> is far from the real 110 pages long proof of FLT that took A.Whiles years to put together
He could've gotten away with the pun "took A. Wile to put together."
Feynman on Fermat's Last Theorem
41–50 of 113 posts
Re: Feynman on Fermat's Last Theorem
#42Earlier quoted context omitted.
I guess every false conjecture can be made to pass it. The trick is to make the set of items searched in large enough. For example, to show that no elephants exist, start with the (infinite) set of all possible chromosome sets. The proportion of them that produces an elephant is zero. QED. Examples from mathematics: The number 42 does not exist (logic: pick an integer. The probability that it equals 42 is zero. QED)…
Naw most of these wouldn't work when actually written out as Feynman did. Example: you can easily give an upper bound to the "chance that N is a prime" that goes to zero as N increases. But you would also need to show that it's sum from 0 to infinity over all N also goes to zero. In fact, there's the classic result that this probability is about 1/log(N) [1], which diverges towards infinity. Hence you would probabili…
That's because you start with a set (the integers) that contains disproportionally many primes. Just pick a larger set of numbers to compare things with. If you look at the reals, the probability of finding a rational already is zero, and there aren't primes that aren't rational numbers.
Mathematicians of course, would not do that. They do use the technique of estimating the solution size to get a feeling for the difficulty of a problem, but always try to pick a reference set that is such that the exercise teaches them anything, and starting with all reals doesn't (how do they know that? Intuition, or they may do it anyway, but realize half-way through that it is silly, or even publish it, and, eventually, get corrected)
Feynman's solution has more good math, but still makes the fatal mistake of stating that "measure zero implies does not occur" (although he probably knew, since he states it was good enough for him)
One can 'prove' the non-existence of any countable infinite set of numbers this way.
Re: Feynman on Fermat's Last Theorem
#43finding that the probability is extremely small doesn't really get you any closer to proving it. for such a hard to prove theorem, it makes sense the probability is small. that's why it was interesting in the first place. I find it hard to believe Feynman concluded from this that the theorem is probably correct. it only takes 1 case among an infinity to make the theorem false.
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Re: Feynman on Fermat's Last Theorem
#44Very cute argument, and very much in his style--he was famous, as the author notes, for heuristic arguments that weren't very formalizable but had a lot of beauty. One story I heard is that a computer scientist tried to explain the P=NP? problem to him; Feymnan couldn't understand why this was a problem. It was obviously true that P != NP, what even needed proving?
Per "Surely You're Joking" and the article, one of the cognitive techniques Feynman used was to keep a physical example, a demo, in his mind that conformed to the math being explained. I wonder if he used that for this, and what his model was.
Re: Feynman on Fermat's Last Theorem
#45Earlier quoted context omitted.
Naw most of these wouldn't work when actually written out as Feynman did. Example: you can easily give an upper bound to the "chance that N is a prime" that goes to zero as N increases. But you would also need to show that it's sum from 0 to infinity over all N also goes to zero. In fact, there's the classic result that this probability is about 1/log(N) [1], which diverges towards infinity. Hence you would probabili…
That's because you start with a set (the integers) that contains disproportionally many primes. Just pick a larger set of numbers to compare things with. If you look at the reals, the probability of finding a rational already is zero, and there aren't primes that aren't rational numbers. Mathematicians of course, would not do that. They do use the technique of estimating the solution size to get a feeling for the dif…
Your arguments are just fundamentally different from Feynman's. We're trying to estimate whether something exists. If you are using a counting measure on the integers, say, then whether or not something exists is whether or not the measure of that set is zero. If you're dealing with, say, a Lebesgue measure on the reals then the measure being zero tells you nothing about whether the set is empty. Put the counting measure on the reals and then you can work, but then the answers you get won't be zero.
Re: Feynman on Fermat's Last Theorem
#46The last quote on the page should be familiar to HN readers as a different version of "fail fast, fail often".
In other words, prefer breadth-first searches to depth-first searches.
Just today it occurred to me that I actually do depth first in some real-life situations I better shouldn't.
Re: Feynman on Fermat's Last Theorem
#47Earlier quoted context omitted.
The meaning here is: pick a positive integer N, what is the chance (aka probability) of it being an n-th power of another positive integer. And you are correct, this probability depends on N.
I don't get it. The probability must depend on how likely I am to select any specific integer, i.e. probability mass function of N. The probability cannot depend on the value of the random variable itself but could involve any parameters that define its distribution. I would consider something like the following a valid question: "Let N be a random integer between 0 and M-1 with uniform distribution. What is the prob…
Re: Feynman on Fermat's Last Theorem
#481e+33 is not such a big number as far as number theory goes (for example: http://mathoverflow.net/questions/15444/examples-of-eventual... , https://en.wikipedia.org/wiki/Skewes%27_number ). It's still a nice exercise. As another example, consider the question of whether there exists a right-angled triangle with rational sides, having an area of 157.
If I understand it right, it's a number with which he could have estimated if it's worth using a computer to perform a brute force search for a possible solution.
Once he had 1e-33 for all n > 100 that could mean that even trying with 1M computers where each tries 10G solutions per second (1e6*1e10) some millions of years could pass without the positive result. Then it's exactly reasonable to say "for my money Fermat’s theorem is true" as in, really not worth trying blindly.
Re: Feynman on Fermat's Last Theorem
#49I don't understand what "the probability of" means here. Does the probability of 10^-31 mean that if we construct the integers in a different way, it might result in the theorem being false? Or is it that if you change the equation there is only at 10^-31 probability that you arrive at a theorem which is false/true? Or something else altogether.
Re: Feynman on Fermat's Last Theorem
#50I don't understand what "the probability of" means here. Does the probability of 10^-31 mean that if we construct the integers in a different way, it might result in the theorem being false? Or is it that if you change the equation there is only at 10^-31 probability that you arrive at a theorem which is false/true? Or something else altogether.
See my other comment here for how I understood it. Note "for any N > N0 = 100" in the article.