I would like to explore this question in a semi-autobiographical way. To narrate the story of why I got into mathematics, the sort of mathematics I do, and then finally from there, address these fundamental existential questions.
I would like to begin my tale with how I landed upon my chosen thesis topic for my mathematics masters course. I was fascinated by the Anglican Bishop and Philosopher, George Berkeley, and his philosophy of "idealism", what today philosophers would call panpsychism, the idea that there is no matter in the world, and it all consists of a system of ideas and sense impressions which God directly feeds into the souls of men, the precursor to "The Matrix". "To be is to be perceived" as Berkeley would. The world consists of nothing more perceptions.
However, this philosophy led me to wonder if projective geometry, rather than the standard Euclidean geometry, might be much more foundational geometry. Projective geometry is the study of geometry "under projection", that is, how geometric objects appear from various perspectives, and how the objects change with changes in perspectives, and what properties remain invariant under change of perspective. This kind of geometry was first studied by the Renaissance artists, trying to determine how 3-dimensional figures appear on a 2-dimensional canvas, but grew into an extremely rigorous geometric system in the 19th century as it became abstracted from its original purpose in art.
My suspicions were confirmed, that projective geometry was truly much more foundational to geometry than a lot of other geometry, as I delved deeper into projective geometry. As the great geometer Felix Klein was reported to have said, all geometry is projective geometry. Doubtless today this would be considered an exaggeration, but the mathematical reason for his declaration is that you can pretty much construct the Euclidean and traditional non-Euclidean geometries, e.g. elliptical, hyperbolic, etc, from within projective geometry.
As I was doing my masters I suddenly had the thought, what if spacetime warping by special relativity was somewhat like the warping and distortion of geometric objects under projection? I decided then that this was what I wanted to do for my thesis: a framing of special relativity within projective geometry. And why not? Every school boy with the barest minimal interest in physics or science would be fascinated by relativity and spacetime bending and all that, that is cool stuff!
I initially set out merely to do an expository thesis on relativity and projective geometry, not an original research thesis. As mentioned, projective geometry has been around since the Renaissance, Einstein's special relativity is about a century old. I thought that someone must have already developed the projective geometric framework for special relativity, and derived special relativity "from the bottom up" from projective geometry.
But the more I researched and looked into it, it became clear that while mathematicians and physicists have noted that the geometry of special relativity, that is, Minkowski geometry, shared certain structures with a kind of projective geometry, nobody had bothered to actually attempted to derive relativistic kinematics/geometry from projective geometry. There are various reasons for this, some of them are technical difficulties, but the most important reason is simply that there was no reason or need to solve these problems. Physicists and mathematicians already have far more powerful mathematical frameworks and devices to derive relativity and to compute the metrics, e.g. differential geometry, for the purpose of physics and even for mathematics, there was simply no pressing need to solve the technical problems for framing it within projective geometry.
So what began as simply a thesis to exposit on what I thought would be prevailing knowledge of the projective framing of special relativity became an original research thesis. I had to use pre-existing techniques from other fields and apply them in a novel way to solve some of the technical problems, and create novel geometric devices to get around some of the difficulties.
Now, I don't claim that these technical problems were insurmountably difficult, certainly not frontier level difficult, nor that my solutions to these problems were some kind of stroke of genius. My solutions to these problems were original for one very simple reason: I cared enough about the problem to solve it, nobody else bothered.
So much for the story of my mathematical journey. I want now to reflect on what effect AI would have on this. And now it should be clear:
AI would take nothing away from this.
I chose to study the specific mathematics I wanted because I was simply fascinated. The part of mathematics I was studying is very well established, trodden by hundreds of greater minds than mine. I did not chose this topic to "challenge myself", to solve a problem no one else could. I chose it because I was curious about the topic and I even thought that someone must have already finished the work. I did original work simply because I was curious to see if it could work and because no one else had that curiosity.
Thus, I would still have chosen to study it even if someone else, whether a human or AI, had already done all the work. Thus, at least for me, that AI could solve a mathematical far faster or better us would take away nothing from those of us who are motivated by sheer curiosity and love for the subject itself. Even if a computer or another human, had already established the field or solve the problem, if you are truly curious about the subject, you would still want to know and learn just for its own sake.
