“We are transitioning from an era of proof scarcity to proof abundance. For thousands of years proofs were hard and… it required understanding and expertise and a bit of luck and talking to people and knowing the literature. It was universally valued because everyone who found a proof of a difficult theorem had to do all this difficult work and this helped grow the field. And now we found the shortcut. We found machines that can get these proofs for us and now we’re drowning in proofs, but we’re not getting all the understanding that we used to get.
“I guess we are seeing the end of the romantic era of mathematics — the age of heroes — which happened in other disciplines centuries ago as early as the industrial revolution. But somehow mathematics held on and we had our idiosyncratic traditions. For example, we all use blackboards, as you can see, and I’ve got this special Japanese-Korean chalk.” – from Terence Tao Talks to Numberphile About AI
Today it’s an uncommon reflection that in industrialization, a discipline’s practitioners change the way they work. We are mostly past that time of industrialization. So the emotional reactions from mathematicians to recent AI proofs speaks to their discipline’s changing norms.
I was struck by the degree to which notions of romance, respect, civility, fear, and more factored into those reactions. I had found similar reactions in other disciplines of the past (see the anti-tech Luddite movement list), but compared to historical accounts, like the impact of the sewing machine on tailors and the tractor on farmers, it is more powerful to witness changes in real time.
Here’s Tao on a call for respect and etiquette:
“I think the broader community has to respect some of our space. If you organize a marathon it’s understood that for the general public to start driving along the same road and beating people at the race – you could do that theoretically, but it’s socially not good etiquette.”
But unless there is something to maintain it, there is always a tendency away from etiquette. 50 years ago we got a taste of a mathematical etiquette break. In 1976 the four color map theory was deemed proved — with a computer’s help.
But there were these comments and many more:
- “The present proof relies in effect on an Oracle, and I say down with Oracles! They are not mathematics” – Paul Halmos, UC Santa Barbara
- “Since the problem had been taken care of by a totally inappropriate means, no first-rate mathematician would now work any more on it, because he would not be the first one to do it.” (from Robin Wilson’s book Four Colors Suffice)
Scoffing at that computerized proof was still common years later when I was in high school.
Beyond that break in etiquette, we also have panic, potentially lost funding, and feelings of persecution.
From the presentation “A Serious AI Alignment Flaw in Mathematics, 28 Days Later”, by Hugo Duminil-Copin“:
“My various PhD students and postdocs are in a state of total panic, and I understand them. And undergraduate, master’s, and bachelor’s students
are also asking these same questions, quite simply. Is there a future for mathematics?
“[P]olicymakers will be making funding decisions in the coming years and we need to be ready to explain our added value. If people spend more time explaining that others are wrong and that their arguments are bad, we simply won’t have any arguments to present to these policymakers… I wrote this text for Proof Prompts… And I had this impression over the summer that there was a kind of imbalance in terms of speaking time and especially who was speaking, sometimes supporting the fact that AI was absolutely wonderful for math, and sometimes the fact that one could fear the consequences of this AI, or the use of AI… And so, I wrote a text that seemed quite measured to me… and I was surprised by the violence of the emails and messages I received following this text.”
Would those policy decisions be effectual in such a situation? How can we explain “added value” adequately, when that value may not be realized for many years, or when the value (in a financial sense) of new learning may be unclear?
Related, this is from the Leiden Declaration on Artificial Intelligence and Mathematics: “Mathematicians have a choice about whether and how to adopt artificial intelligence in the conduct of their research. They also have a responsibility to ensure the continued flourishing of the discipline.”
We have spiritual angst. From a post titled “The Dark Night of Mathematics”: “I am going insane. During the last week or so LLMs have produced a number of counterexamples to significant long-standing conjectures….I am suffering a profound spiritual crisis due to these developments.”
“There is something about mathematical discovery (progress, advancement, creation) which is vital to the spiritual, experiential quality of doing mathematics. The creation (or even the pursuit) of novel mathematics is one way that humans have historically accessed the ineffable and encountered the divine and mystical.”
