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> I struggle to see the beauty in a gargantuan lean proof constructed by 100 different people Why does it need to be beautiful? Once you proved it it's true an

> I struggle to see the beauty in a gargantuan lean proof constructed by 100 different people Why does it need to be beautiful? Once you proved it it's true and you can use its consequences in math, sciences and engineerings.

zerobees 4mo ago
Outside of some niche specializations like cryptography, math isn't practiced because of "consequences". Most mathematicians take pride in their work not having any obvious practical applications. They're also overwhelmingly working in university settings where they're not expected to generate revenue or deliver practical results. We basically subsidize the practice of mathematics as an art form, and if you try to take the artistry away, you might find that the artists don't want to play along. And I guess you can imagine future robo-math production lines without any human involvement, and then LLMs finding applications for the resulting theorems, but it's not possible today.
- setopt 4mo ago
  Are you sure that’s «most» mathematicians? At the universities I’ve been to (as a student and now faculty), «applied mathematics» and «statistics» have been the two largest divisions. But perhaps that’s a bias from engineering-heavy universities?
  - jubilanti 4mo ago
    "Applied Math" and "Statistics" are distinct fields from "Mathematics," not subfields of it. People in those two departments are often closer to Computer Science or the statistics subfield in a domain science field (e.g. biostatistics, econometrics) than to Mathematics in terms of what they actually teach and research.
    - setopt 4mo ago
      That is perhaps fair, is that distinction common internationally? Again, in the universities I’ve been to, «applied math» and «statistics» have generally been placed under the department of mathematics. I myself am a physicist, and consider applied physics, biophysics, etc. to be subfields of physics and not distinct fields of study, but I don’t know what outer physicists think.
- bigmadshoe 4mo ago
  You put it perfectly. And all these AI math startups don't actually care about mathematics. They are just using it as a proxy for general reasoning, with the VC pitch being some kind of world domination after they crack these problems.
- chermi 4mo ago
  Most mathematicians don't take pride in their results having no applications. That's just not true. Maybe some quirky pure logicians or something. But otherwise 90%+* of mathematicians I know would be at least satisfied if not thrilled for their work to be used by others. *Completely made up statistic.
  - limflick 4mo ago
    I've had the opposite experience with all the math nerds that I know. End of the day it's all anecdotal ig https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology
bwestergard 4mo ago
Why prove the Pythagorean theorem rather than just prove 3^2 + 4^2 = 5^2? For any practical application, you are only interested in finite set of concrete identities, so anything beyond that is surplus to requirements, surely?
- spacemanspiffii 4mo ago
  I think you may be interested in more abstract things. In this case, let's say you're creating a program for a 3D printed thing, and you have to fit a diagonal cardboard in a rectangular box, you'd like to be sure that the Pythagorean theorem holds even in cases where you haven't tried it out.
- moregrist 4mo ago
  > For any practical application, you are only interested in finite set of concrete identities I do a lot of numerical work in settings where computational efficiency is useful. In my work, most cases you can do numerically using integration or Monte Carlo sampling or whatever. It’s slow. It often pays to find a closed-form solution. Even if it’s just a starting point that needs refinement. To put in terms of the Pythagorean theorem: Proving the Pythagorean theorem gives you a relationship that’s reliable, fast to evaluate, and general. Proving individual tuples gives you none of this. That doesn’t even touch on how theorems give us a glimpse at deeper structure and truths. Proving a bunch of right-triangle tuples will probably never lead you to the rest of the identities in trig.
- fn-mote 4mo ago
  The current commentators are surely missing the fact that this comment is sarcastic.
- SiempreViernes 4mo ago
  You meant this as satire, right?
- SJC_Hacker 4mo ago
  The cardinality of that “finite set” could end up larger than say, the number of particles in the entire universe
pfortuny 4mo ago
Much (most?) of math consists in transmission of it (according to Thurston [1]), a 1000-page proof with no possibility of transmission is mostly useless. The proof of Fermat's last Theorem is important in itself, and adds much more than the mere result. I am not talking about the supposed "beauty" of a proof (I do not believe in that concept, rather in "elegance", which is not the same), I am talking about the proof itself, and the insights it provides. [1] https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994...
- simianwords 4mo ago
  You are mixing a lot of categories here -- beauty, verbosity, utility, elegance, insights. Why all that when you just need one thing: truth.
- cman1444 4mo ago
  What is the difference between "beauty" and "elegance" of a proof?
  - pfortuny 4mo ago
    "Beauty" is something I cannot define. "Elegance", as I use it, is the use of tools as precisely as possible. It is a technical term, whereas "beauty" I cannot define. Of course, that is my view of it.
    - menaerus 4mo ago
      When writing code I also believed in the "beauty" and "elegance" because IMO it opens up all kinds of different opportunities that involve using or organically growing or improving that code. It turns out that if it doesn't solve a quantifiable problem, (mostly) nobody cares. And the pace of innovating in the field outgrows the pace (by a large) of keeping things "beautiful and elegant".
    - breezybottom 4mo ago
      Maybe English isn't your first language, but these are basically synonyms.
slopinthebag 4mo ago
> Why does it need to be beautiful? “Beauty will save the world”
layer8 4mo ago
You want to understand why it’s true, and that often correlates with beauty.
- simianwords 4mo ago
  How is this relevant here? AI helps you understand the why -- it literally discovers the proof and hands it to you with explanations. It hands you the proof that you would have otherwise not found easily.
  - layer8 4mo ago
    If the proof is hundreds or thousands of lines of Lean, it’s not clear that the AI will be able to provide an insightful “why”, instead of just dozens of microsteps. And if it can provide insightful “whys”, that still correlates with beauty then. Given the slop-like nature of what current generative AI tends to produce, I wouldn’t however count on the latter quite yet.
    - simianwords 4mo ago
      I don't know how you think it only gives you Lean - it gives you everything including the explanation. You can actually ask it explanation using you know.. natural language. > And if it can provide insightful “whys”, that still correlates with beauty then. Yeah it can, you just have to ask it. It has a good interface for it - text! I think you misunderstand how this tech works, its not just spitting out things. It has the understanding also and you can verify it by asking!
  - UcatnapnSula 4mo ago
    [dead]
alasr 4mo ago
> Why does it need to be beautiful? "Beauty", IMO, signifies the idea that you're doing `something` for its own the sake where "its own sake" approximate the idea of getting/being closer to (or in proximity of) `something`/`anything`/`someone` you find "beautiful". > Once you proved it it's true and you can use its consequences in math, sciences and engineerings (sic). The expression "you can use its consequences in ..." suggests that the action is a "just a means" to "something else". However, not everyone is interested in the idea of "something else"; they're interested in the idea itself (in a broad sense) as that's one of the main reason they got started/involved in the first place. --- We all do things as "just a means" to "something else". However, there must be an "end" to this chain of "something else"; otherwise, how do you find any "meaning" (or sense of fulfillment) in this whole enterprise (or chain of "something else"s)?
jvvw 4mo ago
The vast majority of research-level pure mathematics is never going to get used in science or engineering. Obviously it is hard to predict what will be useful, but for the type of mathematics that is unlikely to be, there is a question as to why we care about it, and that almost has to come down to beauty in some sense - some mathematics gives us a new lens to look at parts of the mathematical world and others chip away at problems in more mundane ways in the hope that they inspire or contribute to new parts of mathematics that are beautiful.

