Zeta Zeros and the Changing Nature of Number (4)

We have seen that number has in fact two distinct meanings:

1) The Type 1 (cardinal) interpretation where each number is composed of independent units.

2) The Type 2 (ordinal) interpretation where each number is composed of interdependent i.e. related units.

Now this has a crucial bearing on the nature of the primes.

From the Type 1 (cardinal) perspective, the primes are considered as the unique building blocks from which the natural numbers are formed.

However from the Type 2 (ordinal) perspective, each prime is uniquely defined by its natural number members.

So in the former (Type 1) case, 5 as a prime, constitutes one of the essential building blocks from which the natural numbers are derived.

However in the latter (Type 2) case, 5 is already defined by its 1st, 2nd, 3rd, 4th and 5th natural number members.

So already included in this notion of a prime is the composite natural number 4!

Now because Conventional Mathematics is defined exclusively in Type 1 terms, with ordinal notions - as I have carefully explained - in effect reduced to cardinal, this issue of the necessary two-way interdependence of the primes and natural numbers is completely overlooked!

Because cardinal identity is solely considered in a quantitative manner, an utterly misleading picture emerges, whereby the relationship as between primes and natural numbers is considered to be solely one-way (with the natural numbers unambiguously determined by the primes)


So the first step in moving to a truly coherent dynamic interactive nature of the number system, is to recognise the equal importance of both the Type 1 and Type 2 aspects.

The major issue that then arises is that of mutual conversion of each aspect in terms of the other.

So from one perspective, how do we convert the Type 2 (qualitative) aspect in a consistent Type 1 (quantitative) manner?

Equally from the complementary perspective, how do we convert the Type 1 (quantitative) aspect in a consistent Type 2 (qualitative) manner?

And this is where the Zeta 2 zeros are so important.

Now in general terms for any prime number t, the Zeta 2 zeros are given as the solutions to the finite equation,

1  +  s¹  +  s²  +  s³  + ... +  s^{t – 1}  = 0

These zeros express the truly relative i.e. circular nature of ordinal positions as the t roots of 1, (excluding the default case of 1 where ordinal becomes inseparable from cardinal identity in an absolute manner).

Therefore in the case of our example of the prime 5, the solutions to

1 +  s¹ +  s²  +  s³  +  s⁴  = 0,

express in a  Type 1  quantitative manner, the notions of 1st, 2nd, 3rd and 4th in the context of 5 members.

Again these four solutions are given as .309 + .951 i,  – .809 + .588 i, – .809 + .588 i and .309  – 951 i (correct to 3 decimal places).

Then we combine these 4 values with the default value of 1 (representing 5th in the context of 5) the total sum = 0, expressing the fact, that as these values are expressing qualitative notions of relative interdependence, their collective sum has no quantitative value.

So  (.309 + .951 i) + ( – .809 + .588 i) + ( – .809 + .588 i) + (.309  – 951 i)  + 1 = 0

And remember again these quantitative values represent the conversion of the qualitative ordinal notions of,

1st + 2nd + 3rd + 4th + 5th = 0


We also have the complementary problem of converting our standard Type 1 notion of number consistently in a Type 2 manner!

This in fact represents the same set of values that we have already obtained. However it now requires that these values (the five roots) be understood in a true holistic fashion. This requires moving from a 1-dimensional to 5-dimensional appreciation, which requires a specialisation of intuitive ability that is yet not yet remotely recognised (certainly within Mathematics)! 

However again it is perhaps possible to express what is required with reference to the simplest case.

In other words how do we convert the standard (Type 1) linear quantitative notion of  2 in a coherent Type 2 qualitative manner?

So we start with the 2 roots of 1 i.e. + 1 and – 1. However the task is now to understand these in a genuine holistic qualitative manner. This in turn requires authentic 2-dimensional appreciation which entails the ability to see number reality as representing the interaction of opposite poles, that are positive and negative in relation to each other.

And as I have explained many times before, this relates to the manner in which number "objects" (as external) continually interact in experience with mental constructs (as - relatively - internal).

Thus we no longer view mathematical reality (in 2-dimensional) terms as an abstract objective world (independent of the enquirer) but rather as a dynamic interactive process entailing both external and internal poles that are + 1 and – 1 with respect to each other.

So in raising i.e. transforming through intuitive insight the qualitative nature of these two roots (in a true 2-dimensional fashion) we obtain 1¹ and  1² representing 1st and 2nd (of 2 dimensions).

When we can additionally combine the two-way interactive nature of whole and part, we now have 4 polarities (external/inetrnal and whole/part) that can be viewed like the 4 directions of a compass.

So in general terms all "higher" dimensional appreciation relates to a distinctive manner in which one configures experience with respect to these 4 co-ordinates. However obtaining specialisation with respect to appreciation of such interaction will require considerable evolution in our (unconscious) intuitive abilities that have not yet been remotely tapped!


However the truly important thing to appreciate at this stage is the fundamntal two-way role of the Zeta 2 zeros.

Put simply, they enable the seamless consistent conversion as between both aspects of the primes.

Thus again from one perspective, we are able to convert the Type 2 (ordinal) natural number members of each prime in a consistent Type 1 quantitative manner.

Then equally from the other perspective, we are able to convert the Type 1 (cardinal) notion of each prime in a consistent Type 2 qualitative manner, where its true dimensional nature (as related ordinal members) can be readily appreciated.

However none of this can have any resonance, while we insist on interpreting mathematical reality in the present greatly reduced manner (that solely recognises the Type 1 quantitative aspect).

Clearly a massive revolution is now required with respect to mathematical perspective, for at present through our collective blindness, we are completely misinterpreting the true nature of number, and thereby just about everything else in Mathematics and Science.