In yesterday's blog, I dealt briefly with the holistic interpretation of the prime number notion.
This leads directly to a dramatic change in perspective regarding the nature of prime numbers, whereby one views them inherently in dynamic terms, as entailing the interaction as between both quantitative and qualitative aspects (which are complementary).
This equally entails that one cannot hope to properly understand the primes in merely conscious rational terms, as this gives but a reduced - and thereby distorted - interpretation.
So the implications could not be greater for Mathematics, as it directly implies that we must therefore move to a newer more comprehensive appreciation that formally includes both conscious and unconscious aspects.
It might be useful to also look briefly at the holistic nature of some of the other number types which have a special relationship with the primes.
The natural numbers are in this context very interesting and in holistic terms emerge in experience through a psychological process (that is akin to multiplication).
Once again at the earliest stage of infant development, experience is highly primitive so that object phenomena are of a very fleeting nature. Quite literally, at this stage, the infant is still unable to meaningfully organise the objective (quantitative) with the dimensional (qualitative) aspect, so that both are directly confused with each other.
Put another way, the infant is unable to distinguish cardinal (as independent) from ordinal number notions (as interdependent).
However, through the gradual differentiation of consciousness, a stabler organisation of the child's experience emerges, with natural phenomena thereby achieving greater consolidation. From this perspective, natural phenomena can be seen to represent unique configurations with respect to earlier - merely primitive - i.e. prime impulses.
However this raises a deep question with respect to the conventional mathematical treatment of multiplication, which is interpreted in a merely reduced quantitative manner. For, properly speaking, such quantitative transformation is not possible without a corresponding qualitative transformation in the nature of space and time! So there is a crucial dimensional aspect literally missing from the conventional treatment of multiplication.
Interestingly, primitive type behaviour coincides for the infant with the growth in development of natural type phenomena. However just as with the distribution of the primes among the natural numbers, the primitive aspect tends to then die out with ongoing further consolidation of natural type phenomena. So in general terms the movement from primitive (prime) to natural experience represents the corresponding move from unconsciousness - more correctly confusion of conscious with unconscious - to more mature conscious development.
Transcendental numbers also play a huge role in the subsequent understanding of the primes. For example the famous number e, on which the natural log system is based, is transcendental in nature.
The important question then arises as to the holistic meaning of the transcendental notion.
Perhaps the best way of appreciating this is with reference to the best known transcendental number π.
Now with respect to any circle, π represents the relationship between the (circular) circumference and its (line) diameter.
In corresponding holistic terms, the line is associated with the conventional quantitative approach to Mathematics (in a linear rational manner). The circle by contrast is associated with the corresponding qualitative approach (in an intuitive fashion that seems paradoxical in linear rational terms).
So transcendental in its holistic meaning, pertains very subtly - not to quantitative or qualitative as separate - but rather to the relationship as between both quantitative and qualitative aspects. And this is basically true not just with respect to π, but indeed all transcendental numbers.
Now when one reflects on it, the holistic transcendental is directly complementary with the corresponding prime notion (which entails the confused relationship of quantitative and qualitative).
Therefore it requires the integrated holistic notion of the transcendental (in the relationship of quantitative to qualitative) to unravel - as it were - the corresponding confused prime notion.
Needless to say however the holistic concept of the transcendental cannot be understood in conventional mathematical manner as this would simply reduce the transcendental to merely rational terms!
Thus once again this suggests the true magnitude of what is requires in Mathematics in the need to open up completely new means of understanding mathematical relationships!
The imaginary is another key notion with a vitally important holistic - as well as analytic - meaning.
Basically in holistic terms, the unconscious is dynamically activated through negation of unitary poles of reference.
For example in conventional mathematical terms, interpretation always takes place with respect to such unitary poles of reference i.e. in a 1-dimensional (linear) fashion.
So this is the case for example when we attempt to interpret mathematical relationships from a merely objective perspective (that formally takes place in a conscious rational manner).
However the very activation of the unconscious in understanding requires - literally - the dynamic negation of what has been posited in unitary fashion.
So in holistic terms we represent this unconscious aspect understanding as – 1.
Now just as matter and anti-matter particles fuse in energy, likewise the psychological negation (of what has been posited) results in a 2-dimensional fusion of both positive and negative poles.
Thus the imaginary notion represents the indirect attempt to express in a conventional analytic (i.e. linear) manner the holistic notion of qualitative meaning (that directly represents an energy state in psychological and physical terms).
This is of vital significance for example for appropriate interpretation of the Riemann Hypothesis.
The postulated existence of the Riemann zeros (as all lying on the imaginary line through .5) relates directly therefore (in complementary fashion to the customary analytic interpretation of the primes) to the indirect representation of number meaning that is properly holistic i.e. as energy states.
Thus in approaching the Riemann Hypothesis from the holistic perspective (with which I had immersed myself for several decades) I was confident that I could bring a radical different perspective to bear that would substantially alter the very interpretation of what the Hypothesis is fundamentally about!