could see the tide rising.
My colleagues, generally speaking, did not want to see it. Frankly, there was a remarkable degree of blindness about what was happening.
If you followed developments in the early days, around 2023, AI was quite poor at mathematics. It was almost ridiculous. You could ask it questions and it would answer something completely different.
But little by little, it became less foolish and less incompetent, until it reached the point where it could prove and solve mathematical Olympiad problems. That was already quite an achievement.
When AI began solving Olympiad problems, people said, “Yes, all right, but it can solve Olympiad problems. Surely it won’t be creative like us, solving genuine research-level problems.”
It seemed to me that I could not ignore this subject either. The book discusses it.
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We are now entering a new era: the era of artificial intelligence.
For me, this represents a kind of industrial revolution for mathematics, and not just for mathematics. Whether we like it or not, AI is taking mathematics out of the realm of craftsmanship and promising to produce results on a mass scale.
The first top-level results we have seen from AI systems concern major questions on which mathematicians had been stuck for a long time.
These questions have now been solved, sometimes with proofs that are difficult to follow but extraordinarily complex, and occasionally of impressive beauty.
In other words, we do not have a monopoly on beauty.
Perhaps that sounds a little unsettling.
I think we are living through a dizzying period. Over the past two weeks, the mathematical community has been caught up in this. I have the impression that everyone has stopped to think about little else.
And many of the issues we have discussed today immediately come back to haunt us.
How does all this translate into the age of artificial intelligence?
How does AI replace the sensitivity that draws us towards certain problems, or the intuition that guides us towards solutions?
How does it condense the experience of all humanity at once, potentially replacing our individual experience?
Can AI itself be creative? This is the question all mathematicians are asking.
Will it be able to step outside the established framework and invent entirely new concepts, rather than simply recombining concepts that human beings have already introduced?
Will we be able to understand its proofs? Will we be able to make them our own, so that we can communicate them to others and pass them on as living knowledge?
Will we ourselves change? Will AI-generated proofs change the way we think?
Might we discover new ways of constructing proofs, just as chess and Go players have learned new strategies through playing against machines?
https://www.youtube.com/watch?v=m6JLrjfUL_c