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A Beginning for Mathematics

daniellitt.com

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Re: A Beginning for Mathematics

#31

Post by breezybottom »

gowld wrote: post #5
ksd482 wrote: post #2

> 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…

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'm not sure why an advanced degree is necessary for that at all, besides the pride of a vanity title. It's pretty much what Bill Nye does for science.

Re: A Beginning for Mathematics

#32

Post by wrs »

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. “I dunno, I guess Claude thought this was a good idea” is not a coherent design.

Re: A Beginning for Mathematics

#33

Post by applfanboysbgon »

PeterWhittaker wrote: post #27
chneu wrote: post #22

Earlier quoted context omitted.

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

Examples, preferably academic?

Half of modern academia, for one. Universities have turned into degree mills with the expectation that >50% of the population requires a college degree, regardless of whether they actually have any interest or need for one beyond doing it because it's a prerequisite for a good career, independent of whether said career actually uses the knowledge in any way.

Not that I actually agree that math was at the right level of gatekeeping. It definitely feels intentionally opaque beyond reason, and I think it's why LLMs are able to cut through the obfuscation and solve problems that maybe wouldn't actually have been considered quite so hard if mathematicians did a better job of making their work accessible.

Re: A Beginning for Mathematics

#34

Post by JoeAltmaier »

applfanboysbgon wrote: post #33
PeterWhittaker wrote: post #27

Earlier quoted context omitted.

Examples, preferably academic?

Half of modern academia, for one. Universities have turned into degree mills with the expectation that >50% of the population requires a college degree, regardless of whether they actually have any interest or need for one beyond doing it because it's a prerequisite for a good career, independent of whether said career actually uses the knowledge in any way. Not that I actually agree that math was at the right level…

28% of adult Americans

Re: A Beginning for Mathematics

#35

Post by bayindirh »

PeterWhittaker wrote: post #27
chneu wrote: post #22

Earlier quoted context omitted.

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

Examples, preferably academic?

We have seen the thing in programming. I'm not a gatekeeper, I love seeing more people to code. On the other hand, along the way, we lost the joy of the journey and only fixated at the destination.

Result is more software at lower quality. The reason is statistics. When you increase the population, you increase the population of every kind of programmer, and people who want the result are favored in most competitive sectors because corporations want something somewhat working yesterday.

...and here we are.

Now programmers talking about code quality is stoned en-masse. If it's somewhat working then it's good. Efficiency, resiliency, maintainability and sustainability is an afterthought. Some of my friends who loved debating programming language theory now don't even care about the code. They don't write it, just vibe, and they don't plan to come back to "older, caveman style of development".

Some universities are also adding fuel to the fire: They "prepare students for the job", not teaching the science, but the parts that corporations need for the job only.

Though, hardware was cheap and people were expensive, and now code is cheap and hardware is expensive now. We'll see.

Re: A Beginning for Mathematics

#36

Post by applfanboysbgon »

JoeAltmaier wrote: post #34
applfanboysbgon wrote: post #33

Earlier quoted context omitted.

Half of modern academia, for one. Universities have turned into degree mills with the expectation that >50% of the population requires a college degree, regardless of whether they actually have any interest or need for one beyond doing it because it's a prerequisite for a good career, independent of whether said career actually uses the knowledge in any way. Not that I actually agree that math was at the right level…

28% of adult Americans

66% in Japan. Who said anything about America?

Re: A Beginning for Mathematics

#37

Post by Jblx2 »

Jaxan wrote: post #29
nilkn wrote: post #25

Earlier quoted context omitted.

I actually think it's the opposite: Lean proofs and autoformalization make it very easy to announce proofs alongside proofs of the correctness of those proofs (Lean certificates). It's not an absolutely fool-proof combination (the Lean kernel could still contain bugs), but it does immediately attach a very substantial degree of credibility to the result. And that I think is essential to why some of the world's leadin…

The recent proof of Fermats Last Theorem is interesting: it is (iirc) 13 million lines of lean code. And type-checking takes 5 hours or so on a pretty beefy machine. I cannot independently verify the proof, and I have to take Anthropics word for it that it actually type-checks.

That seems like a red herring. Have you independently verified the human generated proof of FLT? Surely someone else will try to verify Anthropic's formalization on different hardware. Plus, it seems likely that FLT formalizations will improve / get shorter over time, requiring less compute. And computers (and type-checkers) will continue to get faster over time as well. So maybe in 5 years you could own a computer fast enough to verify a/the proof in say a week, instead of 5 hours.

Re: A Beginning for Mathematics

#38

Post by emil-lp »

jkhdigital wrote: post #26

> resulting in the production of an abundance of PDFs. The contents of some of those PDFs may even have important applications. I hope, from the depths of my soul, that the static typeset report format for transmitting knowledge and understanding will finally die and be laid to rest.

As a researcher in theoretical computer science, I love PDFs more than any other format when it comes to mathematics.

There simply is no contender to LaTeX and PDFs.

Lucky for you, almost all research in math, cs, and physics, are put on arxiv, where you can download the source code (.tex) as well as get an HTML render.

Re: A Beginning for Mathematics

#39

Post by emil-lp »

gowld wrote: post #5
ksd482 wrote: post #2

> 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…

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.

Re: A Beginning for Mathematics

#40

Post by Jblx2 »

vld_chk wrote: post #20
robinhouston wrote: post #14

Earlier quoted context omitted.

I don't think that's actually the real problem. Along with the progress in answering mathematical questions, recent progress on AI-powered autoformalisation has been astonishing. All the recent AI discoveries have been accompanied by Lean proofs. And, yes: that doesn't absolutely guarantee correctness. The Lean kernel has had soundness bugs, and may have some still. But it's pretty strong evidence of correctness neve…

I am not that worried, but rather just observing. Humanity is about to enter the phase when we will be using things based on ideas no human ever properly understands. This thought … disturbing, somehow? It is perfectly valid counterpoint to say that we already do it. We everyday use myriad of things, tools, and software we have 0 clue how it operates. But for us as humans it was reassuring that we know that at least…

>Humanity is about to enter the phase when we will be using things based on ideas no human ever properly understands.

Do any one person even understand the humble pencil?

https://dn790006.ca.archive.org/0/items/i-pencil-pdf-2019/I%...

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