61–70 of 75 posts
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> 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" .
>> 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
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> 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" .
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, an…
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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?
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, Ces…
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> The things about which we are totally ignorant are much smaller today, of course One would have to disagree with this idea. The more we study the universe around us, the less we actually know. As a somewhat philosophical point, there are many times when our mental (mathematical) models get in the road of understanding. Very often, people believe that because we have a model that works and appears to give good predi…
"much smaller today" meaning literally smaller: Navier didn't know about atoms, 10^-10 m, but now we know about things at 10^-18 m or so. Sure, our awareness of how much we don't know has grown over time. The point I was trying to make is that theoretical models (like N-S) not only don't have to be perfect to be useful, but more, are useful precisely because they are not complete. By ignoring irrelevant details we ge…
Understanding that a model or theory is useful even when we ignore certain aspects of reality is quite different to the often displayed belief that a specific theory is "gospel" even in the face of anomalies and discrepancies of the real world compared with prediction. Too much of the "theoretical physics" genre (word specifically chosen) is based on the idea that mathematics is the means of finding the "truth".
As I said above, mathematics is a wonderful and useful tool, but it is not a good master. It provides a possible insight into what is going on. However, those insights are not "truth" as such. I have been doing a review of my old mathematics texts for scientists and engineers, as well as other resources. It is interesting that all of them talk of and demonstrate that all the mathematical models are simplified and incomplete. Yet, if one raises the various problems with the various models in use today, one is shouted down. This does not bode well for our advancement in understanding of the universe around us.
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Well, we still don't have the computational power to do things atom-by-atom, even in small things like modeling cells, let alone something like simulating wind shear on a plane or predicting weather patterns. It's not just computational power but memory. There's something like 1E14 atoms in a cell. Storing the x,y,z position with 4 bytes for each dimension (though you'll probably want 8 bytes) comes out to 1.2E15 byt…
> Well, we still don't have the computational power to do things atom-by-atom People do atom-by-atom molecular dynamics simulations of proteins and such.
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Well, we still don't have the computational power to do things atom-by-atom, even in small things like modeling cells, let alone something like simulating wind shear on a plane or predicting weather patterns. It's not just computational power but memory. There's something like 1E14 atoms in a cell. Storing the x,y,z position with 4 bytes for each dimension (though you'll probably want 8 bytes) comes out to 1.2E15 byt…
we still don't have the computational power to do things atom-by-atom That's what puzzles me about this discussion. Of course we don't. The goal here is to model more atoms than we have atoms to model with. Until we get to quantum computers that can represent more information per atom, than information per atom we want to represent, it's simply a matter of objectively & obviously inadequate resources. You can't simul…
So, Navier-Stokes assumes infinite divisibility in the fluid it models, right? Which we know is not the case: atoms exist. It seems obvious to me that Navier-Stokes can't be a perfectly accurate description of reality, so it should come as no surprise if they turn out to blow up or otherwise behave non-physically. Am I missing something? Why isn't it assumed that they're just a very nice approximation, like Newton's…
>> 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…
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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-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 d…
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…
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.
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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…
>> 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
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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 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…
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.
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.