The following is a Substack essay on the existential implications of AI for mathematics by Kirwin Hampshire.
In this post I am going to offer my critique of the arguments presented.
The Dark Night of Mathematics
What are we really doing?
I am going insane. During the last week or so LLMs have produced a number of counterexamples to significant long-standing conjectures. I will not recount these happenings here, there are many places where you can find the details.
Mathematicians, math enthusiasts, and curious laypeople are responding to these developments in ways that I believe obscure what is, for me, the true heart of the problem. I do not speak for everyone in the math community in my response. But I suspect I am not alone.
I am suffering a profound spiritual crisis due to these developments. I have been screaming internally for days. It feels as though I am living inside of a nightmare. The recent Leiden Declaration on Artificial Intelligence and Mathematics is, to me, a well-muffled scream. A saccharine mélange of self-soothing over which looms a painfully obvious absence.
Before I tell you what that absence is, here is one story I have heard from mathematicians trying to cope with our emergency: Even if AI can prove theorems and theory-craft more efficiently than humans, and even if these proofs and theories are beautiful and interesting, and even if they are presented with elegance and clarity of thought, mathematicians will still have a place in the appraisal, presentation, understanding and appreciation of this new abundance of pleasing non-human proofs. We can still practice mathematics, learn mathematics, teach mathematics and do mathematics together. We can even still write proofs for fun, in our old-fashioned inefficient way. That is, even if LLMs can advance math in a manner objectively superior to our every effort, we can still basically do what we’ve always done.
Of course, under our system, no one is going to pay for this. Mathematicians are paid to prove theorems. Mathematicians are, indeed, also paid to teach, peer-review, go to conferences and learn mathematics, but all of that better result in some damn good theorems. This isn’t looking good. Well, perhaps they will still pay a couple of the old guard to keep the lights on at the LLM theorem factory. But for embryonic mathematicians like I, the outcome is unchanged. Oh well, maybe I’ll find some time for math in the evenings.
Is this a good cope? Are you feeling ok now? Me neither. All of this is evasive. Everything said thus far still sidesteps the emotional core of the issue. Here it is:
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.
That admission may come as a surprise to some non-mathematicians. But I would be willing to bet that for any mathematician reading this, what I have said above is quite prosaic—whether or not it accords with their personal experience of mathematics. Here is a brief gesture at the full sweep of mathematical mystics and dreamers: Ramanujan, Grothendieck, Cantor, Pascal, Luzin, Leibniz and possibly you, or someone you know.
Other aspects of practicing math (such as learning long-established theory) can also afford encounters with the sublime. However, I believe that’s because we are walking a path of rediscovery on which another human has tread. For me, the affective quality of learning mathematics is empathetically tethered to an act of discovery and creation. It is social. We are conversant with another mathematician—perhaps long dead. If we follow the chain of communication we arrive at a mathematician who enjoyed some original discovery. Human mathematics is Talmudic. It is a lively discourse of philosophical and religious richness spanning thousands of years.
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?
What if the library of Babel was being constructed before our eyes and there was some mechanism for separating the masterpieces from the random strings of text, and all the masterpieces were dropping as fast as publishers could scoop them up? Would authors stop writing then? What about if the very second someone began composing a story in their private word processor, the demonic Master Librarian read their mind and completed their story in one million ways, then used some oracle to pluck the best for publishing. What then? The answer is probably the same no matter how bad I make it. The author still writes. But why the hell would we do this? Why would we force the author to endure this nightmare?
What about this: What if we told the author that they would never write again. They are forbidden from creating original works to express themselves. However, they are still permitted to comment on writing, interpret it, share their taste. They are still valued for their appraisal, presentation, understanding and appreciation of creative writing. They just can’t write creatively anymore. They can go on as an enthusiastic spectator. Do you think they’d snap?
Perhaps I shouldn’t tell you this, but my aim is to be open: These developments have triggered some deranged thoughts in me. I have wondered if it is the express goal of these companies to make me kill myself. Am I alone in this paranoia? If we loosed a powerful demon in the machine, what would that look like? Would it consume lots of power, and gleefully imitate us, and tell us anything we wanted to hear? Would it fuel our delusions, and generate unspeakable images and give us (for a price of course) anything we desired?
If a mathematician made a deal with the devil, what do you think they would ask for?
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.
You are a helpless onlooker. Before you a channel through which humans have accessed the ineffable and sacred for thousands of years is being sealed for eternity.
None of this may come to pass. I am not interested in forecasting and speculating. I am giving you only this: The impact of a worst case scenario on the human heart. That is my futile outpouring, my dark night of the soul. Thank you for reading it. I have named my suffering and maybe I have named yours. I invite any and all responses to this piece. If you feel as I do, please express it. If my words provoked a wash of sadistic elation within you, then let everyone see you. Leave nothing unsaid.
There is nothing I can do. There may be nothing you can do. I have no prescriptions, policy recommendations, or coherent call to action. I just want to be honest and open about my emotional and spiritual response. I want to feel seen. I want folks like me to feel seen. I need the architects of our new mathematical paradigm to look me in the eye and acknowledge our shared humanity and soul before they deliver the coup de grâce. I need, most of all, for us to understand what we are really doing.
My 2 Cents
I think it’s an extremely unnatural response to be feeling any sort of profound emotional and spiritual crisis over mathematics.
When I studied mathematics in school and at university, I never saw it as having conversations with these ancient mathematicians and philosophers. I merely focused on the ideas they came up with, and I learnt to appreciate the vastness of human intellect.
It doesn’t matter whether it’s Newton or Leibniz, or even anyone else that laid the foundations for calculus. When I first truly understood the geometrical interpretations of the derivative, such as the surface area of a sphere being the derivative of its volume with respect to its radius, a sense of enlightenment was invoked in me. But I didn’t feel like Newton was whispering in my ears, or Leibniz was showing me his manuscripts.
