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The Riemann Hypothesis

golem.ph.utexas.edu

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Re: The Riemann Hypothesis

#71

Post by r_c_a_d »

rat9988 wrote: post #12
liquidise wrote: post #10

Mathematics is a uniquely beautiful field to me. The commutative property has always struck me as special in its own way. 2 x 3 = 3 x 2 feels so obvious, but multiplication is really just addition, and 2 + 2 + 2 = 3 + 3 is far less intuitive, yet states the very same claim. Most fascinating to me is that many theories are effectively 1-way functions. Entire branches of mathematics have been developed to prove otherwi…

>2 x 3 = 3 x 2 feels so obvious You just got used to it because you learned it very young. I'm pretty sure it wasn't that obvious when you learned it.

When my kids started school I was surprised to learn that at age 4 it is not obvious that the number of items on a whiteboard remains constant if all the items are moved to a different part of the board. So what seems obvious, probably is something we just don't remember learning.

Re: The Riemann Hypothesis

#72

Post by croo »

empath75 wrote: post #41
ncmncm wrote: post #14

Earlier quoted context omitted.

The notion that multiplication is ("just") repeated addition is the hardest misconception to overcome blocking progress in mathematics. Repeated addition is an algorithm that works when you have a non-negative integer argument. Imagining that the algorithm defines the operation limits conception.

I think the big conceptual leap in understanding math is getting beyond the idea that it’s about counting concrete objects in general, and realizing that you can create new sorts of abstract structures that may or may not have anything to do with reality and rules for working with them.

True! I was enlightened in class when we defined addition and multiplication on three random characters then checked if the commutative property holds.

Turned out you can do whatever you want... It just that usually you want addition to be commutative and usually want to use an infinite set of symbols called numbers.

Re: The Riemann Hypothesis

#73

Post by zaarn »

sls wrote: post #28
apalmer wrote: post #17

Earlier quoted context omitted.

I would like to understand what multiplication is in your (and i guess advanced mathematics)

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 = 9.7461, remainder: 3.141 * 0.41

shift decimal: 3.141 * 4.1

repeat 3.141 4 times and add; 12.564, remainder: 3.1410.1

shift onto result: 9.87174‬

shift decimal: 3.141 1

shift onto result: 10,053764‬

result: 9,875881‬

check back with calculator: 3.141 * 3.141 = 9,875881

Multiplication can be represented as a number of additions and shifting the results of substeps over the decimal point

Re: The Riemann Hypothesis

#74

Post by commandlinefan »

sls wrote: post #28
apalmer wrote: post #17

Earlier quoted context omitted.

I would like to understand what multiplication is in your (and i guess advanced mathematics)

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…

> What about -3 * -2?

My son actually asked me that a while back (or rather, “why does a negative times a negative make a positive?”), and I honestly didn’t have a very satisfying answer. The best I could come up with was to go back to the definition of multiplication as repeated addition and then start with multiplying/repeated-adding a negative number by a positive number: that would work backward on the number line and give you a negative number. Then, since multiplication is commutative, a positive number multiplied by a negative number must behave the same way: multiplying a positive by a negative causes the negative to move backwards. So, if multiplying a positive by a negative moves it the “other way”, multiplying a negative by a negative must move the first number to the right.

It works, but it’s not as intuitive as I would have liked - I did better with why negative exponents are 1/x^n and why the angles in a triangle add up to 90 degrees.

Re: The Riemann Hypothesis

#75

Post by m11a »

commandlinefan wrote: post #74
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…

> What about -3 * -2? My son actually asked me that a while back (or rather, “why does a negative times a negative make a positive?”), and I honestly didn’t have a very satisfying answer. The best I could come up with was to go back to the definition of multiplication as repeated addition and then start with multiplying/repeated-adding a negative number by a positive number: that would work backward on the number lin…

The following page explains this one pretty well I think, skip to the "A proof" at the bottom: http://mathforum.org/dr.math/faq/faq.negxneg.html

Re: The Riemann Hypothesis

#76

Post by m11a »

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.

