Question for Tarski

[_Tarski]

hmm. wondering if this is good. You know some apologist out there must be considering that if you add up all the numbers from alpha to omega, it sums up to Jesus presiding over the twelve apostles.

ha ha

Well, maybe we should stick with the idea that the asserted equality is a signal that one has used a way of extracting a well motivated answer from a divergent series. It has the advantage of being extremely natural within the context of analytic continuation and also apparently gives physically correct results in certain contexts.

Let me try again to give a feel for it:

Larry: Hey Steven, my analysis of our physical theory has led me to the following series:
1+x^2+x^4+x^6+....

Steven: OK, Great. Is there a problem?

Larry: Yes, I need to plug in the number 2. But then I get 1+4+16+64+.... which doesn't converge.

Steven: I see. Well, doesn't your series converge to 1/(1-x^2) as long as x is between -1 and 1?

Larry: Yes but I need to plug in 2?

Steven: Why not just directly use 1/(1-x^2)?

Larry: Hmmm, well that gives me -1/3 which strangely is just what I expected based on other physical notions.

Steven: Great!

Larry: Hey maybe we could say that in some sense 1+4+16+64+....really equals -1/3.

Steven: Ha ha. Yes, that's a nice way of talking. Did you know Euler actually wrote it down as an equality?

Larry: Oh well, that's interesting.

Steven: Hey Larry, I have my own physical theory about quantum field theory on curved spacetime and it turns out that I also get a divergent series. I just realized that my series is connected to a Zeta function. I wonder if a similar idea might help me there.....

[_Gadianton]

Hi Tarski,

I dug up your book and decided to give it another try. So I got to page 79 and read the first part of what you're trying to explain. I snuck ahead and read the second part, and then re-read your explanation here and I have an inkling of what you're trying to get at, but not quite there yet. I think I will need to read from page 80 to 135 before I say any more except...

...to quote the difference in wording between Krauss and Penrose, it's kind of interesting. Of course, I could be guilty of prooftexting here but for what it's worth:

As these examples show [Hilbert's hotel], adding up infinite numbers of things is a confusing processs, but mathematicians have developed rules that allow one to do so consistently....When considering using appropriate mathematical tools developed to handle infinite series, the sum of the series 1+2+3... can be shown to not equal infinity, but rather -1/12!

Hiding in the Mirror, 132.

From Euler?

1 + 2^2 + 2^4 + 2^6 ...= (1 - 2^2)^-1 = - 1/3

Morever, according to this rigorous treatment [the work of Cauchy], the above equation would be officially classified as 'nonsense'.

Road to Reality, 78

To me, Krauss is saying that as rigor increases, the surprising result is produced with certainty, whereas Penrose is saying the surprising response is incorrect, the more rigorous we take things. Penrose dances around the issue and says there is a "sense" in which it is right (like there is a sense to which there is a square root of -1 when there really isn't) but that sense isn't quite apparent to me yet. We'll see by next Saturday.