I have been reading Matthew Watkins' book "The Mystery of the Primes" (the first volume of a proposed trilogy) recently with much interest.
Though Matthew is clearly a qualified mathematician and therefore well able to deal with issues relating to the primes in the accepted specialised language that fellow practitioners employ, he opts here for a user friendly approach that would be accessible to most lay people (with little or no grounding in Mathematics).
What I like about this approach is that he clearly appreciates how apparently simple mathematical concepts lead to profound problems of a philosophical nature. So philosophical, psychological, religious and even economic observations are introduced early on in a much more wide ranging approach than is conventional, to unveiling the mystery of the primes.
I look forward to following Matthew's trail to see where it will lead in the further two volumes (yet to be published). I am sure that it will be interesting!
Having introduced - what he refers to as - "spiral waves" he then intends to probe more deeply into their relationship to the famed Riemann zeros in the second volume. These, magically have the capacity to eliminate the deviations associated with the generalised prediction of prime number frequency (relating to a continuous function) so that ultimately we can precisely obtain the actual frequency of such primes (which is of a discrete nature).
Such deviations seemingly correspond precisely to the vibrations of some unknown system (with a tantalising relationship to known quantum chaotic systems of a physical nature).
So a fundamental question relates to what in fact is vibrating!
My own considerations on this issue have led to the conclusion that it is the prime numbers themselves that are vibrating.
We are still far too accustomed to think of prime numbers in discrete independent terms as the building blocks of the number system. However, even momentary reflection on the matter reveals that they can have no strict meaning in the absence of the natural numbers.
To resolve this issue one must therefore look at both the prime and the natural numbers in a dynamic interactive manner. Implicit in this observation is the clear recognition that numbers have both quantitative and qualitative aspects. So rather than having a static identity (of an abstract nature) all numbers - from this new perspective - represent dynamic interaction patterns (with both quantitative and qualitative characteristics).
To be precise, the prime number code - which governs subsequent phenomenal interactions - is of a potential kind and thereby already inherent in the physical phenomena that emerge in reality. However the very unfolding of such phenomena in space and time implies some degree of separation of the quantitative from the qualitative aspect of the primes. Once again therefore the ultimate state where quantitative and qualitative (or specific and holistic) aspects are perfectly reconciled (in direct spiritual union of reality) is of an absolute ineffable nature.
And it is to this ultimate state that the Riemann Hypothesis directly points! In other words the Riemann Hypothesis relates to the fundamental condition for maintaining consistency with respect to both the quantitative and qualitative behaviour of the primes. Without such consistency (which unfortunately is the case if the Riemann Hypothesis does not hold) we would have no basis for either trusting the validity of our mathematical axioms or our holistic intuitions as to nature of mathematical truth.
As the condition, to which the Riemann Hypothesis relates, precedes such axioms and intuitions (and is already inherent in their very use) there is no way in which the Riemann Hypothesis can be proved (or disproved) in conventional mathematical terms.
It is therefore truly of the most fundamental nature possible, with its acceptance representing an act of faith in the subsequent consistency of the whole mathematical enterprise!
So in a qualified sense the prime numbers themselves vibrate! More correctly it is the interaction of both the prime and natural numbers that leads to the vibrations corresponding to the Riemann non-trivial zeros. Once again though the starting code is set in an absolute ineffable manner, these vibrations can only become manifest in a phenomenal physical manner (where some degree of separation of both quantitative and qualitative aspects has already taken place).
So the fascinating conclusion that we can draw from this is that manifest physical reality itself represents, at a more fundamental level, vibrations corresponding to number patterns (which are thereby inherent in all physical forms).
Now this might appear ludicrous when we think of numbers with respect to their mere quantitative characteristics. However when we equally recognise the qualitative dimensional characteristics (associated with number) it seems - at least to me - a somewhat obvious conclusion.
And this is the precise nub of the matter. Properly understood the dimensional aspect (i.e. power or exponent) of a number is of a qualitative nature (with respect to its number quantity). However this simple truth is - seemingly - entirely missed in conventional Type 1 appreciation. So when the number a is raised to a corresponding dimensional number b (as power or exponent) i.e. a^b, then - properly understood a and b are quantitative and qualitative with respect to each other.
A simple clue to this truth is given by the fact that when 1 is raised to a dimension that is a rational fraction such as 1/3, a circular number results (i.e. lying on the circle of unit radius in the complex plane). So though 1 and 1/3 are conventionally interpreted as linear quantities (i.e. as points on the real number line) clearly a profound transformation results (from linear to circular form) in relating the base quantity 1 with its dimensional exponent 1/3. So the simple truth is that these numbers - which can indeed be given a quantitative definition in isolation - are actually quantitative and qualitative in relation to each other!
And Type 2 i.e. Holistic Mathematics (which is directly of a qualitative nature) is based on circular rather than linear logical notions!
One important implication of this finding - which I was even dimly aware of as a child - is that one cannot hope to give a coherent explanation of an apparently simple relationship such as the square root of 1 in the absence of Type 2 (qualitative) mathematical understanding!
So remarkably, because of the unquestioned quantitative bias of Conventional Mathematics (which is thereby quite unbalanced) the most obvious problems of mathematical interpretation are persistently overlooked!
Therefore, for example when a number is raised to a whole number dimension (other than 1) in conventional terms a merely reduced quantitative interpretation is given of the result.
Thus 2^3 in Type 1 terms = 8 (i.e. 8^1).
Now by looking at this in a geometrical manner we can easily appreciate that a dimensional transformation of a qualitative nature is equally involved. In other words 2^3 represents cube (3-dimensional) rather than linear (1-dimensional) units.
So, as I have repeatedly expressed, Conventional (Type 1) Mathematics necessarily reduces the qualitative notion of dimension in a merely linear (1-dimensional) fashion.
However, when we begin to appreciate the true qualitative nature of dimension, we perhaps can recognise that physical dimensions - when correctly understood - are intimately related to holistic mathematical notions (of a qualitative Type 2 nature).
I have expressed for some time - see my Integral Science blog - dissatisfaction with the philosophical notion of a string (which has no strict physical meaning).
However once we accept that physical phenomena represent a certain rigid reduction with respect to original vibration patterns of a purely mathematical nature, then we can perhaps appreciate the ultimate nature of strings as the prime number constituents of natural reality with the relationship collectively between strings equally representing the prime number dimensional aspects of this same natural reality.
Reality at its most fundamental level is written in a double binary code (1 and 0) where both quantitative and qualitative aspects interact. This provides a means therefore of potentially encoding all (quantitative) information as is well recognised in the present digital age; however what is not equally recognised is that the same digits when holistically interpreted in a qualitative manner can potentially encode all transformation processes!
Then at the next most fundamental level, reality is written in the prime number code (again with respect to its quantitative and qualitative aspects). And from this code ultimately all phenomenal reality, as we know it, is derived!