At this point you might say, that's well and good for you to study simply just for curiosity sake, but what about other mathematicians who revel in the challenge of solving a problem? My glib answer is first that you can simply join a mathematics competition and literally challenge yourself. My non-trivial answer would be to refer to the example of Alexander Grothendieck, the geometer who invented modern algebraic geometry, an entirely new field of geometry, not an easy feat with mathematics as mature as the 20th century! When Grothendieck was in university, he was dissatisfied with the mathematics education there which treated certain mathematical concepts, I.e. integration and area, with what he thought was insufficient rigor. For three years he isolated himself to formulate the rigorous basis. When he had finished formulating his theory and presented it to senior mathematicians, they informed him that he had basically just re-derived Lebesgue integration, formulated 40 years before him. However far from disappointment, he felt that it validated his abilities as a mathematician, that without books or access to prior knowledge, he could derive from scratch such a fundamental result, and that it showed that he had great mathematical intuition. And further, this independence of mind would later enable him to become the pioneer of one of the great frontiers of mathematics.
Would AI prevent mathematicians from doing a Grothendieck? Grothendieck could isolate himself from books which contained the very thing he wanted to derive, we don't need to use LLMs if we don't want to and we are always free to re-derive everything from scratch if we wanted. AI doesn't force us to use it if we don't want to! Mathematical textbooks come with solutions manuals and answers at the back, if you want to challenge yourself, just don't look at the back!
Thus, I am considerably baffled with all the talk about how AI brute forcing solutions or rapidly reaching the end point somehow would destroy meandering wanderings, playing around and looking at different angles, exploratory discoveries and poking around established theorems to glean new insights. If you want to do all that, how does AI stop you from doing so? Just do it! My own mathematical stint precisely is this.
There are all kinds of questions about the status motivation, how would mathematicians show what humans are capable of if a computer can brute force it? And then there are questions of economic and material incentives for doing mathematics if it can simply be done much more efficiently by computers. How would we justify paying mathematicians for years of works what can be done in a month by someone who doesn't need insurance?
But I speak as someone who is just curious and just loves the subject. It means nothing to me if AI could solve these problems because my primary motivation is that I just want to contemplate and appreciate the actual subject matter and content, I just want to know and understand cool mathematics stuff, how does the fact that AI can solve a problem somehow prevent me from knowing and understanding? How does the fact that AI can brute force a solution prevent me from appreciating the subject matter?
I would like to end off on a religious note.
When I was in junior college (in Singapore it's the stepping stone to university), I was deeply interested in philosophy, being a new Christian and just getting into apologetics. Yet historically philosophy itself was very much integrated with mathematics. So to explore further the deeper cultural and philosophical significance of mathematics, I picked up Morris Kline's Mathematics for the Non-Mathematician, to which I then encountered the following passages:
The mathematicians and scientists of the Renaissance were brought up in a religious world which stressed the universe as the handiwork of God. The scientists whom we shall meet shortly, Copernicus, Brahe, Kepler, Pascal, Galileo, Descartes, Newton, and Leibniz, accepted this doctrine. These men were in fact orthodox Christians. Copernicus was a member of the Church. Kepler studied for the ministry although he did not take orders. Newton was deeply religious and, when late in life he felt too exhausted to pursue creative scientific work, turned to religious studies.
However, in the sixteenth century the new goal in the intellectual world became to study nature through mathematics and indeed to uncover the mathematical design of nature... How then was the attempt to understand God’s universe to be reconciled with the search for the mathematical laws of nature? The answer was to add a new doctrine, namely, that God had designed the universe mathematically... Indeed the work of the sixteenth, seventeenth, and even some eighteenth-century mathematicians was a religious quest, motivated by religious beliefs, and justified in their minds because their work served this larger purpose. The search for the mathematical laws of nature was an act of devotion. It was the study of the ways and nature of God which would reveal the glory and grandeur of his handiwork. The Renaissance scientist was a theologian studying nature instead of the Bible. Copernicus, Kepler, and Descartes speak repeatedly of the harmony which God imparted to the universe through his mathematical design.
It was this passage which led me to study mathematics in university. And no amount of AI brute forcing solutions or solving mathematical problems will ever take this away from me: the rupture of beholding the glory of God in His mathematical design of nature, and His genius in creating mathematics.
Mathematics ultimately does not exist for mathematicians, it does not exist as a jobs program for talented maths brains. Mathematics doubtless has many practical uses, but mathematics is for everyone, and ultimately it exists to manifest the brilliance and wisdom of the Almighty, not to aggrandise the egos of men.
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