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What's a mathematician to do? (2010)

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

#71

Post by pegasus »

rdevilla wrote: post #8
getnormality wrote: post #5

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 appli…

> Everything people have figured out needs to be in living form to carried on. It would appear that LLMs are invalidating this claim. Things can live in synthetic form and carry on just fine. Instead of cultivating a population of learned minds we are just feeding a few dozen egregores of models and training corpuses.

They are not invalidating this claim, and cannot, unless we'd actually try it out for a few generations. Which we shouldn't and won't.

LLMs are quite good at simulating life and living intelligence (in the short term), but they aren't any of that. That's why we call it artificial intelligence. It's true that we can't put our finger on what exactly the difference is, but it's not like reality has ever felt encumbered by our limited understanding.

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

#72

Post by DroneBetter »

zozbot234 wrote: post #47
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…

> 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. This is somewhat misleading, the LLMs' contributions are in a limited niche of highly technical problem solving. They're neat but they're not the first mathematical theorem that gets automatically solved by a computer,…

> done already in the 1990s by human-written programs that iterated through the finite casework that human thought had reduced the theorems to (four-colour theorem, FLT, etc.), which recent developments (eg. LLMs autonomously resolving Erdős problems) seem meaningfully distinct from. > human effort to make the results cleanly understandable well, perhaps loops of "derive proof through reasoning in English, formalise in Lean, use AST size of formal proof as a metric to optimise (via an LLM-guided search), translate back into English" could improve this? a lot of resources are being spent to make frontier LLMs more resistant to hallucinations via Lean, perhaps cogency will increase as a byproduct.

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

#73

Post by whattheheckheck »

getnormality wrote: post #5

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 appli…

But were trapped in only keeping alive that has a reward signal which is can it help pay rent, pay for food, get love, etc

Theres a limited amount of time, space and energy so what's the ideal mechanism to say what to pay attention to or not

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

#74

Post by ashley95 »

I think we also have to be honest and admit that, yes, indeed, there is less novel maths for all of us to be doing. The pioneers came first and discovered a lot of low hanging fruit. There were a lot of geniuses that mined the rest and reached higher in the tree. Now even the smartest mathematicians are left solving abstract puzzles with little utility in the real world. (Don't get me wrong, it's very fun, and sometimes useful too.)

After my PhD in applied mathematics, I decided to leave the field, partly because I feel it really has advanced so far that new discoveries do little to move the needle in the real world. There's enough smart people who obsess over nothing else but maths that I can go and do more practical stuff...

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

#75

Post by _alternator_ »

ashley95 wrote: post #74

I think we also have to be honest and admit that, yes, indeed, there is less novel maths for all of us to be doing. The pioneers came first and discovered a lot of low hanging fruit. There were a lot of geniuses that mined the rest and reached higher in the tree. Now even the smartest mathematicians are left solving abstract puzzles with little utility in the real world. (Don't get me wrong, it's very fun, and someti…

Is this not just a perspective issue? The fields we learn about in school are the ones that are in some sense mature, that have been cashed.

When I was in grad school, we learned about wavelets, but we did research on convex optimization for statistics. The first was an accomplishment of the last generation of mathematicians, and it would be hard to publish something groundbreaking. But nobody had really considered sparsity inducing optimization, so that was our problem.

In many ways, the situation is somewhat better for applied mathematicians because the problem space is wider. Ingrid Daubechies was an applied mathematician, and her work on wavelets was an outgrowth of work that originated in the petroleum exploration community. Wild how these connections get made.

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

#76

Post by whatever1 »

thaumasiotes wrote: post #60
ajs1998 wrote: post #48

Earlier quoted context omitted.

This is why I think Brady Haran is one of the coolest living mathematicians. Numberphile is educating a new generation of young mathematicians for anyone with access to youtube. Accessible math communication is so important. So many cool things are buried in textbooks and papers the average person would never read.

Numberphile doesn't do any education. That's like saying the Discovery Channel is educating a new generation of zoologists.

I believe that I have learned cumulatively double in the past 10 years from YouTube compared to what I learned in 6 years of middle and high school. And I don’t spend 8 hours per day on YouTube.

Plumbing, react, combinatorics, real analysis, python, c++, cad, micro and macro economics, reinforcement learning, to name just a few of the things I learned through YouTube.

We don’t give enough credit to what we take for granted today.

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

#78

Post by beyonddream »

I am glad to see this today - after reading Tim gower's recent post on chatgpt 5.5 pro's phd level research ability I was feeling slightly sad about the future of math research.

Interestingly enough, the moment I saw the title I thought of Bill Thurston's famous article "On proof and progress in mathematics" and the top comment on the OP's thread is from him! Reading his reply sort of gave me the antidote to the temporary blues I felt yesterday.

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

#79

Post by sendes »

netcan wrote: post #18
getnormality wrote: post #5

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 appli…

Living culture is a concept that I think is quite unintuitive to modern minds. Examples of it are all around us... but it's usually blatantly missing from our "big picture" thinking. For example. Take a modern country with a modern economy. Flatten it. Destroy all the factories. Bankrupt all the companies. You can get back to a fully modern economy again quite quickly. WWII demonstrates it. Taking an unindustrialized…

> I think the "living community" thing is the answer to this. It' ecology.

I agree. A body of knowledge, mathematical knowledge being one of them, is a give-and-take between its producers and consumers; a market for ideas. It grows in that ecology of people with its pathfinders, specialists, generalists, historians, educators, etc. Committing to a body of knowledge is becoming part of its living culture.

Where I disagree: I believe some of the loss is inevitable. Keeping in mind the example of a body of knowledge above, as the scale of what's accumulated until now grows, the role each of us play in the sustenance of its culture shrink. This is a direct consequence of the modern developmental process (ie division of labour to the point where it feels like we are all modular parts of a much larger whole).

I can't say whether its better to focus on recovering what's lost, or, trust the process, as it were.

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

#80

Post by fasterik »

beyonddream wrote: post #78

I am glad to see this today - after reading Tim gower's recent post on chatgpt 5.5 pro's phd level research ability I was feeling slightly sad about the future of math research. Interestingly enough, the moment I saw the title I thought of Bill Thurston's famous article "On proof and progress in mathematics" and the top comment on the OP's thread is from him! Reading his reply sort of gave me the antidote to the temp…

Why did it make you sad? I can see why it would make training researchers harder, but once someone attains a postdoc-level skill in mathematics research, wouldn't having a PhD-level AI assistant just boost one's ability to do more ambitious research?
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