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A Beginning for Mathematics

daniellitt.com

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Re: A Beginning for Mathematics

#71

Post by mathisfun123 »

jplusequalt wrote: post #56
mathisfun123 wrote: post #45

Earlier quoted context omitted.

this comparison makes little sense. > we lost the joy of the journey and only fixated at the destination. that's because you can make money with code and so it went the same way as everything else that's capitalized/commodified. here a better comparison would be art which is generally still pursued on its own merits/pleasures.

>here a better comparison would be art which is generally still pursued on its own merits/pleasures. What about the crafts? Carpentry, blacksmithing, stonesmithing, etc. are all commodified yet people still pursue these for pleasure. >this comparison makes little sense. Stop gatekeeping what people find personally fulfilling.

> Carpentry, blacksmithing, stonesmithing

they're literaly not - do you know what the word artisanal means?

> Stop gatekeeping what people find personally fulfilling

wut? i can't even parse this in relation to what i wrote (which is that people who enjoy math will continue to be able to).

Re: A Beginning for Mathematics

#72

Post by bobmarleybiceps »

bonoboTP wrote: post #60

> We already interview faculty hires; we must now do the same for graduate admissions They hire PhD students without hearing them give a talk and then doing interviews? In Germany, the applicant gives a talk (30-40 min) to the research group they want to join, usually presenting their master's thesis, engage in discussion, often share lunch with the group, then do 1 on 1s with individual members of the group and a lo…

1. US a lot of international PhD applicants, so traveling before even being accepted is difficult, and 2. lots of people don't do a masters.

when I applied to PhD programs (not in math) it was basically CV + personal statement + recommendation letters + short chats with interested faculty :shrug: Maybe it was because my CV was "strong" but the chats were more see if interests were aligned, rather than actually interviewing me.

Re: A Beginning for Mathematics

#73

Post by jsrozner »

anyfoo wrote: post #66
jsrozner wrote: post #65

"A computer or monkey could easily start at the axioms of ZFC and iteratively apply deduction rules....simply conjecture all mathematical propositions in alphabetical order...The prospect of automating mathematics by enumerating all conjectures, and all proofs of ZFC, is probably not so disturbing to you." I thought we were going to get at least some brief comment on Godel here?

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 itself.

As I understand, mathematicians are still trying to figure out how much this matters. One of the best examples of its mattering is may be the Continuum Hypothesis: CH is consistent with ZFC, and ~CH (not CH) is also consistent with ZFC. In other words, you have enough flexibility in constructing your ZFC world such that in some ZFC-consistent worlds CH is true, and in others CH is false.

Litt's statement is not wrong; it's just that what he wrote sounds so much like Hilbert's program, that I'm surprised we didn't get some even minor comment on what kinds of truths we could reach if we embarked on such an effort.

Re: A Beginning for Mathematics

#74

Post by bonoboTP »

bobmarleybiceps wrote: post #72
bonoboTP wrote: post #60

> We already interview faculty hires; we must now do the same for graduate admissions They hire PhD students without hearing them give a talk and then doing interviews? In Germany, the applicant gives a talk (30-40 min) to the research group they want to join, usually presenting their master's thesis, engage in discussion, often share lunch with the group, then do 1 on 1s with individual members of the group and a lo…

1. US a lot of international PhD applicants, so traveling before even being accepted is difficult, and 2. lots of people don't do a masters. when I applied to PhD programs (not in math) it was basically CV + personal statement + recommendation letters + short chats with interested faculty :shrug: Maybe it was because my CV was "strong" but the chats were more see if interests were aligned, rather than actually interv…

Remote talks are also a common alternative in such cases. Zoom etc.

I also know that another major difference is that American universities tend to hire without a professors involvement, into a generic "program", then the PhD student seeks an advisor after being accepted, so applicants have to woo some unconnected committee pursuing various goals misaligned from the PIs instead of convincing the PI. In much of Europe a professor basically "owns" a chair and really is boss and decides hiring pretty much alone.

Anyways, if they don't have a masters' yet, they could present their bachelor thesis. But I already think it's a bad model to combine the masters (courses) and the phd (research) into this American hybrid that's the direct-bachelor-to-PhD jump, but that is somewhat unrelated.

Re: A Beginning for Mathematics

#75

Post by jsrozner »

bonoboTP wrote: post #60

> We already interview faculty hires; we must now do the same for graduate admissions They hire PhD students without hearing them give a talk and then doing interviews? In Germany, the applicant gives a talk (30-40 min) to the research group they want to join, usually presenting their master's thesis, engage in discussion, often share lunch with the group, then do 1 on 1s with individual members of the group and a lo…

Unlike Europe, in the US, most PhD applicants do not have a masters. 10 years ago, most had not even conducted substantial senior projects (e.g a semester or year). Though increasingly senior projects / undergraduate "theses" have become more common.

