Saturday, October 30, 2010

Fermat's Last Theorem Revisited

Looking again at the Horizon TV programme on Fermat’s Last Theorem proved a very rewarding experience. Unlike the first time I was able to appreciate much more of the fine detail (e.g. with respect to elliptical curves and modular functions). Also it got me thinking again on a number of levels regarding my own mathematical journey.

Like Wiles as a child of about 10, I too had heard of Fermat’s Last Theorem. The problem seemed so beguilingly simple that in my naïveté I thought I would be able to solve it. However after many hours of futile endeavour I abandoned this quest in failure. Nevertheless as Mathematics remained my favourite pursuit I hoped to major in the subject at College. However after a troublesome first year when I became greatly disillusioned with the mathematical treatment of the infinite, I dropped out of the class.

Many years later I became interested again in Fermat’s Last Theorem from a very different context. I had been paying a great deal of attention to my new pursuit of Holistic Mathematics (where every mathematical symbol can be given a well-defined qualitative as well as quantitative meaning) and was looking for a problem to demonstrate its potential value. So I hit upon “The Pythagorean Dilemma” relating to the irrational nature of the square root of 2. Then to my considerable satisfaction, I felt that I was able to provide a coherent qualitative solution to this problem.

The Pythagoreans would not have been satisfied with a merely quantitative explanations as to why the square root of 2 is irrational. They really wanted a deeper philosophical explanation as to why in qualitative terms such a number can arise!

The implication is that there are really two as aspects to proof (i) (quantitative) analytic and (ii) (qualitative) holistic.
Comprehensive mathematical interpretation then requires both types of proof.

The problem with respect to the square root of 2 arises in the context of the famous Pythagorean triangle so 1^2 + 1^2 does not result in another rational number that is squared. Now as Fermat’s Last Theorem is closely connected with this problem, I decided to look at again (strictly from this new holistic perspective) and came up what I would see as a partial qualitative explanation for its truth.

Central to holistic mathematical appreciation is the notion that mathematical dimensions have a coherent qualitative interpretation whose structure is inversely related to the quantitative root form of those dimensions. And to see this structure we obtain the successive roots of 1!

For example 2-dimensional interpretation is linked to the two roots of 1 which are + 1 and – 1 respectively. Thus qualitative 2-dimensional interpretation is based on the complementarity of opposite poles in understanding (such as external and internal).

Now the significance of all dimensions higher than 2 is that the corresponding root structures will contain both real and imaginary parts.

This means in effect that mathematical interpretation at these dimensions entails both real (conscious) and imaginary (unconscious) aspects with the imaginary expressing the unconscious aspect in an indirect rational manner!

Real understanding in qualitative terms relates to strictly rational type understanding (corresponding to rational numbers in quantitative terms).

Thus in Conventional Mathematics, when irrational numbers are used they are given a strictly reduced interpretation.
However imaginary understanding corresponds to the rational means of conveying true holistic meaning (of a circular kind).

So the qualitative explanation for Fermat’s Last Theorem relates to the basic fact that 3-dimensional interpretation and higher can never be conducted in a merely real rational manner. Likewise in quantitative terms, when we attempt to add two quantities - which represents the simplest form of linear transformation - raised to such dimensional powers, the result likewise cannot be expressed in merely real rational terms.

We can even give a simple geometrical rationale for this.

Clearly from a linear (1-dimensional) perspective, when we add two rational numbers the result will also be rational. The very essence of the linear interpretation is that qualitative considerations are ignored. Therefore no qualitative transformation in the numbers can take place.

2-dimensional interpretation is a half-way house as between linear and circular which can be easily illustrated. Here the two roots of 1 lie on both a straight line diameter and also on its circular circumference. Therefore in quantitative terms when we add two numbers (raised to the power of 2) the answer can be either linear yielding another rational number (raised to the power of 2) or circular (i.e. an irrational number raised to the power of 2).

However for all dimensions greater than 2 the corresponding roots (of 1) cannot lie on a straight line. So both real and imaginary aspects are necessarily involved. The corresponding corollary in quantitative terms is that when we add two rational numbers (raised to such dimensions) then an inevitable qualitative transformation in the nature of the number is involved. So the resulting number (raised to the power of n > 2) must be irrational.

I think it is even possible that Fermat’s “truly marvellous demonstration” of this fact might have related to this simple insight (i.e. that all root structures greater than 2 necessarily entail imaginary as well as real components). Now a demonstration does not amount to a proof and perhaps Fermat subsequently realised how difficult it was to build on such an insight to establish a proof!

Returning to the programme on Wiles, one striking paradox that hit me was how much the very process of his discovery runs counter to the established mathematical notion of rational proof.

So for example Wiles through his early discovery of Fermat’s Last Theorem was inspired by a powerful childhood dream i.e. one day to find the proof to this great puzzle. So his initial motivation relates more to the holistic unconscious (than rational thought). Also his subsequent voyage of discovery in many ways paralleled that of a spiritual contemplative seeking union with God.

Indeed with his thin frame and quiet demeanour he very much fitted the part of the religious ascetic. He then pursued a very uncertain journey in considerable isolation for seven years gaining total immersion in his problem. For much of the time he wandered in darkness, hanging on in faith and hope of an eventual resolution.
Then finally after long and painful endeavour he received a special Euraka moment of illumination when he finally resolved his problem. So great was his emotion at this final revelation that he could not even attempt to describe the feeling but only to say that he would never experience a moment like it again!

Wiles proof is rightly hailed as a truly remarkable mathematical achievement. However the point that I am making is that the actual experience of discovering such a proof entails much more than what is formally recognised.

So Mathematics - especially with truly creative discoveries - entails both intuitive (unconscious) as well as rational (conscious) processes. In particular Wiles’ decisive final insight was of a (holistic) intuitive nature. However this important holistic aspect is screened out totally from formal mathematical interpretation!

Though properly speaking there are two aspects to Mathematics that are quantitative and qualitative, only one is recognised. Thus the interpretations of Conventional Mathematics are of a highly reduced nature (and are only strictly valid within the 1-dimensional mode of qualitative interpretation adopted).

This intimately applies to the nature of mathematical proof.

It is only within a linear (1-dimensional) interpretation that mathematical proof can be given an absolute meaning. So from this perspective Fermat’s Last Theorem is either true or not true. And the popular belief is that Wiles has now finally resolved the matter for once and for all by proving that it is indeed true!

However the actual process by which the validity of his proof was decided shows that such absolute interpretation is not strictly valid.

Indeed when Wiles first presented his “proof” in 1993, it was widely accepted by the mathematical community. It was only later that a referee found a flaw in reasoning at an important juncture. Even when this was pointed out to Wiles, it took him some time for him to recognise the true importance of the difficulty. So more than a year of further investigation was required (with the help of a talented student) before Wiles was finally able to amend his proof.

So in truth “mathematical proof” in such cases – indeed in all cases - represents but a special form of social consensus among the mathematical community and is strictly of a probable nature. In other words as time goes by with no other questions being raised regarding its validity it can be accepted with an ever greater degree of confidence as true. But this truth will still always remain of a merely probable nature.

There is an even bigger challenge with the accepted notion of proof that I am here raising. This relates to the fact that with the passage of time, our very understanding of the nature of “proof” is likely to become considerably more refined.

So when properly understood, each dimensional number in qualitative terms corresponds with a unique mode of interpretation with respect to mathematical symbols.

So there is not just one valid mode - as presently believed - of acceptable mathematical interpretation (but rather potentially an unlimited set).

This understanding can then be applied directly to the status of the proof of Fermat’s Last Theorem. Just as it is not possible to add two rational numbers (raised to the dimensional power of 3 or higher) together to obtain another rational number (raised to the same power), equally it is not possible to maintain a strictly linear interpretation with respect to Fermat’s Last Theorem when understanding takes place in 3 (or higher) dimensional terms.

In other words - in terms of the qualitative understanding of such dimensions - interpretation is subject to the Uncertainty Principle. What this means in effect is that Fermat’s Last Theorem would now be given both a conventional (quantitative) and holistic (qualitative) interpretation. Thus inevitably there would be a trade-off necessary with respect to both types of appreciation. Therefore the more definite the merely quantitative, the more fuzzy would remain the corresponding holistic dimension. Equally the more definite the holistic, the more fuzzy would remain the quantitative aspect.

Now it might be maintained that my own appreciation with respect to the Wiles’ quantitative proof of Fermat’s Last Theorem is still fuzzy; however while accepting this observation, it only helps to confirm the general point I am making with respect to the true nature of mathematical proof in the wider context of interpretation (where both quantitative and qualitative aspects are formally incorporated).

Happily, Fermat’s Last Theorem is at least capable of proof (in the 1-dimensional sense of current mathematical interpretation). However I believe there are other outstanding problems (such as Riemann’s Hypothesis) where even this type of proof (or disproof) will not be possible.

Thursday, October 28, 2010

The Big Bang

It is indeed a pleasure to watch so many beautifully produced programmes highlighting the wonders of the universe. Recently I was viewing "Stephen Hawking's Universe" on Channel 4 and found it fascinating (especially the last episode on the origins of creation).

