Ten is the lovliest number... - by Nicker
demagogue on 9/11/2023 at 03:33
But to answer the question -- well back up, I'm no mathematician or even in a STEM field, so I'm not really qualified, although I did major in philosophy with a big focus on pure logic in undergrad, so maybe I'm more qualified than I'm giving myself credit for -- I tend to be functionalist on these things. There are some situations where seemingly-divergent series do actually converge as they go to infinity where you'd want to use these kinds of techniques just to do the job.
Just speaking of Rama's Summnation, it gets used in quantum physics where you have infinite contributions of discrete sequential elements giving you finite results, cf. the whole problem of renormalization. I can't remember the exact details, but I somehow recall the Casimir effect was an example.
Of course the famous example is the original brand of String Theory for Bosons. In order to get the zero-point energy of one excitation to vanish, you have to have an infinite series of sequential contributions add up to exactly -1/12 -- which in two states calls for 24 dimensions but that's another thing -- and once you find that, like magic modeling a relativistic string will exactly match the physical properties of actual Bosons. It's one of those cases where your hand is forced to have a sum of integers add up to -1/12 or you don't get the right answer. Fortunately for them, they found the theory that lets them have it. One of the founders wrote an article or book chapter explaining how he cracked that problem while driving home one day that was a fascinating story, but I can't recall it just now.
While it's the famous example, the Casimir force example is probably better just because it's settled consensus theory.
demagogue on 9/11/2023 at 05:46
Sure, it's a special regime rule. But if Super String or M Theory is right and we actually live in a 10 dimensional spacetime, with all but 3 of the spatial dimensions curled up, and there turns out to be some experiment that could actually tease that out with 5 sigma confidence, then I'd also recognize that the universe cares more about manifesting abnormal arithmetic to weasel a finite value out of a divergent series than the so-called normal alternative. In that case, forward your complaint to That Guy.
Or again cf. the Casimir effect. It's a real force you can measure. It requires getting a finite value out of a divergent series to properly model it. It's not my fault the universe doesn't respect normal math. ¯\_(公)_/¯
Edit: Anyway, I have to know about the difference for the QM course I'm taking right now because the only Schroedinger Equation solutions that have physical meaning are the ones that converge to zero before or at infinity and that you can normalize, even if, e.g., you need an infinite number of superposed wavelengths to do it for a free particle. So I get it. But I also respect there are special cases that call for special regimes sometimes, and I think it's okay as long as you know when you're in which regime and why.
Qooper on 9/11/2023 at 10:50
Quote Posted by demagogue
Sure, it's a special regime rule.
A non-converging alternating series doesn't have a sum, but there are other ways.
Let
S1 = Grandi's series,
S2 = 1 - 2 + 3 - 4 + ...and finally
S3 = 1 + 2 + 3 + 4 + ...Using Cesàro summation, S1 gives us a value of 1/2. However, S2 isn't Cesàro summable. We can make a generalization of Cesàro sum where, if the plain sum doesn't converge and the arithmetic means don't converge, we just keep going and calculate means of the means until they converge. This way S2 converges to 1/4, and using this generalization doesn't change the previous result of S1, which was 1/2. However, even using this generalization, S3 doesn't yield a sum. So maybe summation isn't the way to go.
Is there an analytic function that we could use to get each of these series and values for them? Well, S1 is a geometric series, and there's a formula for those:
S = ar^0 + ar^1 + ar^2 + ar^3 + ... = a / (1 - r), given that |r| < 1.
In our case a = 1 and r = -1, and thus the formula gives 1 / (1 - (-1)) = 1/2, although this is just outside the valid range for r.
We could use the Dirichlet eta-function, η(s), which just so happens to be able to produce both S1 and S2 with the parameters 0 and -1, respectively.
And what are the results? Well:
η(0) = 1/2
η(-1) = 1/4
Now, what about S3? η(s) doesn't produce that sum for any value s. But Riemann's zeta-function, ζ(s) does. ζ(-1) = S3.
So, what
is the value of ζ(-1)? It is a well-known value, the one and only
-1/12.
