I think that the claim that the "truth" of [1] 1 + 2 + 3 + 4 + ... = -1/12 like that of any other mathematical statement, depends on the theoretical context. Is it true that 1 + 1 = 2, 1, or 0? It all depends if we are working in integer, Boolean, or Z(2) arithmetic. Using standard convergent series, the series 1 + 2 + 3 + 4 + ... Is clearly divergent and has no sum. However, Numberphile derives [1] via formal series manipulations (all of which appear legitimate for convergent series) arriving at the series 1 - 1 + 1 - 1 + ... = 1/2 evaluated using Cesaro summation as opposed to partial sum limits. This suggests that [1] is perhaps a legitimate result in a broader theory of infinite series based on Cesaro sums as opposed to the standard theory of convergent partial sums. My question would be, is this Cesaro-based theory of infinite series coherent? I suspect it is. If physicists routinely use formal manipulations and Cesaro sums such as those shown in the Numberphile video to evaluate classically divergent series, it would be an embarrassment if they had no mathematical foundation to assure them that their evaluations were unique in general or correct in a specific instance. If some particular method evaluating divergent series yields values that appear to work in some particular physics application, this is perhaps evidence, but certainly not proof, that the methods are legitimate and reliable. What is needed is a mathematical theory built on solid mathematical foundations in which [1] and like statements are theorems on a par with the rest of mathematics. Seriously, do physicists routinely use these types of formal manipulations without any mathematical foundation? I'm guessing not. If physicists are using equation like [1] in actual physics, I suspect they must have some theory to justify their methods.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Friday, February 28, 2014 5:30 PM To: math-fun Subject: Re: [math-fun] Fwd: numberphile.com
It appears that MSRI is listed as one of the "sponsors" of the website, and apparently that means the website can use the MSRI logo, but the website claims that it is independent of MSRI.
The website looks very nice - at first glance. But I don't think MSRI would intentionally support a video that claims 1+2+3+ . . . = -1/12, since the research institute seems mainly interested in mathematical truth.
-Dan
On Feb 28, 2014, at 1:33 PM, rcs@xmission.com wrote:
It looks like MSRI is getting into serious outreach. Bob Baillie passed along the url http://www.numberphile.com I haven't looked at the insides yet, but the front page looks tasty.
Rich
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