Wednesday, November 6, 2013

Where Science and Art Coincide (20)

We will wrap up the present discussion with this entry.

As we have seen the proper understanding of number is inseparable from human development, which reflects the inherently dynamic interactive nature of experience.

And just as with the electromagnetic spectrum, we find many distinctive bands of radiation (of which natural light forms just one small band), likewise with the spectrum of development, there are many distinctive bands of which specialised linear understanding (that informs conventional mathematical interpretation) is just one.

So what is recognised as Mathematics in our culture really represents specialised analytic interpretation (of a quantitative kind).

In my overall model of the spectrum of development (comprising 7 bands) this simply represents understanding consistent with Band 2.

Bands 3 and 4 on this spectrum are then concerned with highly refined intuitive type understanding.

The specialised form of this understanding (consistent with Band 4) is then associated with specialised holistic interpretation (of a qualitative kind).

So all mathematical symbols can be given both a Type 1 (analytic) and Type 2 (holistic) interpretation. Thus in relative separation from each other, Type 1 is associated with Conventional Mathematics and Type  2 with Holistic Mathematics respectively.

However the most comprehensive type of understanding entails the combined interaction of both (specialised) analytic and holistic type understanding. Now, the mature attainment of this most developed from of appreciation is associated with Band 6 on the spectrum.


Likewise in mathematical terms therefore we can define Type 3 (comprehensive) mathematical interpretation as the refined dynamic interaction of both its analytic (quantitative) and holistic (qualitative) aspects.

And just as the human perfection in its most complete sense requires the harmonious development of both cognitive (rational), affective (emotional) and volitional (spiritual) capacities, equally this is true of  Mathematics itself when understood in its most comprehensive manner.

Indeed it is especially true of the nature of the number system.

What we sought to demonstrate in the last few blog entries is the remarkable fact that understanding of the Zeta 2 and Zeta 1 zeros are cognitive and affective with respect to each other. This implies that the very ability to properly appreciate the ultimate interdependence of these zeros requires the equal development of both rational and aesthetic abilities.

And the ultimate appreciation of the true mystery of the primes (and their relationship to the zeta zeros) is of a direct ineffable spiritual nature.

One of the great problems with modern Mathematics is the increasing tendency towards specialised understanding of a purely abstract rational nature.

Thus in approaching ever closer to its analytic extreme, recognition of the equal importance of its (unrecognised) holistic aspect is completely blotted out, so much so that I have never found it even possible to discuss such matters constructively with mathematicians.


This was not always the case. For example, with great mathematicians such as Gauss and Riemann, abstract ability would have been closely associated with a strong visual sense. Thus they would never have sought to divorce mathematical from corresponding physical understanding of the world.

Simple mathematical notions can often be given a visual representation that then conveys the (unrecognised) qualitative nature of its symbols.

For example the beautiful fractal images associated with Mandelbrot's set can be derived from a remarkably simple formula (entailing complex number iterations).

So though we may initially consider this formula mathematical  in a merely cognitive (rational) manner, its visual representation then appeals directly to the aesthetic sense (which is of an affective nature).

Thus a more comprehensive appreciation of the formula should properly combine both cognitive and affective appreciation.

At another level the golden ratio (golden mean) again is related to a very simple mathematical formula (with a rational explanation).

However geometrical appreciation of this ratio can be easily seen to likewise require an aesthetic ability.
In fact, properly understood the very equation (for which the ratio, phi is obtained) likewise requires both cognitive and affective appreciation.

And then directly associated with the Zeta 2 zeros are the visual representations (or mandalas) that Carl Jung considered so important as profound archetypes of spiritual integration.

For example the most common patterns Jung found to be related to 4 and 8 respectively.

Now we could attempt to try and explain all this in a merely rational analytic manner as the geometrical representation of the 4 and 8 roots of 1 respectively. However the point is that as Jung rightly observed, these images (or mandalas) now serve - not an analytic - but rather a holistic purpose.

This would of course also suggest that the Zeta 1 zeros, when properly appreciated, again should serve as powerful archetypes - indeed the most powerful of all - of an integral holistic meaning.


