6 ms·

I'm gonna be contrarian and say, yes it does - it's been physically proven! The casimir effect, as other people here have mentioned, depends on this being true,

I'm gonna be contrarian and say, yes it does - it's been physically proven! The casimir effect, as other people here have mentioned, depends on this being true, and it has been experimentally tested. To me it's one of those things where you just go 'damn' because of the perplexing relations that exist between math and physics. If anything hints at what the hell goes on in this universe, to me it's stuff like this.

quantumhobbit 10y ago
The analogy to complex numbers is really useful. Of course sqrt(-1) doesn't exist. But if we just pretend that is does exist, we can build a rigorous theory out of these imaginary numbers. Once we do that we notice that these numbers are really useful for calculating real physical things. So maybe imaginary numbers aren't so imaginary. Same with these infinite sums. By the math you learn in middle school, you can't have infinite sums. But break the rules for just a second and again we have something that is helpful with real physics.
- chowells 10y ago
  I don't know what you mean by sqrt(-1) not existing. What does it even mean to exist? Far better to talk about whether a number is defined in a particular numerical system. In the real numbers, sqrt(-1) isn't defined. But why privilege the real numbers as "existing"? Despite an official-sounding designation, they're very deeply weird. The real numbers are famously uncountable. But any subset of them that can be enumerated is by definition countable. Think about the consequences of that for a moment. No matter what you do, the subset of the reals you can enumerate is countable, meaning the subset you can't enumerate is uncountable. In a rather flippant way, you could describe the real numbers as "mostly useless." Most of them exist to make some theorems work, rather than being a number that you could ever use to describe anything - solely because describing the number would require an infinite amount of information. In a pretty significant sense, it's valid to say that the real numbers are mostly figments of analysts' imagination. If they "exist", might as well say complex numbers exist too. They're actually more useful in physics than real numbers are.
  - quantumhobbit 10y ago
    I'll admit that the concept of numbers "existing" here is quite poorly defined. I was trying to capture the effect of people upon encountering complex numbers for the first time to just sort of shut down and refuse accept that they are "real" ( pun intended). It is hard to remember how weird complex numbers feel after years of middle school math teachers saying that you can't take the square root of a negative. Similarly weird feeling is encountering zeta(-1) = -1/12 after years of calculus teachers telling you to ignore divergent sums because they are infinite.
    - sebastos 10y ago
      I think the point is that if that's true, the concept of complex numbers is being taught incorrectly. If you tell a student that this number is fake but useful, what are they to make of that? It just starts to make mathematics seem spooky and unpredictable. When you're first learning about imaginary numbers in 8th or 9th grade, the answer to "what is sqrt(-1)?" _should_ be undefined. If you claim otherwise, you're pulling the rug out from under their feet, because the number system that they are familiar with indeed has sqrt(-1) undefined. Instead, the teacher should go on to introduce a new system of mathematical objects that have certain rules, and the students could play around with them and see how they have two components, how you can plot those two components in 2 dimensions, how you can think of them as arrows sticking out of the origin, how you can combine their components to rotate each other, etc. Then work backwards