Back in the 20th century there was a goal to ensure maths was consistent, notably by David Hilbert, and that everything could be proven from a finite set of axioms, and he wanted to ensure everything was built on a logical foundation and was rigorous.

In practice it may not change much, but more philosophically speaking, it's dangerous to take something as granted without a proof. If we then have many ideas building off this unproven idea, and it's later shown to be false, a lot of those theorems would be based off a false premise. This doesn't necessarily mean they the proposition is incorrect, but it would mean that they are unproven.

It's more to satisfy the rigour in maths, and also pose as a challenge to mathematicians. In the journey to proving RH a lot of new ideas could be found that could prove useful, too.

Re: The Riemann Hypothesis

#77

Post by j9461701 »

teekert wrote: post #61
j9461701 wrote: post #21

Earlier quoted context omitted.

>monomer-dimer problem Oh hey, I did my undergrad thesis on that! It generates neat looking graphics: https://imgur.com/a/Z6hySAw

I did a Masters in Biophysics, the topic was diffusion of proteins in cell membranes, they usually show random walks (albeit restricted to compartments or showing distinct speeds (bound/unbound?)). That graphs does not look like a random walk, so the Riemann Hypothesis is false?

The graphic comes from certain assumptions that do not hold in the general case. Specifically, that of non-isotropic fragmentation and arbitrary fragmentation depth.

Re: The Riemann Hypothesis

#78

Post by ncmncm »

wmp56 wrote: post #67

Why does everybody think that by virtue of math ought to be nice, such a nice hypothesis ought to be true? Isn't it just a form of the survivorship bias that we observe only nice side of math? What if this hypothesis stands true for all N < 10^10^10^467+17, and then suddenly it doesn't? Perhaps to make a breakthru in math (and physics) we need to consider the possibility that the reality can be ugly and counterintuit…

If it's ugly, mathematicians lose interest. So, only the beautiful stuff remains in the set of things taught by mathematicians.

There could be beautiful stuff concealed by a layer of ugly than mathematics never breaches.

Re: The Riemann Hypothesis

#79

Post by alok-g »

scythe wrote: post #39
apalmer wrote: post #17

Earlier quoted context omitted.

I would like to understand what multiplication is in your (and i guess advanced mathematics)

Multiplication can have a number of meanings, but there are two important ones: - multiplication as function composition, eg the product of two matrices is the linear transform obtained by applying the right matrix and then the left (onto a column vector) - multiplication as “the operation that distributes over addition” — in the theory of rings and algebras, if you have one operator “+” which forms an abelian group,…

I had read somewhere that till the point there's only addition, mathematics can remain consistent and complete both. The moment multiplication operation is introduced, Godel's incompleteness theorems come along. Do I understand correctly? If so, what exactly happens when multiplication comes in?

Re: The Riemann Hypothesis

#80

Post by BlackFingolfin »

wmp56 wrote: post #67

Why does everybody think that by virtue of math ought to be nice, such a nice hypothesis ought to be true? Isn't it just a form of the survivorship bias that we observe only nice side of math? What if this hypothesis stands true for all N < 10^10^10^467+17, and then suddenly it doesn't? Perhaps to make a breakthru in math (and physics) we need to consider the possibility that the reality can be ugly and counterintuit…

I think that's a misconception. Mathematicians are keenly aware that there are plenty examples in history were many people "believed" (hoped? expected?) that some result might be true because it would be "beautiful", but it turned out to be false. Or where numerical evidence suggested something only to turn out to be wrong in the end. And where the first counterexample only exists at huge, numerically infeasible bound (see e.g. https://en.wikipedia.org/wiki/Skewes%27s_number)

And hence a well-known suggestion is that when tackling a hard problem, is to try finding a proof on even days, and a counterexample on odd days...

And indeed, lots of people tried (and still try) to find counterexamples to, or otherwise disprove, the Riemann Hypothesis. However, there are indeed many, many results and heuristics that give a strong suggestion that the RH might be true -- far more than mere numerical results computing zeroes of the Zeta function. Of course none of them constitute a proof; but this really goes far beyond a simple hope for "beauty" in the theory.

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