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What's a mathematician to do? (2010)

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

#41

Post by photochemsyn »

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…

If your motivation is being recognized as the best of the best, winning the competition, yes it’s probably a bleak world. But if you motivation is improving your own capabilities, with the metric being if you’re better know then you were last month, then it’s not a bleak world, there are many more tools available to help you learn and improve now then there were in the past.

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

#42

Post by bee_rider »

GuB-42 wrote: post #29

I think it is like for a programmer to ask "How can one contribute to computer science?", while thinking about people like, Dijkstra, Knuth, or maybe even Carmack. There are some geniuses who do groundbreaking work, but this wouldn't be of much use it it wasn't for the millions of people who do actual work with these theories (applied math), and teachers who train the next generation. In the academia, small discoveri…

I do think this has been less pressing of a question for programmers. For the longest time there was infinite work to do, no matter your depth—implementing business logic, making frameworks more general, making sure things fit into cache lines—that was accessible to us non-geniuses. Maybe in the LLM era this sort of t-crossing and i-dotting will go away.

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

#43

Post by vladislavp »

a) Individualized teaching methodology. We come with different backgrounds, therefore different types of analogies/examples, different levels of background material, different (but systematized) levels of presentation should be used. The same ask should be applied to kids learning through starting at preschool.

b) Readable mathematics papers where the compact notations are abandoned, and narrative, visualizations are introduced, while preciseness is maintained. It is possible that the same paper (or chapter or topic) should be renderable in multiple ways (for professional mathematicians in the field, for a casual reader, for a student, for an individual reader (as for (a) )

c) Mathematical logic / tooling for differentiable data/event computing. Where there are mathematical tools as well as CS implementation of this tools that allow to act on a difference in state, data, actions.

Typical mathematics (with exception of may be time series), does not view time as 'first class citizen' so to speak, be it abstract algebra and category theory or something else. But, I think, when we go to the 'applied world' we must introduce 'time dimension' as first class citizen. So having the mathematical machinery dealing with this dimension in organic way across many of the areas of mathematics -- will be beneficial to the application of this one of the most valuable human tools.

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

#44

Post by zozbot234 »

bee_rider wrote: post #42
GuB-42 wrote: post #29

I think it is like for a programmer to ask "How can one contribute to computer science?", while thinking about people like, Dijkstra, Knuth, or maybe even Carmack. There are some geniuses who do groundbreaking work, but this wouldn't be of much use it it wasn't for the millions of people who do actual work with these theories (applied math), and teachers who train the next generation. In the academia, small discoveri…

I do think this has been less pressing of a question for programmers. For the longest time there was infinite work to do, no matter your depth—implementing business logic, making frameworks more general, making sure things fit into cache lines—that was accessible to us non-geniuses. Maybe in the LLM era this sort of t-crossing and i-dotting will go away.

> Maybe in the LLM era this sort of t-crossing and i-dotting will go away.

On the contrary, prompting LLMs creates a whole lot of newly accessible basic work.

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

#45

Post by lisper »

> One can rewrite their books in modern language and notation or guide others to learn it too but I never believed this was the significant part of a mathematician work

There's yer problem right there. Good pedagogy is hard and highly undervalued. IMHO Grant Sanderson (a.k.a. 3blue1brown) is making some of the most significant contributions to math in all of human history by making very complex topics accessible to ordinary mortals. In so doing he addresses one of the most significant problems facing humankind: the growing gap between the technologically savvy and everyone else. That gap is the underlying cause of some very serious problems.

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

#46

Post by wcfrobert »

I think the answer is to do multi-disciplinary work.

Venture outside of pure theoretical math. Learn some other domain knowledge and combine it with your mathematical ommph. That's the easiest way to make an impact now rather than potentially decades later.

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

#47

Post by zozbot234 »

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, that was done already in the 1990s.

> Maybe by using LLMs as a mighty tool and providing skilled usage and oversight?

Yes, even in the areas where LLMs are at their best, we'll still need a lot of human effort to make the results cleanly understandable. LLMs cannot do this well, even their generated papers have to be rewritten by human experts to surface the important bits.

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

#48

Post by ajs1998 »

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…

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.

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

#49

Post by Lerc »

GuB-42 wrote: post #29

I think it is like for a programmer to ask "How can one contribute to computer science?", while thinking about people like, Dijkstra, Knuth, or maybe even Carmack. There are some geniuses who do groundbreaking work, but this wouldn't be of much use it it wasn't for the millions of people who do actual work with these theories (applied math), and teachers who train the next generation. In the academia, small discoveri…

>I think it is like for a programmer to ask "How can one contribute to computer science?", while thinking about people like, Dijkstra, Knuth, or maybe even Carmack.

I had a conversation a day ago with a couple of high-school students who, were obviously smart (and a bit on the spectrum), but also lacking in broad knowledge, as one would expect from high-school students.

I think it revealed something about that sense of 'magic' or inaccessible talent. One of them mentioned the fast inverse square root function and marvelled about how even anyone could even come up with an idea like that because it seemed to him to be some transcendent feat to be able to realise casting binary floats into ints would be useful. They had looked at the function and couldn't fathom how something like that worked.

We had no computer to hand, but I asked him what 10^100 divided by 10^10 was. Smart kid that he was, answered instantly, correctly. I then asked him what operation he performed in his head to do that, and noted floating point exponents are just using 2 instead of 10. On the spot he figured out how the fast inverse square root worked. The magic vanished and it became accessible. Some people lament the loss of 'magic' in this way, but I think the thing that makes it special, in this instance and in the universe in general, is that it still works, and it isn't magic. It's real and the fact that it be that without invoking some unaccessible property makes it even more special

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

#50

Post by LPisGood »

lisper wrote: post #45

> One can rewrite their books in modern language and notation or guide others to learn it too but I never believed this was the significant part of a mathematician work There's yer problem right there. Good pedagogy is hard and highly undervalued. IMHO Grant Sanderson (a.k.a. 3blue1brown) is making some of the most significant contributions to math in all of human history by making very complex topics accessible to o…

Indeed, pedagogy is important to staving off the end of mathematics.

That sounds dramatic, but it’s really obvious if you think about it. Right now, a person has to study for about 20 years (on average) to make novel contributions in mathematics. They have to learn what’s come before, the techniques, the results, etc. If mathematics continues, eventually it could take 25 years, or 30 years, or even a whole lifetime. At some point, most people will not be able to understand the work that’s been done in any subfield (or the work required to understand a subfield) in a human’s life. I claim this is the logical end of mathematics, at least as a human endeavor.

Now, there will be some results which refine other work and simplify results, but being able to teach a rapidly growing body of literature efficiently will be important to stave off the end of mathematics.

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