into showing that we can call these objects complex numbers for short, because those operations are similar to addition, multiplication, etc. And finally, just as a curiosity, you can see that sqrt(z) = i for z = -1 + 0i. There's no need to introduce this whole concept of an imaginary number line that points off in a direction nobody can see or measure. The whole takeaway should be that you can't just square real numbers get negatives. If you have something that can "multiply" by itself to get its own inverse, then you have either overloaded the multiplication operator with something very very different, or you're dealing with an object that can "rotate" through another dimension. It's an ordinary two dimensional space, and the only difference between the two axes is their name, just like "x" and "y". In my opinion, this lesson should actually be reassuring to a young mathematical intuition: there's only so many ways to skin this cat.
- witty_username 10y ago
  Infinite sums do not require any breaking of rules. Infinite sums are formally defined as the limit of partial sums.
ajkjk 10y ago
I strongly disagree. The Casimir effect's math works, yes, which means that "there exists a sense in which the summation equals -1/12 in this calculation", but I interpret this to mean that the calculation being done isn't the most accurate representation for the physics at hand. QFT is, after all, filled with weird tricks involving infinities - it seems perfectly plausible to me that the Casimir effect's math actually involves something like "-1/12 + <infinity terms>", but we're only measuring the non-infinite term in some sense, so we get away by only using it in our calculations. I think it's Very Not Good to see a totally unplausible mathematical result in physics and take it as anything other than an open problem that needs more work. It's okay to be amazed by it, but it's not okay to say "sure, okay, let's just leave this as it is".
- egjerlow 10y ago
  Are you saying that it's a coincidence that -1/12 ends up in both the casimir effect and in this proof? This coincidence is what I'm referring to..
  - mcv 10y ago
    You could get any result you want from this kind of proof. Someone else here already used some of these steps to prove that 1+1+1+1+1...==0.
  - ajkjk 10y ago
    It's not a coincidence. I'm saying that both calculations - the math we're using to understand that Casimir effect, and the math we use to 'find' the sum equals -1/12, are making the same simplification that 'misses the point' of what's 'actually happening'. In some sense that's hard to put one's finger on.
jiiam 10y ago
With all due respect, you are not really saying that the series 1+2+3+... converges to -1/12 with the obvious metric, are you? Because that's the only reasonable way to interpret 1+2+3+... = -1/12 (and it is false). The fact that there is some physics, that relies on particular results from complex analysis (I guess), which can imply that the sum of the positive integer is "in some sense" -1/12, does not exempt anyone from giving those symbols their true meaning, or they risk looking fool rather than smart.
- egjerlow 10y ago
  Not sure what you mean by the last sentence here, can you elaborate? What I'm saying is that the fact that this strange mathematical result happens to give correct results in a physical experiment must be a pointer to something.
  - jiiam 10y ago
    What I'm saying is that there is no strange mathematical result whatsoever related to the sum of the positive integers, which should be the only meaning reserved to 1+2+3+... There are stuff like this https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7_%C2%B7_%C2%B7 https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7..., but writing 1+2+3+... = -1/12 is purely formal and pretending to deduce it from physics is silly because you are not performing any sum at all.