zerobees 4mo ago

Outside of some niche specializations like cryptography, math isn't practiced because of "consequences". Most mathematicians take pride in their work not having any obvious practical applications. They're also overwhelmingly working in university settings where they're not expected to generate revenue or deliver practical results. We basically subsidize the practice of mathematics as an art form, and if you try to take the artistry away, you might find that the artists don't want to play along. And I guess you can imagine future robo-math production lines without any human involvement, and then LLMs finding applications for the resulting theorems, but it's not possible today.

setopt 4mo ago
Are you sure that’s «most» mathematicians? At the universities I’ve been to (as a student and now faculty), «applied mathematics» and «statistics» have been the two largest divisions. But perhaps that’s a bias from engineering-heavy universities?
- jubilanti 4mo ago
  "Applied Math" and "Statistics" are distinct fields from "Mathematics," not subfields of it. People in those two departments are often closer to Computer Science or the statistics subfield in a domain science field (e.g. biostatistics, econometrics) than to Mathematics in terms of what they actually teach and research.
  - setopt 4mo ago
    That is perhaps fair, is that distinction common internationally? Again, in the universities I’ve been to, «applied math» and «statistics» have generally been placed under the department of mathematics. I myself am a physicist, and consider applied physics, biophysics, etc. to be subfields of physics and not distinct fields of study, but I don’t know what outer physicists think.
bigmadshoe 4mo ago
You put it perfectly. And all these AI math startups don't actually care about mathematics. They are just using it as a proxy for general reasoning, with the VC pitch being some kind of world domination after they crack these problems.
chermi 4mo ago
Most mathematicians don't take pride in their results having no applications. That's just not true. Maybe some quirky pure logicians or something. But otherwise 90%+* of mathematicians I know would be at least satisfied if not thrilled for their work to be used by others. *Completely made up statistic.
- limflick 4mo ago
  I've had the opposite experience with all the math nerds that I know. End of the day it's all anecdotal ig https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology

bwestergard 4mo ago

Why prove the Pythagorean theorem rather than just prove 3^2 + 4^2 = 5^2? For any practical application, you are only interested in finite set of concrete identities, so anything beyond that is surplus to requirements, surely?