To be clear, if we want to be mathematically rigorous, we can't get these values by taking an ordinary sum. If one wants to add and subtract the series to then get a finite value for S3, one first needs to define a special sum that gives 1/2 for Grandi's series. And after that, doing adding and subtracting of the different series no longer means quite the same thing. Once you've gotten S1 = 1/2 in a valid way, you're in a different domain.
heywood on 9/11/2023 at 16:11
Pyrian basically stated my complaint with the "proof". That bit of algebra isn't valid if S is not convergent. S is not a number. It doesn't have a value unless you truncate it. What does it mean to subtract not a number from 1?
1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S is one arbitrary definition
1 − S = 1 − (1 − 1 + 1 − 1 + ...) = (1 − 1) + (1 − 1) + ... = 0 + 0 + ... = 0 is another arbitrary definition
1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 + (− 1 + 1) + (-1 + 1) + ... = 1 + 0 + 0 + ... = 1 is another arbitrary definition
We're outside of the rules of algebra. Summation should be commutative, but we're getting different values here depending on order of operations.
So I think it was really an assertion, not a proof.
Getting back to my point, S doesn't have a value unless you truncate it. So what happens if you truncate it? Then S becomes a finite sequence of length n:
S(1) = 1
S(2) = 1-1 = 0
S(3) = 1-1+1 = 1
...
S = 1,0,1,0,1,0,1,0,1,0,...
1 - S = 0,1,0,1,0,1,0,1,0,1...
Note that 1 - S does not equal S for any value of n out to infinity.
You can treat it as a process with memory: S(n) = 1 - S(n-1). In that case, 1 - S(n) = S(n-1). Or you can treat it as a geometric series of imaginary numbers: S(n) = (1 - i^2n)/2, and in that case 1 - S(n) = (1 + i^2n)/2. But no matter how you slice it, S and 1- S have the same range {0,1} but never have the same value for any finite length of S up to infinity. So what purpose is served by asserting that 1 - S = S in the infinite?
Let me compare it with Schrödinger's cat. We all know that per the design of the experiment, when the box is opened, the cat can only be in one of two states, dead or alive. During the experiment, we model the state of the cat as a two-valued random variable whose pdf varies over time. Prior to ending the experiment, the cat exists in the superposition of both states with different probabilities. Opening the box is equivalent to sampling the random process and that's when state variable acquires a value. But what if we never open the box? If we never observe it, is the state of the cat in the box still constrained by the two possible states it can be observed in? Or can we split the difference and say it's maimed?
I'm somewhat familiar with renormalization. It's been 30 years since quantum mechanics in school, but I recall renormalization techniques being useful for dealing with infinities and singularities. I'm not yet seeing the parallel with these sequences. It's not obvious these special summations can be generalized to other oscillating and divergent series, or what properties a series has to have in order for the method to work.
Strangely interesting topic.
demagogue on 9/11/2023 at 20:09
Just a quick clarification that as far as I understand it renormalization doesn't use these techniques, unless you interpret the String Theory zero-point energy problem as a kind of renormalization problem, but that's not the textbook type even if you did (I think).
I just meant more to handwave towards that kind of world. They both have that flavor of asserting a value by fiat when you have some divergent function, if we're putting it that way. That part I'd agree with. I think the punchline is that it isn't as arbitrary as one might at first think, but I don't know how to defend that intuition well. I'm more banking on the authority of math or physics professors talking about it in that way and thinking they should have a good grounds to do that.
Nicker on 11/11/2023 at 20:04
Quote Posted by Qooper
The real difficulties arise when converting from one base to another. At least for me I'm usually thinking in decimal, even when working with binary or hexadecimal. I convert the value to decimal to "understand" what it is. But some values that can be expressed in some bases cannot be expressed in others. For example decimal 0.3 cannot be finitely expressed in binary. And trinary 0.1 cannot be finitely expressed in decimal.
Is there a point where a base becomes 'universal'? By which I mean, it can express values in all the preceding and proceeding bases?
Or does the problem of limitations in expression continue as the base increases?
And thanks to those still contributing here. I am following along and trying to keep up.
Pyrian on 11/11/2023 at 21:05
Quote Posted by Nicker
Is there a point where a base becomes 'universal'? By which I mean, it can express values in all the preceding and proceeding bases?
This question simplifies to "Is there a 3+ integer whose prime divisors include the prime divisors of all preceding and proceeding positive integers?", to which the answer is "lol no". Like, you can't even do preceding, nevermind proceeding. 3 isn't divisible by 2. 6 is divisible by 2 & 3, but not 5. And it gets worse as you go higher.