The Big Bang about 13.7 billion years ago has now become so commonly accepted as if it is an established scientific reality not to be questioned.

However I always like to take a wider perspective than mere conventional acceptance of present views. Just look at how our worldview has changed so much from even 100 years ago! Is it not reasonable to assume that perhaps even greater changes will take place in the next 100 years making much of what is presently gospel truth seem naive and even foolish!
So even if the Big Bang remains the accepted orthodoxy, I am sure that the manner in which it is understood will have changed considerably.

Indeed it seems to me somewhat ridiculous to attempt to describe in detail what happened during the first nanoseconds of the Big Bang for the simple reason that the linear notions of space and time on which this is all premised clearly could not have existed within the context of the Big Bang itself.

Our very notions of space and time (including the more relativistic notions of Einstein) are always premised on the clear separation of observer from what is observed. Therefore the very registering of space and time requires that the (internal) observer in some way be detached from the outside system (that is observed).

Now clearly in the context of the Big Bang such conditions would not exist, for the evolutionary potential which it contained for eventual emergence of psychic observers (of a physical universe) would have remained indistinguishable from the physical.

Put another way in attempting to travel back in space and time to the supposed beginning of the physical universe we are likewise - though not always realising this fact - attempting to travel back to the beginning of the psychological universe (for clearly our wonderful capacity for conscious investigation ultimately emerged from this initial state).

However once again we can only give a clear meaning to space and time (through detachment of the psychological from the physical aspect) and in the context of what we call the Big Bang this would be clearly impossible. So in attempting to give a linear history in space and time to our universe we are attempting to act as outside observers of an initial event (which clearly is not tenable in the context of that event itself).

Now I would accept that the birth of space and time coincides with the birth of the phenomenal universe. However in this emergent state circular - rather than linear - notions would be more appropriate. Thus any notions of forward movement for the universe would have to be countered by corresponding notions of backward movement. So the best we could say therefore is that the universe emerged out of the present moment which is continually renewed. Furthermore properly understood the very notions of space and time that we use are ultimately purely relative with respect to the present moment. So once again though space and time measurements have a certain validity when we abstract one part of the system from its environment, in the context of the overall universal system they are rendered ultimately paradoxical and meaningless!

Indeed this raises an even more fundamental problem in that the very way we scientifically attempt to interpret the universe is deeply flawed.

This is due to the failure to properly distinguish wholes and parts. Because science is based on mere quantitative type analysis of reality it thereby reduces the qualitative dimension (which is of distinct holistic nature) to the quantitative.

This thereby leads to the notion of one universe composed of many constituent parts.

However when we properly allow for the qualitative dimension, interpretation is much more subtle. Here we view the universe as the intersection of the one and the many.

In other words each part contains the whole (as an individual universe); equally the parts are contained in the whole (as the collective universe).

So what we refer to statically as the universe is in fact a dynamic interaction of many (micro) universes with the one (macro) and equally the reverse interaction of the one (macro) with the many (micro) universes!

And as I have stated on many times to talk about this in correct scientific terms we must incorporate the true holistic meaning of what is real and imaginary (in mathematical terms). Thus from one perspective we have the interaction of many real (individual) universes with the one (collective) universe that is - relatively - imaginary; equally from the other perspective we have the interaction of the one (collective) universe that is now viewed as real with the many (individual) universes that are - relatively - imaginary. Thus the one and the many have both a real and imaginary identity that keep switching through dynamic interaction.

I would also see the very notion of the Big Bang as deeply problematic for the concept of strings.

From the accepted linear viewpoint of the Big Bang, strings could not have existed before this event. Therefore once we accept this we are left with the problem of how they have emerged. And if science attempts to give a phenomenal explanation to this, then clearly it is pointing to something that is even more fundamental than strings.

One alternative would be to accept that strings are created - literally - out of nothing. However scientists would be loath to accept such an explanation which would be inseparable from saying "God created the world".

Of course the best solution - as I have suggested - is to abandon the very notion of strings as having any phenomenal identity. As I would explain it, strings possess merely the inherent potential for both the linear (independent) and circular (interdependent) aspects which necessarily underpin all phenomenal existence.

However properly incorporating such an explanation ultimately requires modifying the very nature of science.

In other words science does not entail mere rational interpretation of reality (at a conscious level); rather it entails the dynamic interaction of both (analytic) rational and (holistic) intuitive processes that operate in both conscious and unconscious terms.

So correctly understood all phenomena (as actually manifest) are in continual dynamic interaction with a fundamental holistic ground (as potential for existence).

So ultimately there is no scientific hope of appreciating the mystery of the origins of our universe without formally incorporating the role of the unconscious in interpretation.

Thursday, September 16, 2010

More on Nature of Strings

I have often stated that the current definition of "a string" is somewhat meaningless from any coherent physical perspective.

A string is viewed essentially as like a thin elastic band - extremely short in length - that is 1-dimensional (with no other spatial characteristic).

However this very definition requires the background existence of space. However it is then also admitted that the dimensions of space and time must in some way arise from strings (as the basic constituent ingredients of everything in the universe). So we clearly have an obvious problem.

Indeed there is a strong parallel here with a similar issue in mathematics relating to the prime numbers.

The prime numbers are conventionally viewed as the basic (independent) building blocks of the natural number system; however equally the general distribution of the primes intimately depends on the natural numbers.

Now I have explained that this issues of the primes in Mathematics ultimately relates to the fact that they can be given both a linear and circular interpretation which are relative.

So it is akin to the simple problem of defining left and right turns with respect to street directions. If for example one defines the direction of a road as either "up" or "down" (as independent) then left and right turns can be given an unambiguous meaning.
However when we treat them as interdependent with a relative meaning, then a turn on a road is both "left" and "right" (with the designation in any given circumstance purely arbitrary).

So with prime numbers when we try to treat the individual existence of prime numbers and the general distribution (overall) of prime numbers as independent, then both issues can be unambiguously studied through conventional mathematical methods.

However when we attempt to properly integrate these two areas, we require both a linear (quantitative) and circular (qualitative) treatment.

And indeed this is the very conclusion I reached with the Riemann Hypothesis which arises through attempting such integration in mere quantitative terms.

And of course it is exactly the same situation with respect to the world of strings.

So "a string" cannot be given an independent existence which already assumes a general background in space; likewise the dimensions of space and time cannot be given an independent existence as they only have meaning with respect to already existing phenomena.

So "a string" strictly has no phenomenal meaning as an independent physical constituent of matter; likewise "a string" has no phenomenal meaning with respect to the dimensions applicable to such reality.

It is only in relation to each other (where both the quantitative and qualitative aspects of matter dynamically interact) that phenomenal investigation of reality can begin.

And again the crucial point to realise is that this necessarily entails both quantitative and qualitative aspects of understanding (pertaining to both linear and circular interpretation of relationships).

Physical science as we know it is still based on a fundamentally false premise i.e. that phenomenal investigation of "specific objects" and corresponding investigation of the general dimensional background of such phenomena can both be interpreted in merely quantitative terms (using the standard rational linear approach).

However as we have seen object phenomena and the dimensions of space and time are in dynamic interactive terms quantitative and qualitative with respect to each other.

So therefore we must use a binary scientific approach combining both linear and circular type appreciation. Whereas the linear aspect corresponds to reason (as conventionally understood), the circular aspect corresponds directly to intuitive type appreciation.
However indirectly this intuitive aspect can be translated in a scientific rational fashion as "imaginary".

Thus a comprehensive scientific approach that can properly deal with the relationship of phenomena to space and time must be in qualitative terms of a complex rational nature (combining both real and imaginary aspects).

Properly speaking a string does not have an actual phenomenal existence (which would already imply space and time). Rather it represents the potential for manifest existence which has the two aspects of (differentiated) independence and (integral) interdependence.

So in a static sense, no distinction can be made as between the linear aspect of the string as (linear) differentiated form and the integral aspect as (circular) emptiness.
Thus in static binary terms therefore with respect to the string, the linear (1) cannot be properly distinguished from the circular aspect as nothingness (0).

However when these two aspects interact in dynamic manner (i.e. through the vibrations of the string) then we enter the world of actual manifest form. The most primitive excitations of the string would correspond in turn with the prime numbers (which in turn would represent the most fundamental particles).

And because at the lower energy levels very few particles become manifest which then serve as the basis for the natural phenomena we recognise, in like manner the majority of natural numbers arise as the composite expression of comparatively few early prime numbers!

So just as there are links in natural terms as between the original numbers (1 and 0), the primes and the natural numbers, there are similar links as between strings (defined in a binary fashion), the fundamental sub-atomic particles and natural phenomena.

However whereas both sub-atomic and natural phenomena enjoy a dynamic interactive actual existence in space and time, strings properly pertain to a mere potential for actual existence that underlies the manifest phenomena.

Thus irrespective of technological developments e.g. with respect to particle accelerators, it will never be possible to experimentally detect strings!

Once again from a more correct physical perspective, what are referred to as strings represent the inherent (linear) capacity of matter for differentiated form (as relatively distinct phenomena) and the equal inherent (circular) capacity for an overall qualitative integral relatedness (as holistic dimension).

