Thursday, 16 August 2012

Little, green ideas

I'm no mathematician but am utterly fascinated by certain aspects of maths, such as the axiomatic set theory, Gödelian logic, etc. - or, the foundational structures of maths and not so much the computational tricks that seem to define maths for so many people (I could hardly mentally calculate two integers if my life depended on it).

In Feynman's book, Surely You're Joking, Mr Feynman - in classic Feynman understatedness - he peppers throughout the book profound mathematical and physics ideas in a rather casual way (many of which he himself initated or had intimate knowledge of). A couple of these are the Tarski sphere (which results in an incredible statement that one may cut a single sphere into infinite, infinitely tiny pieces and reconstitute it back into two spheres of the same size as the original! - that is, if the continuum condition applies), and the other is the Riemann Hypothesis.

The Riemann Hypothesis is a pivotal result in analytic number theory that Riemann used to refine Gauss' analysis of the logarithmic distribution of the prime numbers. The Hypothesis, if ever proven, will not only have profound impact on our confidence in mathematical physics, probability theory and statistics among other very important results that depend upon the Hypothesis being true. The Riemann zeta function extends Euler's landscape (generated by summing the infinite series whose denominators are prime numbers) into the complex plane.

The devastating effects of Gödel's incompleteness and consistency theorems have been felt in the analysis of the Riemann Hypothesis too. Many luminaries of maths believe that the Hypothesis cannot be proved using the current axioms of maths alone.

I think part of the problem lies in the fact that the Riemann zeta function has largely only been regarded as a distributive analytic function rather than looking at it as a question of whether it will always converge to (1/2 + bi). Riemann was a genius of the first order but he was also a geometer of the first order - ie, he did not regard the function in merely analytical terms but also in geometric terms (ie, as a graphing tool).

In analytical linguistics there is a fascinating problem of grammar that in its present state allows the construction of such sentences as 'little, green ideas dream furiously'. I think this happens because we've yet to define the noun function in refined, sophisticated enough terms to preclude such absurdities as 'little, green ideas' constructs - ie, the legalism of analytical linguistics is similar to the analytic distributive treatment of the Riemann zeta function.

The logical implication that Socrates is mortal in a syllogistic premise as "all men are mortal"..."therefore Socrates is a mortal" illustrates the lack of resolution because it treats the noun function without bothering to define its possible abstract and concrete terms - ie, the pattern is too flexible that it allows any premise at all: all ideas are green; Socrates is an idea; therefore Socrates is green.

I suspect this is similar to the problem of the zeta function where Hardy proved that an infinity of Riemann zeros fall into the real part 1/2 but he didn't specify which cardinality the set of these zeros belong. These are not just zeros (ie, not just analytic) but geometric points where the negative values turn to positive values (or vice versa) as plotted in a graph of the Euler function as extended below 1 in the complex plane.

I appreciate that things aren't ever that simple, especially when we view transfinite arithmetic and the axiom of choice, but we are assuming that the "set of all sets" allows sets to be elements in a set. Defining a set as a class doesn't carry us any further. S(x) is not an element, not a set, not a class, not a number but properly a function that "chooses" which elements are allowed/not allowed in a set: the Riemann zeta function converges, but where and how the condition applies rather than where the zeros are distributed might be the more proper formulation as Riemann originally posed the Hypothesis.

I mean, I've read in many authoritative and informative books that Riemann's geometrical intuition is what gave him confidence that his function would always converge to the leyline whose real part is 1/2 though he at the moment of writing his seminal paper didn't pursue it rigorously.


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