> “Mathematics is not just about proving theorems — it’s about a way to interact with reality, maybe.”
This one I like it because in the current trend of trying to achieve theorem proving in AI only looking at formal systems, people rarely mention this.
And this one:
> Just what will emerge from those explorations is hard to foretell. “That’s the problem with originality,” Granville said. But “he’s definitely got something pretty cool.”
When has that been a "problem" with originality? Hahah but I understand what he means.
> another asks whether there are infinitely many pairs of primes that differ by only 2, such as 11 and 13
Is it that many questions were successfully dis/proved and so were left with some that seem arbitrary? Or is there something special about the particular questions mathematicians focus on that a layperson has no hope of appreciating?
My best guess is questions like the one above may not have any immediate utility, but could at any time (for hundreds of years) become vital to solving some important problem that generates huge value through its applications.
So we do know that there are 100,000,000,000! primes that are equidistant from one another, which is neat.
You can find relationships between ideas or topics that are seemingly unrelated, for instance, even perfect numbers and Mersenne primes have a 1:1 mapping and therefore they're logically equivalent and a proof that either set is either infinite or finite is sufficient to prove the other's relationship with infinity. There's little to no intuitive relationship between these ideas, but the fact that they're linked is somewhat humbling - a fun quirk in the fabric of the universe, if you will.
G.H.Hard. Eureka, issue 3, Jan 1940
Knowing that history illuminates the context and importance of problems like the above; but it makes for a long, taxing and sometimes boring read for the unmotivated, unlike the sexy "quantum blockchain intelligence" blurbs or "grand unification of mathematics" silliness in pure math. So, few popularizations care to trace the historical trail from the basics to the state of the art.
If you are fascinated by primes, then you just want to know the answer, independent of any application.
If anybody has a learning path or a primer recommendation for modular forms (assuming formal math education), I'd be very interested in reading more about it.
But as such I doubt it can be more accessible to everybody.
Diamond, Shurman, A First Course in Modular Forms, Springer GTM vol. 228.
Next, and perhaps I shouldn't be suggesting it, Serre's A Course in Arithmetic. Despite it's reputation of terseness it is a great book by one of history's great mathematicians and worth every sweat and tear spent over its short number of pages.
Yet another way is to see the lectures by Richard Borcherds (who won the fields for the moonshine conjecture) on youtube.
Finally, since this hacker news check out William Stein's Modular Forms: A Computational Approach (pdf online)