Alternatively in more scientific language we can say that strings possess both a real and imaginary potential which in dynamic interaction generate both matter phenomena and the dimensions of space and time (which are inextricably linked).

Monday, September 13, 2010

Darwin and Riemann

When doing some research for my articles on the Riemann Hypothesis, I made the interesting discovery that both Darwin's Origin of Species and Riemann's famous article on prime numbers were both published in 1859 (just over 150 years ago).

Indeed the historic connection can be shown to be even closer with the publication date of Darwin's book in November of that year while the full text of Riemann's article also appeared in November (in the monthly reports of the Berlin Academy) though Riemann actually had delivered his address on the contents of that article to the Academy in August, 1859.

However recently I have come to see an even greater significance to this interesting coincidence of publication dates (of what constituted truly ground breaking initiatives in two different fields).

In earlier blogs I addressed the issue that any attempted reconciliation of science and religion would require two key developments.

1) the recognition of an alternative qualitative aspect to science, utterly distinct though of equal importance to the present recognised quantitative aspect.

2) the demythologisation of the manner in which universal spiritual truths are conveyed in the major religious traditions.

It is with respect to the latter of these requirements that Darwin's work is of such enormous significance.
Whereas Newton especially had paved the way early for this change with respect to the natural sciences, Darwin above all has helped to extend it to the biological sciences.

For example in the Christian tradition the evolution of life on Earth, especially with respect to the development of the human species had been shrouded in myth for which no proper scientific basis existed. So in providing a truly coherent scientific explanation for evolution of all life forms, Darwin effectively unmasked the nature of literal Christian beliefs in this regard.
Of course a proper scientific appreciation of the nature of evolution does not affect the legitimacy of spiritual beliefs per se (but rather the manner in which they may be presented in the religious traditions)!

As a child I had already embraced evolution (having dismissed in my own mind any literal basis to the Genesis account of Adam and Eve). However I never saw this as having any direct bearing on spiritual truth (which for me still possessed a powerful significance).
Perhaps because of this early clash with religious orthodoxy I have remained open to the manner in which so many Christian doctrines are still expressed in the form of mythical explanations.
So I do see unquestioning acceptance of the literal meaning of these myths as a major barrier to genuine discourse with the scientific community.

It is with relation to the first requirement above i.e. the need for a qualitative aspect to science, that I now see Riemann's article as being of immense potential significance.

As is well known, Riemann's article was to give rise to the famous Riemann Hypothesis (which still remains unproven from a conventional mathematical perspective).

As I had for many years suspected a hidden qualitative aspect to the Hypothesis in recent years I have given it considerable attention with a view to unravelling the barrier to its resolution.

To my amazement, I eventually was able to conclude that - when properly appreciated - the Riemann Hypothesis is really a statement regarding the basic requirement for maintaining consistency as between both the quantitative and qualitative aspects of mathematical understanding.

One obvious implication of this new understanding is that the Hypothesis has no proof from a conventional perspective (where only the quantitative aspect is recognised). Rather it serves as a more general axiom on which those axioms already used in conventional interpretation depend.

So, the Riemann Hypothesis in fact serves as a powerful expression of the need to incorporate a complementary qualitative with the recognised quantitative aspect of present science.

Thus from my newly adopted perspective, the very basis of the two great revolutions that are required (before science can be be properly reconciled with religion) have already been sown in two major contemporaneous developments in thinking some 150 years ago.

Monday, August 16, 2010

The Uncertainty Principle

Much is made of the Uncertainty Principle in Quantum Mechanics, whereby it is accepted that both the position and momentum of a sub-atomic particle cannot be precisely determined. So there is a trade-off involved with respect to both aspects with ever greater accuracy with respect to one aspect (e.g. position) inevitably being at the expense of the other (momentum). And this is an inherent problem with respect to the behaviour of such a particle (and not due to practical difficulties with measuring devices).

However there is a much wider context to this principle which is not properly recognised (due to the lack of any appropriate qualitative context to Conventional Science).

As I have stated before the very basis of Conventional Science is the use of linear rational logic (reflecting in turn the Middle Band of the psychological spectrum).

However just as electromagnetic energy has many bands (of varying wavelength and frequency) likewise it is true with the modes of possible rational understanding.

So besides the natural mode (which provides the common sense intuitive light informing normal macro understanding of reality) there are many "higher" modes of intuitive energy possible (that are consistent with a more refined circular type of reason).

Now the very essence of the linear mode is that it necessarily reduces (in any context) the qualitative aspect of understanding to mere quantitative interpretation.

Though such reductionism indeed enables an extremely good approximation with respect to the quantitative nature of reality at the macro-level of experience, it begins to break down badly at both "higher" qualitative and "lower" quantitative levels (which are complementary).

In other words to appropriately interpret the quantitative behaviour of a "lower" level of physical reality (e.g. sub-atomic particles) we must use a corresponding "higher" level in qualitative terms of refined rational understanding (with the degree of refinement depending on the quality of the corresponding spiritual intuition required).

So for all other dimensions other than the 1st (i.e. linear understanding) a necessary complementary relationship exists as between what is "objectively" known about reality (in quantitative terms) and a corresponding "subjective" manner of psychological interpretation (in qualitative terms).

So in the deepest sense this inevitable dynamic interaction as between quantitative and (complementary) qualitative aspects is what really underlines the Uncertainty Principle.

There are just two important aspects with respect to this wider Uncertainty Principle that I would like to highlight.

1) When one experiences "normal" reality from the authentic perspective of a "higher" spiritual contemplative stage, the Uncertainty Principle necessarily applies in qualitative terms to all scientific (and mathematical) understanding.

For example this finding intimately applies to the nature of mathematical proof.

At the linear (1-dimensional) level of understanding a mathematical proof (e.g. of the Pythagorean Theorem) is interpreted in a somewhat absolute fashion.
Though it may well be conceded that such a proof represents a "subjective" manner of interpretation (in the necessary psychological use of mathematical constructs) a merely static relationship is maintained with respect to the application of such constructs to "objective" reality. So there is an underlying belief here that the "subjective" mental interpretation absolutely represents the "objective" reality (with respect to the behaviour of right angles triangles).

However at the "higher" levels (where intuition and reason are explicitly involved in understanding) one realises that a dynamic interaction is necessarily involved as between the qualitative means of mathematical interpretation (i.e. the mental "proof") and what it relates to in quantitative terms.

So one's understanding of the proof (in this example of the Pythagorean Theorem) is now understood in a merely relative sense. Thus the truth is strictly approximate and thereby of a merely probable nature.

Thus a mathematical "proof" is now seen to represent but a special form of social consensus. However the degree of understanding which each person will form of the "proof" will vary (perhaps considerably). One could even in some respects form a faulty understanding of the logical connections while confirming a "proof" (without being aware of the fact). Indeed it is quite possible that for a time the mathematical community en masse could form such a faulty understanding and confirm a theorem as true (that is later shown to be false). This famously happened with the 1st "proof" by Andrew Wiles of "Fermat's Last Theorem". Now admittedly this problem was quickly addressed and as time proceeds with no further objections arising then we can assume that the theorem has indeed been proved with an ever greater degree of confidence. But all this merely confirms the probable nature of mathematical proof. So the best that we can hope for is that long accepted "proofs" are true to a high degree of probability!

This qualitative Uncertainty Principle likewise applies to all scientific findings such as physical laws.

I have discussed this previously with respect to Relativity and String Theory. So for example the relativistic nature of space and time does not apply solely to quantitative measurements but equally to the qualitative means by which we strive to interpret such findings - literally - at the "higher" dimensions of understanding.
And then the Uncertainty Principle applies to the dynamic interaction as between both quantitative and qualitative aspects. So once more precise measurement with respect to one aspect inevitably implies diminished measurement with respect to the other. So for example the zenith with respect to qualitative appreciation - which is inherently of an intuitive nature - comes through pure contemplative union with reality (where quantitative measurements lose any specific meaning!)

Indeed the very manner by which Einstein ignored the qualitative aspect of interpretation (in striving to understand Relativity) clearly demonstrates the linear rational nature of the classical paradigm from which he operated!

2) Just as the potential set of all possible numbers is infinite this applies likewise that the set of potential dimensional interpretations is infinite. Therefore the Uncertainty Principle - relating such dimensional interpretations to corresponding reflections of physical reality - has an infinite number of possible expressions.

However it is still true to say that a limited number of these expressions have special relevance. I will return to these in later posts.

Thursday, July 29, 2010

Odd Numbered Dimensions

I have found the odd numbered dimensions more difficult to understand than the even.
As we have seen the even are more properly geared for holistic integral interpretation of reality and as my main concern over the years has been to articulate such an approach in appropriate mathematical terms, it is not surprising therefore that the even dimensions held more resonance for me.

The clue to understanding however of the higher odd dimension (focusing initially on the positive) is the realisation that human development necessarily entails both differentiation and integration which need to be maintained in healthy balance.

So even for one committed to the process of growing in pure contemplative awareness, a certain level of activity at each stage must be maintained.

