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What does the end of mathematics look like?
41–50 of 94 posts
Re: What does the end of mathematics look like?
#42I love the language of this article :-)... it may be florid, but that's quintessentially human. About the substance, I agree that there are fair grounds for concern, and it's not just about mathematics. The best case scenario is rejection and prohibition of uses of AI that fundamentally threaten human autonomy. It is theoretically possible to do so, but since capital and power are pro-AI[^1], getting there requires a…
Thanks for the positive feedback on my writing style! Based on feedback in this thread it seems to be a divisive topic, haha.
Re: What does the end of mathematics look like?
#43The camera didn't kill painting. Neither the bicycle nor the motor-car killed running. There are already subfields of mathematics where it's believed that all the interesting discoveries have been found and no-one is looking except for the occasional amateur - and other subfields where to even have a hope of doing cutting edge research you would need to both do multiple years of postgraduate study and then get accept…
> The camera didn't kill painting But it did. Painter used to be a trade where you could sell your painting skills as, well, a skill applicable for other than purely aesthetic reasons, simply because there were no other ways to document the world around you. It just isn't anymore because of cameras. Professional oil portrait painter isn't a career in 2025.
Well, it is still a career, but it's very niche, and more attuned to 'art' than 'documenting the world'.
Re: What does the end of mathematics look like?
#44Considering that mathematics is, at its core, a language for defining relationships between quantities, and then relationships between those relationships, so on and so forth, I think it's fair to assume that the possible number of such relationships are infinite. Some of these relationships will obviously be useful in the real world, but they don't always have to be. I too, suspect that we can keep on building theor…
> we can keep on building theorems on top of theorems with increasing complexity
This is a somewhat bleak picture of math. We also have the other phenomena of increasing simplicity. Both statements and proofs becoming more straightforward and simple after one has access to deeper mathematical constructions.
For example : Bezout's theorem would like to state that two curves of degree m, degree n would intersect in mn points. Except that you have two parallel lines intersecting at 0 instead of 1.1 =1 point, two disjoint circles intersect at 0 instead of 2.2=4 points, a line tangent to a circle intersecting at 1 point instead of 1.2=2 points. These exceptions merge into a simple picture once one goes to projective space, complex numbers and schemes. Complex numbers lead to lots of other instances of simplicity.
Similarly, proofs can become simple where before one had complicated ad-hoc reasoning.
Feynman once made the same point of laws of physics where in contrast to someone figuring out rules of chess by looking at games where they first figure out basic rules(how pieces move) and then moves to complex exceptions(en passant, pawn promotion), what often happens in physics is that different sets of rules for apparently distinct phenomena become aspects of a unity (ex: heat, light, sound were seen as distinct things but now are all seen as movements of particles; unification of electricity and magnetism).
Of course, this unification pursuit is never complete. Mathematics books/papers constantly seem to pull a rabbit out of a hat. This leads to 'motivation' questions for why such a construction/expression/definition was made. For a few of those questions, the answer only becomes clear after more research.
Re: What does the end of mathematics look like?
#45The camera didn't kill painting. Neither the bicycle nor the motor-car killed running. There are already subfields of mathematics where it's believed that all the interesting discoveries have been found and no-one is looking except for the occasional amateur - and other subfields where to even have a hope of doing cutting edge research you would need to both do multiple years of postgraduate study and then get accept…
> The camera didn't kill painting But it did. Painter used to be a trade where you could sell your painting skills as, well, a skill applicable for other than purely aesthetic reasons, simply because there were no other ways to document the world around you. It just isn't anymore because of cameras. Professional oil portrait painter isn't a career in 2025.
The Royal Society of Portrait Painters might disagree:
https://therp.co.uk/artists/
Re: What does the end of mathematics look like?
#46I love the language of this article :-)... it may be florid, but that's quintessentially human. About the substance, I agree that there are fair grounds for concern, and it's not just about mathematics. The best case scenario is rejection and prohibition of uses of AI that fundamentally threaten human autonomy. It is theoretically possible to do so, but since capital and power are pro-AI[^1], getting there requires a…
Thanks for the positive feedback on my writing style! Based on feedback in this thread it seems to be a divisive topic, haha.
I would say that I am envy that someone can write like that. I can't write in such manner in my native language, let alone in the second one: English.
It is nice to read or hear someone speaking like that, considering we are surrounded by low quality, easy to consume content nowadays.
And I am envy of such skill because I like to think about myself as not entirely being stupid, still I would never be able to write/speak this way because I just do not have an aptitude towards that.
Re: What does the end of mathematics look like?
#47Earlier quoted context omitted.
Thanks for the positive feedback on my writing style! Based on feedback in this thread it seems to be a divisive topic, haha.
I would say that I am envy that someone can write like that. I can't write in such manner in my native language, let alone in the second one: English. It is nice to read or hear someone speaking like that, considering we are surrounded by low quality, easy to consume content nowadays. And I am envy of such skill because I like to think about myself as not entirely being stupid, still I would never be able to write/sp…
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Re: What does the end of mathematics look like?
#48I don’t share the author’s concern about a corporate takeover of mathematics. Most mathematics isn’t of commercial interest. Even when it is, it seems like there would often good reason to share it, like any other source code. Is Lean so different from other programming languages?
Such libraries would need documentation, or nobody would know when to use them, and then sharing is pointless.
If corporations build them, they would have to decide what to contribute to the commons and what to keep private. But that’s no different than any other language.
Re: What does the end of mathematics look like?
#49I think people like author are positive about us, humanity, being able to build AI or something being very close to that.
I am not.
From the energy efficiency perspective human brain is very, very effective computational machine. Computers are not. Thinking about scale of infrastructure of network of computers being able to achieve similar capabilities and its energy consumption... it would be enormous. With big infrastructure comes high need of maintenance. This is costly and requires a lot of people just to prevent it from breaking down. With a lot of people being in one place, there socioeconomical cost, production, transportation needs to be build around such center. If you have centralized system, you are prone to attack from adversaries. In short I do not think we even close to what author is afraid of. We just closer to beginning to understand what is the need to actually start to think about building AI - if ever possible at all.
Re: What does the end of mathematics look like?
#50Earlier quoted context omitted.
Presumably an AI would formalise the proof in a system such as Lean, then you only need to trust the kernel of that proof system. Rejecting a proof would be more complicated, because while for confirming a proof you only need to check that the main statement in the formalisation matches that of the conjecture, showing that a proof has been rejected requires knowledge of the proof itself (in general).
> requires knowledge of the proof itself (in general) Why? If a proof is wrong it has to be locally invalid, i.e. draw some inference which is invalid according to rules of logic. Of course the antecedent could have been defined pages earlier, but in and of itself the error must be local, right?
Human-written proofs are not written in Lean to be checked easily and there'll be potentially many formalizations for written prose and only some of them will be what the author intended. You need to pick the right formalization before you can say that this proof has local errors.