Chuckle. I retired in 2000 as prof of math at a branch of a state university. The research papers I wrote and published, 25 or so, were the results of speculating, then confirming properties of math systems that interested me. Had an AI gone to work on the same subjects and achieved similar results would have been interesting, but would have had no impact on the wonderful experiences I had as I explored and created a tiny new corner of mathematics.
Mathematicians are explorers. AI might fabricate images of machete chopping through the jungle, but nothing takes the place of actually working one’s way through the vines and vegetation of the real thing.
On the other hand I recall a professor assigning problems at the ends of chapters on real analysis that frequently stumped me, and it would have been nice to ask a friendly AI for a little help.
These thoughts arise from an average mathematician, not one of the elites in the article.
That’s not as settles as you seem to think. What counts as making a computation makes a world of difference here. On a sufficiently broad interpretation, Searle’s conjecture might be so; but Chalmers and others have suggested that more is involved. A simple gloss will not do here. The MIT article addresses choice of computation quite directly.
[…] cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet (I have discovered a truly marvelous proof of this, which this margin is too narrow to contain). ~ Pierre de Fermat (Fermat’s Last Theorem)
The stoics required truths (proofs) to be experientially real and forceful. This view ran into the problem of religious experiences. Just yesterday I transcribed the religious experience of this lady, "Jesus came into my room, and lifted me up … ". There’s also the case of a theist who openly admits that he has proof of God but that no one will believe him.
I definitely agree with this. Curry understood math as the science of formal systems. I’d maybe just add that it is the finally informal science of formal systems. I want at least a few other people to “understand” me. So I can’t do without the metalanguage. As much as I love the object language as a thing of beauty, I have to set it up in terms that others will understand.
I largely agree, though I have a soft spot for the most basic algorithms that we learn as children. If I check the details of a pencil-on-paper addition of two large positive integers, then I am basically checking a proof.
But I don’t mean to imply that you were excluding this kind of checking of a calculation.
Going back to the place of AI in Mathematics: if proving some piece of mathematics is a social act, and if AI is outside of the intentionality of such social acts, then it is unclear where an AI proof stands.
Let’s imagine a monkey on a typewriter. The page “just happens to be” a beautiful proof of a beautiful theorem. That page signifies for “conscious” human mathematicians.
Alpha Go made a strange move that fascinated human experts. To me that is sufficient. The monkey-generated theorem is a “genuine” mathematical object. The move by Alpha Go is fascinating to humans as a kind of inhuman intelligence.
AI isn’t a monkey. It is a statistical analysis that generates a suitable next word.
Are you so sure that’s not all you are doing?
Nuh. The point is that Maths is a social activity. Perhaps it would be better to say that neither mathematicians nor AIs “produce” proofs - a proof is produced by a community.
I know. I have studied the math. It’s much much better than random noise.
If even a random string can be meaningful, then of course the output of a sophisticated program can be.
Much to recommend that, but I’m reluctant to ignore the situated-ness of perception and understanding. Sure, we are nodes in a network, but some component of reading/hearing is individual.
“Thought” is significantly distributed (Feuerbach), but if we “think only in signs” and signs are “material,” then perception and yes even “consciousness” matters.
So for me the worry would not be “maybe I’m A.I.”, because I agree with Heidegger that we are mostly bots already. But crucially we are also bots for whom the world is “here.” Can hardware be sentient ? Perhaps. We are more likely to attribute sentience if the bot looks like Alicia Viklander. Or has the shape of human or hound, etc.
I don’t see why that’s unclear. The AI is a tool of the person who uses it, therefore the intentionality is that of the user. In principle, this is no different from the scientist who uses a laboratory instrument in an experiment attempting to prove something. The intentionality is that of the user. The results, i.e. the proof, is then judged by the peers through analysis, repetition, etc., and this constitutes a “social act”. In other words, the essential aspect of this sort of “social act” is peer review, and that’s what exposes the influence of subjective intentionality on the process. AI is not free from such influence.
A golden age of maths is dawning and mathematicians are freaking out
AN article in New Scientist on the topic.
At one point in a group discussion, Tsimerman asked people in the room to indicate whether, in his button-pushing vision of the future, they would want to continue being mathematicians. Only around half raised their hand.
I would say verified, not produced. Once a result has been established in one’s mind, others may examine the argument to determine whether it is acceptable to the community at large. An even larger question is the value of one’s research, determined by some sort of consensus among the community.
A larger threat to PhD mathematicians these days is the growing practice of replacing a retired tenured faculty member with much cheaper part time faculty.
Large institutions require published research, but frequently smaller schools look for teaching ability. So a more profound question is how AI might replace teachers.
A worthy question, which I think we can press further by wondering if the educational institutions will survive.
Schools came into their present form partly as a way of keeping the children of the workers occupied, and perhaps giving them the sort of skills - social and academic - needed to work in a factory. if we no longer need to go to the one place in order to be exposed to academic learning, if we no longer need the master to teach us, and we no longer work in factories, then do we need the institutions?
I retired in 2000 as prof of math at a branch of a state university. The research papers I wrote and published, 25 or so, were the results of speculating, then confirming properties of math systems that interested me. Had an AI gone to work on the same subjects and achieved similar results would have been interesting, but would have had no impact on the wonderful experiences I had as I explored and created a tiny new corner of mathematics.