4 ms·

The biggest project in modern mathematics [video]

yamrzou 4y ago
That was a wonderful video. Thanks for posting.
throwaway81523 4y ago
I hate blind youtube links, and this one even takes a while into the video to say what it is about. Spoiler: the Langlands Program. Video has some nice animations but not much about the math. The author has a writeup here: https://www.quantamagazine.org/what-is-the-langlands-program-20220601/ https://www.quantamagazine.org/what-is-the-langlands-program... See also: https://en.wikipedia.org/wiki/Langlands_program https://en.wikipedia.org/wiki/Langlands_program
- bergenty 4y ago
  It’s explaining context, not all the intricacies of the math. That being said, it does tackle a decent amount of math in that video.
- deleted 4y ago
  [deleted]
- Koshkin 4y ago
  One of the well-known category theorists has said [0], > I’ve never succeeded in understanding the slightest thing about it. [0] https://golem.ph.utexas.edu/category/2010/08/what_is_the_langlands_programm.html https://golem.ph.utexas.edu/category/2010/08/what_is_the_lan...
  - mjreacher 4y ago
    I'm glad Tom was so open about his lack of knowledge here, so often it is easy to assume any professional mathematician should know all this things and it all comes easy to them. However this obviously isn't the case and I doubt things will be any easier in the future as mathematics becomes more specialized.
- Simon_O_Rourke 4y ago
  Thank you for the summary!
- xiphias2 4y ago
  For me it wasn't a spoiler, and also the video contained more math that I understand than the Wikipedia link (that just was a list of lots of mathematical structures that I don't know anything of, and not part of the computer science curriculum, so I don't expect other hackers to understand them either).
- javierga 4y ago
  I guess it is fitting to mention the famous Wandsworth Constant (first 30% of the video can be skipped). In seriousness, having had some exposure to the Langlands program (through the wonderful Love and Math by Frenkel), I was counting the minutes to hear about it. I found the video to have a great layman explanation of what it is about.
- leoc 4y ago
  It does say LANGLANDS on the thumbnail, as well as talk about it in the description.
smelbe 4y ago
The graphic designer behind these incredible animations has my appreciation and respect. In the representation of such intangible concepts, the combination of creativity and deep technical expertise blends perfectly. Total command of the arts and crafts.
superb-owl 4y ago
I've never had someone break down Wiles' proof of Fermat's last theorem so succinctly. I can't speak to its accuracy but I found it super helpful.
- Koshkin 4y ago
  Agree, a wonderful explanation. The Langlands Program, on the other hand, was not explained in any detail at all (except mentioning the word "functoriality").
- Myrmornis 4y ago
  Agreed, as you say, I don’t actually know if it’s a good explanation of Wiles’ proof but it certainly seemed plausible that it was a great explanation!
pmoriarty 4y ago
As a non-mathematician, the last truly big mathematical project that caught my imagination was Hilbert's attempt to found all of mathematics on a firm foundation of logic. That attempt famously and spectacularly failed with Gödel's incompleteness theorems. Since then it seems that mathematicians have lost interest in foundations and are content to search for interesting results, structures, and systems, even if they don't have a solid foundation. More recently I've heard some proposals to revisit the foundational project but with higher-order logics proving the consistency and completness of lower-order ones, which sounds interesting, but I'm not sure how much progress has been made, and to a non-mathematician/non-logician even that attempt sounds a bit like a house of cards. Does anyone here know about this and if there are even any mathematicians around these days who are still interested in it?
- gnulinux 4y ago
  > Since then it seems that mathematicians have lost interest in foundations This is very much not the case. What's closer to truth is that the discussion moved on from a framework laymen can seemingly understand conclusions, to one where conclusions (or their implications on mathematics proper) are a lot harder to explain to laymen. Foundational work is still a thing, but I don't think it affects the nature of mathematics in a way laymen can conceptualize. Take Godel's incompleteness. People say it's something laymen can understand, and people attempt explaining it to masses every day in youtube, reddit etc. But if you truly get into the formal conclusion (i.e. with Rosser's trick, the conclusion is: "a theory cannot be an extension of Q, complete and consistent all 3 at the same time") you'll see that it's already pretty far away from what laymen thought they understood. And modern foundational work exponentially drifted away from this too. I'm not a mathematician so everything in this comment should be taken with a grain of salt.
