What's a Mathematician to Do?

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.

“Comparison is the thief of joy.”

Do the math because you enjoy doing the math and if you do it long enough you may well do something of value to someone else. Same goes for most intellectual and artistic pursuits I think.

I’ve learned for myself that as soon as enjoyment is based on some future achievement or ranking my work against others the day to day satisfaction dries up.

From one of the answers:

> mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification.

Yes! And this applies to all human culture, not just math. Everything people have figured out needs to be in living form to carried on. The more people the better. If math, or any product of human skill, is only recorded in papers or videos, that isn't the same as having millions of people understanding it in their own ways.

Modern culture often emphasizes innovation and fails to value mere maintenance, tradition, and upkeep. This can lead to people like the OP feeling that they have nothing to contribute, when actually, just learning math, being able to do it, being able to help others learn it - all of these are contributions.

We are all needed to keep civilization afloat, in ways we cannot anticipate. We all need to pursue some kind of excellence just to keep human culture alive.

After reading another post about the most recent advances LLMs have made in finding and writing up novel, correct proofs, it sounds like the frontier models are now at the point of PhD student level. I wonder how a math student could contribute today, if they're just starting on the PhD track? Maybe by using LLMs as a mighty tool and providing skilled usage and oversight?

It must feel similar to those who wanted to become chess or go masters after computers surpassed humanity in those games.

If we see our contributions as brownian motion rather than preconceived trajectories, then, rather than focusing on the Gausses, Einsteins, Patons as providing singular progress, they become the the dominant least energy paths to what we recognize as truth. Without negating the individual’s contribution, the ones we see as truly important are the ones that supported by every other’s attempt, finds the path forward. This should provide hope, if we can leave aside our egos and focus on humanity, we can, and do, all contribute even though a few seems to get all the credit.

This also goes for AI, it may be an accelerant in research, but the probability distribution of reality is large, large enough for humans to wonder, ask questions and stumble upon a new path forward, that computers alone don’t find.

Fortunately doing something novel is one of the main things llms can't do.

But unfortunately human knowledge accumulation and advancement over the last many thousand years has been pretty large deep and varied.

Finding something novel for phds or profits or crime or whatever th fk is harder everyday.

At the very foundation, chaining sentences together is what we call logic.

Chaining unrelated sentences is retarded. Chaining sentences like most people is common sense. Chaining sentences airtight is math.

You ask what a true mathematician does. He chains sentences like everyone else but with an effort to make them airtight.