Clarifying Analytic and Holistic Interpretations of "2" (13)

With respect to the Zeta 2 functions we have already looked at the composition of each prime.

Again, using the Peano based additive approach to number, from the (Type 1) cardinal perspective, each prime t, is seemingly made up of  independent homogeneous units (in quantitative terms); however from the (Type 2) ordinal perspective, the composition of these units as 1st, 2nd, 3rd,....,t^th respectively, is unique for each prime (in a corresponding qualitative manner).

Then indirectly these unique ordinal units can be indirectly expressed, in a circular quantitative manner, through the t roots of 1 (excluding 1 which is common in all cases).

The Zeta 2 zeros, representing the unique values for s can then be expressed as the solutions for

ζ₂(s) = 1 + s¹ + s² +  ^…. +  s^{t – 1}  = 0 (where t is prime).

Therefore, the key role of the Zeta 2 zeros, in this context is that they provide the means of converting ordinal notions of a qualitative nature (indirectly) in a quantitative manner.  

And of course once again, we have the seeming paradox that the ordinal nature of each prime is expressed through an orderly sequence of natural numbers!

However we can equally approach number from the multiplication approach, where, again from the (Type 1) perspective, each (composite) natural number is expressed as a unique combination of primes, in a cardinal manner.

So for example 6 = 2 * 3.

However just as before we found that a (hidden) Type 2 explanation (of an inherently qualitative nature) existed, this is equally true in this case.

In other words, when two (or more) primes are multiplied, a qualitative - as well as quantitative - transformation takes place in the units involved. So the Type 2 aspect of interpretation now relates to this (hidden) qualitative aspect of number transformation (arising from multiplication). 

Now I have already used the geometrical representation of number multiplication to hint at this qualitative transformation. 

So once again 2 * 3 can be represented as a 2-dimensional figure comprising 6 square units.

Therefore we can now readily appreciate that while the quantitative result of multiplying 2 * 3 is indeed 6, that this result now represents a transformation from 1-dimensional units (with respect to 2 and 3 separately) to 2-dimensional units (with respect to the combined operation).

Thus again, whereas the quantitative transformation in the units can be directly attributed to their separate independent nature, the qualitative transformation - by contrast - relates to the corresponding interdependent nature of units involved.

And when one reflects on it, the very means by which can recognise 2 and 3 as the (separate) sides of a rectangle, is the corresponding ability to appreciate an orderly relationship in a 2-dimensional manner, as between these respective sides (where they are necessarily understood as interdependent with each other). 

Therefore, whenever we multiply the primes to obtain composite natural numbers, both a quantitative (Type 1) and qualitative (Type 2) transformation is involved.

There is perhaps an even easier way of appreciating this inevitable interdependence that arises from the multiplication of numbers.

The very essence of the manner we view the primes (in a cardinal manner) is their - seemingly - absolute independent nature. This is why the primes are customarily referred to as the "atoms" of the natural number system.

So from this perspective, 2 and 3 as primes are indeed independent "building blocks".

However when we multiply these numbers, they no longer exist in an absolute independent manner (but rather in relationship with each other).

Therefore though 2 and 3 are indeed separate and independent in their cardinal prime state, when multiplied together (i.e. 2 * 3) both numbers now exist as part of a new relationship (where they can no longer be appreciated as merely independent).

Put another way, both 2 and 3 obtain a new qualitative resonance through their unique combination in generating the number 6.

And now we come to the crucial insight! If in fact the primes acquire a qualitative interdependence (though their unique relationship with each other) then they cannot in fact be independent in an absolute manner!

Rather both their (quantitative) independence as individual primes and their (qualitative) interdependence, through their unique relationships in generating the (composite) natural numbers, are thereby necessarily of a relative nature.

The implications of this - though in one way obvious - are quite startling, for it entails that just as the (composite) natural numbers are dependent on the collective nature of the primes, equally the individual primes are dependent on the (composite) natural numbers for their existence.

The great fallacy, perpetuated in conventional mathematical interpretation, is that somehow we can start with the primes as the absolute "building blocks" of the natural number system.

This in turn reflects the merely reduced quantitative empahsis on number that defines such understanding.

However properly understood - in a dynamic interactive manner - both the primes and natural numbers mutually depend on each other for their existence. 
So in fact a paradoxical type relationship exists whereby ultimately they perfectly complement each other (both quantitatively and qualitatively) in an ineffable manner.