This integral meaning is therefore intimately contained in the zeros which then is unlocked through appropriate interpretation.

However the specialised analytic approach that dominates present Mathematics inevitably rules out such holistic interpretation.

So my point once again is that - when comprehensively understood - all mathematical symbols possess both analytic and holistic aspects. However we have to first recognise the equal importance of both aspects in a relatively separate manner, before finally combining each in a dynamic interactive fashion (without each aspect losing its special distinctiveness).


In the end true appreciation of the ultimate nature of the primes is inseparable from the eventual union of the (analytic) conscious and (holistic) unconscious in experience.

Now the involuntary nature of primitive type instincts always betrays a problem in terms of the successful marriage of conscious and unconscious aspects of personality.

Therefore form this perspective, the ultimate resolution of prime (i.e. primitive instinctive) behaviour in psychological terms is inseparable from the ultimate resolution of the nature of prime numbers (in a corresponding physical manner).

So if you want to properly know the nature of prime numbers (externally), you must likewise know the nature of thyself (internally) for both of these aspects are ultimately inseparable.

And herein lies the final message that needs to be clearly realised i.e. that notions of number - and indeed all mathematical notions - properly understood, are dynamically  inseparable from both the physical and psychological aspects of reality (which ultimately are fully complementary).

Tuesday, November 5, 2013

Where Science and Art Coincide (19)

As we have see, in dynamic interactive terms, the successful analytic differentiation of the primes in experience is ultimately inseparable from their corresponding successful integration in holistic terms.

And the former aspect corresponds with the Type 1 and the latter aspect with the Type 2 aspect of the number system respectively. Then the dynamic interpenetration of both aspects corresponds to the Type 3 aspect.

Thus the actual nature of the individual primes (as commonly understood in a quantitative analytic manner) is dynamically inseparable from the corresponding collective set of zeta zeros (as understood in a complementary qualitative holistic fashion).


However just as left and right turns have an arbitrary meaning at a crossroads (depending on the direction from which the crossroads is approached) likewise what is analytic and holistic with respect to the primes and zeta zeros respectively, likewise has an arbitrary distinction depending on the frame of reference adopted.

Thus in a reverse manner, each zeta zero can be understood in an individual and the primes in a collective nature respectively.
Then from this perspective, the successful analytic differentiation of each zeta zero in experience is ultimately inseparable from the corresponding integration of the primes in holistic terms.

This is where the two sets of zeta zeros come in!


What is holistic with respect to one set is analytic with respect to one and what is holistic with respect to one notion of the primes is analytic with respect to the other.

Therefore the tasks of successfully differentiating each prime number (in both Type 1 and Type 2 terms) and likewise integrating with respect to the zeta zeros (Type 1 and Type 2) and then equally differentiating each zeta zero (in Type 1 and Type terms) and integrating with respect to the prime numbers (Type 1 and Type 2) are ultimately totally interdependent in an ineffable manner.

The relative phenomenal aspects of both the primes (with respect to the natural numbers) in cardinal terms and the natural numbers (with respect to each prime) in ordinal terms, only becomes apparent when there is already a degree of separation evident with respect to this original ineffable interdependence.

Equally the phenomenal aspects of the zeros (with respect to the their collective sets) in both Zeta 1 and Zeta 2 terms thereby likewise only becomes apparent due to this degree of relative separation.

We then have with respect to the relationship between the primes and the natural numbers and as between the Zeta 1 and Zeta 2 zeros respectively, differing perspectives with an apparently unambiguous meaning within each (relatively) separate reference frame.

However when we combine these various reference frames together in an interdependent manner, all these interpretations are now seen as paradoxical.


When one appreciates the truly dynamic interactive nature of the number system, then it becomes futile to attempt to understand it coherently in the standard conventional mathematical manner.


We have been accustomed to look at numbers for several millennia now as static absolute entities.

However this view simply reflects the highly reduced - and thereby distorted - nature of the corresponding interpretation adopted.

