Earlier quoted context omitted.
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I don't even understand how you interpreted that to have anything to do with being "smarter or making more money". Get some more coffee dude.
Maybe less coffee :)
Fermat's Last Theorem – how it’s going
91–100 of 216 posts
Re: Fermat's Last Theorem – how it’s going
#92Earlier quoted context omitted.
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.
Would it take too long or is it just that the margin is too small?
Re: Fermat's Last Theorem – how it’s going
#93> 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…
> I don't think this is easily resolved.
It is technically resolvable. Just insist on Lean (or some other formal verifier) proofs for everything.
It looks to me like math is heading that way, but a lot of mathematicians will have to die before its the accepted practice.
Mathematicians who are already doing formal proofs are discovering it have the same properties as shared computer code. Those properties have have lead to most of the code executed on the planet being open source source, despite the fact you can't make money directly from it.
Code is easy to share, easy to collaborate on, and in the case of formal proofs easy to trust. It is tedious to write, but collaboration is so easy that publishing your 1/2 done work will often prompt others to do some of the tedious stuff. Code isn't self documenting unless it's very well written, but even horrible code is far better at documenting what it does than what you are describing.
Re: Fermat's Last Theorem – how it’s going
#94Earlier 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,…
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).
I think this practice happens in many specialized fields. The thing with math is that the main problem is the publications become inaccessible. But when you have the same sort of thing in a field where the formulations assumptions that aren't formalizable but merely "plausible" (philosophy or economics, say), you have these assumptions introduced into the discussion invisibly.
Re: Fermat's Last Theorem – how it’s going
#95Re: Fermat's Last Theorem – how it’s going
#96Earlier quoted context omitted.
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!).…
If the BSD rank conjecture were false, then the simplest counterexample might be an elliptic curve with algebraic rank 4 and analytic rank 2. This could be established for a specific curve by rigorously numerically computing the second derivative of the L-series at 1 to some number of digits and getting something nonzero (which is possible because elliptic curves are modular - see work of Dikchitser). This is a strai…
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Re: Fermat's Last Theorem – how it’s going
#97Earlier quoted context omitted.
> 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…
> my foray into superconducting circuit design
Curious, what do you work on? (I also research superconductivity.)
Re: Fermat's Last Theorem – how it’s going
#98> 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…
Studied math a long time ago and one of my profs was proud about not going into the details. He said "Once you did something 100 times, you can go and say -as easily observable- and move on."
I was an algebra major at the turn of the century and I hated that attitude.
For the prof, yeah, easily observable. What about the students who try to absorb that particular article? You already have to balance the main topic in your brain, and you get these extra distractions on top of it.
Re: Fermat's Last Theorem – how it’s going
#99Richard Feynman, while still a student at Princeton, found an error in some well known proof, and set himself a rule to double check every theorem he would use.
I don't remember the details of the story (I read surely your joking years ago), and remember being amazed by how much time that policy must have cost him.
But now I wonder that he didn't hit dozens or hundreds of errors over the years.
Re: Fermat's Last Theorem – how it’s going
#100Earlier quoted context omitted.
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.
What false assumption did I make? I just pointed out a fact and asked a question. How many false assumptions could I have made with just one statement of fact and two questions? If you don’t have a valid answer to the question then don’t respond. I’m a mathematician and not a computer scientist. The first order PA axioms are recursively enumerable. Hence it’s clearly something of interest to computer scientists. The…
why should computer science be limited to things that are first order PA?