“Suppose any theorem you set out to prove had already been proved in 100 wonderful ways—the companies will pay mathematicians en masse to optimize the weights for “interesting”, “beautiful”, anything you like. Whatever the idiosyncrasy, value-add, or unique synthesis of your approach, it has already been done, or can be done with a mindless prompt in an instant.
“Consider this: If The Library of Babel existed, would authors continue to write books?
“The story of human discovery and the triumph of the human spirit will soon be excised from this discipline. The Dinitz-Garg-Goemans counterexample was the most egregious demonstration. It revealed that the process of prompting novel proofs will be as auraless as ordering doordash. Watch as magic and mystery evaporate. Watch as the sun sets on our heroic age. Is there not something evil in the act of blocking all future generations of mathematicians from the experience of discovery? Forget about accuracy or even attribution. Something fundamental to the experience of mathematics is being taken.”
And we have humility. From “We’re gonna need a lot more mathematicians”:
“The research mathematics community consists largely of those of us who [rarely felt unable to keep up with new concepts]… But we are now entering a time for humility: a time when all of us are going to know what it feels like to be unable to keep up.”
Unironically, the post concludes: “The ideas and opinions presented here are entirely my own, but GPT 6 Astra was instrumental in helping me draft this note.”
Again Tao: “I think we are moving to an industrial era where math will become much more large scale. We will work in big teams more like the other sciences… There was something romantic about the heroic age and hopefully we will keep some of it in our educational system. We will emphasize traditional ways of doing mathematics first before the industrial things just like we teach kids how to do arithmetic by hand before giving them calculators.”
I feel for the mathematicians. But these are not productive feelings, as cruel as that sounds for the end of a “romantic era.”
There’s a Chinese idiom: 生不逢时 or “to be born at the wrong time” that recounts how an old official working in a junior post explained his low rank to the current emperor. He had been a young fighter when a previous emperor favored literature, middle aged when the next emperor favored older experienced advisors, and elderly when the current emperor favored the young.
Hearing this, the emperor promoted the old man for his loyalty across different environments. But in that example, the outcome is individual and not scalable. There is no emperor distributing boons for the mathematicians of today.
You can’t choose when you are born or what environment you are born into. It’s frustrating to dream “I could have been a great explorer had I only been born in 1500 when there were many uncharted lands!” Or to think “I could have been a great mathematician had I been born in 1900, well before AI proofs!”
We embraced the ideal of progress for so long that we forgot that there are still fields of knights charging with lances.
Things change when they scale. But the outcomes of scaling leave us with results that were not quite the original goals. Keeping with the romantic era theme, we saw something change when we moved past chivalrous rules of warfare, the art and community of plowing fields with horses, years spent on an oil painting. Their replacement of early 20th-century trench warfare, GPS-controlled tractors and combines, and pop art gave us more volume, but also something different.
In “Will it be a golden age or a dark age for mathematics (and science)?,” the author admits incomplete understanding to AI-generated proofs, but argues for their potential to advance learning.
“Seizing on that potential will require the careful integration of LLMs into the various STEM ecosystems, with the priorities of empowering a new generation of scientists, developing new STEM expertise, and preserving an open, collaborative, and intellectually independent scientific enterprise: that is the “golden age” scenario in the title. In the “dark age” scenario, scientific and technological output would still increase spectacularly in the short term, but, at the same time, independent human scientific activity would progressively weaken, the production of new knowledge becoming increasingly concentrated in a small number of hands, who become increasingly incapable of understanding the knowledge produced.”
And in Quo Vadis, Mathematics?, we have the analogy of mathematicians as lone explorers navigating dangerous terrain on foot. Meanwhile airplanes come into common use, causing “disorientation, and a sudden loss of purpose. That grief is real, and I won’t pretend otherwise. But it will not last. Mathematics is about to graduate. It will grow into a full-scale industry and become the true infrastructure of all scientific and technological progress, a foundation that every other field stands on rather than a quiet corner of academia. The age of lone climbers is closing, and the age of mapping the whole continent is opening. As for me, I can’t wait to see those vast mountain ranges from my supersonic airplane.”
That chalk that Tao mentioned must be supreme. A colleague of mine bought all he could when he learned that the original manufacturer was shutting down. But the blackboards are already gone.