Thus for example when one returns to active involvement (following the development of the more passive even numbered dimension), the next odd numbered dimension will then unfold.

Thus arriving at the first of the higher dimensions i.e. 2 leads to a new integral appreciation of phenomenal reality as being based on the dynamic complementarity of opposite real poles (e.g. internal and external respectively). This new "higher" rational appreciation of the structure of reality is in turn supported by an enhanced quality of spiritual intuition (not available at the linear level).

However such appreciation is of a holistic nature. So to maintain balance one must equally strive for appropriate appreciation at the more active level of dealing with specific phenomena. So this leads to the unfolding of 3-dimensional understanding.

If one looks at the three roots of unity, the first is + 1, which in a sense stands separate from the other roots (which are paired in a more complementary manner). And this feature is shared by all higher odd numbered roots.

The corresponding qualitative significance of this is that likewise associated with the higher odd dimensions is a certain return in qualitative terms to linear understanding (corresponding with + 1). However unlike the earlier 1-dimensional level, this is circumscribed to a considerable extent by other forms of understanding.

It is easiest to illustrate here with respect to 3-dimensional interpretation and I will do so from my own particular experience.

Having gone through an extended "dark night" lasting about 5 years, which in holistic mathematical terms represented the negation of 2-dimensional understanding, I noticed a distinct lull with respect to customary purgation taking place. This gradually enabled me to turn the focus outwards again and resume many of the activities that I had given up for some considerable time. In one way it seemed like a welcome return to normality. As I came to share again the same interests as others, the preceding "dark night" appeared like a lengthy aberration that had now thankfully passed.

However I slowly realised that things were not quite the same as before and that I had become extremely sensitive to projections of all kinds. Now a projection represents an involuntary eruption of the (repressed) unconscious which then becomes embodied - and thereby confused - with conscious phenomena. So I found myself spending considerable time in the attempted negation of the many projections that had come to interfere with my activities.

Now if one looks at the two other roots (of the 3 roots of unity) one can see that there is a real part (which in both cases is -.5 and an imaginary part that is complementary (i.e. one with a plus sign and the other with a minus).

I began to understand that there is a fascinating holistic mathematical significance here (with intimate relevance for psychological development).

The real part represents the attempt to split dynamic negation of phenomena as between the external and internal aspects of phenomena. However because the unconscious is not yet sufficiently developed, this leads to projections becoming associated with both aspects.

So it is only when the unconscious has matured that such involuntary projections cease and the new integral 4-dimensional appreciation unfolds. Now in a suitably refined manner one clearly realises how the unconscious is necessarily involved in the appreciation of all phenomena so that involuntary projections (which represent a failure of such realisation) largely cease.

So in a valid sense the odd numbered dimensions can be seen as transition points between successively higher integral stages (represented by the even numbers).

In this way a new "higher" stage of integration requires a corresponding preceding stage of differentiation before it can successfully unfold.

Indeed this qualitative explanation would also help to explain why in the Riemann Zeta Function, pi values do not conveniently arise where odd numbered dimensions are concerned.

As we have seen integration is associated with pure circular understanding (based on the complementary of opposites). However for odd numbered roots we never obtain matching complementarity. And likewise with odd numbered dimensions, which represent a new form of differentiated experience, likewise we do not obtain full complementarity. Thus in a sense with each new stage of differentiation there is inevitably symmetry breaking with the circle therefore incomplete (until integration with the next even numbered dimension can be restored).

However it is the negative odd dimensions that are perhaps the most perplexing!

Once again if we go directly to the Riemann Zeta Function, we have the unusual finding that the value of the Function always results in a rational number (where the value of the dimensional power i.e. s, is odd and negative). Also this value keeps alternating as between positive and negative with successive "higher" (absolute) values of s.

So when for example s = - 1, the value of the Function = - 1/12 and then when s = - 3 the value = 1/120.

Now once again - though this is not yet recognised - such values (where s is odd < 0) possess a holistic mathematical rather than conventional linear meaning.

Indeed there is a famous account of when Ramanujan first wrote to Hardy, he included the finding 1 + 2 + 3 + ......= - 1/12. Both Hardy and his collaborator Littlewood thought initially that perhaps Ramanujan was mad and then realised that this actually represented the Riemann Zeta Function for s = - 1. So quite independently of Riemann, Ramanujan without any formal training had been able to derive a key result in complex analysis. Not surprisingly their opinion as to Ramanujan's talent quickly changed.

Now from a holistic mathematical perspective, I grappled for years to find why this value is rational. And then eventually it dawned on me, providing in the process much greater clarity regarding the nature of spiritual development.

As we have seen the negative even dimensions are required through a necessary form of passive purgation (or cleansing) so that the spiritual light can shine as pure intuition without any phenomenal attachments remaining. In a scientific context, this would entail the removing of all rigid rational interpretation from such dimensions (whereby the spiritual light to a degree becomes reduced to the phenomenal structures used in representation).

However - though St. John does not specifically refer to it - there is an equal need for a new form of active purgation following each "higher" stage of differentiation. So in a more healthy - and balanced - view of the spiritual journey, passive does not replace active purgation at the more advanced contemplative stages, but rather such advanced stages require both active and passive purgation. Now of course the degree that is appropriate at each stage will largely depend on personality factors and special circumstances.

In this regard I remember being impressed by a lengthy account given by Eveylyn Underhill in her Chapter "Dark Night of the Soul" (from "Mysticism") on the Dominican monk Henry Suso. There is little doubt that he traversed an authentic spiritual journey (entailing a lengthy "dark night") that was very arduous. However many of his key trials - even at the more advanced stages - were decidedly of an active kind (i.e. relating in holistic mathematical terms to negative odd dimensions). So this served in my mind to counteract somewhat the account of St. John in his "Dark Night" where the emphasis is very heavily on passive purgation (corresponding in turn to the negative even dimensions).

We can now provide a fascinating qualitative reason as to why all the negative odd dimensional values of the Riemann Zeta function are associated with rational values.

Again in an intellectual context we explained how in the case of the even numbered dimensions, negation with respect to any (rigid) rational attachment would be required so as to enable purely intuitive understanding (which is literally 0 in phenomenal terms) to arise.

Now an exactly counter experience is required in order to sharpen reason. Reason is cleansed by, in turn, removing all unrefined intuitive elements.
Normally the use of reason is facilitated by a ready supply of intuition (like oil in an engine) which is taken for granted. However it becomes much more difficult to use the rational faculties when this intuitive light is taken away. What then happens is that one is forced to greatly economise on reason by concentrating on what is truly essential with respect to every task.

Now without such radical purgation, most people never use their minds efficiently, constantly wasting intuitive energy on distractions. However when one is required to do tasks in faith (with all supporting consolation of the light taken away) reason is greatly sharpened.

So intuition is refined and purified (through negating rational support); in counter manner, reason is likewise refined and purified (through negating intuitive support).

So in an intellectual context in this sense, pure (linear) reason is developed through the negation of the odd numbered dimensions (in the "dark night" of active involvement).

It has always been my position that reason and intuition are of equal importance (and essential for the proper working of either aspect). However what I did not realise until recently is that the holistic mathematical appreciation of the Riemann Zeta Function has an intimate and precise bearing on the manner through which "higher" spiritual psychological development unfolds.

We can in addition provide a holistic mathematical explanation of some of the other aspects of the negative odd dimensional values.

For example why do the zeta values successively alternate as between positive and negative?

Well the reason is this! Whereas the even numbered integral dimensions relate to the complementarity of both poles, the odd numbered relate to the temporary separation of the positive from the negative pole. Now it is in the very nature of differentiation therefore that either the external or the internal is given temporary prominence. So when negation takes place it tends to be one-sided. So typically at one stage the external aspect will be more dominant before being negated; then at the next stage the internal will now likewise undergo - relatively - more development and be negated etc. Therefore through the continual positing and negating of each pole separately as with an iteration one gradually moves to the position where both are equally refined.

Another fascinating aspect relates to the fact that the absolute value of the Function keeps falling until s = - 5 where it reaches - 1/252, before then continuing to increase, gradually at first and then more and more rapidly.

The qualitative significance here relates to the fact that with contemplative development, rational type use of faculties tends to diminish to be used very sparingly (as the emphasis is primarily on intuitive awareness at this stage). Even before the 8th dimension is reached, a certain rebalancing is required so that the holistic transcendent drive to pure intuitive union with reality does not become too extreme. However by the time that the radial stages unfold, the value has reached 1. This would imply that only then has the emphasis of form been brought back into balance with (intuitive) emptiness. Put in spiritual terms, both the transcendent and immanent aspect would now be in substantial balance. And then the progressive increase in the value would signify an ever greater ability to use the rational faculties, as the radial stages proceed.

Now I can see why I laid so much emphasis on the dimensions up to 8 in my earlier approach (as these are the ones indeed most important in terms of holistic development). I had already worked out that a lengthy transition phase would be necessary to reincorporate the rational faculties (which I had identified with the dimensions from 8 to 16). So it is remarkable that the actual value for the Zeta Function only passes one, for the negative dimensional numbers of s between 15 and 17.

