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The Riemann Hypothesis

golem.ph.utexas.edu

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Re: The Riemann Hypothesis

#41

Post by empath75 »

ncmncm wrote: post #14
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…

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.

Re: The Riemann Hypothesis

#42

Post by v64 »

lonelappde wrote: post #40
mlthoughts2018 wrote: post #7

Earlier quoted context omitted.

I disagree with this. I don’t think the Riemann Hypothesis is “harder” than various other unsolved problems like Yang-Mills or Navier-Stokes. Most professional mathematicians would succeed just as much or as little as the particular subset who actively work on the Riemann Hypothesis, if they chose to study it. I think it’s more that in terms of instrumental rationality / goal achievement, working on the Riemann Hypot…

The other unsolved problems at frightening as well, insofar as such an informal emotional word has any consensus meaning, perhaps "if I had funding to spend my whole life on this problem, there's a god chance I'd make very little progress."

Agreed, I think frightening carried a connotation that I didn't intend. My intention was to state that problems like the Riemann hypothesis are of a level of difficulty such that mathematicians believe these problems cannot be solved without a major fundamental leap forward in our understanding. Erdos expressed a similar frustration toward the Collatz conjecture when he stated "Mathematics may not be ready for such problems".

So with respect to the original comment I was replying to, there are other areas of mathematics that one can study that are well understood and not so opaque as the Riemann hypothesis.

Re: The Riemann Hypothesis

#43

Post by khawkins »

arkano wrote: post #11
teh_infallible wrote: post #2

Every now and then I try to delve into the frightening world of math. Then I see something like this, and start to feel very tired. Then I think, “My hair is already falling out. Do I need something like this to accelerate the process?”

As a mathematician, I have noticed that people that like to build something that, for example, can fly, start from a paper plane; they don't get discouraged because they can't yet build a 737 aircraft. However, in math, you need a lot of experience before you can even judge whether a problem is in fact a 737 and not a paper plane (and even then, you can be mistaken). I often see students discouraged because of this a…

Honestly, I think the near constant exaltation of problems like the Riemann Hypothesis, P=NP, Fermat's last theorem, etc. is more damaging to the field than good. Many of these theorems have dubious application to anything practical were the theorems unquestionably proven. Subsequently, it frequently gives the impression to a lay observer that mathematics is all about number theory and tackling pointless puzzles.

Going into undergrad I was briefly discouraged from going into mathematics because this was the impression I got. They're interesting to think about, but I didn't want my future to be firmly situated in inapplicable theory.

I say this knowing there is plenty of work to be done in the applied mathematics, especially in trying to simplifying the understanding of complex problems. I'd like to see more of the glorification of moderately hard problems which take more time to explain but are well within the grasp of people who start working on it, than easy to explain problems which will likely never be in the grasp of anyone.

Re: The Riemann Hypothesis

#44

Post by wodenokoto »

Cybiote wrote: post #13

Here's a quite friendly elucidation by the inimitable Avi Wigderson, from: https://www.ias.edu/ideas/2009/wigderson-randomness-pseudora... > Let’s elaborate now on the connection (explained on the cover of this issue) of the Riemann Hypothesis to pseudorandomness. Consider long sequences of the letters L, R, S, such as > S S R S L L L L L S L R R L S R R R R R S L S L S L L . . . > Such a sequence can be thought of a…

> (e.g., t=18, which is divisible by 32)

Did they mean 3x3x2 instead of 32? There is no way 18 is divisible by 32 in any common sense of divisible.

> This explicit sequence of instructions, which is determined by the prime numbers, causes a robot to look drunk, if and only if the Riemann Hypothesis is true!

So, do they look drunk for large walks? That sounds like something that is easily computed for tens of thousands of steps.

Re: The Riemann Hypothesis

#45

Post by emilgouliev »

wodenokoto wrote: post #44
Cybiote wrote: post #13

Here's a quite friendly elucidation by the inimitable Avi Wigderson, from: https://www.ias.edu/ideas/2009/wigderson-randomness-pseudora... > Let’s elaborate now on the connection (explained on the cover of this issue) of the Riemann Hypothesis to pseudorandomness. Consider long sequences of the letters L, R, S, such as > S S R S L L L L L S L R R L S R R R R R S L S L S L L . . . > Such a sequence can be thought of a…

> (e.g., t=18, which is divisible by 32) Did they mean 3x3x2 instead of 32? There is no way 18 is divisible by 32 in any common sense of divisible. > This explicit sequence of instructions, which is determined by the prime numbers, causes a robot to look drunk, if and only if the Riemann Hypothesis is true! So, do they look drunk for large walks? That sounds like something that is easily computed for tens of thousand…

They meant 3^2 = 9.

Re: The Riemann Hypothesis

#46

Post by emilgouliev »

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 + 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.

Re: The Riemann Hypothesis

#47

Post by Double_Cast »

apalmer wrote: post #17
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 would like to understand what multiplication is in your (and i guess advanced mathematics)

The problem is that there's different types of multiplication. The fact that multiplication over the Reals can be thought of as "iterated addition" is a consequence of commutativity. But mathematicians often generalize multiplication to various contexts as an arbitrarily-defined transformation, which may or may not be commutative. E.g. multiplication over the Reals is commutative; multiplication over Complex Numbers is commutative; multiplication over Quaternions is non-commutative.

Incidentally, "exponentiation is just repeated multiplication", right? Try using iteration to solve 0^0.

  lim{x -> 0} (0^x) = 0
  lim{x -> 0} (x^0) = 1
The "iteration mindset" doesn't always generalize cleanly.

Re: The Riemann Hypothesis

#48

Post by Gene_Parmesan »

apalmer wrote: post #17
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 would like to understand what multiplication is in your (and i guess advanced mathematics)

Limited to a numeric sense, multiplication is an operation which began life as repeated addition, and which had to then be extended in various ways as we extended our number systems. We know, for instance, that integral multiplication x * y provides the area of a rectangle with sides of length x, y. This applies without modification to the rationals and reals as well, so we know from this that multiplication remains consistent without having to rely on the concept of repeated addition.

Similar extensions happened with negative arguments. At a certain point, pure intuition vanishes, and instead you are left with the task of simply creating a set of rules that define a consistent mathematics. Often times, the approach you take is limited by this very requirement; our mathematics is only consistent if a negative multiplied by a negative results in a positive, for instance.

This topic hints at the gap between applied mathematics rooted in concrete reality, as it were, and theoretical mathematics, which of course deals purely with abstract structures.

Re: The Riemann Hypothesis

#49

Post by zaroth »

jacobolus wrote: post #22
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.

> hardest misconception to overcome This seems like an exaggeration.

More accurately, it’s a colloquialism.

Re: The Riemann Hypothesis

#50

Post by chii »

wodenokoto wrote: post #44
Cybiote wrote: post #13

Here's a quite friendly elucidation by the inimitable Avi Wigderson, from: https://www.ias.edu/ideas/2009/wigderson-randomness-pseudora... > Let’s elaborate now on the connection (explained on the cover of this issue) of the Riemann Hypothesis to pseudorandomness. Consider long sequences of the letters L, R, S, such as > S S R S L L L L L S L R R L S R R R R R S L S L S L L . . . > Such a sequence can be thought of a…

> (e.g., t=18, which is divisible by 32) Did they mean 3x3x2 instead of 32? There is no way 18 is divisible by 32 in any common sense of divisible. > This explicit sequence of instructions, which is determined by the prime numbers, causes a robot to look drunk, if and only if the Riemann Hypothesis is true! So, do they look drunk for large walks? That sounds like something that is easily computed for tens of thousand…

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