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The Riemann Hypothesis

golem.ph.utexas.edu

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Re: The Riemann Hypothesis

#81

Post by emilgouliev »

chii wrote: post #51
emilgouliev wrote: post #46

Earlier quoted context omitted.

2 + 2 + 2 = 3 + 3 becomes much more obvious when you think of a 3x2 rectangle and rotate it by 90 degrees to a 2x3 rectangle.

but then isn't this just a tautological argument? why does multiplying the sides of a rectangle give you the area?

you don't need to invoke a concept of area to justify this. you can just think of a grid of dots. since multiplication is repeated addition, each row represents one step of the repeated addition.

2x3 = 3 + 3

...

...

3x2 = 2 + 2 + 2

..

..

..

since rotation doesn't change the number of dots, we expect 2x3 and 3x2 grid of dots to contain the same number of dots, which proves that multiplication is commutative.

Re: The Riemann Hypothesis

#82

Post by ncmncm »

zaroth wrote: post #49
jacobolus wrote: post #22

Earlier quoted context omitted.

> hardest misconception to overcome This seems like an exaggeration.

More accurately, it’s a colloquialism.

The number of people who fail to overcome it is close to the number of people who never get a grasp of what mathematics is about. Ask them. It comes up all the time.

Re: The Riemann Hypothesis

#83

Post by jacobolus »

ncmncm wrote: post #82
zaroth wrote: post #49

Earlier quoted context omitted.

More accurately, it’s a colloquialism.

The number of people who fail to overcome it is close to the number of people who never get a grasp of what mathematics is about. Ask them. It comes up all the time.

Yes, but your inference that having a concept of multiplication as repeated addition leads people to “not grasp mathematics” is backwards.

I have seen little evidence that people who become interested in mathematics have an unusually difficult time “overcoming” this viewpoint.

Re: The Riemann Hypothesis

#84

Post by nullc »

Yajirobe wrote: post #19

So if RH is proven, what actually changes? As far as I know, there are tons of theorems that already presuppose RH to be true There wouldn't suddenly be an insight into how to find larger primes, for example.

Proving some things may become easier (as an example of that there are already more than a few papers with proofs that rest on RH ... and then additional effort to go and find RH-less proofs.

A silly example (from memory) using another rather non-obvious proof.

1. Assume 2^(1/4) = X/Y for integers X/Y (Assume the quartic root of 2 is rational)

2. Raise both sides to the 4th power: 2 = X^4 / Y^4

3. Multiply both sides by Y^4: Y^4 + Y^4 = X^4

4. QED, quartic root of 2 is irrational, by contradiction with Fermat's last theorem.

Re: The Riemann Hypothesis

#85

Post by sls »

jacobolus wrote: post #56
sls wrote: post #54

Earlier quoted context omitted.

I don't agree that a definition of multiplication that includes transcendentals such as pi must be numerical. In fact, to the extent that numerical approximations are approximations _of_ something, the thing they would be approximating is the actual value of the operation of multiplication being applied to pi and pi. The only reason we're talking about algorithms at all was to distinguish between them and definitions…

Well, that’s a philosophical rather than mathematical question. I don’t really believe in a concept of “actual value” outside of the context of computations (though I don’t mind conceding it as a matter of convenience and social convention, since the distinction almost never matters for practical purposes). I am not an expert, but mathematicians have investigated this, https://en.wikipedia.org/wiki/Computable_analysi…

I am familiar with the constructivist family of ideas. Obviously I agree this a question of philosophy, specifically the philosophy of mathematics. (So not "rather than", since foundations of mathematics is a branch of mathematics.) And because adopting a constructivist approach to mathematics means adopting different ideas about what a mathematical definition _is_ or _can be_, I have to say that it's a bit disingenuous to introduce this context only after engaging in the above discussion.

For example, if someone asks you to explain Euclid's proof of the infinitude of the primes, and you say that Euclid did not provide any such proof and nothing more, I think it's quite disingenuous. It would be more proper to say, from a constructivist view, the argument Euclid made isn't a valid proof, and then either explain the proof in the logical context in which it was made or decline to.

In this case, the point of discussion was separating the definition of multiplication from an algorithm implementing it. It's quite unfair to silently take a position that a mathematical definition without an algorithm isn't valid or meaningful and then on that basis argue that only numerical approximations to transcendentals have meaning.

So many common mathematical concepts such as "the integers" have no meaning in a constructivist approach that it's not sensible to engage in mathematical discussion without establishing that one's fundamental basis of approach varies so widely from the common one.

Re: The Riemann Hypothesis

#86

Post by sls »

zaarn wrote: post #73
sls wrote: post #28

Earlier quoted context omitted.

