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Fermat's Last Theorem – how it’s going

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Re: Fermat's Last Theorem – how it’s going

#51

Post by jerf »

gedpeck wrote: post #25
dboreham wrote: post #16

Earlier quoted context omitted.

And Mathematics is Computer Science. Took me most of a lifetime to realize this.

There are statements provably true about the natural numbers that can’t be proven in first order PA. Are such statements part of computer science? If so, how?

That question contains so many false or unnecessary assumptions that it would take far longer to unpack them than it took you to type them, so I will limit myself to the observation that we do not even remotely confine ourselves to first order anything in computer science, nor should we.

Re: Fermat's Last Theorem – how it’s going

#52

Post by enugu »

bee_rider wrote: post #7

This hurts my engineer-brain. Just a reminder than the gap between us and mathematicians is about the same as the gap between us and everybody else (edit: in terms of math skills), just in the other direction, haha. Oh well, hopefully when they get the machines to solve math, they’ll still want them to run a couple percents faster every year.

Don't think you should be intimidated just by reading the article itself. It is using several domain specific terms. But you would encounter that in many other contexts - for instance a group discussing an intricate board game you are seeing for the first time.

However, unlike board games where the concepts can be explained to you in a few minutes (usually it becomes clear only when you play), a lot mathematics especially algebraic geometry/number-theory can have many layers of dependencies which takes years to absorb.

It would be interesting to compare it to understanding a large open source project like the Linux kernel, well enough to contribute. I would say it is not so conceptually deep as mathematics of the article (while still having a few great ideas). But understanding the source would require familiarization with 'tedious details' which incidentally, what this article is also about.

So the issue, stated this way, is not so much raw talent as time and effort. This leads to the topic of motivation - finding something great in an idea can lead to investment in the subject. For those more talented, the journey might be easier.

Alan Kay's maxim is crucial - a change of perspective can simplify things relative to 80 extra IQ points. A long sequence of technical steps can be compressed into a single good intuitive definition/idea like the complexity of navigating a path becomes clear from a higher viewpoint.

Grothendieck, considered by many to be the best mathematician of the past century, made the point that there were several colleagues who were more proficient at symbolic manipulation. But, he was able to look at things with a more foundational perspective and could make great contributions.

Here's a good essay by Thurston "Proof and Progress in mathematics". https://arxiv.org/pdf/math/9404236

He discusses this problem of being intimidated by jargon.

Re: Fermat's Last Theorem – how it’s going

#54

Post by vouaobrasil »

impendia wrote: post #48
vouaobrasil wrote: post #24

> The experts are in agreement that the important ideas are robust enough to withstand knocks like this, but the details of what is actually going on might not actually be where you expect them to be. Past researcher in pure math here. The big problem is that mathematicians are notorious for not providing self-contained proofs of anything because there is no incentive to do so and authors sometimes even seem proud to…

Speaking as a current researcher in pure math -- you're right, but I don't think this is easily resolved. Math research papers are written for other specialists in the field. Sometimes too few details are provided; indeed I commonly gripe about this when asked to do peer review; but to truly provide all the details would make papers far, far longer. Here is an elementary example, which could be worked out by anyone w…

> but to truly provide all the details would make papers far, far longer.

I think what we need in math is that eventually, a collection of frequently cited and closely related papers should be rewritten into a much longer book format. Of course some people do that, but they hardly get much credit for it. It should be much more incentivated, and a core part of grant proposals.

Re: Fermat's Last Theorem – how it’s going

#55

Post by ykonstant »

impendia wrote: post #48
vouaobrasil wrote: post #24

> The experts are in agreement that the important ideas are robust enough to withstand knocks like this, but the details of what is actually going on might not actually be where you expect them to be. Past researcher in pure math here. The big problem is that mathematicians are notorious for not providing self-contained proofs of anything because there is no incentive to do so and authors sometimes even seem proud to…

Speaking as a current researcher in pure math -- you're right, but I don't think this is easily resolved. Math research papers are written for other specialists in the field. Sometimes too few details are provided; indeed I commonly gripe about this when asked to do peer review; but to truly provide all the details would make papers far, far longer. Here is an elementary example, which could be worked out by anyone w…

The example you gave, however, is obvious to every graduate student in any field that touches analysis or asymptotics. That is not the real problem; the real problem is proof by assertion of proof: "Lemma 4.12 is derived by standard techniques as in [3]; so with that lemma in hand, the theorem follows by applying the arguments of Doctorberg [5] to the standard tower of Ermegerds."

