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>> So, Navier-Stokes assumes infinite divisibility in the fluid it models, right? Which we know is not the case: atoms exist. Atoms have structure, they consis

>> So, Navier-Stokes assumes infinite divisibility in the fluid it models, right? Which we know is not the case: atoms exist. Atoms have structure, they consist of... whatever parts. And these parts have structure (like... quarks and whatnots). And... are we sure that quarks do not have structure themselves? Whatever the case is, it is not something that one should declare in a HN thread. Even more interesting... 1+2+3+4... = -1/12 https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B... And that result actually pops up in physics. That kind of indicates continuity/infinite divisibility.

twiceaday 9y ago
There is a standard definition of what '=' means and your equation does not satisfy it. https://www.youtube.com/watch?v=YuIIjLr6vUA https://www.youtube.com/watch?v=YuIIjLr6vUA
- carterehsmith 9y ago
  You say "your" and yet, this has nothing to do with me. I pointed to the wikipedia article. BTW this result is not exactly new, and it has specific confirmation in physics. You cannot argue that with high-school math. There is more to it.
  - twiceaday 9y ago
    The bit on Wikipedia that follows the equation is part of the equation. It is the context that explains how to compute it. You need that bit because the equation uses extremely common syntax as short-hand for an operation that is different from the common one. If you write it in a way that doesn't rely on redefining common syntax like so "ζ(-1) = -1/12" it seems a lot less interesting doesn't it? Perhaps syntax confusion is the only interesting thing to it.
    - carterehsmith 9y ago
      There is more to it - the number -1/12 actually pops up in physics when doing something awfully similar to 1+2+3... This may be useful: https://motls.blogspot.ca/2014/01/sum-of-integers-and-oversold-common.html https://motls.blogspot.ca/2014/01/sum-of-integers-and-overso... " The real problem is that the definition of the sum involving the limit of partial sums – limits that way too often "diverge" or "refuse to exist" – isn't the only definition or the best definition or the most natural definition that may be connected to the sum. There exist better definitions of the infinite sum – numerous definitions that turn out to be more natural in physics applications – and they generally produce the result −1/12−1/12. It is no trick or sleight-of-hand. The value −1/12−1/12 is really the right one and the rightness may be experimentally verified (using the Casimir effect). "
      - kazinator 9y ago
        The ordinary sum of any collection of positive integers is not a fraction, let alone a negative one. If we take the set { 1, 2, 3, ... 4 } the summation or partial summation of no subset of this set, finite or not, converges on any fraction or negative number. No matter what order we choose for traversing that set for generating a series, we never see anything resembling -1/12 as a partial sum or limit. The ordinary arithmetic sum of any collection of positive integers is an integer which is strictly greater than each the integers. The pages you're referencing are all crackpottery.
        carterehsmith 9y ago
        >> The pages you're referencing are all crackpottery. So... I can trust Ramanujan and Abel, that published results on these things, and Terrence Tao that has a nice writeup, and a bunch of others, or I can trust HN user "Kazinator", who... published some middle-school algebra "proof" on HN. Guess who is crackpot here. >> https://en.wikipedia.org/wiki/Ramanujan_summation https://en.wikipedia.org/wiki/Ramanujan_summation
        kazinator 9y ago
knight-of-lambd 9y ago
> Even more interesting... 1+2+3+4... = -1/12 What.. No. This sum diverges. It does not equal -1/12. Now if you plug -1 into the Riemann zeta function, you get -1/12. One could interpret that to mean the sum 1 + 2 + 3 + .. can be mapped to -1/12. But the sum has never, and will never equal -1/12. It diverges, simple as that.
- carterehsmith 9y ago
  What is the argument here? There are certain ways of assigning a number to the divergent (by partial sums) sum. If they are well-defined, they all arrive at the same number. In this case, -1/12. There is nothing to argue here, that shit has been around for 200 years or more. And, there are some results in physics that actually measure close to the number -1/12, from something that looks like a sum of 1+2+3... and that's kind of telling us that the whole construction is not just some math sleight-of-hand, it actually has some meaning in the real world. Example: Casimir effect So my point is... that kind of suggests the smoothness, or continuity, or differentiability, or whatever we want to call it, of the underlying function. The opposite of discrete. Is what my point was.
  - twiceaday 9y ago
    Your equation is only correct if you assume that the left hand side is computed using the analytic continuation of the Riemann zeta function. This assumption completely changes how the equation is interpreted and therefore verified. But you don't state this assumption so how are you surprised that people are taking the equation at face value and telling you it's wrong?
    - carterehsmith 9y ago
      Just to clarify, the above is not "my" equation. It is something that mathematicians (not me) figured out, repeatedly, many times over. Of course, the -1/12 number is preposterous, but it does turn out to have a physical meaning. There are many ways of making the sum, including limit of partial-sums/summation by parts (the one we use most of the time), but there is also Abel summation, Borel summation, Ramanujan, Cesaro, and more. And frankly there is no reason to think that "summation by parts" is the "right" way. It is surely not "right" in physics. Example: why do we start summation from 1? Is the first element somehow more important than others? No, that is just our (ie human) arbitrary pick.
kazinator 9y ago
> 1+2+3+4... = -1/12 Only through a derivation which involves some cute but unsound pseudo-algebra on infinite series. See, in the same page, the remark "Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums".
- carterehsmith 9y ago
  >> some cute but unsound pseudo-algebra on infinite series. Sure it looks like that, but, amazingly, there is a physical meaning to that number. I replied here: https://news.ycombinator.com/edit?id=16164915 https://news.ycombinator.com/edit?id=16164915
  - deleted 9y ago
    [deleted]
- jcranmer 9y ago
  Just because you don't understand doesn't mean it's "unsound pseudo-algebra." There are ways to properly manipulate infinite series, and Wikipedia does actually explain some of them on the page, even if it relies on a generally-unsound way to illustrate the intuition to a lay reader. It is very well-known and long-recognized that our natural intuition is very wrong when it comes to infinities and infinite objects, and mathematics had a crisis in trying to come to grips with it.
  - deleted 9y ago
    [deleted]
effie 9y ago
It does pops up in physics calculations, but how does that 'indicates continuity/infinite divisibility'? Many things pop up in physics calculations, most of them are just tools to represent a model.
- carterehsmith 9y ago
  Good question, I guess I should have made that more clear. Terence Tao has a nice lecture published... trying to find it, brb. In any case, the gist of it was: long as we treat the numbers in the sum as integers (ie discrete), we will get ill-defined sums, contradictions, infinities, and so on. Once we switch to real numbers (and there is an underlying function that is differentiable), things "can" be made to work and converge. To -1/12 always. So that's it. In particular, if we decided that our smallest unit of measure is... the size of atom, or whatever finite value, many of these techniques will just fail to work, and they won't match what we measured. Which means something is wrong. Maybe things really are continuous. Or maybe they are not, it is just that our math is not sophisticated enough. Obviously this is an open question.