The comment to which you replied says that thinking about multiplication as "just" repeated addition is problematic, so let's look at that. Consider 3 x 2. If we take that approach, it seems ok - we understand it to mean "add together 3 2's" - 2 + 2 + 2, which gives the correct answer of 6. What about -3 * -2? What does it mean to add a negative number of times? What about pi * pi? What does it mean to add something…

Easy. -3 x -2, multiply signs first: +, remainder: 3x2 repeat 2 3 times and add; 2 + 2 + 2 = 3, remainder: 0 3.141 * 3.141, multiply signs first: +, remainder 3x2 repeat 3.141 3 times and add; 3.141 + 3.141 + 3.141 = 9,432, remainder: 3.141 * 0.141 shift decimal: 3.141 * 1.41, remainder: 3.141 * 1.41 3.141 * 1.41, repeat 3.141 1 times and add: 3.141 = 3.141, remainder: 3.141 * 0.41 shift onto result: 9.432 + 0.3141 =…

No one is arguing that one cannot implement multiplication algorithms using addition. In fact, one can do it with just NAND. But the definition of multiplication isn't in terms of NAND or addition.

Re: The Riemann Hypothesis

#87

Post by ncmncm »

jacobolus wrote: post #83
ncmncm wrote: post #82

Earlier quoted context omitted.

The number of people who fail to overcome it is close to the number of people who never get a grasp of what mathematics is about. Ask them. It comes up all the time.

Yes, but your inference that having a concept of multiplication as repeated addition leads people to “not grasp mathematics” is backwards. I have seen little evidence that people who become interested in mathematics have an unusually difficult time “overcoming” this viewpoint.

Their difficulty letting go of repeated addition leads them to consider mathematics opaque, and they lose interest.

Re: The Riemann Hypothesis

#88

Post by posterboy »

teekert wrote: post #62
chii wrote: post #50

Earlier quoted context omitted.

what does "look drunk" actually mean tho? it's a bit of a weird property...

The drunken man is moving away from where he started (any point in time can be labelled as "start") at a speed of about the square root of his linear speed (speed from his point of view). And the direction is random. This can also be 1D motion. Actually in 1 and 2D it is likely that the drunken man hits his starting point again at some point, it goes to 0 fast in 3D.

For 1D It's trivial to see that a side scroller with side-strafing controls (Space Invaders not Asteroids) will always return to the center if L and R appear with equal probability, in euclidean geometry

Re: The Riemann Hypothesis

#89

Post by schwurb »

drdeca wrote: post #57
schwurb wrote: post #27

Earlier quoted context omitted.

> P=NP in that a solution can be used to attack RSA encryption. Note that 1) P=NP does not necessarily give raise to any polynomial algorithm that solves a NP problem. The proof would prove the existence of one such algorithm, but it might well never be found (which is the current status quo) 2) even if it would be polynomial, it could still run longer than the heat of the universe. O(n) = n^10000000 would still be a…

Isn’t there some (highly impractical) algorithm which dovetails through different Turing machines, in a way that has an asymptotically optimal runtime for a given problem, just with really terrible constants? I thought we knew an algorithm that, if P=NP, would solve NP problems in P time, (but with absurd constants), and otherwise solves the problems is worse than polytime. But I could be remembering this totally wro…

> (highly impractical)

Forget everything practical - we are in deep theoretic waters here! There are thousands of algorithm with even a polynomial solution where you still go for the heuristic because the polynomial version is way to slow.

> Isn’t there some (highly impractical) algorithm which dovetails through different Turing machines, in a way that has an asymptotically optimal runtime for a given problem, just with really terrible constants?

Optimal might be, as long as optimal does not mean polynomial. Otherwise you would read about it in the newspapers ;) I don't know of such algorithm, but "trying out different turing machines" gives me a strong gut feeling of "not polynomial".

> I thought we knew an algorithm that, if P=NP, would solve NP problems in P time, (but with absurd constants), and otherwise solves the problems is worse than polytime.

Is that the algorithm you are refering to? Sounds like what Turing proposed once. The interesting branch is the P=NP, since then you could answer really really interesting things in P. Theoretically - and if indeed P=NP ;)

Re: The Riemann Hypothesis

#90

Post by drdeca »

schwurb wrote: post #89
drdeca wrote: post #57

Earlier quoted context omitted.

Isn’t there some (highly impractical) algorithm which dovetails through different Turing machines, in a way that has an asymptotically optimal runtime for a given problem, just with really terrible constants? I thought we knew an algorithm that, if P=NP, would solve NP problems in P time, (but with absurd constants), and otherwise solves the problems is worse than polytime. But I could be remembering this totally wro…

> (highly impractical) Forget everything practical - we are in deep theoretic waters here! There are thousands of algorithm with even a polynomial solution where you still go for the heuristic because the polynomial version is way to slow. > Isn’t there some (highly impractical) algorithm which dovetails through different Turing machines, in a way that has an asymptotically optimal runtime for a given problem, just w…

this doesn't say the exact algorithm (I haven't found it), but this stack overflow answer talks about it : https://stackoverflow.com/questions/5107140/what-is-meant-by...
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