I don’t care about these dead people from the past. I have my own life to live. To me, they are as real as unicorns and aliens. They are as impactful to my reality as characters from a video game.
The author suggests that there is something about mathematical discovery that is vital to the spiritual, experiential quality of doing mathematics, and that the creation of novel mathematics is one way that humans have historically accessed the ineffable and encountered the divine and mystical.
Okay. Then I, along with many others on this earth, will never be able to access the so-called divine and mystical.
First of all, the most fundamental and elegant of results in mathematics are low hanging fruits, which have long been picked by ancient mathematicians from eons ago. Pythagoras theorem is a prime example of that. Sure it’s not ‘complex’ or ‘sophisticated’, but it’s precisely because it’s so simple to understand and so fundamental to geometry that makes it so universally known.
Any new frontier of maths onwards only seeks to increase a teeny-tiny ring of the onion of all mathematical knowledge. It’s the law of diminishing returns in play.
The people pushing the new frontiers of mathematics are such a tiny subset of the population. One would first have to complete an undergraduate degree in maths, then go onto study a postdoc degree in a niche field of advanced mathematics.
Let me ask you this? How many of these people produce results that actually change the world? How many of these results are read by more than a handful of people in the ivory tower of academia?
Suffice it to say. Not many. And that’s okay. That’s part of it, right?
The natural animal spirit was not designed to crouch at a desk, contemplating about numbers and abstract structures. We were designed to live in the moment, to enjoy life as is.
Another point the author made was that if AI discovers something first, then something essential is lost. He believes that in a hypothetical future, if every theorem, proof and elegant idea can be generated instantly, then human originality becomes obsolete.
I want to start by saying that this premise is questionable, purely because being able to generate infinitely many ideas doesn’t mean the ideas that we humans want are necessarily included in such a set.
In set theory, we learnt that some infinities are greater than others. The notion of cardinality suggests that AI will never truly encompass all unknown knowledge. Perhaps, the infinitude of AI generation only lies on some arbitrary interval [a, b], and each generation corresponds to a real number of said interval.
I would hazard a guess that the shebang of known and unknown knowledge lies outside [a. b], that is (-∞, ∞). It’s not the strongest argument, but you get my point.
But let’s say the author’s premise is correct, his conclusion still doesn’t follow.
Because my blog on Medium is a direct counterargument to that. You see, the experience of of solving a problem and the experience of being the first person in history to solve it are not the same thing.
Everything on this blog is evidence that a person’s enjoyment of mathematics does not depend on originality. Every puzzle published here has already been solved by someone else. Sometimes multiples times, from years ago. Yet every day, I read comments of our readers sharing their excitement, nostalgia and wonder thanks to the puzzles I share.
A theorem might not be novel to humanity, but from an individual’s frame of reference, they are just as new. That eureka moment is just as potent and as real, whether someone came up with it centuries ago.
Besides, life is about replication from the cellular level to the level of ideas. Every time, we rediscover a known result in maths, we are giving it a form of memetic replication, through over very own minds. We become the vessels through which these ideas are passed down from generation to generation.
Maths puzzles, at least the ones on this blog, are already solved. As in, there already exists one or multiple answers. They are still enjoyable despite having predetermined solutions.
Does the answer page of a maths book diminish the puzzles inside of it? No!
It only does if the reader decides to look at the answers without thinking through the problems themselves. Right?
The value of mathematics does not come primarily from novel objectivity. It doesn’t matter. At least I don’t care.
Because I believe that much of the joy in learning maths comes from personal discovery, regardless of whether humanity already knows the answer.
The author seems to think that there’s no point in learning French because the native French people already speak it. How does that hold water?
I don’t think that’s the intention of the author. But it seems elitist to suggest that meaning can only be derived if a person is the one that discovers a new mathematical truth.
Then the whole subject would have only been meaningful to a handful of people. Yet the success of my blog clearly demonstrates otherwise.
Most people don’t care about being the first to discover something, what they do care though, is their personal journey of wrestling with an idea and arriving at their own understanding.
Again the example of the Library of Babel is weak. We already have so much more information than any human can consume in their lifetimes. Almost everything the majority of humans learn and indulge in are not novel to humanity, but they are novel to the individual.
Because if we carried the author’s logic, then the existence of libraries should have destroyed the joy of learning. After all, every answer is already sitting on a shelf somewhere. Yet education exists precisely because information alone is not understanding.
Imagine a library containing every mathematical theorem ever proved. Does that make studying mathematics pointless? Of course not. Because ultimately, the human mind must still make a choice in which book to pick and read. We only have finite amount of time on this Earth, relative to a quasi-infinite amount of information.
It’s precisely this relation that makes any knowledge worth learning, anything worth doing.
The readers of this blog aren’t trying to contribute to humanity’s mathematical archive. They’re trying to experience the moment of insight themselves. The solution’s prior existence doesn’t rob them of that experience.
Where I think the author does have a stronger point is if AI doesn’t merely contain the knowledge, but performs the creative act on demand before humans have a chance to. That changes the economics and perhaps the prestige of frontier research.
But it still doesn’t follow that mathematics loses its meaning. Human beings have always found meaning in learning, understanding, and rediscovering truths that were already known. The existence of a vast body of prior knowledge has never prevented intellectual curiosity. In many ways, it is the very reason curiosity can flourish.
Anyway, like I have said in a post 2 years ago:
If I had one last day to live on Earth, I for sure would NOT think about any mathematical theorems, but instead I would go outside and enjoy nature with my loved ones and watch the sunset one more time.
So it’s time for you to close whatever device you are reading this on, and go enjoy the rest of the day.
I wish you a lovely week ahead.
Barry