Though this interpretation is indeed extremely useful within a limited quantitative perspective, ultimately it is completely lacking in overall coherence.

And the good news is that we can provide this much needed coherence in a uniquely distinctive manner that opens up limitless further possibilities in terms of the nature of the number system (and by extension Mathematics and all the Sciences).

And the limited quantitative aspect - that has become falsely synonymous with Mathematics - will still of course continue but now with the benefit of being seen as forming just one aspect of an altogether more comprehensive perspective.


From the limited quantitative standpoint, Mathematics is seen solely in scientific terms (as the indispensable tool for all the other Sciences).


However from the comprehensive developmental perspective, Mathematics, in its fullest expression as expression,  represents the culmination of successful development in cognitive (rational), affective (emotional) and spiritual (intuitive) terms.

In particular the very developmental process corresponding directly with the understanding of the Zeta 1 (Riemann) zeros is of an affective - rather than cognitive - nature.

Though these zeros can indeed be given a cognitive form - that indirectly corresponds with reason - in a direct sense their appreciation comes from artistic (aesthetic) rather than scientific (rational) development.

Thus the culmination in understanding of the great mystery of the primes is inseparable from appreciation of  the great mystery of life itself, both of which unfold through the human development process.

Indeed this mystery of the primes (at the heart of the number system) is directly central to the nature of  all phenomenal evolution.

In this sense - as Hilbert intuited - it serves as key issue that is the basis of everything we can know and understand.

Saturday, November 2, 2013

Where Science and Art Coincide (18)

The transcendent (top-down) aspect of integration generally unfolds through the refined use of reason in disciplining - what may be perceived as - the "lower" senses (in the form of unconscious primitive impulses).

As we have seen the Zeta 2 zeros are directly associated with the holistic nature of such transcendent development.

However the immanent (bottom-up) aspect of integration unfolds in a reverse complementary manner in the refined use of emotional feeling through unconscious projection, that gradually empties the unconscious of all such involuntary primitive impulses.

And as we have seen the Zeta 1 zeros are likewise directly associated with such immanent development.

This means in effect in this partial relative context, that the Zeta 2 and Zeta 1 zeros are directly associated with both cognitive (rational) and affective (emotional) aspects of understanding respectively.


So we started in Type 1 (conventional mathematical) terms, by considering number interpretation in a solely rational manner.

Then in Type 2 (holistic mathematical) terms, we extended number interpretation to likewise include an intuitive - as well as rational - aspect.

However, now at the Type 3 (radial mathematical) level, we must include the additional fact that numbers reflect both cognitive and affective aspects in relationship to each other.

Thus if we look at the Zeta 2 zeros as representing the masculine (cognitive), then the Zeta 1 zeros thereby represent the complementary feminine affective principle.

Of course ultimately in Type 3 terms - where the interdependence of both Zeta 1 and Zeta 2 zeros is fully recognised - both sets will now be seen as equally reflecting cognitive and affective aspects.


As we have seen the Zeta  2 zeros provide the means (though the non-trivial roots of 1) of differentiating natural number ordinal positions (with respect to a given prime).

So for example for the prime number 3, (where 3 represents a group of individual members) we can thereby uniquely differentiate 1st, 2nd  and 3rd in this context. This is achieved in an indirect quantitative manner through the 3 roots of 1 (of which all except 1 are non-trivial)  However this equally implies that we can successfully integrate these 3 members in qualitative manner (expressed indirectly through the sum of the 3 roots = 0).


The Zeta 1 zeros then provide the corresponding means of differentiating primitive impulses (with respect to the confused composite elements of the unconscious).

Primitive impulses reflect the confused activity that results due to qualitative being directly identified with quantitative appreciation.

So the very process of resolving such confusion requires the successful (analytic) differentiation of the quantitative aspect equally combined with successful (holistic) integration at a qualitative level.

In an exactly similar manner, resolving the true nature of the prime numbers requires the successful (analytic) differentiation of each prime (in cardinal terms) equally combined with their successful collective (holistic) integration in a qualitative manner.

