Do Numbers Evolve? (7)

In all that has been said so far, it is plain that we have in fact two distinct notions of number, which dynamically interact in experience.

For example the number 3 can be given an independent (cardinal) existence or alternatively an interdependent (ordinal) type definition.

Thus, from the first perspective, 3 is viewed in quantitative terms, which can be expressed in terms of its component units as 1 + 1 + 1 .So the independent (absolute) units are all homogeneous in nature (thereby lacking any qualitative distinction).

However 3 can also be given an interdependent ordinal type definition, which is expressed in terms of its component members as 1^st + 2^nd + 3^rd. Thus the units here are of an interdependent (relative) nature (thereby lacking quantitative distinction).

I have dealt with this latter ordinal (Type 2) notion of number in my previous entries. This established the vitally important fact that associated with it is a simple function (Zeta 2) with a corresponding set of zeros. These in effect show how to convert Type 2 qualitative type meaning in a (reduced) Type 1 quantitative manner. So the various prime roots of 1 (excluding 1) can thereby be indirectly used to uniquely express its various ordinal members.

So again the 3 roots of 1 can be used to express the unique ordinal nature of 1^st 2^nd and 3^rd members (in the context of 3). However since all ordinal type relationships necessarily entail the fixing of position with respect to one member in an independent fashion, one solution i.e. 1 is thereby trivial in this respect!  

Once again therefore the importance of the (unrecognised) Zeta 2 zeros is that they enable the unique conversion of the  qualitative (Type 2) ordinal nature of number in an indirect quantitative (Type 1) manner.

It is very important to appreciate this fact, as the Zeta 1 (Riemann) zeros can be then shown to play a direct complementary role with respect to number.

The general Zeta 2 equation can be expressed as follows:

ζ₂(s) =  1 + s¹  + s²  + s³  +….. + s^{t – 1}   (with t prime). However ultimately this can be extended to all natural numbers.

Now this can equally be written as

ζ₂(s) =  1 + s^{– 1}  + s^{– 2}  + s^{– 3}  +….. + s^{– (t – 1)}  

Therefore the zeros for this function are given as

ζ₂(s) =  1 + s^{– 1}  + s^{– 2}  + s^{– 3}  +….. + s^{– (t – 1)}   = 0.

The corresponding Zeta 1 equation, i.e. the Riemann zeta function can be expressed by the infinite equation.

ζ₁(s) =   1^{– s}  + 2^{–  s} + 3^{– s}  + 4^{– s}   +…..

And the zeros for this function are given by

ζ₁(s) =   1^{– s}  + 2^{–  s} + 3^{– s}  + 4^{– s}   +…..  = 0.

Notice the complementarity as between both expressions!

Whereas the first expression represents a finite, the second represents an infinite series of terms.

Then whereas the natural numbers 1, 2, 3, …. represent dimensional powers in the Zeta 2, they represent base quantities with respect to the Zeta 1 and in reverse fashion whereas the natural numbers represent base quantities with respect to the Zeta 1, they represent dimensional powers with respect to the Zeta 2.

Now in the case of the Riemann (Zeta 1) function we start with the notion of numbers, expressed as the unique product of primes (representing base quantities) as independent entities.

However this begs the obvious question of how to express the corresponding interdependence of these numbers in their consistent interaction with respect to the overall number system!

In other words, in dynamic interactive terms, quantitative independence (with respect to individual numbers) implies the opposite notion of qualitative interdependence (with respect to their overall relationship with each other).

Thus the Riemann (Zeta 1) zeros represent an infinite set of paired numbers that uniquely expresses in Type 2 fashion the qualitative interdependent nature of the number system.

So just as the Zeta 2 zeros, as we have seen convert (Type 2) qualitative ordinal type natural number notions in a (Type 1) quantitative manner, in reverse fashion, the Zeta 1 zeros convert (Type 1) quantitative cardinal type prime notions in a (Type 2) qualitative manner.

Indeed we could rightly say - though Conventional Mathematics has no way of adequately interpreting this notion - that the Zeta 1 zeros express the collective ordinal nature of the primes (with respect to the natural number system).

Indeed in even simpler terms, the Zeta 1 zeros express the holistic basis of the cardinal number system.