Comment by aurareturn
1 day ago
Humans only invest in solving problems that matter one way or another.
I also disagree that none of them solve "real" problems. They clearly do. Solving them have implications on real world problems.
1 day ago
Humans only invest in solving problems that matter one way or another.
I also disagree that none of them solve "real" problems. They clearly do. Solving them have implications on real world problems.
If we are talking about pure/theoretical mathematics, then the vast majority of the problems people pose and solve have at best tangential relationship with applications, and a big part even is only related to other math problems. Of course quite a bit of mathematics historically emerged as this kind of intellectual endeavour to find applications later, but there is neither a way to predict which ones are that and how to get them, nor is there indication of this thing going on to the same proportion nowadays as it was, considering the mathematical production is much higher. In mathematics human mathematicians have to decide which problems matter, it does not come from somewhere.
Solving "real" problems in theoretical mathematics (as in problems directly related to applications) is a very small proportion compared to the vast majority of math work that does not. So if we are discussing about the future of mathematics as a field, we have to understand what the field of theoretical mathematics is actually about.
So my question is this:
Would this slow technological progress? Or make it go faster?
That math problems are found and solved as we run into real physical problems.
"The FT’s Gillian Tett reported that a senior financier’s New York firm now seeks out humanities students, because “AI-native” Stem graduates are entering the job market with “alarmingly shallow ideas”."*
I don't advocate for the dichotomy of stem and humanities. A good counterecample from the 20th century being Ernst Mach (Mach-speeds are named after him) and his work in phenomenology ("bodies do not produce sensations, sensations produce bodies")
Your incatation of contextless 'technological progress' still kinda calls for a quote like the above
*https://www.theguardian.com/books/ng-interactive/2026/aug/08...
Technological progress is not bottlenecked by most of the millennium prize problems or erdos problems per se, or most of the rest open problems in theoretical math, ie that merely knowing the solution of them will help applications in some manner. I doubt the solution of such problems has any direct effect on technology progress at all, at least in any deterministic, foreseeable manner.
In fact, the relationship between theoretical mathematics and "real physical problems" is bidirectional, as in "real physical problems" informs to some degree some problems that may be interesting to research on in theoretical math, and at the same time pure mathematical research that is developed completely independent may find applications at some point. And even theoretical mathematicians working close to applications are mostly dealing with problems not directly addressing applications. Eg maybe they study properties of a certain function that arises often in application without any view to solving a specific "real physical problem" with it, and somebody after may find that useful for some application after some point, but that could be one out of 50 papers (random number) and it is hard to predict that. There is of course some work more related to specific real problems, but that's most often not what theoretical math is about, and not what these new developments with erdos problems, navier stokes etc are about.
So what could (in a chaotic sense) have effect in application is mathematical theories developed along the way of solving these pure math problems, which brings us back to the question of what happens if we remove this friction and if AI can do more than construct examples and proofs, ie actually build theories (autonomously or humans+AI). If anything, it is through building theories that mathematical progress germinates applied sciences, as this is the process that develops mathematical tools that can be taken up later, including whole mathematical fields. Building mathematical theories is a heavily social process, and it is the community that basically decides which directions are important to follow.
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>Humans only invest in solving problems that matter one way or another.
Fermat's Last Theorem was one of the most famous open problems in math for centuries, and it has no direct applicability to any tangible problems here in the physical world.
Oh some few. No right triangle with rational sides has area equal to a perfect square depends on N=4 for instance. And it can be used to form other theorems that are terribly actionable. Every elliptical curve over Q is modular, which has consequences throughout number theory.
But yes, none of those are very tangible, until applied to problem solutions that are tangible.
What real-world consequences are implied by a solution to Navier-Stokes?
That depends on the solution, right?
Not really. I can't imagine a realistic solution that would affect, say, how we actually model a real fluid. Hypothetically we could find out that a whole class of real fluids are not modelled by Navier-Stokes at all, but I don't think that's remotely likely, nobody familiar with the problem expects it.