Interestingly where pure intuitive development reaches its zenith, one would expect that the positive and negative polarities of rational experience would be in substantial balance. And this indeed is the case for - 1/252 is followed by 1/240. So the absolute ratio of the two numbers is nearly 1 here (and much closer than the ratios of any other successive zeta values for negative odd dimensions).

All in all therefore, though it is more difficult to precisely "nail" the reasons for the precise magnitude of the vales of the Zeta Function (for negative odd values of s) we can once again show strong associations with spiritual psychological development amply demonstrating that these dimensional values (as with the even) have a precise holistic mathematical significance. In fact the meaning of all the values of the Function for negative dimensional values is - directly - of a holistic mathematical nature (as their values cannot be defined here in standard linear terms).

Wednesday, July 28, 2010

Negative Even Dimensions

We have looked at the positive even numbered dimensions. These are the most suited for pure integral interpretation of reality and are always based on the matching complementarity of opposites. Within the even numbered dimensions 2, 4 and 8 would command a special importance (as suited for integral interpretation of reality).

However so far we have only considered the positive numbered even dimensions!
So our next task is to give a meaning to the corresponding negative numbers.

I have mentioned the philosopher Hegel as proving initially inspirational in terms of formulating the holistic mathematical meaning of 2-dimensional understanding.

However I gradually became disenchanted with Hegel's approach. It seemed to me that that he was led into a fundamental error in elevating the mere formal rational interpretation of his logic above the very spiritual intuition that it really illustrated. In other words though one can formulate in rational terms the basic principle that all phenomenal interactions are based on the dynamic complementarity of opposites, the true understanding of this principle can only be attained in pure intuitive awareness (where the separate identity of either pole is no longer seen to exist).

So I gradually realised that the paradoxical rational formulation itself is properly designed to lead one on to a purer spiritual intuitive realisation of the meaning it contains. And quite literally this requires the negation of any rigid attachment to the intellectual formulation.

I then discovered in St. John of the Cross a person who had addressed these very issues with a profundity that is perhaps unmatched in the Western tradition.

St. John principally in "The Ascent of Mount Carmel" and "The Dark Night" deals in detail with these problems.

Firstly he deals with the problem of the standard linear attachments that arise from normal day to day duties. So to remove such rigidities the "active nights" of sense and spirit are required. This entails the dynamic negation of linear i.e. 1-dimensional, understanding (or more correctly rigid attachment to such understanding). Sense in this context would relate to the more superficial empirical type of understanding while spirit would relate to more deep-rooted concepts and volitional capacity.

All going well in the contemplative life this leads on to the illuminative stages.
Now this would be characterised in our terms by 2-dimensional understanding with a much more spiritually intuitive worldview properly consistent with the both/and logic of the complementarity of opposites.

However St. John warns strongly of the dangers of secondary attachment to such illuminated understanding. In an intellectual context this would imply undue attachment to the rational forms by which 2-dimensional understanding is mediated (without corresponding development with respect to appropriate intuitive recognition).

So this leads on to need for the passive nights of sense and spirit. So once again whereas the active nights relate directly to linear type understanding, the passive nights relate to more refined circular appreciation (of which 2-dimensional understanding represents the earliest kind).

Thus in holistic mathematical terms, the passive nights refer to the negation of "higher" even dimensional understanding. And the goal of all this purgation is, as St. John emphatically stresses, "nada" (or nothing). In other words the purpose of passive purgation is to remove any undue attachment to circular type understanding (that mediates the spiritual light).

Now there is an extremely important - though completely unrecognised - connection here with the Riemann Zeta Function.

In the Riemann Zeta Function the values for all even valued dimensions is defined. And all cases the resulting values are expressed in terms of corresponding powers of pi.
As we have seen in the previous post, in holistic mathematical terms this rational interpretation expresses the relationship as between circular and linear understanding. So the paradoxical nature of such understanding arises from the attempt to convey circular type relationships through - the necessary use of - linear language.

So when undue importance is attached to this new rational logic, it leads to an inevitable reductionism of its true meaning (that is ultimately spiritual) in rational terms. So in the end Hegel tends to elevate philosophy (in the expression of such rational forms) over religion!

However when rigid attachment is negated with respect to these paradoxical rational forms, their true meaning is revealed in a purely intuitive manner (= 0 in phenomenal terms).

And the remarkable finding here is that in the Riemann Zeta Function, when the even dimension is negative, the value of the function is always = 0.

Now these are referred to misleadingly as the trivial zeros of the Function.
However what is entirely missing from present understanding is that these values actually relate to holistic rather than conventional mathematical interpretation.

Now if one for example tries to calculate the value of the Riemann Zeta Function when s (i.e. the value of the dimension) = - 2 in the standard conventional manner i.e. 1 + 4 + 9 + 16 + ..., the series will diverge (i.e. infinite in common expression).

However in the way the Riemann Zeta Function is defined, the value that arises from the Function = 0. So what is not explained - or indeed is even capable of explanation in standard terms - is to give such values an intelligible meaning.

However what is actually happening is that for negative values of dimensions that the Riemann Zeta Function switches from linear interpretation (where values are infinite) to holistic higher-dimensional interpretation (where values are finite).
Once again conventional mathematical interpretation is based on the default dimension of 1 (i.e. where all calculations are reduced in terms of their quantitative values in this dimension).
And for example in the Riemann Zeta Function whenever the value of s (representing the power or dimesnion involved) > 1, a finite value for the series emerges (using the standard default dimension of 1). However remarkably when the Function here is now calculated - not in terms of dimension 1 - but rather in terms of the actual qualitative dimension arising in each case, a finite result emerges.
Thus we have two methods of calculation a) the conventional linear manner (based on a default dimension of 1) and (b) the holistic mathematical calculation based directly on the qualitative dimension actually involved.

Now interestingly when s = 1, the Zeta Function diverges in conventional terms (with no finite value). And because we are here using the Function is already defined in terms of the default dimension of 1, the holistic mathematical calculation is exactly the same. And this is the one point on the Riemann Zeta Function where its value cannot be defined in finite terms (in either the standard or holistic mathematical fashion).

So when the value of s > 1) quantitative interpretation on the RHS (of the real line) can take place for values of the zeta function in the conventional manner. However qualitative interpretation of the holistic mathematical kind is required on the LHS for s < 0).
And indeed this relationship is even enshrined (again without its true significance being recognised) in the Riemann Transformation Formula!

So enhanced integral understanding actually requires two related aspects. In rational terms it requires appreciation of the higher-dimensional forms that I have been illustrating on these blogs. Though the 2-dimensional has indeed been well articulated in various spiritual and philosophical cultures, without holistic mathematical appreciation it is not really possible to formulate higher even dimensions in a satisfactory manner.
The second aspect relates to appropriate intuitive recognition of what these rational forms are supposed to represent.
Quite simply this cannot be attained without journeying through the various stages of contemplative spiritual development. So initially the rational structure of a certain dimension is posited through initial illumination. However sustained attainment of the appropriate intuition associated with this structure requires a very deep purgation or cleansing i.e. negation of the rigid phenomenal aspects of such understanding. And this is the process that spiritual writers such as St. John have already well documented.

Tuesday, July 27, 2010

Update on Dimensions

As one may perhaps appreciate, the qualitative use of numbers (as dimensions) plays a key role in holistic mathematical understanding.

So I have spent at this stage more than 40 years in the slow - and often painful - task of unravelling the hidden meaning contained in all numbers (as dimensions).

In the attempt here to explain my present position, I will relate how this understanding actually unfolded in development.

The first key insight was in recognition of the linear nature of the rational paradigm which underpins interpretation of both mathematics and science.

So in qualitative terms, such understanding is defined in holistic mathematical terms by the number 1 (i.e. 1-dimensional interpretation). As we have seen this entails in Conventional Mathematics that all number quantities are ultimately defined in terms of a (default) power of 1. For example 2 ^ 2 = 4 (i.e. 4 ^ 1). In physics it implies for example that object phenomena are unambiguously posited in just one (external) direction.

The next key step was the recognition of the nature of 2-dimensional understanding. This arose through a deep interest for a time in Hegelian philosophy. Rather than a linear use of reason Hegel favoured a more circular form based on the complementarity of opposite poles (popularly represented as thesis and antithesis).

The key insight here was to see in holistic mathematical terms a direct connection between such poles and the square root of a number.

This can be generalised by saying that there is a direct connection between the n roots of the default number 1 in quantitative and the corresponding n dimensions in qualitative terms. However whereas linear either/or logic applies in the former case, circular either/or logic operates in the latter.

For example in quantitative terms the two roots of 1 are given as either + 1 or - 1.

However corresponding qualitative 2-dimensional interpretation is both + 1 and - 1.

What this means is that all understanding now entails a dynamic interaction as between the knower (as internal subject) and what is known (as external object) with a merely relative meaning so that these two poles keep switching in experience.

Though it may not be articulated in a holistic mathematical manner such understanding represents a normal stage with respect to authentic contemplative development. Here the rigid barriers as between the world (as objective) and the self (as subjective) gradually break down with neither having any absolute identity (in isolation).

