Friday, May 7, 2010

Transcendental Numbers

Transcendental numbers especially e and pi play an important role with respect to prime number behaviour.

For example the simplest version of the prime number theorem, providing a general means of calculating the frequency of primes uses the natural log of n (which is based on e).

Also the sum of terms for even number dimensional values in the Euler Function - which again can be shown to have an intimate relationship with the primes - contains neat numerical expressions involving powers of pi.

We have seen that all number types (as quantities) can be given a corresponding holistic meaning (in qualitative terms). In this context we have already looked at the primes, (algebraic) irrational and imaginary numbers.

Though transcendental numbers are also classed as irrational, they differ in an important way from other (algebraic) irrational numbers such as the square root of 2.

All (algebraic) irrational numbers can ultimately be derived as solutions to higher dimensional polynomial expressions (with rational coefficients).

For example, in this context the square root of 2 arises as the solution of,

x^2 - 2 = 0.

So the irrational number here serves - literally - as the reduced expression of a number that is rational with respect to a higher dimension (or combination of dimensions) but then becomes irrational when interpreted in a reduced linear manner.

So the number 2 in this context has a rational meaning when expressed with respect to the 2nd dimension (i.e. where x is raised to the power of 2) which then becomes irrational when given a reduced linear value with respect to the 1st dimension (where x is raised to the power of 1).

Now the corresponding qualitative interpretation here is very revealing.
Interpretation according to 2-dimensional understanding is indeed rational within the same dimensional perspective.

As we have seen, such 2-dimensional understanding represents both/and logic that is based on the complementarity of opposite polarities. However when we attempt to express such interpretation through the medium of standard either/or linear logic (relating to the 1st dimension) it appears as irrational (i.e. paradoxical).

In fact Hegel is very interesting in this context as identifying "true" reason with 2-dimensional (rather than standard 1-dimensional logic). Such reason then appears very circular and paradoxical (irrational) from the conventional linear standpoint.

One of the problems that frequently arises in the spiritual contemplative life is that intuitive type awareness - properly pertaining to higher dimensional understanding - inevitably gets reduced to a degree to the standard 1-dimensional understanding (that governs so much of practical affairs).

St. John of the Cross carefully distinguishes lower-level attachments (with respect to conventional understanding) and higher-level attachments (with respect to intuitively inspired consciousness).

The first - with respect to linear (1-dimensional) understanding he terms active and the second type - with respect to higher dimensional understanding he terms passive.

So the famous dark nights that he depicts are with respect to the need for cleansing the spiritual aspirant of all passive attachments (so that spiritual intuitive illumination is not confused with the conscious rational symbols through which it is mediated).

And the direct relevance of this for Mathematics is that strictly this same contemplative process is necessary to avoid confusing - in any relevant context - infinite with merely (reduced) finite notions. And unfortunately, conventional mathematics is rife with such reductionism (that is not even recognised as such!)

Now all of this sets up the proper context of appreciating the true qualitative significance of what is meant by a transcendental number.

A major clue comes from consideration of the geometrical nature of pi (the best known of all transcendental numbers).
Pi as is well-known expresses the ratio of the circumference of a circle to its line diameter.
In corresponding qualitative terms, a transcendental number expresses the relationship as between circular and linear interpretation (with respect to such a number).

Now the distinction here from the earlier (algebraic) irrational numbers is that transcendental numbers cannot be reduced in the same linear manner.
In quantitative terms this entails that transcendental numbers cannot arise as the reduced 1-dimensional expression of a variable (with respect to polynomial equations with real rational coefficients).

In other words because the emphasis is now - neither on linear or circular interpretation as separate - but rather on what connects both, (qualitative) transcendental understanding cannot be expressed through either aspect (as independent).

Thus there is something more elusive about both the nature of transcendental numbers (as quantities) and corresponding transcendental numbers (as qualitative understanding).

In spiritual terms such appreciation would relate to a more refined contemplative state where finite symbols can mediate the pure spiritual light (with little conscious attachment). Realistically some degree of attachment would remain (at an imaginary unconscious rather than real conscious level).

The work of Cantor was to point to this elusive nature of transcendental numbers. Though they are amazingly dense with respect to the number system, remarkably few are directly known. Indeed Cantor was able to point to their existence without directly identifying any particular example!