And once again the Zeta 1 zeros directly correspond with this latter holistic aspect of the primes (representing number dimensions).


However we can now quickly demonstrate the requirement for integration of both sets of zeros.

From the Zeta 2 perspective, we must assume the existence of each prime before differentiating its group members (in a natural number ordinal manner).

From the Zeta 2 perspective, we must assume the existence of each (composite) natural number (before differentiating its individual members as primes).

Thus the Zeta 1 and Zeta 2 zeros are mutually implied by each other.

This even helps to clarify a notable feature regarding my own experience of these developmental stages.


With respect to the Zeta 1 zeros, for many years, I was more concerned with the even numbered dimensions (especially 2, 4 and 8) than odd (except for 3).

This ultimately reflected an undue emphasis on the transcendent aspect of contemplative development. It is only much later I realised that this corresponding lack of immanent type development (with which the Zeta 1 zeros - and the successful differentiation and integration of the primes - are associated) was impeding my ability to appreciate the nature of the prime numbered dimensions (associated with the Zeta 1 zeros) .

In dynamic interactive terms, the (analytic) differentiation of the individual primes in cardinal terms is ultimately inseparable from their corresponding (holistic) integration in a collective manner.

The Zeta 1 zeros can then be seen as relating directly to this latter collective integral aspect  (that perfectly shadows the individual nature of the primes).

Likewise from a dynamic interactive perspective, the (analytic) differentiation of the individual natural numbers in ordinal terms (i.e. the members of each prime grouping)  is ultimately inseparable from their corresponding (holistic) integration in a collective manner.

The Zeta 2 zeros can then be seen as relating to this latter integral aspect aspect (that perfectly shadows the individual nature of each natural number).

So ultimately the primes and natural numbers are thereby seen as perfect mirrors of each other that are ultimately identical (i.e. fully interdependent) in an ineffable manner.

However the fact that the zeta zeros now mutually reflect both cognitive and affective aspects of interpretation raises the issue as to whether Mathematics reflects scientific or artistic appreciation.

We will return to this important issue in the next blog entry.

Friday, November 1, 2013

Where Science and Art Coincide (17)

There is a further unexpected surprise with respect to the interpretation of the Zeta 1 zeros.

However to appreciate this we must, as always, place our discussion in the dynamic interactive context of development.

I have contrasted before the nature of top-down development (where "lower" dimensions are integrated with respect to the "higher") and bottom-up development (where "higher" dimensions are integrated with respect to the "lower" ).

The former reflects the transcendent aspect of development (with which the Zeta 2 zeros are directly associated); the latter represents the corresponding immanent aspect (with which the Zeta 1 zeros are likewise directly associated).

The transcendent aspect, on the one hand, represents the attempt to achieve the "higher" integration of conscious development (which implies the unfolding of an increasingly important intuitive element).

This ultimately continues until the stage where conscious awareness is so transparent and dynamically interactive, that phenomena no longer appear to arise in experience. This seamless conscious activity can then be integrated with an ever-present pure intuitive awareness (of an unconscious kind).

So this represents the transcendent goal of spiritual development.


In practice severe limitations are likely to affect the attainment of this goal.

Higher level transcendent development is often based on a mistaken hierarchy, where the cognitive activity of reason is ranked as superior to that of affective sense, with pure spirit seen as the apex of the hierarchy.

This in the attempt to control the "lower" senses (in the pursuit of purer contemplative awareness), reason in effect is used to a degree to repress such activity (especially at a primitive unconscious level).

Therefore without this problem being properly addressed, the transcendent goal itself cannot be attained.


So there is a complementary immanent approach to development which inverts this hierarchy of reason as superior to the senses.

Whereas the transcendent aspect is based on the attempted "higher" integration of consciousness, the immanent aspect, in an inverse complementary manner, is based on the corresponding "lower" differentiation of the primitive unconscious.

Put another way, whereas  the transcendent aspect represents making the conscious increasingly unconscious, so that all its linear type rigidity is ultimately eroded, the immanent aspect represents the making of the unconscious increasingly conscious in experience, so that all the primitive instincts of the - initially unrecognised - shadow are ultimately fully brought into the conscious light.


