[bloaf]

So I've got a gut feeling that math (like human languages (like programming languages)) is best learned in service of some greater end.

I look at some truly impressive projects like CLASP which sprang into existence not because of someone noodling around, but because they had a bigger goal which required the team build it.

So my advice to any mathematician who feels lost, like they don't know what to work on, would be to go collaborate with someone who has an actual goal, to look for inspiration in the kinds of math they need.

Today, there are a lot of opportunities to jump forward that only get capitalized on through coincidence (e.g. two people bump into each other at a conference, or researcher happens to have a colleague working on a related problem through the lens of a different discipline). If AI does nothing but guarantee that everyone will have such a coincidence by serving as that expert from a different discipline, that will still be a massive driving force for progress.

The question of "whats a mathematician to do" is still clear: you need to find and curate and clearly express interesting and valuable problems.

[nimonian]

It's a delightful counterintuition that your gut feeling is mostly wrong: https://webhomes.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf

Far from being motivated by some applications, the most useful discoveries in mathematics are usually discovered "for their own sake" and their application is only discovered later. Sometimes centuries later!

[skybrian]

If so that seems like an opportunity for people who want to work on applied math? There’s a big backlog of techniques that so far have not been useful.

[mswphd]

I've seen this floated as a response to the current anxieties over LLMs in math. Namely in applied math, LLMs being good at pure math may actually allow the import of pure math techniques. Unclear if that will pan out, but it's interesting to consider.

[kcexn]

Absolutely! The backlog is enormous though, and much of mathematics requires a great deal of work to understand it to the depth required before a novel application becomes apparent.

[transitorykris]

Parents reads as a comment on the usefulness of applying mathematics to problems in the world (applied mathematics) and discovering mathematical problems that push mathematics forward (pure mathematics) in the process. Pure mathematics is incredibly important, but I’d hardly count it as useful if we need to wait centuries.

[aleph_minus_one]

> but I’d hardly count it as useful if we need to wait centuries.

This is not the fault of the mathematicians.

[dennis_jeeves2]

>are usually discovered "for their own sake"

Like prime numbers? (used in cryptography)

[jvans]

Lots of fun counter examples to this. Complex numbers were introduced in the 1600s with no practical application for almost 300 years until they were used in electromagnetism and quantum mechanics.

[lmm]

> Complex numbers were introduced in the 1600s with no practical application for almost 300 years.

On the contrary, complex numbers were introduced to make the cubic formula work.