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A Beginning for Mathematics

daniellitt.com

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Re: A Beginning for Mathematics

#61

Post by youoy »

As with the rest of the domains, AI/LLMs will do syntax and search better than any human.

In code, any developer whose differentiaton was clean code and knowledge of different technologies is now average.

In math, any mathematitian whose differentiation was to manipulate formal systems and know tricks of different domains will be average.

Fortunately, humans do more than syntax and search.

The bad news for developers is that if you know what the output of your program should be (which happens most of the time), almost all of the job is syntax and search to build the code that reproduces the output.

The good news for mathematitians is that for the majority of problems you never know the output, or you just know the output is either "True" or "False". There are some cases where you need something else, for example "a solution that blows up in finite time". For those cases AI will outperform you easily (see new Navier-Stokes solution)

So if as a mathematitian you were doing more than syntax and search, then keep doing that and use AI just for what its best.

Re: A Beginning for Mathematics

#62

Post by tuckerman »

waynecochran wrote: post #42
chneu wrote: post #22

Earlier quoted context omitted.

Lowering the bar for the masses isn't always a good thing

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.

Re: A Beginning for Mathematics

#63

Post by jplusequalt »

Another mathematician waxing poetically about a future that will not come to happen. Call me a pessimist if you'd like, but capital has no incentive to ensure mathematicians maintain their current status in society.

If you're a mathematician you are in the same boat as the software engineer, and the Dodo.

Better learn a trade buddy /s

Re: A Beginning for Mathematics

#64

Post by boccaff »

Jun8 wrote: post #10

Excellent 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.

Re: A Beginning for Mathematics

#65

Post by jsrozner »

"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?

Re: A Beginning for Mathematics

#66

Post by anyfoo »

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?

Re: A Beginning for Mathematics

#67

Post by ksd482 »

emil-lp wrote: post #39
gowld wrote: post #5

Earlier quoted context omitted.

Why is "Doctor of Philosophy" the correct certificate of "becoming expert in a topic"? That's a radical departure from "PhD" being a certificate that someone is qualified to produce new research. What you describe is more like a Masters Degree.

I think you have misunderstood. PhD has nothing to do with expertness. If you have a PhD, you have completed some kind of research training. That's all there is. Says nothing about knowledge or whether or not you're a genius. You cannot conclude anything else, and nobody claims that you can. If someone has a PhD, they have some training in doing research.

Looks like @gowld is agreeing with you.

Re: A Beginning for Mathematics

#68

Post by mathisfun123 »

bayindirh wrote: post #55
mathisfun123 wrote: post #52

Earlier quoted context omitted.

Have people completely forgotten how to read? I'm saying your analogy between software and math is completely, irreconcilably flawed. I don't need to read the rest of your screed because this is a thread about math not software.

I see. Somebody said that lowering the bar is not good. Somebody else asked for examples, and I provided an example. So, if you want to be pedantic, that doesn't track well, because what I answered was not about mathematics. If you want something about mathematics, computation is mathematics, as software is. So, my example tracks the same way in mathematics. Finding solutions without understanding its parts or the pa…

> computation is mathematics, as software is. So, my example tracks the same way in mathematics.

brother i already addressed this literally in my first response to you: the reason software went to shit is because it became commodified (ie a thing produced in a factory) not because it's computational nor because the bar got lowered. seriously read my whole original reply again and see whether you really disagree before just re-asserting your rant on SWE.

> Finding solutions without understanding its parts or the path

you didn't read the post at all did you? it's emphatically about deep understanding over blind results.

> Using AI too much blunts critical thinking skills in some, as I read.

irony

Re: A Beginning for Mathematics

#69

Post by abdullahkhalids »

jltsiren wrote: post #51
wrs wrote: post #32

The 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. “…

An oral defense is ultimately a meeting, with the same weaknesses as other types of meetings. In particular, if new information comes up in a meeting, you cannot reasonably expect to get useful responses from the other participants. First impressions maybe, and plausible-sounding bullshit from those prone to generating that. If you want anything more, you need to provide the information in advance or schedule another…

You are missing the entire point of the oral defense while almost stating it. The existence of the oral defense is what extrinsically motivates the supervisor to do their job well. If the student fails, the supervisor loses faces in front of their peers, which obviously they try to avoid. Removing the oral defense would massively reduce the quality of PhDs because supervisors will not do their job.

Also in my observational experience in physics, a fair number of students (probably like 10%) get some sort of major corrections to their thesis. After writing this, I did a search, and in the UK, the number across all fields is 16% [1].

[1] https://www.lexacademic.com/blog/how-common-is-passing-with-...

Re: A Beginning for Mathematics

#70

Post by bonoboTP »

I think this goes in the right direction. You have to rethink the role of human work in math, can't put your head in the sand and cling to your comfy institutions and system just because you got to know it's ins and outs and just want it to be like that forever.

But the bigger picture is: while I understand the author know his field best and wants to keep the post focused, the same issue will hit many more fields. We need to also have a broader discussion that involves more fields of knowledge work, largely academic scholarship but also regular office work, then it will come to engineering design, medicine, it's already coming for 3d modeling and vfx, software dev, it will come for a lot of middleman services. Not at the same rate, but we have to understand that it's not just that math will change.

Change will be the default. Everything will change. It will be much easier to make math change because all things will change. You shouldn't worry and imagine that funding criteria will be like today or that journals or academia or politicians expectations will be like today. No, everything will adjust with some timing differences of course but it won't be a static world and then math changing and having to justify and fight to explain the change to other actors who are baffled. They won't be baffled they will themselves have to change.

The world will transform as much as it did when society moved from feudal agrarian to urban capitalist industrial, or from the vast majority doing physical labor to a service economy with a huge amount of desk jobs. I can't tell how it will change exactly but it will be bigger than what we have seen in the last couple of generations or maybe more.

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