Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Some stuff I found interesting about number theory research

twitter.com

Post reply on HN
Print view View original

61–70 of 118 posts

Re: Some stuff I found interesting about number theory research

#61

Post by webmaven »

rocqua wrote: post #28
themodelplumber wrote: post #8

One of my takeaways: When I see "stitch together black boxes," to me that says that aspects of academic math are becoming similar to opportunistic software development. When such an effect happens in group psychology it is typically part of the "grounding" effect happening. There is a high supply of theory without the energy to go into analyzing so many individual black boxes anymore, and a correspondingly high deman…

I think the reason for this black-boxing in software and mathematics is quite different. In top-level mathematics its that you have no choice but to treat results as black-boxes because the complexity otherwise is just too high. Its the only way forward, and mathematicians would prefer not to take this approach. In software development, black-boxing is a way of accelerating development. It allows people to abstract a…

But "accelerating development" and "lowering complexity" are, if not synonymous, then at least highly correlated.

Re: Some stuff I found interesting about number theory research

#62

Post by hnfong »

laichzeit0 wrote: post #13
renewiltord wrote: post #10

I wonder if they have the equivalent the engineers who bemoan the fact that some React developers make functioning websites without “knowing anything about how CPUs or memory models work”. Bourbaki was a thing so probably. And we know the joke about Bourbaki and Lang. Though the reference is diminished by the latter’s relationship with the former.

It’s higher than that. People build software calling .search() and .sort() after a 4 month nanodegree and don’t know how to implement either. I’d say they are even using http requests without knowing what a TCP socket connection is, maybe even what an http request header is. The barrier to entry in the dev world is extremely low. To relate it to maths: It would be equivalent to someone doing math research without kno…

I'm not a mathematician, but in the rare occasion I have to use the Pythagoras theorem, I use it without knowing how to prove it.

And that's OK.

(Mathematicians are probably too polite to moan about amateur plebs like me who didn't even learn primary school mathematics properly)

Re: Some stuff I found interesting about number theory research

#63

Post by webmaven »

linspace wrote: post #26
throwaway81523 wrote: post #18

Earlier quoted context omitted.

There is mathoverflow.net and its meta site as math researcher hangouts, plus ncatlab.org for category theory, and various sites, blogs, mailing lists etc. for other fields. You might also like Bill Thurston's famous article "On Proof and Progress in Mathematics", which says not so much that math is social, but that human mathematicians want personal understanding of mathematical truths. So black box computer-checked…

A little offtopic: It's telling how you can tell the article was written by a mathematician, apart from the obvious fact that it's about mathematics. I'm talking about the structure. For example: "How do mathematicians prove theorems? This question introduces an interesting topic, but to start with it would be to project two hidden assumptions: (1) that there is uniform, objective and firmly established theory and pr…

There are variatons on this joke that are even more apropos to HN, for example:

A man is flying in a hot air balloon and realizes that he is lost. He reduces height and spots a man down below. He lowers the balloon further and shouts, "Excuse me. Can you help me? I promised a friend I would meet him half an hour ago, but I don't know where I am."

The man below says, "Yes. You are in a hot air balloon, hovering approximately 30 feet above this field. You are between 40 and 42 degrees north latitude, and between 58 and 60 degrees west longitude.

"You must be an engineer" says the balloonist. "I am", replies the man. "How did you know?"

"Well..." says the balloonist. "Everything you told be was technically correct, but I have no idea what to make of your information and the fact is I am still lost."

The man below says, "You must be a Manager"

"I am", replies the balloonist. "How did you know?"

"Well..." says the man. "You don't know where you are, or where you are going. You made a promise which you have no idea how to keep, and you expect me to solve your problem. The fact is you are in the exact same position you were in before we met but now it is somehow my fault."

Re: Some stuff I found interesting about number theory research

#65

Post by pramodbiligiri »

mhh__ wrote: post #58
jkhdigital wrote: post #45

Earlier quoted context omitted.

Are you familiar with homotopy type theory? One of its proponents, Fields medalist Vladimir Voevodsky, has stated that his contributions are part of a “personal mission” to bring mathematics into a new age of formal verification [1]. I’d be curious to know how this compares to LEAN. [1] https://www.ias.edu/ideas/2014/voevodsky-origins

I am kind of sceptical of these verified-mathematics projects in the sense that my litmus is whether I could easily verify my workings rather than the proofs I rely upon, but nonetheless I have enormous respect for those doing this work and really hope it comes to fruition. I really hope this work trickles down to our programming - state machine problems are basically "solved" (for my needs at least), but more comple…

When you say that state machine problems are basically solved, are you referring to any tools that you're using to reason about them?

Re: Some stuff I found interesting about number theory research

#66

Post by andai »

caddemon wrote: post #59
hyperbovine wrote: post #41

Earlier quoted context omitted.

