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.