Consider the most familiar base to us, 10. It's divisible by 5 and 2. You can take any rational number of the form 1/n, and figure out how many digits it will take to express in decimal by the greater of how many 2's and/or 5's the n divides into; and any
other prime multiples makes it infinite. 1/2? 0.5, one digit. 1/8=1/(2*2*2)? 0.125, three digits. And so on. Throw in any other prime, e.g., 1/3 or 1/7, (nevermind 1/11) and it goes infinite.
This holds for any other base. Base 6 can handle 1/2 and 1/3, but not 1/5. Base 30 can handle 1/2, 1/3, and 1/5, but not 1/7. Base 15 can handle 1/3 and 1/5, but not 1/2.
No base can handle
any prime number
greater than that base.
demagogue on 11/11/2023 at 21:43
I especially get a kick out of base e, which is useful for exponential functions & natural log but I doubt it's clean in any other base you'd want to use. But as a base it's far from arbitrary.
Qooper on 11/11/2023 at 23:02
Quote Posted by heywood
That bit of algebra isn't valid if S is not convergent. S is not a number. It doesn't have a value unless you truncate it. What does it mean to subtract not a number from 1?
1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S is one arbitrary definition
1 − S = 1 − (1 − 1 + 1 − 1 + ...) = (1 − 1) + (1 − 1) + ... = 0 + 0 + ... = 0 is another arbitrary definition
1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 + (− 1 + 1) + (-1 + 1) + ... = 1 + 0 + 0 + ... = 1 is another arbitrary definition
Very true, S is not a number. But I think we can assign mathematical meaning to such algebra in certain cases, as long as the path is well defined.
Since in this case everything depends on Grandi's series, we must get a proper value to it in some way. The Dirichlet eta-function gives us that value at 0: η(0) = 1/2. The eta-function is defined for all complex values, so it's also defined at 0. To be clear, this does not mean that the sum of Grandi's series is 1/2, it just means we have a value for it. If we use this value while doing algebra with other series, it means we're no longer in the classical domain of series summation, we're somewhere else. So using this to get a value for 1 - 2 + 3 - 4 + ... means that this value is also not a sum, but something else. It's a valid value, just not a sum.
The real problem is that getting -1/12 for the sum of all positive integers using Riemann's zeta-function goes outside the defined range of the function. ζ(s) is defined only if Re(s) > 1, and the sum of all positive integers is ζ(-1). However, there is a way! Analytic continuation. ζ(s) can be defined in terms of η(s), and since η(s) was defined everywhere, so is this analytic continuation for ζ(s).
Quote:
Let me compare it with Schrödinger's cat. We all know that per the design of the experiment, when the box is opened, the cat can only be in one of two states, dead or alive. During the experiment, we model the state of the cat as a two-valued random variable whose pdf varies over time. Prior to ending the experiment, the cat exists in the superposition of both states with different probabilities. Opening the box is equivalent to sampling the random process and that's when state variable acquires a value. But what if we never open the box? If we never observe it, is the state of the cat in the box still constrained by the two possible states it can be observed in? Or can we split the difference and say it's maimed?
I'd say before observation we can't say it's one or the other, and it definitely cannot be a mixture of the two where it's a little bit an alive cat and a little bit dead. The paths of reality are discrete and do not mix, and before observation both possibilities exist distinctly without mixing into each other. The analogy works really well for Grandi's series, since it is a type of on/off/which-one-is-it-neener-neener -type of series. But what about for example 1 - 2 + 3 - 4 + ... = 1/4, what does this mean? Does the analogy end here, or does 1/4 also encode some aspect of a quantum system? Maybe not superposition, but something else?
Quote:
Strangely interesting topic.
I find it fascinating too. I'm feeling a bit poetic:
1 is where infinity and the finite kiss
1 is where we try so hard yet miss
1 is where infinity comes to tell us whether the cat is alive or dead
1 is where we get a glimpse of infinity but dare not further treadOkay it's a lousy poem. Shoot me.
Quote Posted by Pyrian
Base 15 can handle1/3 and 1/7, but not 1/2.
I think you meant 1/5 instead of 1/7?