The importance of this "higher" 2-dimensional understanding is that it is replicated at a corresponding "lower" level by the behaviour of sub-atomic particles. So the provision therefore of a coherent interpretation of quantum behaviour in qualitative holistic terms therefore requires (at a minimum) 2-dimensional understanding. What this means in effect is that in terms of such understanding quantum behaviour intuitively readily resonates with esperience (corresponding to such circular rational interpretation).

2-dimensional understanding represents the first of the truly integral approaches (based on the complementarity of opposites). I usually refer to this as the Integral 1 approach. Once again it is based on the complementarity of opposite real (conscious) poles

4-dimensional understanding is the next truly important integral approach (Integral 2).

Just as the earlier approach establishes the complementarity of internal and external aspects (the knower and what is known) this likewise establishes the complementarity of whole and part which is perhaps the most fundamental of all relationships in physics.

Standard linear interpretation cannot properly maintain the key qualitative distinction of whole and part. Instead it simply reduces (in any context) the whole to the parts in quantitative terms.

Not surprisingly, gross reductionism pervades the conventional approach to science and mathematics (which for the most part is not even recognised as such by its practitioners).

Precisely because the whole - when properly understood - is qualitatively distinct from the parts one cannot maintain such a distinction without incorporating the qualitative dimension of science on an equal basis with the quantitative.
Of course wholes and parts can be given a quantitative interpretation. However in the dynamics of recognition by which the mind switches from wholes to parts (and parts to wholes) an unconscious aspect is necessarily involved.

Properly understood therefore in the recognition of any object phenomenon an unconscious (as well as conscious) aspect is involved. Indeed even in popular language this is often recognised. For example one may speak of a "dream" house. So here the house has a local (conscious) identity; however equally it possesses a holistic (unconscious) identity as the embodiment of a deeper meaning.

However, strictly this necessarily applies to all phenomena. Indeed the very desire of scientists to study certain phenomena often speaks strongly of this holistic (unconscious) meaning (which cannot ultimately be separated from associated recognition of a conscious kind).

The key breakthrough here (which owed much to an interest in Jungian Psychology) was the recognition that the "embodied" unconscious aspect of understanding is "imaginary" in a precise holistic mathematical sense.
Now the negative direction (by which phenomena are literally negated in experience) represents the unconscious direction of understanding.Inherently this leads to a 2-dimensional form (combining positive and negative polarities). So to express such understanding in a (reduced) linear form we take the square root!

Thus "imaginary" interpretation is the every means through which the qualitative - as opposed to quantitative - aspect of understanding is expressed in a rational manner.

Mathematics recognises that numbers (as quantities) incorporate both real and imaginary members.
The holistic mathematical corollary of this is that science contains both "real" and "imaginary" aspects (i.e. as quantitative and qualitative aspects respectively).

So this whole blog on "Integral Science" is thereby designed to elaborate in various ways the nature of the unrecognised "imaginary" aspect of science.
And a fully comprehensive approach to science - which I term radial - would be "complex" combining both "real" and "imaginary" aspects (as equal partners).

As we have seen in the previous post, one important application of this relationship as between "real" and "imaginary" relates to object phenomena and dimensions (which are "real" and "imaginary" with respect to each other. Thus a clear implication of this is that we cannot hope to properly understand String Theory without incorporating a corresponding holistic qualitative aspect. And as I was mentioning all the important concepts in String Theory can be given corresponding holistic interpretations with a comprehensive understanding then relating to the relationship as between both aspects.

The next truly fundamental integral approach - which I term Integral 3 - relates to 8-dimensional approach. As well as the 4 dimensions (of Integral 2) this opens up 4 new dimensions (of a complex kind). However in geometric terms the diagonal lines representing the corresponding additional roots (on the circle of unit radius in the complex plane) can be represented as null lines.

The psychological implication of this is that pure spiritual attainment (that is nothing in phenomenal terms) is approximated when both conscious (real) and unconscious (imaginary) aspects are fully harmonised.

Remarkably the corresponding physical implication relates to the very nature of forces. For example light is often in Relativity Theory represented as a null line. So the deeper implication is that light (in itself) represents the complete harmonisation of phenomenal and dimensional characteristics.

Even in the Biblical account in Genesis, the World is created out of light. So the phenomenal and dimensional characteristics of the created Universe can be seen to emerge from a common origin!

For many years 8-dimensional interpretation represented the limit of what I felt could be achieved in integral terms.
Indeed I formulated A Holistic Theory of Everything based on such understanding!

Then shortly after that when studying the holistic mathematical nature of Jungian Personality Types I saw how to extend 4 to 24-dimensional interpretation through obtaining all possible permutations of the existing 4. This - as I have related elsewhere - then led to an extension of the Myers-Briggs system (to include 8 missing types). The realisation that each of these Personality Types represented a unique circular dimension (in the characteristic manner of configuring space and time) led me a similar notion within physical reality (which I now see as the means of interpreting dimensions in String Theory).

So just as with String Theory, I was left with 5 main integral models for holistic qualitative interpretation of reality.

It is only in more recent years - largely through association with the Riemann Hypothesis - that I have been able to significantly extend this understanding.

After some time I realised that 16-dimensional interpretation (and all subsequent powers of 2) could be explained like a compass. So as with the compass we have 4 main coordinates (applying to 4-dimensional interpretation. Then with higher powers of 2 we are able to define our directions ever more accurately. Likewise it is similar with higher dimensional understanding where 2 is raised to 0, 1, 2, 3, 4,...etc.

Then all other even numbered dimensions likewise have a direct integral significance.

The basis of integral understanding is the complementarity of opposites in understanding (with again 2-dimensional interpretation offering the simplest example).

However all other even based dimensions can likewise be represented by the complementarity of opposites (replicated in quantitative manner by the format of the roots of every even number).

Indeed there is an intimate connection here with the Riemann Zeta Function.
As is well known for all even numbered integer dimensions (powers) of the Function, an expression involving pi is involved.

Now pi represents in quantitative terms the relationship i.e. ratio, as between the circular circumference and its line diameter.

In like manner in qualitative terms all the even based dimensions (as interpretative models) represent the direct relationship as between circular and linear understanding!

Monday, July 26, 2010

String Theory Again!

I have already contributed a number of posts to this personal blog on the subject of String Theory.

Though most of the unease within the conventional physics community relates to the difficulties in empirically testing String Theory (for some considerable time to come) I have been concerned with a deeper problem!

For in terms of offering a coherent philosophical view of the nature of physical reality the present position is utterly impoverished.

Now this failure to properly recognise a key weakness, relates to the traditional bias of science i.e. that is geared merely to quantitative analysis - rather than qualitative synthesis - of reality.
Though it is certainly possible that ingenious ways of an indirect nature may be found to remedy the empirical testing deficit, little hope however exists of offering any coherent explanation as to what the theory is supposed to represent.

Now scientists may immediately retort that they are not in the business of qualitative (i.e. philosophical) speculation! But I would reply by saying that the very failure to incorporate a qualitative dimension has led to a growing problem of overall (i.e. holistic) incomprehensibility with respect to modern theories of physical reality.

This has raised therefore a very interesting paradox. As physicists move ever closer to what they hope will be the "Theory of Everything", they find themselves ever less capable of offering any intelligible explanation of what the TOE is supposed to represent.

This problem when properly understood highlights a divide in science (and equally mathematics) that properly should never have arisen i.e. the pursuit of a merely quantitative approach to meaning (without emphasis on its equally important qualitative aspect).

At the normal level of macro investigation of reality this problem is not apparent. Here scientific rational explanations of reality (based on linear logic) correspond readily with everyday qualitative intuitions with respect to such reality.
Because this is so, science conveniently ignores the important role of supporting intuition in all interpretation, misleadingly identifying physical behaviour with mere rational type explanation.

However as with the electromagnetic spectrum there are many other types of spiritual light besides that corresponding to natural light!

So for example, where authentic contemplative development unfolds, various bands of intuitive energy (with their own unique characteristics) unfold that do not correspond to conventional (i.e. linear) rational interpretation. However these bands do indeed correspond with higher-dimensional rational structures!

Such "higher" contemplative states (with their supporting higher dimensional rational structures) have a vital relevance for interpretation of physical reality.

Just as horizontal conplementarity exists at each level as between physical and psychological reality (i.e. where physical reality acts as a mirror of a corresponding manner of psychological interpretation) likewise vertical complementarity likewise operates. What this entails is that corresponding to every "higher" contemplative level of dimensional understanding exists a corresponding "lower" level of physical reality.

In other words as we delve deeper into the "lower" levels of sub-atomic matter, we need the corresponding "higher" levels of dimensional understanding for appropriate qualitative understanding.

So this immediately points to the difficulties that physics faces in providing for example a philosophically satisfying interpretation of quantum mechanics (which appears counter intuitive in terms of everyday understanding of macro reality).

Quite simply, normal macro reality corresponds intuitively with linear (1-dimensional) understanding. However quantum reality belongs to a "lower" level of sub-atomic reality where at a minimum 2-dimensional interpretation is appropriate.