Also the paradoxes of infinity that Cantor was able to demonstrate, whereby different number sets exhibit different degrees of infinity, I would see as an ultimately unsatisfactory way of understanding the relationship as between finite and infinite. From a spiritual perspective, infinite degrees of infinity would refer to the fact that each object (as finite) can uniquely reflect the spiritual light (that ultimately transcends and yet is potentially immanent in every object).
In this context - of dynamic experiential interaction - the notion of one homogeneous infinite concept makes no sense.

Though Mathematics uses the linear rational approach it is quite clear, in the very names we use to describe the main number types, that their inherent nature is far from rational. For example as well as the primes with connotations of primitive, we have the (algebraic) irrationals, the imaginary, the transcendental and ultimately transfinite numbers.

In spiritual contemplation, as one moves away from the rational world, there are various hierarchies or degrees of more intuitively inspired understanding. likewise with respect to number types as we move away from the rationals they exhibit greater and greater degrees of subtlety (not directly amenable to rational interpretation). Ultimately any remaining finite quality ultimately disappears (as with the transfinite numbers).

Again what is missing from conventional appreciation is that these number types require distinctive means of qualitative interpretation (that ultimately transcends all rational understanding).

In this way the Pythagoreans were indeed correct that - properly appreciated - mathematics should offer an extremely important means of attaining ultimate contemplative awareness of reality.

We have already looked at the inherent nature of pi (which in qualitative terms entails the relationship as between circular and linear understanding).

It is fruitful to also look at the inherent nature of e (which plays perhaps an even more significant role with respect to the primes).

As we know differentiation and integration play very important roles in Mathematics; equally they play an extremely significant role with respect to psycho spiritual development.

If we differentiate the simple function y = x^2 we get dy/dx = 2x.

So what has happened here is that we have reduced the higher dimensional expression to a lower dimensional form (where the dimension 2 as qualitative becomes transformed to 2 as quantitative number).

Differentiation in psychological terms is somewhat similar and entails a form of reductionism whereby object phenomena can become identified in quantitative terms.

Now integration (in this context) in both mathematical and psychological terms entails the reverse procedure whereby the lower quantitative understanding becomes transformed in higher dimensional terms. So when we integrate 2x we obtain x^2 (with number as quantity now becoming the corresponding number as qualitative dimension).

In psycho spiritual terms (and indeed in all biological life processes) it is very similar. Integration here essentially entails moving the quantitative to the qualitative perspective (where various quantitative parts can assume a coherent qualitative whole identity).

Now the simple function y = e^x is unique in the sense that when we differentiate (and once again integrate), its value remains unchanged.

Therefore from a qualitative mathematical perspective, the number e plays a unique role in that it represents that very point whereby differentiation becomes inseparable from integration in development.

In psycho spiritual terms, this would entail a highly refined state where the differentiation of phenomena becomes so rapidly interactive that they no longer even appear to arise in experience. Therefore in this state, differentiation (in the generation of discrete phenomena) would become inseparable from continuous integration in maintenance of a stable spiritual equilibrium.

Now we have already seen that the inherent nature of a prime number entails the extremely close relationship as between both linear and circular modes of behaviour.

Not surprisingly therefore prime number behaviour (in its general distribution) is linked very closely with e (which in its qualitative nature entails the reconciliation of both aspects).

The ultimate goal in terms of psycho-spiritual development is the complete unravelling of all primitive impulses. This is achieved when a purely continuous intuitive state, combined with the rapid unattached generation of discrete phenomena in experience, unfolds.
This further entails the final reconciliation of both the linear and circular modes of interpretation.

In this way the prime (i.e. primitive) problem is eventually solved (though in truth human experience can always only approximate to this goal).

In a very similar fashion the prime number problem is likewise eventually solved through obtaining the complete reconciliation of the linear and circular methods of behaviour (and corresponding interpretation).

This reconciliation is already implicit in a general distributional manner in the prime number theorem (that incorporates the use of e). However errors still remain here with respect to the exact prediction of the number of primes.

However, as we shall see the full reconciliation of linear and circular aspects is in fact implied by the Riemann Hypothesis.

Indeed, once again, in this context the Riemann Hypothesis simply operates as the necessary condition to ensure the full reconciliation (or consistency) of both linear and circular aspects of quantitative prime number behaviour (together with reconciliation with respect to the corresponding linear and circular aspects of qualitative interpretation).

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