  - Jenz 4y ago
    Noteworthy to me, is that it seems foundational work is to a significant degree, not considered mathematics or something people calling themselves “mathematicians” would spend much of their time on; but it’s left for logicians and more philosophically inclined people.
nestorD 4y ago
Interesting, to me the biggest project in contemporary[1] mathematics is the work done on theorem provers. The goal being not to be able to prove new things automatically, but to check all existing proofs automatically and make it possible for mathematicians to do their work using those tools. We are still far from those goals (it is a big project for a reason) but the work of professional mathematicians could look very different in a hundred years if those efforts succeed. [1]: And the distinction between contemporary and modern might be relavant.
- mbrodersen 4y ago
  I agree. Mathlib (LEAN) is a great example of this.
s-xyz 4y ago
Maybe I am the only one, but I do miss sometimes Bachelor’s level (probably High school level in some countries) math. Have not applied it anywhere since, but there was something about those classes and exercises that were rewarding. This video brought back old memories and feelings.
jnash 4y ago
I am not a mathematician. But long term I think the #1 most important project is something like mathlib: https://leanprover-community.github.io/ https://leanprover-community.github.io/ I think it is the future of mathematics. Yes I know that 99% of mathematicians will disagree. That's not unusual for a fundamental change to how people think. And yes I know it will take a very long time for this change to manifest itself. That is also not unusual for a fundamental change to how people think. But don't underestimate the exponential power of real formal mathematics. What mathematicians are currently calling "formal" proofs aren't. They are just informal proofs with lots of details. Again, 99% of mathematicians will strongly disagree. However eventually we will see math journals dedicated to formally proven correct mathematics. And those proof will start to be regarded above and beyond the current informal proofs. It will take a long time. But IMHO it is inevitable.
- destring 4y ago
  Can’t help but laugh at the phrase “Real formal mathematics”. Mathematics are as formal as required to be understood by the audience. After all, math is just a way to communicate abstract arguments and reasoning. Computers require more help to understand abstract concepts, doesn’t mean that formulation is better than our current one. See Principia Mathematica on why too much formalism is not a good idea.
  - jnash 4y ago
    Can't help but laugh at your comment. Mathematics is an informal social activity where humans try to convince other humans that certain abstract structures have certain properties. Having to turn those proofs into formal machine checked proofs makes a huge difference. Principia Mathematica was a big step forward and directly leading to Type Theory. It didn't succeed in what the authors tried to accomplish, but it very much had a massive impact on the world of mathematics. That's why we both know about it and discuss it today.
muahahahah 4y ago
Hi, non-native english-speaker here. I read much of 'but hu ?' on the (in this thread) linked websites !? Will make a comic about what i thought reading and concluding this topic (another non-mathematic here): 1st) you have to generate a 'numbers-room' So, when you multiplicate two 'one-digit' numbers, the 1st calculation with a two digits result is 2 * 5 2 * 6 2 * 7 2 * 8 2 * 9 3 * 3 3 * 4 3 * 5 3 * 6 3 * 7 3 * 8 3 * 9 4 * 4 4 * 5 4 * 6 4 * 7 4 * 9 5 * 5 5 * 6 5 * 7 5 * 8 5 * 9 6 * 6 6 * 7 6 * 8 6 * 9 7 * 7 7 * 8 7 * 9 8 * 8 8 * 9 9 * 9 10 * 1 10 * 2 10 * 3 10 * 4 10 * 5 10 * 6 10 * 7 70 * 8 10 * 9 and 12 * 9 is the first sum with a 3-digits result. > But there was 10 * 10 a '4-digits' multiplication with onla a 3-digits result. And if you took physics -acustics, you know there are primary- and secondary waves spreading... ...must be my humor, but... hm?* (-; edited: (asterixes, readability ^^)

That was a wonderful video. Thanks for posting.

