Friday, January 30, 2015

I've been had by Mr. Bender's lectures

Bamboozled, duped, hoodwinked, well, you get the idea. I drank his cool aid on (lack of) mathematical rigor and I bought his idea of summing infinite series like

\( 1+ 2+ 3+ \cdots = -1/12\)

What I did wrong was swallowing hook, line, and sinker his postulates:

  • Rule 1: Summation property:
    • \(S(a_0 + a_1 + a_2 + \cdots )= a_0 + S(a_1 + a_2 + \cdots )\)
  • Rule 2: Linearity:
    • \(S(\sum (\alpha a_n + \beta b_n) )= \alpha S(\sum (a_n)) + \beta S(\sum (b_n))\)

    Why? Because in addition to assuming a unique result, they are inconsistent. If:

    \(1+2+3+ \cdots = -1/12 = A\)

    Then for example consider this:
    \(A-A = (1+2+3+ \cdots ) - (1+2+3+ \cdots )\) 
    which by rule 1:
    \( 0 = (1+2+3+ \cdots ) - (0+1+2+3+ \cdots ) \) 
    such that
    \(0 = 1+1+1+1+ \cdots\) 

    Aha! This now implies that
    \(0=1 + (1+1+1+\cdots ) = 1+0\) and so \(0 = 1\)!!!!

    The two rules work for alternating sums, but when the sign of the sum terms is the same the two rules are clearly inconsistent. 

    But does this mean that \(1+2+3+ \cdots \) is not \(-1/12\) ? Not at all. The result is still valid due to deeper reasons: analytic continuation of Riemann zeta function.

    It is not easy to find why things like this work in math, but in general physics intuition is a very good clue that there must be a solid and rigorous foundation. It is just that physicists' focus is on solving the practical problems and not on the deeper mathematical theory. One may say that a physicist to a mathematician is like an engineer to a physicist :) This is not that bad though: the engineers make more money than physicists, and physicists make more money than mathematicians. 

    I still regard Mr. Bender's lectures as outstanding, but I should have trusted my mathematical intuition more and not disregarded the alarm bells of mathematical rigor. The inconsistent argument above is due to David Joyce and I ran across it on Quora where the -1/12 result is discussed often.

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