Number theorist fears many proofs widely considered to be true are wrong • Hacker News Board

Re: Number theorist fears many proofs widely considered to be true are wrong

#71

Post by knzhou » Sun, Sep 29, 2019, 6:51 PM UTC

thanhhaimai wrote: post #32

knzhou wrote: post #25

Earlier quoted context omitted.

That doesn’t mean that it’s wrong, it means that you need to read more. In a lot of fields, undergraduate material doesn’t qualify you to read papers, a PhD does. If all papers actually were written to undergraduate level, many would be 500 pages long. Who has time to write a book-length exposition of the basics of their field every time they publish? I mean, have you ever published a technical work? If so, did you g…

The above statement is true with the assumption that the topic being discussed is truly above the level of understanding for undergraduates. For the majority of the papers, that assumption is not true though. Papers are hard to read has more to do with the idea that if it's not filled with jargons, its chance of getting published is greatly reduced. Papers are not rewarded for being easy to understand. People/organiz…

Depends on the field. Many CS papers can be understood by undergrads because CS is a very young and very broad field; the path to the frontier in any direction is short. Theoretical physics and math are not like that. They have been building cumulatively for centuries.

Re: Number theorist fears many proofs widely considered to be true are wrong

#72

Post by lolc » Sun, Sep 29, 2019, 6:53 PM UTC

chongli wrote: post #59

hannob wrote: post #38

Earlier quoted context omitted.

So I find this interesting, because it seems to confirm a suspicion I held. I "only" did some undergrad math during my computer science (but due to a lack of specialization at my university back then it was "real math", i.e. we took the same courses as the math students). My impression often was that a lot of this stuff isn't so hard if you "get to it", but it's clouded in a lot of complicated language that makes it…

The problem with math is that the only strategy we’ve discovered (with any success) for making a concept easier to learn is to connect it with a concrete, real-world analogy. This works because people find it easier to learn something if they can connect it to something else they already know. The Achilles heel of this approach is that it discards the abstract form of the concept such that the full generality is lost…

People struggle with monads because it's a two-step process to learn them: Learn their mathematical properties, then learn how to apply them as a pattern. That's not a quick process you can do in half an hour. It doesn't help that the concept is alien to everyday processes. So yeah, analogy doesn't get you far with monads.

Re: Number theorist fears many proofs widely considered to be true are wrong

#73

Post by Hasz » Sun, Sep 29, 2019, 6:53 PM UTC

knzhou wrote: post #25

Hasz wrote: post #9

Headline is waaaay overblown, but not all of it is total hyperbole -- checkout out arxiv and try to read the abstract from just about any paper. It's basically totally impermeable. I'm almost done with an undergraduate degree in math and basically have no idea what ~90% of the research is about at anything other than a topical level. This is fine, it's written for specialists in the field (hopefully), but damn, for m…

That doesn’t mean that it’s wrong, it means that you need to read more. In a lot of fields, undergraduate material doesn’t qualify you to read papers, a PhD does. If all papers actually were written to undergraduate level, many would be 500 pages long. Who has time to write a book-length exposition of the basics of their field every time they publish? I mean, have you ever published a technical work? If so, did you g…

Not trying to imply it's wrong -- if anything, research should be written to a professional level. I also don't think the background should take you from high school calc to topology or differential geometry in one fell sweep.

Re: Number theorist fears many proofs widely considered to be true are wrong

#74

Post by dorchadas » Sun, Sep 29, 2019, 7:03 PM UTC

imglorp wrote: post #55

Hasz wrote: post #9

Headline is waaaay overblown, but not all of it is total hyperbole -- checkout out arxiv and try to read the abstract from just about any paper. It's basically totally impermeable. I'm almost done with an undergraduate degree in math and basically have no idea what ~90% of the research is about at anything other than a topical level. This is fine, it's written for specialists in the field (hopefully), but damn, for m…

Software is very much in crisis because of this exact problem. Most of us build on top of a mountain of existing work; sometimes just setting a pebble on top and we have a new thing. But no human alive has seen, could remember, or could understand, each and every behavior down in that mountain, from pixels on the screen to pulses on the phy all the way to gates inside their processors. So you read about the APIs you…

In this case, it likely applies to every aspect of modern day life. Man is no longer a generalist, and we are forced to build upon the foundations others have put, and hope they work.

Re: Number theorist fears many proofs widely considered to be true are wrong

#75

Post by thethirdone » Sun, Sep 29, 2019, 7:10 PM UTC

hyperbovine wrote: post #35

knzhou wrote: post #25

Earlier quoted context omitted.

That doesn’t mean that it’s wrong, it means that you need to read more. In a lot of fields, undergraduate material doesn’t qualify you to read papers, a PhD does. If all papers actually were written to undergraduate level, many would be 500 pages long. Who has time to write a book-length exposition of the basics of their field every time they publish? I mean, have you ever published a technical work? If so, did you g…

A B.A. in math gets you to the forefront of mathematical research as it was circa 1912. (Source: https://slate.com/human-interest/2003/05/is-math-a-young-man... ) This is purely my impression as somebody who switched away from pure math early in their career, but: amazingly, in pure math, unless you are really advanced, you are not even that close to engaging independently in cutting-edge research until well after yo…

> The same could be said for physics and various other "classical" scientific fields.

I think mathematics is in a different class than physics or other scientific endeavors because there is meaningful mathematics going back to ancient Greece whereas I am not aware of any physics that was known thousands of years ago that is still relevant.

Re: Number theorist fears many proofs widely considered to be true are wrong

#76

Post by LolWolf » Sun, Sep 29, 2019, 7:35 PM UTC

knzhou wrote: post #69

LolWolf wrote: post #67

Earlier quoted context omitted.