Too many papers follow this pattern, especially for the more tedious parts. The two problems are that it makes the lives of students and postdocs torture, and that the experts tend to agree without sufficient scrutiny of the details (and the two problems are intrinsically connected: the surviving students are "trained" to accept those kinds of leaps and proliferate the practice).

Frustratingly, I am often being told that this pattern is necessary because otherwise papers will be enormous---like I am some kind of blithering idiot who doesn't know how easily papers can explode in length. Of course we cannot let papers explode in length, but there are ways to tame the length of papers without resorting to nonsense like the above. For example, the above snippet, abstracted from an actual paper, can be converted to a proof verifiable by postdocs in two short paragraphs with some effort (that I went through).

The real motive behind those objections is that authors would need to take significantly more time to write a proper paper, and even worse, we would need actual editors (gasp!) from journals to perform non-trivial work.

Re: Fermat's Last Theorem – how it’s going

#56

Post by brendanyounger »

bell-cot wrote: post #41
boothby wrote: post #27

This reminds me of a fun experience I had in grad school. I was working on writing some fast code to compute something I can no longer explain, to help my advisor in his computational approach to the Birch and Swinnerton-Dyer conjecture. I gave a talk at a number theory seminar a few towns over, and was asked if I was doing this in hopes of reinforcing the evidence behind the conjecture. I said with a grin, "well, no…

> The crowd went wild; I've never made a group of experts so angry as... Also not a number theorist...but I'd bet those so-called experts had invested far, far too many of their man-years in that unproven conjecture. All of which effort and edifice would collapse into the dumpster if some snot-nosed little upstart like you, using crude computation , achieved overnight fame by finding a counter-example. (If I could gi…

To put a little color on the BSD conjecture, it states that the rank (0, 1, 2, 3, etc.) of rational points on an elliptic curve is related to the residue (coefficient of 1/q) of the L-function for the curve. There are some additional multiplicative factors, in particular the size of the Tate-Shafarevich group.

No one knows how to compute the size of that group in general (in fact no one has proved that it's finite!). Computing the rank of a curve via non-analytic means is more akin to a bespoke proof than a straightforward computation (see Noam Elkies' work).

So saying you're going to disprove BSD with blind computation is rather naive unless you're sitting on several career-defining proofs and not sharing them.

Re: Fermat's Last Theorem – how it’s going

#57

Post by zozbot234 »

impendia wrote: post #48
vouaobrasil wrote: post #24

> The experts are in agreement that the important ideas are robust enough to withstand knocks like this, but the details of what is actually going on might not actually be where you expect them to be. Past researcher in pure math here. The big problem is that mathematicians are notorious for not providing self-contained proofs of anything because there is no incentive to do so and authors sometimes even seem proud to…

Speaking as a current researcher in pure math -- you're right, but I don't think this is easily resolved. Math research papers are written for other specialists in the field. Sometimes too few details are provided; indeed I commonly gripe about this when asked to do peer review; but to truly provide all the details would make papers far, far longer. Here is an elementary example, which could be worked out by anyone w…

> there exist constants C, X > 0 such that for every real number x > X, we have

>> log(x^2 + 1) + sqrt(x) + x/exp(sqrt(4x + 3)) These problems are only "uninteresting" to the extent that they can be "proven" by automated computation. So the interesting part of the problem is to write some completely formalized equivalent to a CAS (computer algebra system - think Mathematica or Maple, although these are not at all free from errors or bugs!) that might be able to dispatch these questions easily. Formalization systems can already simplify expressions in rings or fields (i.e. do routine school-level algebra), and answering some of these questions about limits or asymptotics is roughly comparable.

