Wednesday, December 21, 2011

Spiral Waves

I have referred before to Matthew Watkin's book "The Mystery of the Prime Numbers".

One of the features that I especially like about this book is that he successfully converts the standard natural log notion (which has such an important bearing on the general distribution of the primes) into the much more expressive form of an equiangular spiral.

Now in geometric terms equiangular spirals are very suggestive as they combine both linear and circular notions in a systematic ordered manner.

Indeed at the two extremes of such spirals we get - what he refers to as degenerative spirals - of both the straight line and the circle.

Matthew also extends this notion of spirals into the treatment of the famous deviations in the Riemann Prime Counting Function which are intimately related in turn to the non-trivial zeros of the Zeta Function!

I look forward very much to an extended treatment of these "spiral zeros" in the second volume.

It is also worth noting that considerable attention has been given to the Prime Spiral (Ulam's Spiral) where when natural numbers are entered on a grid in spiral fashion that the prime numbers then tend to fall along diagonal lines through the spiral! However my intention here is to focus on the qualitative significance of the relationship of prime numbers to spiral wave forms!

Just as linear and circular notions have a well defined quantitative meaning in conventional (Type 1) mathematical terms, equally they have a well defined qualitative meaning in holistic (Type 2) mathematical terms. However this latter type of interpretation is totally ignored by the mathematics profession.

Thus I would strongly contend that it is the absence of this vital qualitative dimension that is preventing recognition of the true nature of prime numbers.

In other words prime numbers combine both specific (independent) aspects in their individual nature with holistic (interdependent) aspects in their overall distribution.

Actual experience of mathematical reality equally combines specific and holistic elements through the interaction of (conscious) reason and (unconscious) intuition. However once again in conventional terms the qualitative (intuitive) aspect is reduced in a quantitative (rational) manner.

Proper understanding of the Riemann Hypothesis thereby requires both linear (either/or) logic based on the clear separation of polar opposites such as external/ internal and circular (both/and) logic based on the corresponding complementarity of such opposites.

Indeed ultimately the Riemann Hypothesis relates to the vital condition necessary for the consistent relationship of both types of logic!

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