> He responded that it was because people secretly "proved" everything to themselves in multiple ways, but only wrote up one proof. I don't do pure math, but I write the occasional theory paper, and this resonates. So much ends up on the cutting room floor--usually you proved the key result three or four different ways before finding a proof that is actually incisive/aesthetically pleasing/whatever to justify signing…

As a student working on some theory papers, I wish this kind of thing would go in an Appendix somewhere. Is it left out because it's not worth the effort to include compared to the number of people that would actually read it? Or is it something that runs the risk of negative perception by those established in the field?

It does seem like including a few alternative proofs would greatly aid in understanding.

Re: Some stuff I found interesting about number theory research

#67

Post by jkhdigital »

mhh__ wrote: post #58
jkhdigital wrote: post #45

Earlier quoted context omitted.

Are you familiar with homotopy type theory? One of its proponents, Fields medalist Vladimir Voevodsky, has stated that his contributions are part of a “personal mission” to bring mathematics into a new age of formal verification [1]. I’d be curious to know how this compares to LEAN. [1] https://www.ias.edu/ideas/2014/voevodsky-origins

I am kind of sceptical of these verified-mathematics projects in the sense that my litmus is whether I could easily verify my workings rather than the proofs I rely upon, but nonetheless I have enormous respect for those doing this work and really hope it comes to fruition. I really hope this work trickles down to our programming - state machine problems are basically "solved" (for my needs at least), but more comple…

I'm a PhD student in computer science, and after spending some time on this subject I'm pretty well convinced that parts of mathematics and CS are converging towards a more unified foundational framework that will benefit both fields tremendously.

In a nutshell, the key challenge is taking the core of complexity theory (which is easy to formalize in terms of state machines) and migrating it into the language(s) of category/type theory (which is much more modular/compositional).

A couple interesting recent works in this area:

Categorical Complexity: https://www.cambridge.org/core/services/aop-cambridge-core/c...

Dusko Pavlovic's Monoidal computer series: 1. https://arxiv.org/pdf/1208.5205 2. https://arxiv.org/pdf/1402.5687 3. https://arxiv.org/pdf/1704.04882

Re: Some stuff I found interesting about number theory research

#68

Post by jkhdigital »

alexilliamson wrote: post #50
jkhdigital wrote: post #46

Earlier quoted context omitted.

This is an explicit goal of one of of the creators of homotopy type theory, Fields medalist Vladimir Voevodksy: https://www.ias.edu/ideas/2014/voevodsky-origins Apparently he now uses Coq in his everyday work.

He's been dead for four years?

Oh snap, I had no idea that he died! Wow... that's a profound loss for the field of mathematics.

Re: Some stuff I found interesting about number theory research

#69

Post by wbl »

hnfong wrote: post #62
laichzeit0 wrote: post #13

Earlier quoted context omitted.

It’s higher than that. People build software calling .search() and .sort() after a 4 month nanodegree and don’t know how to implement either. I’d say they are even using http requests without knowing what a TCP socket connection is, maybe even what an http request header is. The barrier to entry in the dev world is extremely low. To relate it to maths: It would be equivalent to someone doing math research without kno…

I'm not a mathematician, but in the rare occasion I have to use the Pythagoras theorem, I use it without knowing how to prove it. And that's OK. (Mathematicians are probably too polite to moan about amateur plebs like me who didn't even learn primary school mathematics properly)

Behold! http://www.geom.uiuc.edu/~demo5337/Group3/Bhaskara.html

It's so pretty everyone should know it.

Re: Some stuff I found interesting about number theory research

#70

Post by Hermitian909 »

RajuVarghese wrote: post #15

In a book on mathematics by Alex Bellos he says that any contemporary cutting-edge research mathematics paper can be understood by no more than 100 mathematicians. The field is getting bigger and bigger and there are less and less mathematicians that have background knowledge sufficient to _understand_it much less critique it. The bar for the word 'understand' may be much higher in mathematics compared to other field…

I think the main thing missing from this analysis is how much of the mathematical corpus ends up being "uninteresting" over time.

Take Algebraic Geometry, a field primarily concerned with the question "what are the zeroes of multivariate polynomial equations" e.g. x^2-y+z^3. Papers that could be considered part of the field were published as early as the 16th century but very little is worth reading from before the work of Alexander Groethendieck in the mid 20th century which really formalized the field in the form it is today. The vast improvement in abstractions meant old proofs could be rewritten in a much terser, easier to understand manner with the new machinery.

Similarly, now that we've solved the classification of finite simple groups[0] a lot of the deep expertise mathematicians had surrounding techniques to classify finite groups is no longer needed. A new researcher can learn a small, curated subset of these techniques and fully grasp the field. Since there's no more work to be done, there's no need to build deep intuitions.

More generally a lot of papers in mathematics are attempts to build the machinery to solve some bigger problem and most of those end up not being useful when the field cracks said problem. At which point, all those papers effectively get "trimmed away" since the result they were meant to support does not depend on them, and so no one need read them.

[0] https://en.wikipedia.org/wiki/Classification_of_finite_simpl...

Post reply on HN