spacemanspiffii 4mo ago
I think you may be interested in more abstract things. In this case, let's say you're creating a program for a 3D printed thing, and you have to fit a diagonal cardboard in a rectangular box, you'd like to be sure that the Pythagorean theorem holds even in cases where you haven't tried it out.
moregrist 4mo ago
> For any practical application, you are only interested in finite set of concrete identities I do a lot of numerical work in settings where computational efficiency is useful. In my work, most cases you can do numerically using integration or Monte Carlo sampling or whatever. It’s slow. It often pays to find a closed-form solution. Even if it’s just a starting point that needs refinement. To put in terms of the Pythagorean theorem: Proving the Pythagorean theorem gives you a relationship that’s reliable, fast to evaluate, and general. Proving individual tuples gives you none of this. That doesn’t even touch on how theorems give us a glimpse at deeper structure and truths. Proving a bunch of right-triangle tuples will probably never lead you to the rest of the identities in trig.
fn-mote 4mo ago
The current commentators are surely missing the fact that this comment is sarcastic.
SiempreViernes 4mo ago
You meant this as satire, right?
SJC_Hacker 4mo ago
The cardinality of that “finite set” could end up larger than say, the number of particles in the entire universe

pfortuny 4mo ago

Much (most?) of math consists in transmission of it (according to Thurston [1]), a 1000-page proof with no possibility of transmission is mostly useless. The proof of Fermat's last Theorem is important in itself, and adds much more than the mere result. I am not talking about the supposed "beauty" of a proof (I do not believe in that concept, rather in "elegance", which is not the same), I am talking about the proof itself, and the insights it provides. [1] https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994...

simianwords 4mo ago
You are mixing a lot of categories here -- beauty, verbosity, utility, elegance, insights. Why all that when you just need one thing: truth.
cman1444 4mo ago
What is the difference between "beauty" and "elegance" of a proof?
- pfortuny 4mo ago
  "Beauty" is something I cannot define. "Elegance", as I use it, is the use of tools as precisely as possible. It is a technical term, whereas "beauty" I cannot define. Of course, that is my view of it.
  - menaerus 4mo ago
    When writing code I also believed in the "beauty" and "elegance" because IMO it opens up all kinds of different opportunities that involve using or organically growing or improving that code. It turns out that if it doesn't solve a quantifiable problem, (mostly) nobody cares. And the pace of innovating in the field outgrows the pace (by a large) of keeping things "beautiful and elegant".
  - breezybottom 4mo ago
    Maybe English isn't your first language, but these are basically synonyms.

slopinthebag 4mo ago

> Why does it need to be beautiful? “Beauty will save the world”

layer8 4mo ago

You want to understand why it’s true, and that often correlates with beauty.

simianwords 4mo ago
How is this relevant here? AI helps you understand the why -- it literally discovers the proof and hands it to you with explanations. It hands you the proof that you would have otherwise not found easily.
- layer8 4mo ago
  If the proof is hundreds or thousands of lines of Lean, it’s not clear that the AI will be able to provide an insightful “why”, instead of just dozens of microsteps. And if it can provide insightful “whys”, that still correlates with beauty then. Given the slop-like nature of what current generative AI tends to produce, I wouldn’t however count on the latter quite yet.
  - simianwords 4mo ago
    I don't know how you think it only gives you Lean - it gives you everything including the explanation. You can actually ask it explanation using you know.. natural language. > And if it can provide insightful “whys”, that still correlates with beauty then. Yeah it can, you just have to ask it. It has a good interface for it - text! I think you misunderstand how this tech works, its not just spitting out things. It has the understanding also and you can verify it by asking!
- UcatnapnSula 4mo ago
  [dead]

alasr 4mo ago

> Why does it need to be beautiful? "Beauty", IMO, signifies the idea that you're doing `something` for its own the sake where "its own sake" approximate the idea of getting/being closer to (or in proximity of) `something`/`anything`/`someone` you find "beautiful". > Once you proved it it's true and you can use its consequences in math, sciences and engineerings (sic). The expression "you can use its consequences in ..." suggests that the action is a "just a means" to "something else". However, not everyone is interested in the idea of "something else"; they're interested in the idea itself (in a broad sense) as that's one of the main reason they got started/involved in the first place. --- We all do things as "just a means" to "something else". However, there must be an "end" to this chain of "something else"; otherwise, how do you find any "meaning" (or sense of fulfillment) in this whole enterprise (or chain of "something else"s)?

jvvw 4mo ago

The vast majority of research-level pure mathematics is never going to get used in science or engineering. Obviously it is hard to predict what will be useful, but for the type of mathematics that is unlikely to be, there is a question as to why we care about it, and that almost has to come down to beauty in some sense - some mathematics gives us a new lens to look at parts of the mathematical world and others chip away at problems in more mundane ways in the hope that they inspire or contribute to new parts of mathematics that are beautiful.