Thus once again, the integration of both the (analytic) conscious and the (holistic) unconscious in development, requires this approach from two opposite directions. Thus from the transcendent aspect, the conscious is made fully compatible with respect to the life of the unconscious; from the immanent perspective, in reverse manner, the unconscious is made fully compatible with respect to corresponding conscious activity.

So it is exactly similar with respect to the nature of the number system.

The Zeta 2 zeros reflect this former aspect of integration, where analytic appreciation of the number system (in a quantitative manner) ultimately is made fully compatible with corresponding holistic appreciation (of a qualitative nature). So we move from a linear (1-dimensional) notion of the quantitative nature of number to an increasingly circular (higher dimensional) notion of its qualitative nature.

And this higher dimensional appreciation is directly provided through the holistic interpretation of the Zeta 2 zeros.


The Zeta 1 zeros however reflect the latter aspect of integration where initial confused unconscious interpretation of the number system (in a qualitative manner)  ultimately is made fully compatible with corresponding analytic appreciation (of a quantitative nature). So here we move from circular confusion to an increasingly linear (1-dimensional) appreciation i.e. where qualitative appreciation can be fully compatible with linear quantitative interpretation.

So in one case (Zeta 2 zeros) we move from the linear to the mature circular i.e. where quantitative is made fully compatible with the qualitative understanding of number; in the latter case (Zeta 1 zeros) we move from the (confused) circular to mature linear i.e. where in reverse manner the qualitative is made fully compatible with the quantitative nature of number.

This indeed is the very reason why all the Zeta 1 zeros are compelled to lie on a straight line!

These zeros in fact thereby represent the fully differentiated appreciation of the unconscious i.e. holistic nature of the natural number system.

So we have differentiated conscious interpretation in our customary quantitative interpretation of the natural numbers as represented by the real number line.

We then at the other extreme of understanding, we have differentiated unconscious interpretation in this new (unrecognised) appreciation of all the Zeta 1 zeros, as represented by the imaginary number line. And remember in holistic terms, the imaginary notion is used to express indirectly what is unconscious in origin!

Now initially when the primitive impulses of the shadow unconscious are projected into experience they are directly confused with phenomena of a quantitative nature. However, all going well, through a growing holistic qualitative awareness, inappropriate rigid  attachment to such quantitative phenomena is gradually eroded. Success in this regard thereby enables one to confront the shadow in an ever-deeper manner with new (formerly unrecognised) unconscious elements continually brought to light.

In this way ultimately all such shadow elements can (in principle) be successfully differentiated.

So - quite literally - all primitive (i.e. prime) elements that have been successfully differentiated in an analytic manner, are thereby now equally successfully integrated  in a corresponding holistic fashion.


Remarkably it is exactly similar in dynamic interactive terms with respect to the nature of prime numbers.

Thus the very differentiation of the prime numbers (in a quantitative analytic manner) is ultimately inseparable from their corresponding integration (in qualitative holistic terms).

In other words the Zeta 1 zeros represent the holistic integrative aspect with respect to the prime numbers.

Thus the very ability of the primes to maintain their unique status (as independent building blocks of the natural number system in cardinal terms) is inseparable from the corresponding unique status of the Zeta 1 zeros (which seamlessly maintain the overall interdependent nature of the primes with respect to the natural number system).

And this seamless integration is then demonstrated from a different perspective by the Zeta 2 zeros which now serve as the unique independent build blocks of the ordinal number system.

Thus ultimately the Zeta 1 and the Zeta 2 zeros are themselves totally interdependent.


However what is especially interesting is that the relationship of both sets of zeros is as cognitive to affective (and affective as to cognitive) respectively.

Thus if we initially fix the Zeta 2 zeros with customary cognitive aspect of understanding, then the Zeta 1 correspond with the affective aspect.

Now this will initially appear very strange as up to now, Mathematics - even in ts dynamic interactive  operation - has been viewed as a cognitive (rational) type discipline.

So we will address this important issue further in the next blog entry.