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A Beginning for Mathematics

daniellitt.com

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Re: A Beginning for Mathematics

#81

Post by robotpepi »

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…

Code has a very different purpose than math tough. The development in basic science follows a different motivation system.

Re: A Beginning for Mathematics

#82

Post by ComplexSystems »

JW_00000 wrote: post #77
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 se…

> But for math: what is the point of a proof if no one will read it and no one uses its result?

What is the point of writing software if nobody will run it?

> 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.

So they ask the AI to explain the proof.

Re: A Beginning for Mathematics

#83

Post by kenjackson »

JW_00000 wrote: post #77
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 se…

If the point of math is understanding then maybe our incentive structure is wrong. Maybe it should be teaching the concepts to as many people as possible rather than just continuing to write papers that 10 people in the world understand, which is the current state of a lot of math.

Re: A Beginning for Mathematics

#84

Post by cman1444 »

JW_00000 wrote: post #77
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 se…

I keep seeing people repeat that the goal of math is "understanding".

I do think that is one goal of math but I don't think it's the only one.

I think an additional goal is simply "truth", which can be found without understanding as we've seen with these human-incomprehensible proofs.

Yet another is practical applications. While there's less of these in pure mathematics than in most domains, they do still exist.

Re: A Beginning for Mathematics

#85

Post by bobmarleybiceps »

bonoboTP wrote: post #74
bobmarleybiceps wrote: post #72

Earlier quoted context omitted.

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…

I agree the combined masters + phd is weird tbh. I think almost everyone treats it like "just focus on research and spend as little time on course work as possible." I would prefer it if courses were more flexible.

I'm not a big fan of the US application setup. IDK how it is in Europe, but in the US, it feels like there's a lot of not-very-meritocratic "secret" stuff you need to know to up your chances.

Re: A Beginning for Mathematics

#86

Post by Tazerenix »

waynecochran wrote: post #18

As 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.

I think you're a little naive if you think this is because of gate-keeping by mathematicians rather than the essential complexity of mathematics. Mathematicians individually and as a group would love nothing more than a world which is capable of understanding their work more deeply.

In the exact same way that LLMs allow anyone to vibe code an app but do not replace real understanding of system design due to its essential complexity, non-mathematicians will quickly learn that asking an LLM to pump out advanced mathematical statements to you, even if they are correct (and even if you could verify them) does not constitute understanding, and that the human brain is the bottleneck either way.

It is only if the LLM is super-human at simplification and explaining that a difference will be noted. This would be excellent for mathematics but its not a foregone conclusion (and the argument of most mathematicians, such as Terence Tao, is that this distillation process is one of the key parts of doing mathematics, and that LLMs so far seem to be going in the opposite direction. I suspect its probably user error and leveraging the tools better will produce different outcomes, but mathematicians are only just starting the journey that software developers have been going through, so patience is needed).

Re: A Beginning for Mathematics

#87

Post by waynecochran »

tuckerman wrote: post #62
waynecochran wrote: post #42

Earlier quoted context omitted.

It is not lowering the bar to find better ways to demystify and explain things. In fact, I would say those who can explain it well understand it the best. Richard Feynman would be my best example.

Do you have any examples of places you felt like there was a lot of gatekeeping? Perhaps having studied mathematics I am a bit blind to the issue here and would like to learn more. I agree making simple things sound complicated to appear more impressive is bad but there are limits. Even with Feynman he could only go so far, e.g. his interview about why questions and magnetism.

I am not thinking of intentional gatekeeping, but more of the kind where mathematicians build themselves their own island of concepts and notation with no thought of building a passageway for others to engage and make use of the theory. I would find modern algebraic geometry highly useful as someone who works in computer graphics and computer vision, but much of the theory is akin to learning a new language and I don't get the sense that those who publish their work in this field care if I enter their world.

Re: A Beginning for Mathematics

#88

Post by joe_the_user »

waynecochran wrote: post #18

As 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.

I too have a degree in math and it's easy to see some of what's happening as too much of an emphasis on intellectual chest thumping and to little emphasis on showing why understanding the concepts is useful for everyone. I will say the problem may well increased here by the American education system which denigrates such understanding and isn't influenced by mathematicians.

But also, with things rapidly changing, perhaps in ten years chat programs will not only do superhuman but make their proofs marvelously accessible and provide incredible tutoring sufficient to bring any curious up to a super high level quickly. Then what can you say and what can you complain of.

But I think "do everything machines" are necessarily inevitable but the situation does make it uncertain where the limits are.

Re: A Beginning for Mathematics

#89

Post by curt15 »

cman1444 wrote: post #84
JW_00000 wrote: post #77

Earlier quoted context omitted.

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 se…

I keep seeing people repeat that the goal of math is "understanding". I do think that is one goal of math but I don't think it's the only one. I think an additional goal is simply "truth", which can be found without understanding as we've seen with these human-incomprehensible proofs. Yet another is practical applications. While there's less of these in pure mathematics than in most domains, they do still exist.

The primary goal of all basic sciences is human understanding. "Truth" is no more a goal for mathematicians than the physical laws are a goal to physicists; they simply exist in nature. The goal is rather to develop useful language and conceptual frameworks for reasoning and communicating. That understanding underpins all practical applications.

Re: A Beginning for Mathematics

#90

Post by augment_me »

What is the incentive for a person to sit though seminars and evaluations? People already hate redundant meetings. What is the incentive to change the system from the existing one to one that rewards this verification somehow? Who benefits from this change?
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