I think the GP is trying to say that, once you get past all of the definitions and proofs, the underlying idea is almost always relatively simple. I do agree that it's pretty hard to be rigorous without setting up all of this edifice, but I've also found that it's very easy (at least, as an applied mathematician) to take an idea and continue generalizing it until the original application becomes a very small corollar…

People will complain about that too. The introduction is "redundant" or "trivial". The book is a long, bloated slog. (Since what you're proposing will unavoidably increase the page count.) The author intentionally confuses students by alluding to some "main theorem" but waiting a hundred pages to say what it is, instead of putting it in front where it belongs. For every complaint about a textbook, there is an equal a…

I mean, perhaps; though I've found it does quite the opposite. The idea is proposed front and center, sure (this approach does not preclude that), but far less time is spent explaining the motivation and doing mathematical yoga to clarify steps in a proof than is spent providing clear examples and sensible constructions.[1]

I am not, of course, saying that all textbooks (or papers) will be better served by this structure, but I think a surprisingly large amount are. This may also be a consequence of the field I'm in, as well, where most statements I've made have several natural generalizations which are all relatively straightforward to prove, but presenting them all under one large umbrella with a very general construction makes the end result much harder to understand.

Either way, both ways have their merits, but I've (along with several of my peers and editors) found one much harder to work with and read than the other. Though, I will again mention that this may be due to the specific fields I work in.

--

[1] I want to be clear that I'm also not advocating for the "big reveal finish" which many papers (and some terrible textbooks) fail miserably at. Neither of these mediums is a novel. But I am saying that some approach motivating the construction and building the picture in a natural way (instead of a here's the theorem, let's explain why everything is a trivial corollary of it) lends itself to more clear expositions.

Perhaps either presentation can be made clear, but I've found it's very, very difficult to make the latter work in a way that readers can easily understand, while the former falls almost naturally from most research notes I've written and keeps people more engaged.

Re: Number theorist fears many proofs widely considered to be true are wrong

#77

Post by YeGoblynQueenne » Sun, Sep 29, 2019, 7:36 PM UTC

higherkinded wrote: post #31

Suggestion to rely on AI for proof verification is just laughable. Neuron weights instead of formal definitions. So reliable.

Proof assitants have nothing to do with neural nets or optimisation. They are entirely determinitsic and return an absolute "true" or "false" result, not an approximation.

https://en.wikipedia.org/wiki/Proof_assistant

This is Good Old-Fashioned AI, through and through.

Re: Number theorist fears many proofs widely considered to be true are wrong

#78

Post by [deleted] » Sun, Sep 29, 2019, 7:42 PM UTC

robinhouston wrote: post #40

Ceezy wrote: post #28

"I believe that no human, alive or dead, knows all the details of the proof of Fermat’s Last Theorem. But the community accept the proof nonetheless," Buzzard wrote in a slide presentation[...] That s 100% clickbait. People knows the demonstration and the demonstration helped to create entire new fields. By the way there are new and shorter proof of the theorem.

There’s a great discussion of this on Buzzard’s blog. https://xenaproject.wordpress.com/2019/09/27/does-anyone-kno... Read the comments too.

[deleted]

Re: Number theorist fears many proofs widely considered to be true are wrong

#79

Post by ncmncm » Sun, Sep 29, 2019, 7:53 PM UTC

mathgenius wrote: post #57

hannob wrote: post #38

Earlier quoted context omitted.

So I find this interesting, because it seems to confirm a suspicion I held. I "only" did some undergrad math during my computer science (but due to a lack of specialization at my university back then it was "real math", i.e. we took the same courses as the math students). My impression often was that a lot of this stuff isn't so hard if you "get to it", but it's clouded in a lot of complicated language that makes it…

Absolutely. It's a problem with academia in general. Academics don't get rewarded for explaining things in an easy to understand way. Moreover, it is often a huge amount of work to distill results down like this. And if you succeed the response you may get is "well that is obvious". Anyway, this is my counter argument to the "formalization of mathematics": why don't we incentivize people to bring some clarity to the…

In other fields, writing survey articles is (or was, recently) an appreciated activity.

Survey articles are explicitly supposed to bring people up on the important results and current state of a subject.

Re: Number theorist fears many proofs widely considered to be true are wrong

#80

Post by chongli » Sun, Sep 29, 2019, 7:59 PM UTC

lolc wrote: post #72

chongli wrote: post #59

Earlier quoted context omitted.

The problem with math is that the only strategy we’ve discovered (with any success) for making a concept easier to learn is to connect it with a concrete, real-world analogy. This works because people find it easier to learn something if they can connect it to something else they already know. The Achilles heel of this approach is that it discards the abstract form of the concept such that the full generality is lost…

People struggle with monads because it's a two-step process to learn them: Learn their mathematical properties, then learn how to apply them as a pattern. That's not a quick process you can do in half an hour. It doesn't help that the concept is alien to everyday processes. So yeah, analogy doesn't get you far with monads.

I think most monad tutorials skip over the math and try to teach people the applications and I think this is why people don’t get very far. But then people are probably not ready for the math if they haven’t gone beyond high school algebra. Very few high school students learn about abstract algebra topics such as group theory unless they’re studying on their own.

Aside: as with monads, group theory has many disparate applications, from solving Rubik’s cubes [1] to particle physics [2] to crystallography [3]. Trying to teach someone group theory using only a Rubik’s cube may not be the best idea, in light of the abstract nature of the structure.

[1] https://en.wikipedia.org/wiki/Rubik%27s_Cube_group

[2] https://en.wikipedia.org/wiki/Particle_physics_and_represent...

[3] https://en.wikipedia.org/wiki/Crystallographic_point_group