I hate blind youtube links, and this one even takes a while into the video to say what it is about. Spoiler: the Langlands Program. Video has some nice animations but not much about the math. The author has a writeup here: https://www.quantamagazine.org/what-is-the-langlands-program-20220601/ https://www.quantamagazine.org/what-is-the-langlands-program... See also: https://en.wikipedia.org/wiki/Langlands_program https://en.wikipedia.org/wiki/Langlands_program

bergenty 4y ago
It’s explaining context, not all the intricacies of the math. That being said, it does tackle a decent amount of math in that video.
deleted 4y ago
[deleted]
Koshkin 4y ago
One of the well-known category theorists has said [0], > I’ve never succeeded in understanding the slightest thing about it. [0] https://golem.ph.utexas.edu/category/2010/08/what_is_the_langlands_programm.html https://golem.ph.utexas.edu/category/2010/08/what_is_the_lan...
- mjreacher 4y ago
  I'm glad Tom was so open about his lack of knowledge here, so often it is easy to assume any professional mathematician should know all this things and it all comes easy to them. However this obviously isn't the case and I doubt things will be any easier in the future as mathematics becomes more specialized.
Simon_O_Rourke 4y ago
Thank you for the summary!
xiphias2 4y ago
For me it wasn't a spoiler, and also the video contained more math that I understand than the Wikipedia link (that just was a list of lots of mathematical structures that I don't know anything of, and not part of the computer science curriculum, so I don't expect other hackers to understand them either).
javierga 4y ago
I guess it is fitting to mention the famous Wandsworth Constant (first 30% of the video can be skipped). In seriousness, having had some exposure to the Langlands program (through the wonderful Love and Math by Frenkel), I was counting the minutes to hear about it. I found the video to have a great layman explanation of what it is about.
leoc 4y ago
It does say LANGLANDS on the thumbnail, as well as talk about it in the description.

smelbe 4y ago

The graphic designer behind these incredible animations has my appreciation and respect. In the representation of such intangible concepts, the combination of creativity and deep technical expertise blends perfectly. Total command of the arts and crafts.

superb-owl 4y ago

I've never had someone break down Wiles' proof of Fermat's last theorem so succinctly. I can't speak to its accuracy but I found it super helpful.

Koshkin 4y ago
Agree, a wonderful explanation. The Langlands Program, on the other hand, was not explained in any detail at all (except mentioning the word "functoriality").
Myrmornis 4y ago
Agreed, as you say, I don’t actually know if it’s a good explanation of Wiles’ proof but it certainly seemed plausible that it was a great explanation!

pmoriarty 4y ago

As a non-mathematician, the last truly big mathematical project that caught my imagination was Hilbert's attempt to found all of mathematics on a firm foundation of logic. That attempt famously and spectacularly failed with Gödel's incompleteness theorems. Since then it seems that mathematicians have lost interest in foundations and are content to search for interesting results, structures, and systems, even if they don't have a solid foundation. More recently I've heard some proposals to revisit the foundational project but with higher-order logics proving the consistency and completness of lower-order ones, which sounds interesting, but I'm not sure how much progress has been made, and to a non-mathematician/non-logician even that attempt sounds a bit like a house of cards. Does anyone here know about this and if there are even any mathematicians around these days who are still interested in it?

gnulinux 4y ago
> Since then it seems that mathematicians have lost interest in foundations This is very much not the case. What's closer to truth is that the discussion moved on from a framework laymen can seemingly understand conclusions, to one where conclusions (or their implications on mathematics proper) are a lot harder to explain to laymen. Foundational work is still a thing, but I don't think it affects the nature of mathematics in a way laymen can conceptualize. Take Godel's incompleteness. People say it's something laymen can understand, and people attempt explaining it to masses every day in youtube, reddit etc. But if you truly get into the formal conclusion (i.e. with Rosser's trick, the conclusion is: "a theory cannot be an extension of Q, complete and consistent all 3 at the same time") you'll see that it's already pretty far away from what laymen thought they understood. And modern foundational work exponentially drifted away from this too. I'm not a mathematician so everything in this comment should be taken with a grain of salt.
- Jenz 4y ago
  Noteworthy to me, is that it seems foundational work is to a significant degree, not considered mathematics or something people calling themselves “mathematicians” would spend much of their time on; but it’s left for logicians and more philosophically inclined people.