Re: A Beginning for Mathematics

#77

Post by JW_00000 »

ComplexSystems wrote: post #58

I admit I just don't understand this attitude. People had the same complaints that the code produced by early coding models was messy, lazy, poorly commented, had terrible architecture and so forth. The central complaint was that it was just too difficult for humans to review. The answer is just to improve the models and move on. Similarly now we're getting AI doing math. The proofs compile but are a mess. So just ma…

There's a fundamental difference between the goal of code and math. The goal of code is to produce software that does something useful. As long as the code does what it's supposed to do, arguably, it's good to ship. (As you imply, we want the code to be good enough to also be reasonably certain there are not too many bugs, that it is maintainable and can be extended etc. This is what early models failed at but now seems broadly fine.)

But for math: what is the point of a proof if no one will read it and no one uses its result? To quote the article, "AI systems will [...] result in the production of an abundance of PDFs. The contents of some of those PDFs may even have important applications." But if there's no one reading the PDFs, what's the point - no matter how good your AI model.

The point of math is understanding. So mathematicians should feel free to use AI as much as you want, but in the end, they should've gained some understanding on what happened.

Re: A Beginning for Mathematics

#78

Post by bonoboTP »

jsrozner wrote: post #75
bonoboTP wrote: post #60

> We already interview faculty hires; we must now do the same for graduate admissions They hire PhD students without hearing them give a talk and then doing interviews? In Germany, the applicant gives a talk (30-40 min) to the research group they want to join, usually presenting their master's thesis, engage in discussion, often share lunch with the group, then do 1 on 1s with individual members of the group and a lo…

Unlike Europe, in the US, most PhD applicants do not have a masters. 10 years ago, most had not even conducted substantial senior projects (e.g a semester or year). Though increasingly senior projects / undergraduate "theses" have become more common.

In my opinion, that's not a good time to really know you want to pursue academic research. A masters gives you a little taste at least and allows to show aptitude. Doing a student research assistant job, working on a thesis, maybe even a paper submission, diving deep into specialized courses. Then you can know if it's for you as well as the PI can know if it's for you. Straight from a bachelor and them interleaving the phd with coursework is a bad model I think, but of course I'm from Europe so it's what I'm used to.

Re: A Beginning for Mathematics

#79

Post by jsrozner »

ComplexSystems wrote: post #58

I admit I just don't understand this attitude. People had the same complaints that the code produced by early coding models was messy, lazy, poorly commented, had terrible architecture and so forth. The central complaint was that it was just too difficult for humans to review. The answer is just to improve the models and move on. Similarly now we're getting AI doing math. The proofs compile but are a mess. So just ma…

One funny thing is that in order to tune the models to make what they're doing explainable to humans, you need to have humans involved in the RL pipeline to indicate which explanations are good.

You can understand this as learning a mapping between the model's internal "world" (i.e., 'meaning,' which is hopefully coherent and consistent -- but definitely not always! see, e.g., https://arxiv.org/html/2505.11581v1) and language (i.e. 'form') that reflects that world.

For this to work, you need both coherent / consistent internal model worlds, and also good mappings onto human language. Supervision by mathematicians has provided the signal for both internal coherence (though this can also come from interacting with a proof oracle) and for good explanations. If models exceed human capacities, you could imagine that aligning their explanations potentially becomes harder (though not necessarily). Also, humans naturally have to do the same thing: as researchers we must find analogies to make our work legible to collaborators or laypeople. Often in doing this, we further clarify our own understanding!

More deeply I think the "end of the world" vibe arises not only from the practical need to have models that explain, but also Litt's (and many other fields' researchers) grappling with being relegating to not mattering.

Re: A Beginning for Mathematics

#80

Post by asa123 »

While I broadly agree with the premise of re-directing the “purpose” of math, I quite detest the idea that judgement might be primarily based upon some in person discussion, or oral presentation, and the claim that written mathematics that is not orally communicated might be less worthwhile in some sense (i know this isn’t the exact statement of the authors proposition).

There are a good deal of people, whom, falter much more in oral discussions, whether this be for a psychological thing, stage fright, or difficulty explaining things on the spot. There are also certainly brilliant people, who can’t give an informative, discussion inviting talk to save their lives, but given enough time, can formalize their thoughts in writing at the highest levels of their field, and that writing is likewise very enlightening (sometimes).

It’s not clear to me, that, AI as is, could not pose successfully in an oral discussion of a topic. I mention this because it seems that one implication of the article is that AI might write things that are logically correct, but devoid of understanding. I suggest rather that 1) it is not extremely improbably that AI is incapable of generating mathematics that furthers human understanding and if 2) it is indeed highly likely that they cannot generate mathematics that furthers human understanding in a textual format, then surely one could also differentiate between human and AI on a textual level, and judge the contribution of a human, without the need of oral discussion?

I suppose another aside is, one might claim that the existence of AI means people have much much more text to filter for, and so, it becomes difficult to find one person’s good writing amidst a sea of, logically correct, yet understanding devoid textual content. But by and large much or mathematical academia certainly operates off of some reputation/vouching system presently anyways, that already serves as a “filter” in some sense. Perhaps the existence of such a system/culture is not a good thing, but oral discussions/seminars certainly aren’t immune from such predilections.

Perhaps I’m babbling like an idiot, but the entire and sole purpose of this comment is just to say: for the love of god please don’t let the standard be judged by oral presentation

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