Wednesday, April 30, 2014

The Limits of Conventional Logic

Marcus du Sautoy in the "Music of the Primes" refers to the Continuum Hypothesis which was the first on Hilbert's famous list of 23 great unsolved mathematical problems.

In 1963, Paul Cohen proved that this represented one of Godel's undecidable propositions. In other words on the basis of the accepted mathematical axioms it was not possible to prove (or disprove) this proposition as to whether another set of infinite numbers exists between - as it were - the rational fractions and the real numbers.

In fact Cohen was able to construct two axiomatic worlds where the proposition could be proven true in one and false in the other!

He then goes on to argue that the Riemann (8th on Hilbert's list) is distinct from the Riemann Hypothesis in a very important sense.

So according to du Sautoy if the Riemann Hypothesis is undecidable then two possible outcomes exists
(1) it is either true and we can't prove it or (2) it is false and we can't prove it.
However if it is false there is a zero off the critical line which we can use to prove it is false. So it can't be false without us being able to prove that it is false.

So therefore according to this logic, the only way that the Riemann Hypothesis can be undecidable is that it is indeed true (without us being being able to prove that it is true).

However, I would strongly question the validity of this use of logic. From my perspective, the Riemann Hypothesis - by its very nature - already transcends our accepted mathematical axioms.

Once again, I would see the Hypothesis as relating ultimately to an a priori assumption that both the quantitative (as independent) and qualitative aspects (as interdependent) of mathematical interpretation can be consistently combined with each other.

However the reduced nature of conventional mathematical proof implicitly assumes such consistency to start with. In other words the very use of its axioms thereby assumes the truth of the Riemann Hypothesis (as an a priori act of faith).

Therefore there is no way that such an a priori assumption can be either proven (or disproven) through the use of such axioms.

So du Sautoy maintains that if the Riemann Hypothesis is in fact false, that we can find a zero off the critical line (and thereby prove that it is false).

However even if one was to accept in principle that a zero might lie off the critical line, this does not infer that we can thereby automatically show that it is off the line.

For example from one valid perspective, it could be so beyond the range of finite magnitudes that can conceivably be investigated that it would remain practically impossible to experimentally detect it!

However there is a much more crucial difficulty which du Sautoy has overlooked.

If my basic premise is correct that the the Riemann Hypothesis is an a priori assumption that underlines the very consistency of conventional axioms (which cannot be proven or disproven through use of these axioms), then if the Hypothesis is indeed false then we no longer have a sufficient basis for trusting the ultimate consistency of any of our mathematical procedures.

Therefore if a zero was somehow to be experimentally verified as existing off the critical line, this would imply that the whole mathematical edifice is ultimately built on inconsistent premises.

Therefore in such a circumstance we could not use the subsequent emergence of a zero off the critical line to disprove the Riemann Hypothesis (in a conventional manner) as this assumes the inherent consistency of our mathematical approach.

Now, realistically I do not expect that a zero will ever be found off the line!
So acceptance of the Riemann Hypothesis is already built into our assumptions of how the mathematical world operates. However this acceptance strictly exists as an act of faith (rather than logic).
However this does imply that uncertainty is fundamentally an inherent part of Mathematics, with the possibility always remaining that this act of faith (in what is implied by the Riemann Hypothesis) is ultimately unwarranted.

There is also the interesting case of the Class Number Conjecture (referred to by du Sautoy).
In 1916, a German mathematician Erich Hecke, succeeded in proving that if the Riemann Hypothesis was true then Gauss's Class Number Conjecture was also true.
Then later three mathematicians Max Deuring, Louis Mordell and Hans Heilbronn succeeded in showing that if the Riemann Hypothesis is false, this could also be used to prove that the Class Number Conjecture was also true.

The significance of this finding for me really points to the inadequacy of using conventional linear logic to interpret the nature of the Riemann Hypothesis.

Conventional Mathematics is 1-dimensional in nature solely based on quantitative notions of interpretation.

However viewed more comprehensively Mathematics properly entails the dynamic interaction in a relative manner of both quantitative (analytic) and qualitative (holistic) aspects of interpretation. And this is the truth to which the Riemann Hypothesis directly relates.
Therefore if we insist on just one pole of interpretation (in an absolute manner) then the very nature of the Hypothesis is rendered paradoxical.

Thus is the the quantitative aspect (as objective) is true in absolute fashion, the qualitative aspect (as mental interpretation) is thereby rendered absolutely false; in turn if the qualitative aspect (as interpretation) is absolutely true, then the quantitative aspect (as objective) is absolutely false.

This in this restricted (linear) logical sense, the Riemann Hypothesis can be accepted as absolutely true and absolutely false at the same time.

Thus the "proof" of the Class Number Problem really depends on the acceptance that the Riemann Hypothesis transcends conventional notions of logical understanding.

However strictly all mathematical proof is based on the same supposition  (which itself cannot be proven or disproven).

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