quantumhobbit 10y ago

The analogy to complex numbers is really useful. Of course sqrt(-1) doesn't exist. But if we just pretend that is does exist, we can build a rigorous theory out of these imaginary numbers. Once we do that we notice that these numbers are really useful for calculating real physical things. So maybe imaginary numbers aren't so imaginary. Same with these infinite sums. By the math you learn in middle school, you can't have infinite sums. But break the rules for just a second and again we have something that is helpful with real physics.

chowells 10y ago
I don't know what you mean by sqrt(-1) not existing. What does it even mean to exist? Far better to talk about whether a number is defined in a particular numerical system. In the real numbers, sqrt(-1) isn't defined. But why privilege the real numbers as "existing"? Despite an official-sounding designation, they're very deeply weird. The real numbers are famously uncountable. But any subset of them that can be enumerated is by definition countable. Think about the consequences of that for a moment. No matter what you do, the subset of the reals you can enumerate is countable, meaning the subset you can't enumerate is uncountable. In a rather flippant way, you could describe the real numbers as "mostly useless." Most of them exist to make some theorems work, rather than being a number that you could ever use to describe anything - solely because describing the number would require an infinite amount of information. In a pretty significant sense, it's valid to say that the real numbers are mostly figments of analysts' imagination. If they "exist", might as well say complex numbers exist too. They're actually more useful in physics than real numbers are.
- quantumhobbit 10y ago
  I'll admit that the concept of numbers "existing" here is quite poorly defined. I was trying to capture the effect of people upon encountering complex numbers for the first time to just sort of shut down and refuse accept that they are "real" ( pun intended). It is hard to remember how weird complex numbers feel after years of middle school math teachers saying that you can't take the square root of a negative. Similarly weird feeling is encountering zeta(-1) = -1/12 after years of calculus teachers telling you to ignore divergent sums because they are infinite.
  - sebastos 10y ago
    I think the point is that if that's true, the concept of complex numbers is being taught incorrectly. If you tell a student that this number is fake but useful, what are they to make of that? It just starts to make mathematics seem spooky and unpredictable. When you're first learning about imaginary numbers in 8th or 9th grade, the answer to "what is sqrt(-1)?" _should_ be undefined. If you claim otherwise, you're pulling the rug out from under their feet, because the number system that they are familiar with indeed has sqrt(-1) undefined. Instead, the teacher should go on to introduce a new system of mathematical objects that have certain rules, and the students could play around with them and see how they have two components, how you can plot those two components in 2 dimensions, how you can think of them as arrows sticking out of the origin, how you can combine their components to rotate each other, etc. Then work backwards into showing that we can call these objects complex numbers for short, because those operations are similar to addition, multiplication, etc. And finally, just as a curiosity, you can see that sqrt(z) = i for z = -1 + 0i. There's no need to introduce this whole concept of an imaginary number line that points off in a direction nobody can see or measure. The whole takeaway should be that you can't just square real numbers get negatives. If you have something that can "multiply" by itself to get its own inverse, then you have either overloaded the multiplication operator with something very very different, or you're dealing with an object that can "rotate" through another dimension. It's an ordinary two dimensional space, and the only difference between the two axes is their name, just like "x" and "y". In my opinion, this lesson should actually be reassuring to a young mathematical intuition: there's only so many ways to skin this cat.
witty_username 10y ago
Infinite sums do not require any breaking of rules. Infinite sums are formally defined as the limit of partial sums.

ajkjk 10y ago

I strongly disagree. The Casimir effect's math works, yes, which means that "there exists a sense in which the summation equals -1/12 in this calculation", but I interpret this to mean that the calculation being done isn't the most accurate representation for the physics at hand. QFT is, after all, filled with weird tricks involving infinities - it seems perfectly plausible to me that the Casimir effect's math actually involves something like "-1/12 + <infinity terms>", but we're only measuring the non-infinite term in some sense, so we get away by only using it in our calculations. I think it's Very Not Good to see a totally unplausible mathematical result in physics and take it as anything other than an open problem that needs more work. It's okay to be amazed by it, but it's not okay to say "sure, okay, let's just leave this as it is".

egjerlow 10y ago
Are you saying that it's a coincidence that -1/12 ends up in both the casimir effect and in this proof? This coincidence is what I'm referring to..
- mcv 10y ago
  You could get any result you want from this kind of proof. Someone else here already used some of these steps to prove that 1+1+1+1+1...==0.
- ajkjk 10y ago
  It's not a coincidence. I'm saying that both calculations - the math we're using to understand that Casimir effect, and the math we use to 'find' the sum equals -1/12, are making the same simplification that 'misses the point' of what's 'actually happening'. In some sense that's hard to put one's finger on.

jiiam 10y ago

With all due respect, you are not really saying that the series 1+2+3+... converges to -1/12 with the obvious metric, are you? Because that's the only reasonable way to interpret 1+2+3+... = -1/12 (and it is false). The fact that there is some physics, that relies on particular results from complex analysis (I guess), which can imply that the sum of the positive integer is "in some sense" -1/12, does not exempt anyone from giving those symbols their true meaning, or they risk looking fool rather than smart.

egjerlow 10y ago
Not sure what you mean by the last sentence here, can you elaborate? What I'm saying is that the fact that this strange mathematical result happens to give correct results in a physical experiment must be a pointer to something.
- jiiam 10y ago
  What I'm saying is that there is no strange mathematical result whatsoever related to the sum of the positive integers, which should be the only meaning reserved to 1+2+3+... There are stuff like this https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7_%C2%B7_%C2%B7 https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7..., but writing 1+2+3+... = -1/12 is purely formal and pretending to deduce it from physics is silly because you are not performing any sum at all.