## Sunday, January 22, 2012

### The mystery of e

For some 40 years or so I have been attempting to develop to my own satisfaction a distinctive type of qualitative Mathematics i.e. Holistic Mathematics. Recently I have referred to this greatly neglected aspect as Type 2 Mathematics. So just as Conventional (Type 1) Mathematics is devoted to the specialised development of the quantitative, equally in a balanced approach Holistic (Type 2) Mathematics would be correspondingly devoted to the specialised development of the (neglected) qualitative aspect.

Then finally, we would have the Comprehensive (Type 3) Mathematics where both quantitative and qualitative aspects would be dynamically combined in a highly productive and creative manner.

Though initially I spent many years developing holistic mathematical concepts in considerable isolation from conventional mathematical pursuits, recently I have begun to see in a much clearer manner, how a merely reduced interpretation is given in Type 1 Mathematics of fundamental relationships.

Ultimately both Type 1 and Type 2 aspects - because of their mutual interdependence - can only be fully appreciated in the context of comprehensive (Type 3) understanding.

For example I have pointed out elsewhere how the notion of the infinite cannot be successfully incorporated in standard (linear) rational terms; also I have pointed to the fact that when we raise a number to a (dimensional) power or exponent, that the number representing the exponent is properly speaking of a qualitative nature (in relation to its base number quantity). So this begs the obvious question of how even a simple procedure such as the squaring of a number (which entails both quantitative and qualitative aspects) can be properly interpreted in Conventional (Type 1) terms!

Yet another important example of the same problem is provided by irrational numbers. The very point regarding an irrational number such as the square root of 2 is that it combines both finite and infinite aspects of behaviour. So in quantitative terms the number can be approximated in discrete rational terms to any required degree of accuracy. However in qualitative terms its exact (quantitative) value can never be known with a decimal sequence that continues indefinitely in no fixed order.

In fact there is a very deep reason why the Pythagoreans found the discovery of the the existence of the square root of 2 so devastating. They could see - much more clearly than in modern times - that the nature of this number literally transcended their (Type 1) rational paradigm of Mathematics. In other words there was no way using this paradigm that the qualitative aspect of an irrational number could be successfully accommodated.

In subsequent Western Mathematical history, this issue has not been solved but rather avoided through a continual specialisation in the process of quantitative type reductionism.

The well known number e is an irrational (transcendental) number that combines both quantitative and qualitative aspects in a very special manner. And it is the appreciation of the nature of this quantitative/ qualitative type identity of e that is key to understanding the true nature of prime numbers.

Now it is well known in simple calculus that if y = e^x, that dy/dx = e^x.

Now differentiation and integration also have a qualitative holistic significance as is apparent in psychological development. So basically in experience, the differentiation of meaning pertains to the (rational) conscious, whereas integration pertains to the (intuitive) unconscious. In other words differentiation is discrete and quantitative in nature; by contrast integration is continuous and qualitative in nature.

Before development commences, in a sense differentiation and integration are identical (in total confusion) as mere potential for existence. Then - as is portrayed well in the mystical literature - again differentiation and integration once again approach identity with each other in pure spiritual union (where discrete phenomena are now so short-lived and refined that they no longer appear to even arise in experience!

What is truly remarkable is the intimate connection which e has with the prime numbers.

So for example the probability that a number is prime is given as 1/log n.
Alternative the average gap (or spread) between primes in the region of n is given as log n.

Now if e^x = n then log n = x.

So if we allow x to take on the value of any positive real number > 1 then x is telling us the average gap between primes in the region of n (i.e. e^x). And the accuracy of this estimate improves steadily for larger n! (Of course because prime numbers are positive integers we would need to round the value of n to the nearest integer!)

Now we have seen that because of the invariant characteristics of e^x relating to differentiation and integration both with respect to (conventional) Type 1 and (holistic) Type 2 interpretations, that e uniquely combines quantitative and qualitative characteristics in its own identity.

Likewise the fundamental nature of prime numbers is that they too combine both quantitative and qualitative characteristics in a unique manner. So this is the precise reason why e has such an intimate relationship with the primes!

So the Riemann Hypothesis - when correctly interpreted - represents a fundamental condition for maintaining the consistent identity of both the quantitative and qualitative aspects of the primes!

And this identity clearly points to the ineffable state in physical terms before creation begins where the prime number code still potentially awaits unveiling. Once we begin to differentiate distinct prime numbers then it is no longer possible to maintain total identity of discrete primes with the corresponding continuous general distribution of primes. So in a sense it is only before differentiation takes place in a phenomenal manner that an exact identity can be maintained of the discrete primes with their general distribution in a potential manner. It also points to the corresponding ineffable state spiritually at the peak of mystical union where the full mystery of the nature of the primes can finally be actualised.