Friday, December 30, 2011

Changing Our Ideas on the Primes

I have continually asserted the fact that the way we fundamentally look at the primes is very misleading.

Arising from the linear (1-dimensional) nature of Conventional Mathematics, the primes are viewed as the basic (independent) building blocs of the natural number system.

However even momentary reflection on the matter would immediately lead to the realisation that we cannot even begin to think of the primes in the absence of the natural numbers.

For example as soon as we try to rank primes we automatically are required to use the natural numbers in an ordinal sense. So 2 is the 1st, 3 the 2nd, 5 the 3rd, 7 the 4th prime and so on.

But the ordinal natural numbers in turn implies the cardinal use of these numbers. So we cannot give 1st, 2nd, 3rd for example a meaning in the absence of the cardinal numbers 1, 2, 3!

Thus the prime and the natural numbers are mutually interdependent.

This requires adopting the radical view that just as the natural numbers (from one perspective) are determined by the primes, that the primes in turn are determined by the natural numbers.

Now the clue to this new realisation comes from recognition that there are in fact two key logical systems. The linear logical system based on unambiguous either/or distinctions (on which Conventional Mathematics depends) can be represented as + 1.

However the alternative circular logical system based on relative both/and can be represented as 1 - 1 = 0.

Clearly before phenomena can unfold, both the linear and circular systems are identical i.e. as undifferentiated form (which equally is emptiness).

Now the conventional notion that natural numbers are derived from the primes is based on the linear logical approach.

However the reverse view that the primes are derived from the natural numbers springs from appreciation of the circular approach.

Now initially the circular is represented as 1 - 1 = 0.

However with the emergence of natural numbers it is represented as n - 1 (where n is a natural number).

The point about this is that all new primes can be represented uniquely by a natural number (that has been derived from earlier primes - 1).

For example 29 (which is a new prime) can be uniquely derived as 30 - 1.

30 in this case is the composite result of multiplying the earlier primes 2, 3 and 5.

Thus 29 can be uniquely represented as (2 * 3 * 5) - 1.

So the number system starts with 1 and 0 with 0 itself represented as 1 - 1.

Now the origin of the next number 2 (leading to the birth of duality) arises from portraying the circular system in linear terms. Thus rather that preserving the dynamic complementarity of + and - signs, both are represented as +.

Thus instead of 1 - 1 = 0, we now have 1 + 1 = 2.

All subsequent natural and prime numbers are ultimately derived from 2 (and 1).

So 3 the next prime number = (2 * 2) - 1 = 3.

Then the next prime number 5 arises from (2 * 3) - 1 = 5.

So remarkably we have two systems here!

The first (linear) system is based on multiplication of primes (to derive the natural numbers).

The second (circular) system is based on subtraction of 1 (from the composite natural number result of multiplying primes) to uniquely define new prime numbers.

So in actual reality (and experience) both systems are involved. The natural numbers and primes are built up in conjunction with each other in a dynamic interactive manner.

In a very valid sense neither the primes nor natural numbers preexist this dynamic interactive process that takes place in space and time.

Thus the prime numbers and the natural can never be fully reconciled with each other in a formal phenomenal procedure. Such reconciliation takes place in an ineffable manner (as the secret code pre-existing phenomenal creation) and equally the ultimate spiritual realisation of the true mystery of the primes (and natural numbers).

Incidentally the true circular nature of the number 0 is revealed by the fact that is invariant with respect to either positive or negative signs. So + 0 = - 0.

What this really means is that 0 properly embraces both positive and negative signs in a circular manner! And the very symbol we use for 0 signifies this circular nature (just as the symbol we use to represent 1 likewise symbolises its linear nature)!