However to properly appreciate this, the quantitative (Type 1) and qualitative (Type 2) aspects of both the primes and natural numbers must be emphasised in equal balance.

Clarifying Analytic and Holistic Interpretations of "2" (12)

Before moving on further, I wish to emphasise how closely the Zeta 2 function is linked with the famed Euler Identity.

Basically this (finite) function provides the (t – 1) roots of 1 (with the default root of 1 excluded).

Then the actual solutions for these roots are estimated through the Euler Identity.

So e^iπ  = – 1.

Therefore e^2iπ  =  1 (in what I refer to as the fundamental Euler Identity(.

Then according to De Moivre's Theorem e^ix =  cos x + i sin x.

Therefore when x = 2π (radians),

e^2iπ =  cos 2π + i sin 2π  =  cos 360⁰ + i sin 360⁰


There is a direct connection as between the fundamental Euler Identity and the Type 2 aspect of the number system.

So e^2iπ = 1¹ and in more general terms,

e^2kiπ = 1^k = cos k(360)⁰ + i sin k(360⁾⁰

Therefore for the two roots of 1

s² = 1¹ and s² = 1² respectively, 

so s = 1^1/2 and s = 1^2/2 = 1.

So eliminating the default root 1,

s = 1^1/2 represents the solution for the general Zeta 2 function,

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

Thus, as in this case, t = 2,

 1 +  s¹  = 0.

So s, i.e.  s¹ = 1^1/2  = cos (360/2)⁰ + i sin (360/2)⁰  

= cos 180⁰ + i sin 180⁰ =  – 1 + i (0)  =  – 1.

So once again, this represents the first - and in many ways the most important - of the Zeta 2 zeros.

And once again the importance of these zeros is that it enables us to give an indirect quantitative expression to ordinal notions of a strictly qualitative nature.

So 1st (in the context of 2 units) is indirectly expressed as  – 1 (which thereby literally negates or excludes  the 2nd member).

Now it would not need the Euler Identity (through De Moivre's Theorem) to calculate the non-trivial root in this case.

However it quickly becomes extremely difficult to do so (in the absence of De Moivre's Theorem).

So for example if we were to satisfy the Zeta 2 function for - say -  t = 37, we would follow the same general procedure setting k (with respect to the general expression of De Moivre's Theorem) = 1/37, 2/37, 3/37,....., 36/37 whereby we could thereby calculate the 36 zeros (in this case) for the Zeta 2 function. 

And these values in turn would then give indirect quantitative expression to the notions of 1st, 2nd, 3rd,......., 36th (in the context of 37 unit members).


Thus the Euler Identity (especially in the form of the fundamental Euler Identity) is extremely important because of its direct connection with the Type 2 aspect of the number system.

In this way, it is directly linked to the qualitative - as opposed to the quantitative - nature of number.

An though respecting the equal importance of the Type 2 aspect (with the Type 1) we can properly clarify many issues of interpretation, that are not properly dealt with in conventional mathematical terms.

For example in conventional mathematical terms,

e^2iπ = 1;
Also, e ^{– 2iπ} = 1;
Finally, e⁰ = 1.

This might then seem to suggest that 2iπ  = – 2iπ  = 0, which is clearly nonsensical from a quantitative point of view!

However, the clue to clarification of this seeming mathematical dilemma is to define the RHS of the equations more precisely in terms of the Type 2 aspect of the number system.

So e^2iπ = 1¹,

e ^{– 2iπ} = 1 ^{– 1}, and

e⁰ = e^2iπ  * e ^{– 2iπ}  = 1¹ * 1 ^{– 1} = 1⁰.

So in fact each of the three equations is associated with 3 distinct results when expressed in terms of the Type 2 aspect of the number system!

Finally, it is also customary in conventional mathematical terms to maintain that

e^2iπ  = e^4iπ = e^6iπ =.... e^2kiπ  = 1, where k = 1,2,3,.....

However, properly interpreted, these all relate to distinct numbers with respect to the Type 2 aspect of the number system.

So e^2iπ = 1¹,   e^4iπ = 1²,  e^6iπ = 1³ and  e^2kiπ  = 1^k.

Likewise, as we have seen, the various roots of 1 for example, refer to distinct aspects of the Type 2 aspect of the number system.

So, again with respect to the two roots of 1, 

+ 1 corresponds to 1^2/2 = 1¹, and – 1 corresponds to 1^1/2  respectively.