So what does this mean? Well in linear terms, "objective" reality is viewed as independent of the "subjective" observer.

However in 2-dimensional interpretation both objective (external) and subjective (internal) poles are considered as complementary. So strictly speaking where quantum events are concerned what is observed as objective cannot be considered as independent of the observer.
Though this fact is recognised by physicists - though in practice largely ignored - the appropriate intuition for ready comprehension (with corresponding 2-dimensional rational explanation) does not fit in with the accepted conventional paradigm which is decidedly linear. Therefore, even though physicists are able to deal with the merely probable quantitative aspects of quantum behaviour to a remarkable degree of accuracy they are unable to provide an appropriate qualitative explanation (which explains in a satisfactory manner why such behaviour arises).

Now again many physicists might try and retort that they are not concerned with philosophical aspects of understanding. However such a position is extremely short-sighted and ultimately untenable for without appropriate holistic understanding of a qualitative nature, overall interpretation of theory becomes incomprehensible.

Indeed this is major problem at present with String Theory. Though undoubtedly there are good reasons for excitement regarding the mathematical relationships that have been discovered, no coherent explanation has been given by physicists as regards the very nature of the reality that they are attempting to interpret.

For example one major problem attaches to the nature of dimensions in which the strings supposedly operate. In various approaches this has required more than the customary 4-dimensions with 26, 10, and now 11 being used.

However once again this is calling for the kind of multi-dimensional interpretation that the qualitative approach can provide.

We have already seen that in the 2-dimensional approach, external and internal aspects of reality dynamically interact in a relative manner (corresponding to positive and negative polarities that are understood in a real conscious manner).

However in the 4-dimensional approach, not alone do internal and external interact but also whole and parts in the form of object phenomena and dimensions of space and time. So properly understood phenomena and associated dimensions are now real and imaginary with respect to each other!

If we take these 4 poles (representing both real and imaginary coordinates with positive and negative directions) we can permutate them in various ways. So each permutation now represents a unique configuration with respect to space and time.

The need for higher dimensions in String Theory essentially points to this fact.

At this level of reality it is meaningless to try and consider space and time as separate from the object phenomena they contain. So a dimension now represents a unique configuration with respect to possible arrangements of what are 4 independent dimensions in linear understanding.

So the key to proper appreciation of what is going on is the recognition that one must now employ a circular (as opposed to linear) notion of dimension.

And as we approach closer to the origin of life - because of the greater interdependence both of internal and external aspects and also object phenomena and related dimensions - appropriate understanding of reality, in qualitative terms, becomes ever more circular. Put another way the standard linear approach of Conventional Science becomes ever more inappropriate in providing a coherent overall explanation of the dynamic relationships involved. So what we are left with is attempted quantitative interpretation of relationships completely devoid of any proper holistic context.

So just as the Riemann Hypothesis lies at the leading edge of Mathematics where both quantitative and qualitative aspects intersect, likewise this is also true of String Theory. Having attempted for some time to decode the philosophical meaning of the various concepts used, I made the amazing discovery that every notion that is currently used in the quantitative sense can be given an equally important qualitative interpretation. So the real truth that String Theory is pointing to is the inevitable conclusion that our quantitative explanations of physical reality have no meaning independent of the qualitative interpretations that we use as viewing lenses. So the origin of life cannot be quantitatively understood (nor qualitatively understood) in isolation. Rather its mystery is revealed when quantitative and qualitative aspects of understanding are fully reconciled (which represents pure spiritual awareness). And this experience is inseparable from the contemplative mystical desire to discover the ultimate goal or destiny of the Universe!
So realisation of the origin and goal of the universe are revealed in the same spiritual experience as the present moment continually renewed. Only here is any remaining gap as between the knower and what is known finally dissolved where both realise their common identity (in spirit).

And it is not really surprising that both the Riemann Hypothesis (relating to the nature of prime numbers) and String Theory are so closely related in this manner. For the most fundamental particles - derived from what we might call strings - are in fact the prime constituents of all natural physical reality!

Saturday, July 24, 2010

Holistic Mathematics and Science

In the last post I addressed the fact that corresponding to each number in Holistic Mathematics is a unique scientific interpretation of reality.

So the range of possible interpretations is infinite with Conventional Science based on just one of these numbers (i.e. 1).

This I believe is a truly remarkable finding - which if even remotely grasped - should end all notions of science having reached its zenith!

Indeed we can make further remarkable statements based on holistic mathematical interpretation. For not alone does every number possess a unique qualitative dimensional significance (as indicated) but equally every symbol and relationship - with an already established significance in Standard Mathematics - can likewise be given a unique holistic mathematical explanation with intimate relevance for psychological interpretation of reality.

And those theories and hypotheses that already have been shown to have a special importance in standard quantitative terms would possess an equally important significance in holistic qualitative terms.

Indeed to demonstrate this point to my own satisfaction I spent several years examining the Riemann Hypothesis in the attempt to establish its true qualitative psychological significance. Then as an altogether unexpected bonus this paved the way for - what I am confident - is a true resolution of the hypothesis.

So I established in the end that the famed hypothesis is actually expressing a fundamental condition for maintaining consistency as between both the quantitative and the qualitative interpretation of mathematical symbols.
Of course one clear implication of this is that it cannot be solved in standard mathematical terms (as it is based on recognition of the quantitative aspect only).

So correctly understood - as I would see it - the Riemann Hypothesis serves as one expression of a vital axiom in a more comprehensive radial mathematical approach i.e. that explicitly entails both quantitative and qualitative aspects of interpretation.
So not alone is the standard approach to Mathematics inadequate for the task of solving the Riemann Hypothesis, it is even inadequate in terms of understanding its true nature!

However it would be mistaken to believe that the relevance of Holistic Mathematics (and related Integral Science) is confined to qualitative interpretation of a psychological kind.
For once one departs from the default linear (1-dimensional) interpretation of Conventional Mathematics (and Science), a two- way complementary relationship exists as between physical and psychological understanding. Thus at all other levels, what we "see" with respect to physical reality inevitably reflects in considerable measure the psychological manner by which it is interpreted.

Thus the physical and psychological aspects of (interpreted) reality are mirrors to each other so that the structure revealed in one is the same structure pertaining to the other.

The clear implication of this therefore is that not alone do we have an infinite set with respect to potential interpretations of reality, but equally we have an infinite set of (corresponding) physical realities.

So the great error that is committed by adherents merely of the standard linear approach is the belief that there is just one physical reality "out there" with just one valid manner of overall interpretation.

Though most scientists may not fully subscribe to the pure representation view (i.e. that our chosen mental constructs reveal reality as it actually is) in practice they behave as if this is exactly the case!

However once again when we recognise the legitimacy of the other dimensional numbers (as interpretative structures) we are faced with the inevitable conclusion that we cannot possibly view what is "out there" for in truth a potentially infinite set of interpretations (and corresponding physical realities) exist.

Putting this another way, the inevitable conclusion of adopting this more comprehensive approach is an acceptance that science can only attempt to grapple with secondary phenomenal appearances that serve in an infinite variety of ways to mask what is truly primary and essential (i.e. spirit).

Existing scientists are slowly coming to accept that physical reality in its deepest workings behaves very differently from what is seen at the normal macro level of observation.

So the 1-dimensional method of Conventional Science is best equipped to deal with normal macro reality.

Unfortunately while now recognising that sub-atomic reality conforms to different rules, essentially the same linear method is still being used for interpretation.

It is hardly surprising therefore that quantum reality for example appears so strange and paradoxical (as it is continually viewed through an inappropriate interpretative lens).

At an even deeper level string theory in the same attempt to pursue a merely quantitative type appreciation has rendered itself virtually meaningless as a coherent philosophical (qualitative) explanation of reality.

And this is where the new interpretative models based on higher dimensional numbers) come into play which lead among other things to a new appreciation of dimensions.

I will return to this briefly again in a further post!

Thursday, July 22, 2010

More on Nature of Holistic Mathematics

Just as the standard analytic approach to science is ably served by its corresponding mathematical tool i.e. Conventional Mathematics, likewise the (qualitative) integral approach to science is likewise potentially served by its own respective mathematical tool i.e. Holistic Mathematics.

A great barrier however that I have continually faced is the recognition that Holistic Mathematics - though using the same symbols - is radically different from what most people understand as Mathematics. So what certainly is not intended here is the standard use of Mathematics to deal with holistic type problems (which represents the old reductionist approach)! Rather it requires a new interpretation of mathematical symbols that inherently depends on corresponding appropriate level of intuitive insight for correct usage.

To put this in context, one must appreciate that just as there are many bands on the electromagnetic spectrum (with natural light comprising just one), likewise it is true with respect to the potential for human psychological development. Such development comprises many bands where the nature of the intuitive light - that necessarily informs all understanding - varies greatly.

So what informs rational interpretation - dictating conventional approaches to mathematics and science - is a natural light (in the normal intuition that underpins the use of such reason). However potentially there are many other bands on the spectrum with very different forms of intuitive energy so that here "normal" intuition is no longer appropriate.

