Tapestry of the Number System

I mentioned yesterday, in another of my blogs (Pop Memories) my appreciation of Don McLean’s songs (especially with respect to the quality of the lyrics that these songs contain).

In particular I singled out his song “Tapestry” in this regard. This provides both a wonderfully poetic description of the holistic interdependence of creation and the grave threat that is now posed to our environment through economic mismanagement of the planet’s resources.

Now you might think that this has nothing whatsoever to do with Mathematics, but certainly from my perspective you would be fundamentally wrong in this regard.

The current mindset that dominates economic thinking (with its emphasis on the exploitation of natural resources) is firmly rooted in our limited view of science (that attempts to view the environment in a detached objective manner).

And this view of science in turn is rooted in an ultimately distorted appreciation of Mathematics that misleadingly attempts to divorce quantitative from qualitative reality.

And this likewise leads to the attempted separation of the arts and the sciences (which again from an overall holistic perspective is most unhealthy).

Though I would maintain that the arts and sciences represent separate domains corresponding to distinctive methods of understanding, they do so in a merely relative sense. So ultimately from an overall human perspective they represent complementary pursuits (that are necessary for each other's integrated development).

Thus when we adopt an experiential perspective that is truly holistic, we no longer attempt to compartmentalise disciplines in a rigid manner.

Therefore in this spirit, I am suggesting to you that obtaining the fundamental insight to appreciate what the Riemann Hypothesis is truly about, can perhaps more easily come directly from artistic rather than scientific - or indeed direct mathematical - appreciation.

Indeed in this regard I suggest that you listen with a fresh mind to a song like “Tapestry”.

In connection with the Riemann Hypothesis, mathematicians often use the analogy of the music of the primes. However because of their compartmentalised training, they never seem to make the connection that music represents essentially a qualitative - rather than quantitative - mode of appreciation.

Put another way, it represent a holistic appreciation of the qualitative interdependence of various constituent notes, chords, instruments etc. rather than the strict analytic interpretation of each individual item in a quantitative manner.

However as it stands, Conventional Mathematics is totally lacking a qualitative dimension (in formal terms). Alternatively it is totally lacking a genuine holistic means of appreciating its relationships.

And as the mystery of the Riemann Hypothesis essentially relates to the marvellous  interdependent aspect of the number system, Conventional Mathematics remains simply incapable of recognising its true nature.

So it is like in a detective story where all the evidence is available to solve the crime, but no one can make the obvious inferences.

Now, though admittedly written in an artistic – rather than scientific – context, the opening lines of “Tapestry” are very revealing.

“every thread of creation is held in position

By still other strands of things living."

So in the context of our holistic appreciation of the number system, the threads here refer to the prime numbers.

Now these prime numbers are held in position by the (non-trivial) zeta zeros. And I have been continually making the point that there are in fact two complementary sets of zeta zeros (Zeta 1 and Zeta 2).

So these zeta zeros play a true holistic function in perfectly shadowing the prime numbers (which - relatively - are of an independent analytic nature).

In other words they ensure that the primes are locally distributed in just one optimally possible manner, so that their (relatively) independent nature at this local level, is yet fully compatible with their overall holistic interdependence in collective terms.

So in this way we have individual unique notes (the primes) comprising a cosmic number symphony (with not one note out of place). And this is because the precise positioning of the primes is holistically determined by the zeta zeros.

We could equally maintain from the reverse perspective that there is a holistic nature to the primes (as a collective whole) which ensures that not one of the non-trivial zeros (as relatively separate) is out of place!

As I have continually emphasised in my blogs, the holistic nature of the primes comes from their dimensional nature (which indirectly is expressed in a circular manner).

So, for example, we can view the prime number 3 in two ways:

1) in a linear cardinal manner. Here 3 =  3¹ (where the quantitative number 3 is raised with respect to 1 as representing its default qualitative dimension).

2) in a circular ordinal manner. Here 3 = 1³  (where the default quantitative value 1 is raised to 3 as representing its varying qualitative dimension). Indirectly this qualitative dimensional notion of 3 is expressed in a circular quantitative fashion through the 3 roots of 1).

So the primes (which must always be considered in two-way relationship to the natural numbers) keep dynamically switching as between cardinal (quantitative) and ordinal (qualitative) meanings.

Now the lyrics of Tapestry refers to “other strands of things living”.

This leads to the greatest surprise of all regarding the number system i.e. that from an authentic dynamic perspective, it is indeed living!

The reason for this remarkable property is that properly understood, numbers do not exist in some abstract world, but rather as the fundamental inherent encoding of all living phenomena.