twiceaday 9y ago

There is a standard definition of what '=' means and your equation does not satisfy it. https://www.youtube.com/watch?v=YuIIjLr6vUA https://www.youtube.com/watch?v=YuIIjLr6vUA

carterehsmith 9y ago
You say "your" and yet, this has nothing to do with me. I pointed to the wikipedia article. BTW this result is not exactly new, and it has specific confirmation in physics. You cannot argue that with high-school math. There is more to it.
- twiceaday 9y ago
  The bit on Wikipedia that follows the equation is part of the equation. It is the context that explains how to compute it. You need that bit because the equation uses extremely common syntax as short-hand for an operation that is different from the common one. If you write it in a way that doesn't rely on redefining common syntax like so "ζ(-1) = -1/12" it seems a lot less interesting doesn't it? Perhaps syntax confusion is the only interesting thing to it.
  - carterehsmith 9y ago
    There is more to it - the number -1/12 actually pops up in physics when doing something awfully similar to 1+2+3... This may be useful: https://motls.blogspot.ca/2014/01/sum-of-integers-and-oversold-common.html https://motls.blogspot.ca/2014/01/sum-of-integers-and-overso... " The real problem is that the definition of the sum involving the limit of partial sums – limits that way too often "diverge" or "refuse to exist" – isn't the only definition or the best definition or the most natural definition that may be connected to the sum. There exist better definitions of the infinite sum – numerous definitions that turn out to be more natural in physics applications – and they generally produce the result −1/12−1/12. It is no trick or sleight-of-hand. The value −1/12−1/12 is really the right one and the rightness may be experimentally verified (using the Casimir effect). "
    - kazinator 9y ago
      The ordinary sum of any collection of positive integers is not a fraction, let alone a negative one. If we take the set { 1, 2, 3, ... 4 } the summation or partial summation of no subset of this set, finite or not, converges on any fraction or negative number. No matter what order we choose for traversing that set for generating a series, we never see anything resembling -1/12 as a partial sum or limit. The ordinary arithmetic sum of any collection of positive integers is an integer which is strictly greater than each the integers. The pages you're referencing are all crackpottery.
      - carterehsmith 9y ago
        >> The pages you're referencing are all crackpottery. So... I can trust Ramanujan and Abel, that published results on these things, and Terrence Tao that has a nice writeup, and a bunch of others, or I can trust HN user "Kazinator", who... published some middle-school algebra "proof" on HN. Guess who is crackpot here. >> https://en.wikipedia.org/wiki/Ramanujan_summation https://en.wikipedia.org/wiki/Ramanujan_summation
        kazinator 9y ago

