There still seems to be a difference between useless pure math research and useless science or useless philosophy. Science, even useless science, still has a subject matter that is relevant to us independently of science, the real world. And philosophy studies concepts (like "knowledge") that occur in natural language and thought, and those concepts are relevant to us independently of philosophy. But pure math is entirely self-referential. Pure math abstractions are used only in pure math. Pure mathematics is relevant exactly to pure mathematics and those who study it.
I doubt that pure mathematics can claim the success of cryptography. For example, no result from advanced mathematics is required to know that factoring the product of two large prime numbers is slow.
The fact that mathematicians cannot devise a method to factor that large product, despite large amounts of money and research time, is what’s valuable!
But even then your argument breaks down when you consider more complicated cryptographic structures.
> The fact that mathematicians cannot devise a method to factor that large product, despite large amounts of money and research time, is what’s valuable!
Well, programmers couldn't devise such a method either. You seem to assume that mathematicians are better than them at finding efficient algorithms, which isn't very plausible.
> But even then your argument breaks down when you consider more complicated cryptographic structures.
Even if help of mathematics is needed there, applied — rather than pure — math might still be sufficient.
"useless" pure math often turns out to have scientific applications.
And surely you don't mean all pure math, so your argument ends up being circular --- useless math is useless. You happen to think that this math is useless, but you might be, or turn out to be, wrong.
Also, you're misusing the term "self-referential". You seem to mean that it's a closed system ... but it's not, since mathematicians interact with it.
Finally: so what? We do all sorts of enjoyable activities with no benefit other than the enjoyment. Solving Erdős problems seems at least as justifiable as finding the trillionth digit of pi or writing an Apple ][ emulator in Brainfuck or playing those video games that you demonize by calling them addictive. (Compare to, say, compulsively reading Britannica's The Great Books of the Western World ... it's snooty judgments all the way down.)
Postscript to finally: This framework has wider application ... it's not limited to Erdős problems or anything else that someone happens to consider to be useless.
> "useless" pure math often turns out to have scientific applications.
I don't think that's true for a reasonable interpretation of "often". I'm pretty sure the vast majority of pure math research is and remains useless.
> Also, you're misusing the term "self-referential". You seem to mean that it's a closed system ... but it's not, since mathematicians interact with it.
Then which better term do you propose? The point was that its subject matter lies within itself, which is very different from science and philosophy.
> so your argument ends up being circular --- useless math is useless.
It's not circular: You can replace "pure" with "useless" and the argument stays the same. I contrasted useless math with useless science and useless philosophy for a reason.
> Finally: so what?
I'll just point out that "so what" is not a counterargument. If you agree with my point but find it unimportant: that's fine with me.
> Solving Erdős problems seems at least as justifiable as finding the trillionth digit of pi or writing an Apple ][ emulator in Brainfuck or playing those video games that you demonize by calling them addictive.
The difference between achieving a new record for a video game, and pure math research, is that nobody is confused about what the former is: It's a game, or a sport. Speed runners or chess champions or pi digit calculators don't confuse themselves with noble researchers advancing the frontier of human knowledge.