Re: Fermat's Last Theorem – how it’s going

#58

Post by jhanschoo »

graycat wrote: post #28

This thread seems to be about good writing for math. Okay, for some decades, I've read, written, taught, applied, and published, in total, quite a lot of math. Got a Ph.D. in applied math. Yes, there are problems in writing math, that is, some math is poorly written. But, some math is quite nicely written. (1) Of course, at least define every symbol before using it. (2) It helps to motivate some math before presentin…

This isn't just about good writing for math; the post author was trying to verify FLT as is developed in the literature, and along the way they discovered that a lemma underpinning a whole subfield is untrue, as was used. They nevertheless have confidence that the subfield is largely salvageable, by virtue of the faith that if it were bogus, someone would have already found negative results.

But now they had to find a suitable replacement to underpin the field.

Re: Fermat's Last Theorem – how it’s going

#59

Post by boothby »

bell-cot wrote: post #41
boothby wrote: post #27

This reminds me of a fun experience I had in grad school. I was working on writing some fast code to compute something I can no longer explain, to help my advisor in his computational approach to the Birch and Swinnerton-Dyer conjecture. I gave a talk at a number theory seminar a few towns over, and was asked if I was doing this in hopes of reinforcing the evidence behind the conjecture. I said with a grin, "well, no…

> The crowd went wild; I've never made a group of experts so angry as... Also not a number theorist...but I'd bet those so-called experts had invested far, far too many of their man-years in that unproven conjecture. All of which effort and edifice would collapse into the dumpster if some snot-nosed little upstart like you, using crude computation , achieved overnight fame by finding a counter-example. (If I could gi…

> All of which effort and edifice would collapse into the dumpster if some snot-nosed little upstart like you, using crude computation, achieved overnight fame by finding a counter-example.

Not at all. In fact, if I had found a counterexample, it would cause a flurry of new research to quantify exactly how wrong the BSD conjecture is. Such a finding would actually be a boon to their career! That's why my response is curiosity, and not to sneer at them for protecting their extremely secure tenured careers.

Edit 1: And if you think you've found a counterexample to a long-standing conjecture with a computation, you'd better be damned sure that your computation is correct before opening your mouth in public. And that takes a ton of work in the case of the BSD conjecture, because you've almost certainly identified a bug in the extremely complex code underlying that computation. If I ever thought I was holding onto such a counterexample, I'd approach a human calculator like Ralph Greenberg as my first step (after internal checks: re-running the code on another computer to rule out cosmic bit flips, and perhaps running more naive, unoptimized implementations).

Edit 2: This attitude pervades my software development career, and I've brought it to my foray into superconducting circuit design: a bug report brings joy to my life, and I aim to shower the reporter with praise (which may involve chocolate). There is nothing more satisfying than being proven wrong, because it helps us collectively move toward greater truths.

Re: Fermat's Last Theorem – how it’s going

#60

Post by impendia »

ykonstant wrote: post #55
impendia wrote: post #48

Earlier quoted context omitted.

Speaking as a current researcher in pure math -- you're right, but I don't think this is easily resolved. Math research papers are written for other specialists in the field. Sometimes too few details are provided; indeed I commonly gripe about this when asked to do peer review; but to truly provide all the details would make papers far, far longer. Here is an elementary example, which could be worked out by anyone w…

The example you gave, however, is obvious to every graduate student in any field that touches analysis or asymptotics. That is not the real problem; the real problem is proof by assertion of proof: "Lemma 4.12 is derived by standard techniques as in [3]; so with that lemma in hand, the theorem follows by applying the arguments of Doctorberg [5] to the standard tower of Ermegerds." Too many papers follow this pattern,…

I strongly agree with you, in the sense that many papers provide far fewer details than they should, and reading them is considered something of a hazing ritual for students and postdocs. (I understand that this is more common in specialties other than my own.)

The blog post seems to be asserting a rather extreme point of view, in that even the example I gave is (arguably!) unacceptable to present without any proof. That's what I'm providing a counterpoint to.

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