knight-of-lambd 9y ago

> Even more interesting... 1+2+3+4... = -1/12 What.. No. This sum diverges. It does not equal -1/12. Now if you plug -1 into the Riemann zeta function, you get -1/12. One could interpret that to mean the sum 1 + 2 + 3 + .. can be mapped to -1/12. But the sum has never, and will never equal -1/12. It diverges, simple as that.

carterehsmith 9y ago
What is the argument here? There are certain ways of assigning a number to the divergent (by partial sums) sum. If they are well-defined, they all arrive at the same number. In this case, -1/12. There is nothing to argue here, that shit has been around for 200 years or more. And, there are some results in physics that actually measure close to the number -1/12, from something that looks like a sum of 1+2+3... and that's kind of telling us that the whole construction is not just some math sleight-of-hand, it actually has some meaning in the real world. Example: Casimir effect So my point is... that kind of suggests the smoothness, or continuity, or differentiability, or whatever we want to call it, of the underlying function. The opposite of discrete. Is what my point was.
- twiceaday 9y ago
  Your equation is only correct if you assume that the left hand side is computed using the analytic continuation of the Riemann zeta function. This assumption completely changes how the equation is interpreted and therefore verified. But you don't state this assumption so how are you surprised that people are taking the equation at face value and telling you it's wrong?
  - carterehsmith 9y ago
    Just to clarify, the above is not "my" equation. It is something that mathematicians (not me) figured out, repeatedly, many times over. Of course, the -1/12 number is preposterous, but it does turn out to have a physical meaning. There are many ways of making the sum, including limit of partial-sums/summation by parts (the one we use most of the time), but there is also Abel summation, Borel summation, Ramanujan, Cesaro, and more. And frankly there is no reason to think that "summation by parts" is the "right" way. It is surely not "right" in physics. Example: why do we start summation from 1? Is the first element somehow more important than others? No, that is just our (ie human) arbitrary pick.

kazinator 9y ago

> 1+2+3+4... = -1/12 Only through a derivation which involves some cute but unsound pseudo-algebra on infinite series. See, in the same page, the remark "Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums".

carterehsmith 9y ago
>> some cute but unsound pseudo-algebra on infinite series. Sure it looks like that, but, amazingly, there is a physical meaning to that number. I replied here: https://news.ycombinator.com/edit?id=16164915 https://news.ycombinator.com/edit?id=16164915
- deleted 9y ago
  [deleted]
jcranmer 9y ago
Just because you don't understand doesn't mean it's "unsound pseudo-algebra." There are ways to properly manipulate infinite series, and Wikipedia does actually explain some of them on the page, even if it relies on a generally-unsound way to illustrate the intuition to a lay reader. It is very well-known and long-recognized that our natural intuition is very wrong when it comes to infinities and infinite objects, and mathematics had a crisis in trying to come to grips with it.
- deleted 9y ago
  [deleted]

effie 9y ago

It does pops up in physics calculations, but how does that 'indicates continuity/infinite divisibility'? Many things pop up in physics calculations, most of them are just tools to represent a model.

carterehsmith 9y ago
Good question, I guess I should have made that more clear. Terence Tao has a nice lecture published... trying to find it, brb. In any case, the gist of it was: long as we treat the numbers in the sum as integers (ie discrete), we will get ill-defined sums, contradictions, infinities, and so on. Once we switch to real numbers (and there is an underlying function that is differentiable), things "can" be made to work and converge. To -1/12 always. So that's it. In particular, if we decided that our smallest unit of measure is... the size of atom, or whatever finite value, many of these techniques will just fail to work, and they won't match what we measured. Which means something is wrong. Maybe things really are continuous. Or maybe they are not, it is just that our math is not sophisticated enough. Obviously this is an open question.