I have to associate the mod account as primary to have blue rights
brb trying to figure this out
@Shog9 I seem to be having the same issue that Madara-sama has with his account where his Anime account got replaced as his primary account by his highest rep account
@Mysticial With that sort of "proof" you still have to assume that the computer performs the calculation algorithm as you expect it to. Computer-assisted proofs like this are controversial among mathematicians because you never know exactly what is going on internally.
I'm not the sort of person who is opposed to computer-assisted proofs, but there are some professional mathematicians don't accept any computer-aided results because they can't verify them.
@Mysticial From a mathematical standpoint, fault-tolerance, no matter how good, does not constitute a proof. A proof is a string of sentences, each of which is either an axiom or a consequence of the preceding sentences via one of the rules of inference.
@LoganM yep, and neither do humans with eyes and hands.
it basically boils down to a matter of math vs. engineering.
Oh and I'm an engineer.
@Eric It's really complicated. At the high-level is to use two different formulas and see if they match at the end. If they don't match, then you know something went wrong.
The probability of the algorithm returning the wrong answer is about the same as hitting in the tail-end of a normal distribution about 30 or so deviations above the mean.
@Mysticial Humans are capable of producing proofs, as are computers. There are theorem-proving programs which work fully in axiomatics and generate what are purported to be mathematically correct proofs. Most mathematicians do not have a problem calling such a result a proof. Human mathematicians are the same way. However, no physical system can verify a proof with 100% accuracy, if for no other reason than quantum randomness.
What you've done isn't quite the same thing. I agree that for all practical purposes computing $\pi$ to 50 decimal places via whatever algorithm you choose is sufficient, but mathematics was never about being practical. It's about generating proofs.
@LoganM I understand that. But when you have a 100 page proof to the Fermat's Last Theorem, the probability that the thousands of mathematicians missed something and there is in fact a flaw, is not 0%.
Also, I'll just add that for most algorithms that compute $\pi$ to arbitrary precision, the amount of effort required to prove that the method indeed gives you $\pi$ is quite a bit less than evaluating the simple integral in question.
@Mysticial I agree wholeheartedly. In my view the problem is even deeper than that. Quantum mechanically, there's no way I can even communicate a string of sentences (or anything else) to you exactly. To do so, you need to observe a quantum mechanical system (admittedly, it's very close to being classical, but it is still somewhat quantum), and there is always some extremely small probability you observe a different string of sentences.
So no proof can ever be checked with 100% certainty simply because no message can ever be transmitted exactly.
Exactly, and my argument, is that an unreliable method can be made more reliable than a reliable method by simply pushing the probability of error to less than background noise.
Which is why a lot of mathematicians hate my ass, but the engineers are fine with it.
That fact is a big part of the reason I'm in Physics and not in Math. I realized that if I believe in quantum mechanics, which I do, then trying to prove theorems which are always true (the goal of mathematics) is a bit hypocritical since the proofs can't be checked exactly within our universe anyway.
@Mysticial The trouble is, a proof has nothing to do with errors or exactness. I agree that you've demonstrated that $\pi < 22/7$, but I don't think you've proven it in the mathematical sense of the word prove.
@LoganM If yeah. The word "mathematical sense", is gonna be a fuzzy line at some point. Obviously no (human) is gonna be able to verify that the 10 trillion digits I computed are correct in the mathematical sense.
A computer-assisted proof (by means of multiple algorithms) can make you very confident that they are correct. But if they want that 100%. It's never gonna happen.
I love how the "vote for Stackoverflow mod" thing shows up in the Anime site bulletin. lol
Of course that doesn't mean I don't care about proofs. But a proof says that: If the proof is correct. Then this assertion holds with 100% confidence.
@Mysticial I don't think it's a fuzzy line. A proof is just a sequence of sentences. Most mathematicians don't write out their proofs in that form, but in principle they could be written as such. Nothing about a proof requires that it is verified or even verifiable in principle. But a computer calculation is a physical process, not a mathematical one, and it can't be converted to such easily.
@Mysticial Only insofar as the computer you are using is an idealized system. In reality it's a physical system, just like everything else, and it's only accurate insofar as we understand the physics of the system. Which admittedly is very well, but not perfectly.
How do you prove that 1.41421356 is the first 10 digits of sqrt(2)?
rigorously
If someone asked me that on an exam. I wouldn't even know how to approach it other than to show all the calculations by hand. (which would be insanely long)
In principle, there's no problem checking 1.41421356^2 on a computer which generates a valid proof of the value, preferably checkable by hand. But there aren't many languages equipped to do that.
You could probably do the same thing with Pi, by plugging into sin(x). (leaving aside the fact that evaluating sin(x) is much more difficult than computing Pi...)
@Mysticial That's basically what I figured. Both of those methods require significantly more advanced efforts to demonstrate that they actually give pi than the integral here. I don't understand many of the technical details in those methods, but I could teach the integral to a high school student without any trouble.
So from that standpoint, I'd say the integral proof is easier.
How do you prove that 1.41421356 is the first 10 digits of sqrt(2)?
this one I do know the continued fraction for
so I could use it - the approximations p/q it gives have quality 1/q^2 (at least)
I might be wrong :(
so I have to get this far: 1+1/(2+1/(2+1/(2+1/(2+1/(2+1/(2+1/(2+1/(2+1/(2+1/(2+1/2)))))))))))) = 19601/13860 to get a denominator of size at least 100000 which gives me approximation 19601/13860 = 1.4142135642135642135642135642135642136 good to a few places
You could probably do the same thing with Pi, by plugging into sin(x). (leaving aside the fact that evaluating sin(x) is much more difficult than computing Pi...)
I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in http://arxiv.org/abs/0911.1929.
When I try it I get $$\frac{\sin(r)}{\cos(r)} = \frac{r}{1 + r^2\cdot\frac{\frac{...
@JNat It may be impossible to reword that question without significantly altering its intent. It is basically asking for things which we consider "fun".
I cannot see how that question can be made to fit in SE. It is good for chat though, but she does not have 20.
which kinda creates a chicken-and-egg situation. New users (without the 20) tend to not read the FAQ before posting. If we close their question, they never come back to get the 20 and chat.
I want to setup a chatroom for online students learning HTML, CSS & JS (I encourage everyone learning programming to get a StackOverflow account for Q&A). Stack Overflow Chat is BRILLIANT. But unfortunatly new members can't participate right away. The ability to "talk" is only available o...
Are there any anime that would be consider either Noir or neo Noir? Any that are usually considered to be in the top 90% based on reviews would be excellent as well.
@MadaraUchiha it is but I'm willing to cut the user some slack and assume he's asking a list question about top few series (by popularity) in a specific genre
I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in http://arxiv.org/abs/0911.1929.
When I try it I get $$\frac{\sin(r)}{\cos(r)} = \frac{r}{1 + r^2\cdot\frac{\frac{...
Are there any anime that would be consider either Noir or neo Noir? Any that are usually considered to be in the top 90% based on reviews would be excellent as well.
