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What's a mathematician to do? (2010)

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Re: What's a mathematician to do? (2010)

#31

Post by dennis_jeeves2 »

nimonian wrote: post #21
bloaf wrote: post #16

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 b…

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!

>are usually discovered "for their own sake"

Like prime numbers? (used in cryptography)

Re: What's a mathematician to do? (2010)

#32

Post by skybrian »

nimonian wrote: post #21
bloaf wrote: post #16

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 b…

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!

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.

Re: What's a mathematician to do? (2010)

#33

Post by transitorykris »

nimonian wrote: post #21
bloaf wrote: post #16

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 b…

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!

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.

Re: What's a mathematician to do? (2010)

#34

Post by [deleted] »

r_lee wrote: post #25
DoctorOetker wrote: post #17

Earlier quoted context omitted.

Those are very strong claims, do you really believe an LLM can't be trained to solve Rubik's Cubes? Can you imagine what if feels like to be a LLM? Can one LLM have a better sensation of what it feels like to be a different LLM (say one that score a little better?)? You design circularly defined criteria...

honestly I'm pretty sure opus could solve a rubiks cube if you just gave it the layout if the sides and looped until it would solve it or even just take a picture of the thing, since they can digest visual input now

[deleted]

Re: What's a mathematician to do? (2010)

#35

Post by sdenton4 »

4qshT wrote: post #27
Schlagbohrer wrote: post #2

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 b…

The Mathoverflow question was asked 15 years ago. The top answer says that the human community part is very important and spreading knowledge an critical thinking is valuable. The most recent advances are stunts by a handful of famous prompters who are funded in various ways by the LLM industrial complex. How many theorems are proven by mathematicians each year? Let's guess 10000. Then the Erdos toy proofs with unkno…

...And in 1900, how many carriages were horseless?

Re: What's a mathematician to do? (2010)

#36

Post by photochemsyn »

Today, even understanding what new mathematics is being done in a particular zone of the mathematical universe appears to require a four-year graduate program of constant study just to be able to follow some mathematician’s original work - and that’s only going to give you a window into a rather narrow subsection of mathematics. The days when people of great talent like Euler and Gauss could contribute to many areas of mathematics are long gone.

But mere mortals can still derive great satisfaction from following along in the footsteps of past pioneers, possibly adapting their work to new problems in a minor way, or just creating educational visualizations and tools that help other people understand things like Galois theory, Poincare phase space or Markov chains, which can be applied to quantum mechanics, orbital dynamics, or protein sequence analysis. That’s valuable, even if no Fields Medals will be coming your way.

For the core discipline, though, I’d mostly worry about lack of opportunities for serious mathematicians to practice their craft in the USA due to the trends of academic budget cuts, anti-intellectual rhetoric, insistence on profit generation as the only rationale for doing anything, etc. Looks a bit 1930s Germany to me, at least here in the USA.

Re: What's a mathematician to do? (2010)

#37

Post by ppll_12 »

sdenton4 wrote: post #35
4qshT wrote: post #27

Earlier quoted context omitted.

The Mathoverflow question was asked 15 years ago. The top answer says that the human community part is very important and spreading knowledge an critical thinking is valuable. The most recent advances are stunts by a handful of famous prompters who are funded in various ways by the LLM industrial complex. How many theorems are proven by mathematicians each year? Let's guess 10000. Then the Erdos toy proofs with unkno…

...And in 1900, how many carriages were horseless?

In 2026, how many people X-ray their feet at the shoe store or have watches with radium paint?

Ironically, there is a shoe company pivoting to AI. My taxi driver told me buy the stock:

https://www.bbc.com/news/articles/c98mrepzgj7o

Re: What's a mathematician to do? (2010)

#39

Post by [deleted] »

r_lee wrote: post #25
DoctorOetker wrote: post #17

Earlier quoted context omitted.

Those are very strong claims, do you really believe an LLM can't be trained to solve Rubik's Cubes? Can you imagine what if feels like to be a LLM? Can one LLM have a better sensation of what it feels like to be a different LLM (say one that score a little better?)? You design circularly defined criteria...

honestly I'm pretty sure opus could solve a rubiks cube if you just gave it the layout if the sides and looped until it would solve it or even just take a picture of the thing, since they can digest visual input now

[deleted]

Re: What's a mathematician to do? (2010)

#40

Post by generic92034 »

morkalork wrote: post #22
generic92034 wrote: post #4

Earlier quoted context omitted.

I wonder if AI is one means to overcome the natural limits of human knowledge aggregation [0]. On the other hand, in the very long run, what does it mean if a talented human being does not have enough years of life to fully analyze and understand an extremely advanced proof created by AI? [0]: https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/

Perhaps it will become like those cathedrals that took centuries and many generations of humans to build.

Yes, but you (as a human) can still understand the cathedral (the building). This is not guaranteed for advanced AI work in mathematics in the future. If so, are we/they are really still adding to human knowledge, at this stage?
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