For example higher bands of intuition unfold through the process of contemplative spiritual development and in the various mystical traditions their associated states have been documented to a considerable level of detail.

However what has been largely missing is the attempt to develop in systematic manner the implication of such states for understanding of science. For quite simply, a radically distinct approach is needed so as to maintain true compatibility with authentic contemplative understanding.

In my initial quest to deal with this issue I realised that a new type of holistic or qualitative science was needed so as to be consistent with the various stages of contemplative development.

I then further recognised that just as the most developed type of intellectual development would combine both reason and contemplative insight in a refined manner that likewise the most comprehensive kind of science - which I term radial - would likewise combine both analytic (quantitative) and holistic (qualitative) aspects in a balanced interactive fashion.

However I have since come to realise with respect to the qualitative holistic aspect that an infinite variety of interpretations potentially exist.
So what this all mean?

Using holistic mathematical language, I always term the basic approach of science (what one might refer to as its metaparadigm) as linear i.e. 1-dimensional.

Now customarily we think of numbers 1, 2, 3 as quantities. But as we know in Mathematics these same numbers can also be used to represent dimensions (as powers or exponents).

However, in re-opening a major misgiving felt even as a child, I realised that that when numbers are used to represent dimensions in Conventional Mathematics they are given but a reduced i.e. quantitative meaning.

For example when we raise the number 2 to the power of 2
i.e. 2 ^ 2, a qualitative as well as quantitative change takes place. So a table with each (straight) side = 2 metres would have an area of 4 square metres. Thus the 2-dimensional area here (in square units) is qualitatively distinct from the 1-dimensional measurement of each side (in linear terms).

However remarkably in Conventional Mathematics, the result of
2 ^ 2 is provided in a merely reduced quantitative manner as 4 i.e.
4 ^ 1. So this illustrates clearly the truly linear nature of Mathematics where the result of numerical calculations is ultimately expressed in a 1-dimensional manner (so that implicitly this result is raised to the power of 1!)

So - quite literally - Conventional Mathematics is confined purely to a 1-dimensional manner of interpretation (with the qualitative dimension thereby reduced in a merely quantitative fashion).

However associated with every number is a unique qualitative interpretation.
Therefore the default standard interpretation - in what we misleadingly call Mathematics - represents just one of a range of qualitative interpretations that is potentially infinite!

You might well ask what possible bearing this can have on scientific investigation!

In fact when appropriately perceived it has very important consequences.

We can illustrate the standard linear manner of interpretation in science very directly.

When one for example observes an object it is - literally - posited in experience as distinct. Now in holistic terms such positing is denoted by the mathematical symbol + (this time with a holistic meaning). Likewise implicit in the recognition of any phenomenon as distinct is its unitary nature (i.e. where it is literally identified as a unit). And here we have the holistic meaning of the symbol 1.

So standard scientific investigation is thereby one-directional which can be defined here in holistic terms as + 1.

Now if we take the 1st root of 1 (which is the same as raising to the power of 1) the answer remains 1.
More precisely the answer is + 1. So again in 1-dimensional terms, no distinction can be made as between quantitative and qualitative interpretation.

However when we obtain the two roots of 1 something strange happens in that the answer splits quantitatively in two possible directions i.e. + 1 and - 1.

The corresponding implication in qualitative terms is that when we raise the number 1 to the power of 2 that the answer now splits again in two directions + 1 and - 1.

As one would expect - because of its 1-dimensional bias - there is no recognition of this in Conventional Mathematics (which merely reduces the result in quantitative terms). So 1 ^ 2 is treated in precisely the same manner as 1 ^ 1 (= 1).
However in Holistic Mathematics when we raise 1 to the power of 2, it corresponds in qualitative terms to the taking of two roots.

So very importantly raising 1 qualitatively to D (as dimension) structurally corresponds to raising 1 to (1/D) in quantitative terms. However the qualitative inetrpretation is based on circular (both/and) rather than linear (either/or) logic!

So to make a long story short, 2-dimensional - as opposed to standard 1-dimensional interpretation - in a scientific observational context entails giving every phenomenon two directions that are + (positive) and - (negative) with respect to each other.

This in fact refers to the inescapable fact what all experience of reality (in any possible context) inevitably entails a dynamic interaction as between the knower (as subject) and what is known (as object). So from a 2-dimensional perspective an "object" phenomenon has no strict meaning outside this two-way dynamic interactive context.

So we now can only speak relatively of every phenomenon with two directions involved. Thus if we associate + with the external (objective) aspect, then we must associate - with the corresponding internal (subjective) aspect; however if we equally in any arbitrary polar context fix + with the internal, then in relative terms we must then associate - with the external. So in dynamic interactive terms positive and negative signs keep switching in a relative manner.

This type of circular understanding based on the complementarity of opposite poles, thereby enables one to deal more subtly and accurately with the actual dynamics of scientific experience.

Considerably more refined experiences are possible (and in the appropriate scientific contexts extremely important). For example with 4-dimensional understanding (which is especially significant) every object phenomenon would be given a 4-directional interpretation with both two real and two imaginary aspects that are each positive and negative with each other. This again would correspond, from a structural perspective, with the 4 roots of 1 (in quantitative terms).

What this means in effect is that as well as recognising the dynamic nature of the external/internal dialogue in experience, we also recognise the equally important interaction of whole and part (which inevitably defines all possible experience).

This leads on to the key insight that the imaginary symbol i corresponds directly with the unconscious recognition of an object (in holistic terms). Thus switching from whole to part (and part to whole) in experience always entails the interaction of conscious and unconscious. And in holistic mathematical terms whereas conscious understanding is "real" unconscious - by contrast - is "imaginary".

So the qualitative implication of the complex number system is the recognition that both rational (conscious) and intuitive (unconscious) aspects must be formally recognised and that correctly understood in 4-dimensional terms in any appropriate context the whole is always - relatively - "imaginary" with respect to the parts that are "real". So when we fail to recognise this fact (as in standard 1-dimensional interpretation which is solely "real") the whole is inevitably reduced to the parts.

So again we can characterise the limitation of the present accepted approach as one that formally recognises solely the "real" aspect of understanding.

Thus what I am attempting to address here is the need for recognition of the equally important "imaginary" aspect of mathematical and scientific understanding (i.e. Holistic Mathematics and Integral Science respectively).

Then the radial approach to both - combining them as both equal in importance - can thereby constitute in qualitative terms the complex rational approach. And when both aspects are equal they can be represented as null lines, likewise such an approach also represents the most simple (based on pure spiritual intuition).

We have not time here to deal with further dimensional interpretations (which are ever more demanding in terms of appropriate intuitive recognition).

However we can hopefully generalise what is involved in a meaningful manner.

Just as all the possible roots of unity can be represented by sets of complex numbers on the circle (in the complex plane), likewise all possible dimensions of understanding can be understood in circular logical terms as representing various configurations of both "real" (conscious rational) and "imaginary" (unconscious intuitive) understanding. (Once again what is "imaginary" in this context strictly relates to an appropiate indirect rational means of expressing holistic intuition!)

Monday, July 19, 2010

The End of Science?

I have been reading again "The End of Science" by John Horgan which I find interesting on several levels.

Firstly, whether one agrees or not with its conclusion it puts forward a most provocative hypothesis i.e. that the great era of theoretical scientific discovery is at an end with diminishing returns with respect to further development now to be expected.

Secondly it attempts to cover a wide range of different scientific fields conveying in the process some flavour of the rich developments that have already taken place.

Finally - and most notably - Horgan managed to do an impressive amount of research in interviewing a significant number of the biggest names associated with these fields (at least at the time of writing in the mid 90's). What I like most about his approach is a certain innate scepticism which prevents him from ever appearing unduly awestruck with their strongly held beliefs.

My own approach would be somewhat different to Horgan's and more optimistic that even within the confines of - what is presently accepted as - science, further impressive developments in many fields (theoretical and empirical) can be expected for some time to come.

However in the quest to obtain final definitive answers to life's greatest mysteries e.g. the origin of life, science is in fact straying way beyond present boundaries without seemingly recognising the significant limitations of its present methods.

So in the opinions of many of those interviewed there is much reductionist thinking masquerading as scientific wisdom.

What is more likely is that we are now witnessing the peaking of the golden age with respect to a particular form of science i.e. analytic science geared to quantitative interpretation of reality. However by its very nature such science is not suited to deal with qualitative issues (except in a grossly reduced manner). Likewise it is leading to significant fragmentation with respect to knowledge already accumulated.

It has become increasingly obvious to me over the past 40 years or so that what we now need is a truly massive revolution in the accepted nature of science itself which will allow for the proper inclusion of an equally important - though utterly distinctive - holistic qualitative aspect. And then with gradual development of a more specialised appreciation of the nature of such qualitative science, we will be in a position to unleash its true potential in a comprehensive approach that combines both quantitative and qualitative aspects as equal partners.

So I would say with considerable confidence that we are not witnessing the end of science. Rather we are perhaps witnessing the end of the total domination of just one aspect of science i.e. its analytic quantitative aspect.
However in terms of a truly comprehensive worldview, science is still very much in its early infancy.