So through incorporating the qualitative with the quantitative aspect of number, one’s whole appreciation of the nature of number radically switches to viewing it as the intrinsic encoded nature of all living systems. And from this perspective, everything in nature - from the most minute sub-atomic particle up - is understood as necessarily comprising part of a living process!

So perhaps we can now appreciate a little better the tapestry of the number system (as a dynamic living system with twin analytic and holistic manifestations which can switch depending on context).

This of course also implies that the number system is an intelligent system.

Again the very reason why we find this so difficult - even abhorrent - to appreciate is because we have become so used to looking at number in a distorted (merely quantitative) fashion.

So again the mystery of how the primes can be so perfectly distributed throughout the number system, points directly to its holistic intelligence (manifested through the zeta zeros) which ensures such synchronicity. Thus in a very important sense, the fundamental nature of the holistic intelligence that pervades all creation is thereby enshrined in the zeta zeros. 

I will quote part of another verse from “Tapestry” which may be instructive.

“Where the birth and the death of unseen generations
are interdependent in vast orchestration
and painted in colors of tapestry thread.
When the dying are born and the living are dead."

Now again in the context of the number system, the unseen generations here can be taken to refer to the unlimited nature of the zeta zeros.

Though many billions of these have now been detected (and inevitably will keep increasing), this represents but a very small set of all zeros.

So the interdependence of the primes with the number system is in fact dependent on all these unseen generations of zeta zeros which act to ensure perfect synchronicity.

So again this is how we can locate the “tapestry threads” i.e. the individual primes  (through the background holistic activity of the zeta zeros).

Now the last line is very instructive here:

“when the dying are born and the living are dead”

This essentially points to the ultimate paradoxical nature of the relationship of the primes to the natural numbers.

The very appreciation of interdependence implies paradox (from a dualistic perspective). If the primes and natural numbers are truly interdependent - which indeed ultimately they are - this implies that the primes in a sense are contained in the natural numbers and the natural numbers in turn contained in the primes. So from the cardinal perspective each natural number represents a combination of prime factors; however equally in reverse from the ordinal perspective each prime represents a combination of natural number members! 

So in this sense the primes (ultimately) are seen as perfect mirrors of the natural numbers and the natural numbers as perfect mirrors of the primes. 

And the only way of truly seeing this relationship is through avoiding any rigid identification in understanding with either perspective (in isolation).

So in this highly dynamic appreciation, the primes, as mirrors of the natural numbers, only can live briefly in experience through letting the alternative complementary perspective (of the natural numbers as mirrors of the primes) momentarily die.

Then the corresponding dynamic appreciation of the natural numbers as mirrors of the primes can in turn only live briefly through again letting the alternative perspective (of the primes as mirrors of the natural numbers) momentarily die.

Thus ultimately the pure experience of this mutually interdependent relationship is ineffable!

I will finish up by quoting the last verse of “Tapestry”

“Every fish that swims silent, every bird that flies freely,
every doe that steps softly.
Every crisp leaf that falls, all the flowers that grow
on this colourful tapestry, somehow they know.
That if man is allowed to destroy all we need.
He will soon have to pay with his life, for his greed.”

This clearly is pointing to the holistic nature of intelligence that pervades, in a fully natural manner, all plant and animal life on the planet.

The remarkable assertion here is that human beings uniquely possess the capacity to behave in an unnatural manner, thereby having the potential to destroy all that is in nature.

Now the intellectual roots of this unnatural behaviour lie deep in the nature of our distorted mathematical appreciation. And it is such appreciation that directly informs in turn our scientific and economic thinking!

And as we have seen, though again most readers again might find this abhorrent to accept, the ultimate reason for our capacity for such unnatural behaviour lies in this deeply confused interpretation of the nature of the number system (and by extension all mathematical understanding) where we have attempted - wrongly - to completely divorce qualitative from quantitative type appreciation.

So we have lost any concept of the global holistic nature of intelligence (as the collective unconscious of all life forms).

And again this is because we have attempted to portray Mathematics and the sciences - misleadingly - as representing merely conscious activity.

We are thereby reaching a true crisis now on our planet where due to this great imbalance, we are in increasing danger of being  swamped through ever more frequent economic and environmental disasters.

Much as we might hate to admit it, the key intellectual cause of all this problem lies in our extremely limited interpretation of the nature of the number system (and by extension all mathematical relationships).

Many continue to look on Mathematics as the one area of truth that can remain safely insulated from the contamination of global change.

In fact, properly seen, Mathematics (or rather our false understanding of Mathematics) is the root intellectual cause of this growing contamination!