Does anyone have a rough estimate of how many research mathematicians there are in the U.S. or world?
A Beginning for Mathematics
101–110 of 110 posts
Re: A Beginning for Mathematics
#102As someone who has a degree in math, I still can't help but think mathematicians are getting a little bit of a comeuppance. In a lot of areas of mathematics there had been little effort to make the work understandable and leaves numerous folks who could benefit from the knowledge on the outside looking in. Now AI comes along and do the same to mathematicians. Makes me chuckle a little bit.
Kind of makes me think of Arnold's, "On Teaching Mathematics": https://www.maths.tcd.ie/pub/Maths/Courseware/ProblemSolving...
> Mathematics is a part of physics
Re: A Beginning for Mathematics
#103As someone who has a degree in math, I still can't help but think mathematicians are getting a little bit of a comeuppance. In a lot of areas of mathematics there had been little effort to make the work understandable and leaves numerous folks who could benefit from the knowledge on the outside looking in. Now AI comes along and do the same to mathematicians. Makes me chuckle a little bit.
Not everyone needs to be a communicator.
Why should we expect everyone to be both a great researcher and communicator? The obvious result is that it is as effective as engineering managers. Sure, there's some amazing ones, but most aren't. Though that also doesn't mean no expertise in the field (i.e. non-engineering manager) is any better. It is just that managing/communicating is a different and orthogonal skill.
What needs to happen is we need to make it okay for people to specialize in more things. More nuance to this rather than trying to throw everyone into nice easy to manage buckets. Those buckets are just unrealistic abstractions filled with hope, denial, and laziness. Reality is surprisingly complex. Math can do a really good job helping you understand that, but it's a sufficient condition, not a necessary one
Re: A Beginning for Mathematics
#104Excellent optimistic post in a sea of negativity, and with actual suggestions, too. After reading, my mental image is this: think of Olympiads in Ancient Greece. * A weightlifter was only awarded a laureate if he were able to lift a heavy stone (have no idea what they were lifting, for illustrative purposes only :-) * Along comes Archimedes who invents what we would call an exoskeleton. Now any regular guy can lift t…
I don´t think so. For programming agents can run code, check compiler output, etc. For mathematics, it is almost the same once you factor in the usage of lean. For the reality, you can´t close the loop that fast, or with that precision. You will have to slow down by several orders of magnitude.
Doesn’t have to. There are petabytes of experimental physics data that can be fed to AI to extract additional insights. The only holdup is this is slightly harder to than math.
With bio, you’re right, generally designing and conducting an experiment goes hand in hand and theoretical biologist is not a common label.
Re: A Beginning for Mathematics
#105Earlier quoted context omitted.
Gödel effectively says ZFC must be incomplete, otherwise it would not be sound, but does that stop you from listing all mathematical propositions it can generate in some well-defined order?
That's right. You could list all those propositions and search for proofs of them. In fact this is very similar to Hilbert's very program to which Godel's First Incompleteness Thm was a response ( https://en.wikipedia.org/wiki/Hilbert%27s_program ). Godel showed that there are true statements that cannot be proven, and also that among the unprovable statements from within the system is the consistency of the system i…
People always bring up the Continuum Hypothesis as though it has something to do with Gödel's incompleteness theorems, but it doesn't really. The Continuum Hypothesis being neither proven nor disproven by some particular axioms is a similar phenomenon as that the group axioms neither prove nor disprove commutativity, the ordered field axioms neither prove nor disprove the existence of a square root of 2, etc. There's no particular reason to expect any particular formal system to be complete, sans some demonstration that is.
Gödelian incompleteness is the specific kind established by Gödel's proof, where theory T can't prove Con(T) without being inconsistent. But the inability of ZFC to consistently decide the Continuum Hypothesis is established in a completely different manner, with the Continuum Hypothesis not consistently decided by ZFC + Con(ZFC) either, or any such thing.
Re: A Beginning for Mathematics
#106The author argues for evaluating Ph.D. candidates based more on the oral thesis defense than on the actual thesis. By essentially the same reasoning, I’ve been arguing for prioritizing in-person design/code reviews over code-only async PR comments. The important thing is to verify that the human has a coherent design in mind and can demonstrate that it got implemented, regardless of who or what was at the keyboard. “…
I can see how this trend is going to go. Manager: “Why isn’t feature X available?” Person B: “key pieces are delayed due to the developer not understanding all of the LLM doesn’t and implementation.” Manager: “does it work? What are the risks?” Person B: “well yes it works for now but we’re accumulating tech debt due to a lack of understanding and potential flaws that haven’t been thought out yet” Manager: “they want…
The simulated dialog you have provided is very much representative of what many of us have heard first-hand. However, I disagree with:
Most businesses don’t care about later risk or any future
planning beyond the quarter horizon, they’re not concerned
about how it will effect their performance in 3 quarters or
lead to instability or issues, those are future problems
for a future person and we’re here for money now.
Individuals within an organization do not unless it will specifically affect their bonuses/promotions.
Re: A Beginning for Mathematics
#107Earlier quoted context omitted.
Kind of makes me think of Arnold's, "On Teaching Mathematics": https://www.maths.tcd.ie/pub/Maths/Courseware/ProblemSolving...
> Mathematics is a part of physics This feels backwards. I frequently joke "physics is the subset of mathematics that reflects the observable world". Math can be as abstract as it wants, but physics has a constraint. It must model an observable world (related, this us part of why people say String Theory is math and not physics)
What about the next 3,900 words after the first sentence?
Re: A Beginning for Mathematics
#108I’m not a mathematician, but if I were I would imagine the main attraction would be the act of thinking hard and solving problems. Maths felt like the ultimate profession for someone who loved puzzle solving. If you delegate the thinking, any percent of it to an LLM, I don’t think it’s maths anymore. Might as well join any other job and earn better money now. A completely ludicrous proposition, that maths will no longer involve mathematical problem solving.
Re: A Beginning for Mathematics
#109> I propose the following reconceptualization of the goal of a mathematics PhD: to become a world expert on some interesting, deep topic, and to be able to convey that interest and understanding to others. Part of operationalizing this might be a thesis, but the degree would be awarded primarily on the basis of a rigorous defense, in which the student explains the topic to their examiners until they are satisfied. I…
> a refreshingly forward looking idea What? "a rigorous defense, in which the student explains the topic to their examiners until they are satisfied." is exactly how PhDs were awarded for hundreds of years. Even my BSc in Applied Physics (1977) had a viva voce that was a substantial fraction of the final exam.
I am not implying that this is a new idea and I don't believe the author did either. If you read the full article, the context is clear.
The author is re-emphasizing the importance of thesis defense old school style and highlighting that being familiar with one's own material rigorously should still be a requirement.
Everything that leads up to the thesis defense perhaps changes signficantly.
Re: A Beginning for Mathematics
#110My worry is that the frontier of math is too far away for most humans to reach. AI is only going to make that worse. If today it takes twenty years of math study to reach the frontier (in a narrow field), what's it going to be like when it takes forty years? Or four-hundred years? Will the fields just get narrower and narrower to accommodate finite human intelligence?