There are two ways in which we can look at the bi-directional relationship between the primes and natural numbers, which we can usefully refer to as the internal and external approaches respectively.
We will look here in this blog entry at the internal approach.
In accordance with this approach, when we look within each prime, we find that it can be expressed in two ways, which in dynamic interactive terms are complementary.
First we have the cardinal definition where each prime is defined as composed of homogeneous units in a merely quantitative manner.
Therefore in our much used example 3 (as prime) = 1 + 1 + 1.
Then secondly, we have the alternative ordinal definition, where each prime is defined by a unique set of ordinal members in a qualitative manner.
Therefore in this context 3 = 1st + 2nd + 3rd.
Both of these, considered as separate, constitute analytic interpretation that appears unambiguous representing in fact equivalent statements..
However when considered holistically in dynamic interactive terms as complementary, direct paradox arises.
(This again parallels the example of a crossroads where both left and right turns at the crossroads appear unambiguous when approached from just one direction (N or S); however when when both N and S directions of approach are viewed simultaneously, left and right turns are now rendered paradoxical).
In the first (cardinal) case each prime is considered as an independent "building block" of the natural number system; however in the second (ordinal) case, each prime is already composed of an interdependent set of ordinal natural number members! Thus from this dynamic perspective, each prime and its ordinal set of natural numbers are mutually interdependent with each other in a manner ultimately approaching total synchronicity
Now I have stated that the two complementary perspectives - with respect to the cardinal and ordinal nature of the primes - entail the corresponding complementarity of the cognitive and affective aspects of understanding.
Thus when the cardinal numbers are understood in conventional cognitive terms, true ordinal recognition (i.e. that is not of a reduced nature) entails corresponding affective appreciation
And then when reference frames are reversed, when a cardinal number is understood in affective terms (where the individual uniqueness of the number is recognised) then ordinal appreciation entails corresponding cognitive appreciation.
The conventional quantitative notion of 3 is properly expressed (in Type 1 terms) as 31.
The corresponding qualitative notion of 3 is then expressed (in Type 2 terms) as 13.
Holistic appreciation then entails bringing these two aspects together whereby the qualitative dimensional notion of 3 indirectly finds its conventional (1-dimensional) expression. This then entails obtaining the 3 roots of 1 (i.e 11/3, 12/3 and 13/3) respectively.
In this way we can indirectly convert qualitative Type 2 notions in a quantitative manner.
In other words the ordinal notion of 1st (of 3) is given as – .5 + .866i; the ordinal notion of 2nd (of 3) is then given as – .5 – .866i and the ordinal notion of 3rd (of 3) as given as 1. And these three positions are fully interchangeable with each other depending on the criterion chosen for (relative) ordinal rankings..
We can now perhaps appreciate precisely how ordinal notions are reduced in conventional mathematical terms. When we speak of 1st, its meaning is fixed as the last of a group of 1 item
(i.e. 11/1). Then when we speak of 2nd, its meaning is fixed as the last of a group of two items
(i.e. 12/2). and then when we speak of 3rd, its meaning is fixed as the last member of 3 items (i.e. 13/3).
So here 1st, 2nd and 3rd in turn are reduced to 1 (as the last fixed unit in a series).
Therefore, in this context 1st + 2nd + 3rd in ordinal terms becomes indistinguishable from 1 + 1 + 1 (= 3) in a cardinal manner.
However the true holistic meaning of 1st, 2nd and 3rd have a potential (interchangeable) rather than actual (fixed) meaning.
Therefore, when attempting to rank number members, each in turn can be 1st, 2nd or 3rd (depending on the relative context). So what might for example be ranked 1st (according to one criterion, might be ranked 2nd according to another and then 3rd with respect to yet some other criterion). And this likewise applies to the other two members.
So the true relative interdependence of the 3 members in ordinal terms is reflected by the fact that each can be ranked 1st, 2nd or 3rd (depending on arbitrary context).
And this reflects the holistic meaning of the number 3, where each unit member potentially shares all three possible positions (1st, 2nd and 3rd). And this qualitative nature of "threeness" is directly appreciated in an intuitive manner that then indirectly can be expressed in a circular rational fashion through the 3 roots of 1.
So the quality of 3 (i.e. as "threeness") arises when one intuitively appreciates the interdependence - rather than independence - of the three unit members of a number This entails - as we have seen - that each member shares the potential ordinal quality of 1st, 2nd and 3rd respectively.
With such ordinal rankings, there is always one less degree of freedom that the number of units involved as when s – 1 positions are ranked the remaining position is automatically determined.
We can refer to this then as a trivial, with the other positions constituting non-trivial solutions!
So if we refer to the case where the sth root = 1, this concurs with analytic rather than holistic appreciation. Therefore to isolate the roots - indirectly corresponding to holistic understanding - we divide sn –
1 = 0 by s – 1 = 0 or in preferred form,
1 – sn = 0 by 1 – s = 0, to obtain
1 + s + s2 + ...... + sn – 1 = 0 (where initially s is prime).
Again it can be easily seen that as the s –
1 roots (other than 1) are unique for each prime, this means that the non-trivial natural number ordinal positions are thereby likewise unique for every prime.
This is what I refer to as the Zeta 2 function which complements in an internal fashion the better known Zeta 1 function (Riemann zeta function) that expresses the external relationship as between the primes and zeros.
In other words the very purpose of the Zeta 2 zeros (as solutions to this finite equation) is to express the holistic notion of the qualitative interdependence of the unique set of natural number ordinal members of each prime, indirectly in a quantitative manner.
In this way the Zeta 2 zeros give quantitative expression to the qualitative notion of the uniqueness of all non-trivial ordinal positions (1st, 2nd, 3rd,........., s –
1th) with respect to each prime.
So in the case of 3 (which of course is prime),
1 + s + s2 = 0.
Therefore the two solutions for s, i.e. – .5 + .866i and – .5 – .866i give unique quantitative expression to the qualitative notions of 1st and 2nd (in the context of 3 members).
In this manner, therefore unique quantitative expression can be indirectly provided for the qualitative nature of all the (non-trivial) natural number ordinal members of each prime.