nestorD 4y ago

Interesting, to me the biggest project in contemporary[1] mathematics is the work done on theorem provers. The goal being not to be able to prove new things automatically, but to check all existing proofs automatically and make it possible for mathematicians to do their work using those tools. We are still far from those goals (it is a big project for a reason) but the work of professional mathematicians could look very different in a hundred years if those efforts succeed. [1]: And the distinction between contemporary and modern might be relavant.

mbrodersen 4y ago
I agree. Mathlib (LEAN) is a great example of this.

s-xyz 4y ago

Maybe I am the only one, but I do miss sometimes Bachelor’s level (probably High school level in some countries) math. Have not applied it anywhere since, but there was something about those classes and exercises that were rewarding. This video brought back old memories and feelings.

jnash 4y ago

I am not a mathematician. But long term I think the #1 most important project is something like mathlib: https://leanprover-community.github.io/ https://leanprover-community.github.io/ I think it is the future of mathematics. Yes I know that 99% of mathematicians will disagree. That's not unusual for a fundamental change to how people think. And yes I know it will take a very long time for this change to manifest itself. That is also not unusual for a fundamental change to how people think. But don't underestimate the exponential power of real formal mathematics. What mathematicians are currently calling "formal" proofs aren't. They are just informal proofs with lots of details. Again, 99% of mathematicians will strongly disagree. However eventually we will see math journals dedicated to formally proven correct mathematics. And those proof will start to be regarded above and beyond the current informal proofs. It will take a long time. But IMHO it is inevitable.

destring 4y ago
Can’t help but laugh at the phrase “Real formal mathematics”. Mathematics are as formal as required to be understood by the audience. After all, math is just a way to communicate abstract arguments and reasoning. Computers require more help to understand abstract concepts, doesn’t mean that formulation is better than our current one. See Principia Mathematica on why too much formalism is not a good idea.
- jnash 4y ago
  Can't help but laugh at your comment. Mathematics is an informal social activity where humans try to convince other humans that certain abstract structures have certain properties. Having to turn those proofs into formal machine checked proofs makes a huge difference. Principia Mathematica was a big step forward and directly leading to Type Theory. It didn't succeed in what the authors tried to accomplish, but it very much had a massive impact on the world of mathematics. That's why we both know about it and discuss it today.

muahahahah 4y ago

Hi, non-native english-speaker here. I read much of 'but hu ?' on the (in this thread) linked websites !? Will make a comic about what i thought reading and concluding this topic (another non-mathematic here): 1st) you have to generate a 'numbers-room' So, when you multiplicate two 'one-digit' numbers, the 1st calculation with a two digits result is 2 * 5 2 * 6 2 * 7 2 * 8 2 * 9 3 * 3 3 * 4 3 * 5 3 * 6 3 * 7 3 * 8 3 * 9 4 * 4 4 * 5 4 * 6 4 * 7 4 * 9 5 * 5 5 * 6 5 * 7 5 * 8 5 * 9 6 * 6 6 * 7 6 * 8 6 * 9 7 * 7 7 * 8 7 * 9 8 * 8 8 * 9 9 * 9 10 * 1 10 * 2 10 * 3 10 * 4 10 * 5 10 * 6 10 * 7 70 * 8 10 * 9 and 12 * 9 is the first sum with a 3-digits result. > But there was 10 * 10 a '4-digits' multiplication with onla a 3-digits result. And if you took physics -acustics, you know there are primary- and secondary waves spreading... ...must be my humor, but... hm?* (-; edited: (asterixes, readability ^^)