I doubt it, because the ‘problem space’ of philosophy is qualitatively different to that of mathematics. Even though mathematics is a vast subject, it is bound by rules in a way that philosophy need not be. Philosophers deal with questions of meaning and value and often grapple with questions that lack the kinds of precise definitions that are the subject of mathematics. Maybe if the mathematicians can’t out-do AI in their chosen field, they may elect to pursue philosophy — for philosophical reasons, of course.
Is that the place for the mathematician of the future? Hence “Journeymen” but never “Master”?
The mention of “proof by intimidation” is interesting; presumably the tone adopted by the AI could be less threatening - perhaps the voice of Uriah Heap would be appropriate…
If the proof is too difficult to be understood, what do we do? Set two AI’s against each other?
Though I agree that philosophy is a different ballgame, I’ve seen AI do well on philosophical topics, capturing much of what some might say is only humanly possible.
I’ve done some math with AI and from what I could gather they can easily replace a high school teacher in the class room. Rudimentary or introductory philosophy may no longer be the exclusive domain of humans. I base this on an extremely conservative grading of AI capabilities, bracketing out all those times when AI clearly outperformed human experts.
The future is uncertain. It’s not the same as computing the trajectory of a rocket or the calories in a chocolate bar. Fear mongering is as old as the hills.
Fear is the path to the dark side. Fear leads to anger, anger leads to hate. Hate leads to suffering ~ Master Yoda
To some extent it would flatten the field: We trust the authority of professional mathematicians, they trust the authority of o4-mini…
Except that one cannot trust the AI.
And this might lead us to question the very notion of “proof”. Is a proposed solution a “proof” if it has not been checked by a mathematician? It’s a proof if it convinces you, and your peers… is that now enough? And note that providing a proof is a social act; an AI generated proof that no one can follow is not a proof.
Platonism is the idea that mathematicians discover proofs that are already out there, somewhere. The alternative is some form of constructivism, were mathematics is putting symbols together in interesting patterns.
Now o4-mini is an LLM; it works by statistical pattern matching.
If mere statistical pattern matching can indeed construct proofs and conjectures, then are those proofs and conjectures more than interesting patterns? In what way?
When the world chess champion played Deep Blue and lost, it was quite clear our champion didn’t see what was coming down the pike for him. I’m not sure but I think he lost consecutively. Deep Blue was what we call an expert system, a one trick pony as some would say. The AI we have now is different. It can beat you at chess for as many times as you want and then send you flowers of condolence to nurse your hurt ego.
The basis of everything that a particular a.i. can possibly do is delineated in its hardware and software design. It has been said that a.i. engineers don’t fully understand how their inventions behave, but this is true only within strict limits. What limits an a.i. is the fact that it functions on the basis of a plan which is grounded in a particular era of philosophy.
Today’s most advanced a.i.’s are essentially applied instantiations of the 18th century ideas of Leibnitz and Hume, since that is the way today’s engineers and tech ceo’s understand how the mind works. Tech polyannas like Sam Altman are convinced these devices think the way we do, or even better than we do. But in fact current a.i. reflects the way leading philosophers from 300 years ago believed we think. A lot has changed in philosophy since then, but the capabilities of a.i. won’t reflect these changes until someone programs them into it.
From the article in “Scientific American”: “If AI reaches that level, the role of mathematicians will undergo a sharp change. For instance, mathematicians may shift to simply posing questions and interacting with reasoning bots to help them discover new mathematical truths, much the same as a professor does with graduate students. Ono predicts that nurturing creativity in higher education will be a key in keeping mathematics going for future generations.” It is implied that in the AI era mathematicians will primarily be shaping a horizon of enhanced mathematical research. Currently, it seems unlikely that AI is pinpointing the most significant problematic domains or inventing new frameworks. However, the article’s prediction has another aspect. One must undertake an apprenticeship to become a leading researcher. But individuals might not gain basic knowledge and skills in a learning environment supported by AI.
It’s worth recalling that during the original railroad madness we also got all sorts of hype about steam powered calculating machines that were going to replace mathematicians.
You’d think that if the world melting capabilities of “the latest model” were all they were cracked up to be these companies could use it to learn how to turn a profit instead of needing an endless hype loop to keep to keep the speculative bubble going.
The paper describes how an AI was configured to make “large moves”, in which “one reinforcement-learning model suggests new moves and a second model checks to see if those moves help”. This playing of one AI against another is described as “thinking outside the box”.
At any step a human can interact and direct the process, ‘Let me look down this little alley. Oh, I found something!’
And pivotally, “…it’s one thing for a computer to play Go at a superhuman level and another thing for the computer to invent the game of Go.”
The emerging picture is of a mathematics in which the grunt work is done by an AI, under the direction of a Master mathematician.
…which is where your suggestion comes in, in that we might need to reconsider what it is to be a Master Mathematician - someone who recognises the results of the calculating, choosing the direction of exploration… and again we are back to Platonic metaphors.
There are few contemporary mathematicians more deserving of respect than Terence Tao, and he’s an advocate for (and active participant in) the use of AI in mathematical research. But he’s not a naïve AI enthusiast. He’s also one of the most clear-headed critics of exaggerated claims regarding AI mathematical abilities, and he actively explores the limitations of AI tools. Faced with the question of what it is that might happen if AI agents come to be more productive than human beings currently are in discovering new mathematical truth and finding proofs for current conjectures, Tao highlights that mathematicians will still play an important role in deciding which results are interesting and which are not, and in explaining why.
Personally, I think LLM-based AI agents already are fairly good at explaining mathematical and scientific results and concepts, and also at highlighting why they are interesting and beautiful as assessed by competent human mathematicians and scientists. So, there remains an essential role for human inquirers, which is to learn enough about the relevant fields to establish what is useful for us and/or valuable by our own lights, lest there be nothing whatsoever for the AI to explain to us or make useful for us.
The error in that time period was thinking that if you could do computations really fast, you could essentially do anything. This was the era of the cosmos as a heat engine—Leplace’s Demon still ruling the roost.
But, as would later be demonstrated, any physical system of sufficient complexity can be said to be instantiating any computation. Everything can be said to be a computer of sorts (indeed this has been very in vouge in physics since the 1990s). And so it turns out that knowing what is worth computing is 99% of the battle, and having a fast computer doesn’t really help all that much with that (except in a roundabout way).
The article suggests AI could potentially replace a typical mathematician with a PhD. But there will remain a need for advanced researchers for ‘discovering new mathematical truths’. This situation may endanger the whole academic environment because just its minor segment develops novel fields of inquiry. Most researchers are part of the academic group dedicated to maintaining current knowledge and frameworks and facilitating conditions for a possible paradigm’s change. Also, it is unlikely that one can jump directly to the highest rank, bypassing the usual path of a scholar or a disciple. The structural imbalance might lead to a tendency that goes against the article’s claim that ‘nurturing creativity in higher education will be a key in keeping mathematics going for future generations.’ Maybe this is the reason for the panic in mathematics?
