Indeed as the true implication of the Riemann Hypothesis are so fundamental and far reaching for Mathematics, in effect the present has been used in many respects to overlap with material covered in that latter blog.
Though in some senses a bit vague, the word “spectrum” is now widely used in physics to refer - in any context - to a variety of related phenomena,
Perhaps its best known use is in connection with the “spectrum of light” whereby in passing through a prism, the various wavelengths of natural light can become manifest as different colours. And of course the colours of the rainbow produce a beautiful natural example of this phenomenon!
Then when it was realised that natural light represented just one small band with respect to electromagnetic energy, this spectrum notion was widened to include all its possible bands from very high frequency gamma radiation at one end to extremely long frequency radio waves at the other.
The notion of light in physical terms invites comparison with the corresponding notion of light in a psycho-spiritual context so that human development itself represents a spectrum of all possibilities with respect to the capacity for enlightenment.
Such enlightenment is directly related to the holistic aspect of intuition.
As we have seen mathematical activity itself, in dynamic terms, necessarily entails an interaction between reason and intuition. Indeed it will be readily admitted by mathematicians, that where especially creative work is involved, that intuition necessarily plays an important role.
However in formal terms, mathematical results are then presented in a merely reduced rational manner (with the important holistic aspect of intuition edited completely out of the story).
What is not readily recognised however is in a somewhat complementary manner to the electromagnetic spectrum, that the full range of intuitive possibilities allowed by the psycho-spiritual spectrum is much greater than generally imagined.
Indeed just as natural light forms just one small band of the electromagnetic energy spectrum, likewise - what we might call - natural intuition (that supports the linear use of reason in Conventional Mathematics) comprises just one small band of the full range of intuitive possibilities available.
Detailed explorations of higher intuitive states have been undertaken in all the major spiritual contemplative traditions.
However what has been almost completely missing has been the attempt to study the relevance for Mathematics of such psycho-spiritual development.
So from a very early age, I could somehow see this area as my special goal in life, i.e. to marry the holistic possibilities opened up through spiritual contemplation, to the mapping of a new mathematical understanding (consistent with such development).
I then realised that inherent in the very notion of number, is a marvellous way of precisely detailing all possible stages of development in a holistic mathematical manner.
Some 20 years ago I wrote an online book “The Number Paradigms” that basically outlines my earlier findings on this matter.
What struck me forcibly at the time is this!
From a quantitative perspective, they are now indeed many recognised number types that play a huge role in Conventional Mathematics.
So I outlined firstly what I termed the original numbers 1 and 0. Next on the list we have the primes, then the natural numbers and the integers (positive and negative).
These are followed by the important rational numbers.
Then we move on to irrational number (algebraic and transcendental), the imaginary and complex nos. and the transfinite.
Now this is the key point!
In conventional mathematical terms all these are understood within what may be correctly referred to - qualitatively speaking - as the rational paradigm.
So, while we recognise a wide range of number types in quantitative analytic terms, we recognise just one i.e. rational in a qualitative holistic manner. And as I have repeatedly stated in my blog entries the very nature of this rational paradigm is to reduce all qualitative meaning in merely quantitative terms!
Thus we have sectioned off just one very small restricted band of possible development (relating to specialised rational understanding) and defined Mathematics exclusively in terms of this band.
It is quite akin to sectioning off the small band corresponding to natural light and refusing to recognise the validity of all other energy wavelengths on the electromagnetic spectrum!
Indeed the parallels are very striking! It would be extremely difficult for one identifying exclusively with the present mathematical paradigm to even appreciate what is being said here.
When one literally “sees” exclusively in accordance with the natural intuition that informs conventional mathematical understanding, any appeal to different types of mathematical reality (based on holistic intuition relating to other bands on the spectrum) literally has no resonance.
Though my thinking on these matters have become considerably more refined in the last 20 years or so, I still would see the basic position taken in “The Number Paradigms” as fully sound.
Putting it simply, when we adopt a true holistic (i.e. qualitative) mathematical perspective, then all of human development - and in complementary terms physical evolution - can be fruitfully seen as representing a number spectrum.
So giving a brief outline we have firstly the lower stages of development.
The basic point here is that these are marked by confusion as between conscious and unconscious processes.
So the first task is to successfully differentiate conscious from unconscious understanding. The earlier number types mentioned, original, prime, natural and integers in qualitative holistic terms are then associated with these stages.
The middle stages represent the specialisation of differentiated conscious understanding (free of holistic unconscious confusion). Conventional Mathematics - par excellence - represents an extreme in terms of this understanding which of course in qualitative terms is associated with the rational numbers.
The higher stages then represent the process of now developing in a mature way the unconscious holistic aspect of understanding, which is qualitative terms is associated with irrational (algebraic and transcendental) and imaginary numbers.
One of the interesting findings I made at the time was that the most refined form of holistic understanding (where phenomena are now of a highly transparent and fleeting nature) are associated with transcendental structures of an imaginary nature.
As these directly relate to the very nature of the Zeta 1 (non-trivial) zeros, let me briefly elaborate.
Imaginary in a qualitative holistic sense, refers to the indirect expression (in a refined conscious) manner of what pertains directly to unconscious meaning.
So when we are understanding in a refined imaginary manner we can quickly recognise all phenomenal projections (though invested in consciously recognised objects) as directly expressive of holistic (unconscious) meaning.
Transcendental refers to a highly refined form of understanding where objects are neither interpreted exclusively in a linear (conscious) nor circular (unconscious) manner, but in terms of the common relationship - which is ultimately ineffable - as between both.
So the ability to immediately interpret projections (as imaginary) in a transcendental manner, requires the most refined spiritual development possible (consistent with experience in the phenomenal realm).
Now the significance of this is that the zeta zeros necessarily unfold in the very first instants of phenomenal creation.
When number is appropriately understood in a dynamic interactive manner, it is no longer considered in abstract (merely rational) terms, but rather as the most intrinsic encoding of all phenomena (both in physical and psychological terms).
Without this encoding, enabling both quantitative and qualitative aspects of reality to coherently interact, no phenomena could ever become manifest.
Thus to properly appreciate what is - literally most primitive or prime - with respect to earliest creation, requires ultimately the most developed spiritual experience possible.
In the truest sense the zeta zeros come closest in the phenomenal realm to serving as the first bridge from the nothingness preceding physical evolution.
Therefore their mature understanding likewise requires reaching as close as possible to that final spiritual bridge to the ineffable.
The zeta zeros in fact represent the holistic (unconscious) counterpart to the analytic (conscious) aspect of the primes.
We look on the primes as perhaps the most rigid and immutable in absolute terms.
In fact, as is the nature of all experience, they cannot be totally absolute, though yet coming closest in the phenomenal realm to reaching this standard.
However as the zeta zeros represent the holistic (unconscious) shadow of the primes, they approximate closest at the opposite extreme to the most purely relative - and inherently dynamic - expression possible of the number system.
So the number system is the original and most important spectrum (with respect to both physical and psychological aspects). And of course it is an inherently dynamic spectrum that necessarily vibrates through the medium of physical and psychological interactions)..
The idea that this system is some fixed abstract entity frozen in space and time is just the product of a false and limited type of understanding.
In fact the number system represents all the possibilities for the evolution of phenomenal creation (as its inherent encoding in quantitative and qualitative terms).