Addition and Multiplication as Complementary Facets of Same Phenomenon

In the last blog entry I demonstrated how the Euler Product Formula can be expressed with respect to either the Zeta 1 or Zeta 2 Functions indicating in fact that these represent complementary expressions.

Once again The Euler Product Formula beautifully demonstrates the relationship as between addition (with respect to the natural numbers) and multiplication with respect to the primes.

From a deeper level of appreciation it expresses the complementary relationship as between the quantitative and qualitative aspects of the number system, which are dynamically mediated through the interaction of the primes with the natural numbers (and the natural numbers with the primes) respectively.

Each prime number can be uniquely represented in two ways.

(i) through addition as the sum of its individual members (in quantitative terms). Thus from this perspective the prime number 3 = 1 + 1 + 1. This represents the Type 1 definition of 3 which more fully is represented as 3¹.

(ii) through multiplication as the product of its individual members (in qualitative terms). Thus from this perspective the prime number 3 = 1 * 1 * 1. This represents the Type 2 definition of 3 which is more fully represented (in an inverse manner) as 1³.

Now it is important to point out in this context that what represents multiplication from a Type 1 perspective represents addition from the Type 2 (and vice versa).

So 1 * 1 * 1 = 1³ = 1^{1 + 1 + 1}

So the Type 1 and Type 2 aspects of the number system again dynamically represent - in the interaction of both cardinal and ordinal aspects - how the notion of number is always quantitative as to qualitative (and qualitative as to quantitative) respectively.

Thus both Type 1 and Type 2 aspects of the number system are defined in their pure form with respect to the default number 1. In the case of the Type 1 aspect, 1 represents the (default) dimensional value, to which the varying base or ground value is raised; with the Type 2, 1 represents the (default) base or ground value which is raised to a dimensional value that varies.

In this way we can clearly distinguish the quantitative aspect (with respect to the Type 1) and the qualitative aspect (with respect to the Type 2) respectively.

Of course when we remove  this restriction with respect to base and dimensional values remaining fixed respectively as 1, both quantitative and qualitative transformation is involved with respect to any number expression.

So 3² and 2³, which again are the inverse of each other (with respect to both base and dimensional values) from a dynamic interactive perspective, involve both quantitative and qualitative transformation.

And as both the Zeta 1 and Zeta 2 Functions involve number expressions of this kind clearly both quantitative and qualitative aspects (from complementary perspectives) are involved with respect to both Functions! And this is why the Euler Product Formula - entailing the relationship as between addition and multiplication - as a corresponding relationship as between the natural and prime numbers - can be defined  in the context of both the Zeta 1 and Zeta 2 expressions!


Now we can attempt to convert the pure (linear) qualitative Type 2 notion of number in an indirect (circular) quantitative manner.

So for example the  Type 2 dimensional notion of 3 represents in ordinal terms its 1st, 2nd and 3rd members which inversely are associated in quantitative terms with the corresponding 3 roots of 1¹, 1² and 1³ respectively.

So each of these roots now possesses an individual quantitative identity (on the circle of unit radius in the complex plane). However their holistic qualitative identity is now expressed through the addition of the 3 root values = 0.

Thus whereas from the linear number perspective, addition of the individual units of 3 relates to the cardinal (quantitative) aspect of number, from the circular perspective, addition of the 3 roots of 1, which indirectly represents the ordinal identity of these three individual members, relates to the holistic (qualitative) aspect of number.

Therefore to represent the ordinal aspect of number in cardinal terms we must switch from a linear to a circular (Type 2) representation of number; likewise to represents the cardinal in ordinal terms we must switch from a linear to a circular (Type 1) representation of number!

The same inversion applies to the multiplication aspect. From the Type 1 perspective, pure multiplication such as 1¹ * 1¹ * 1¹ represents the qualitative aspect of number transformation.

However when we multiply the three roots of 1 (as the corresponding circular expression) the answer is always 1 (with 1 the very symbol of its linear counterpart). So multiplication = 1 (for prime roots except where 2 is a factor) now expresses a quantitative holistic relationship in circular terms, whereas it was qualitative in linear terms.

In like manner the addition of roots = 0, expressed a qualitative holistic relationship in circular terms (where it was quantitative from a linear perspective).

Thus when one recognisies both Type 1 and Type 2 aspects of the number system (with their related Zeta 1 and Zeta 2 Functions) both addition and multiplication can be expressed in a complementary fashion, where addition in the context of Type 2 represents multiplication in the context of Type 1 (and vice versa).

Likewise what represents multiplication in the context of Type 2 represents exponentiation in the context of Type 1 (and vice versa).

Once again the relationship between addition (with respect to the natural numbers) and multiplication (with respect to the primes) as embodied in the Euler Product Formula, can be expressed in either Type 1 of Type 2 terms through the Zeta 1 and Zeta 2 Functions respectively.

Now both of these, like a left turn at a crossroads, enjoy a - relatively - unambiguous interpretation within their respective reference frames (as the direction from which the crossroads is approached).

However when we attempt to properly combine both interpretations as interdependent in Type 3 terms through the Zeta 3 Function (representing their combined dynamic interaction) deep paradox results, as what is addition from one context in now multiplication from the opposite; and what is multiplication from one context is likewise addition from the opposite.

This again is exactly analogous to the situation at a crossroads where - simultaneously applying both reference frames - a left turn is equally a right; and a right turn is equally a left.

So in Type 3 terms, addition and multiplication are understood as but complementary facets of the same phenomenon and as ultimately identical in an ineffable manner.
We could of course equally say that the quantitative (cardinal) and qualitative (ordinal) aspects of number in dynamic terms represent complementary facets of the same phenomenon which again are also identical in an ineffable manner.

And here lies the true nature of number that resides ultimately in total mystery! So from this perspective, the phenomenal forms that become manifest in space and time - through the relative separation of both quantitative and qualitative aspects - can be seen as but the alluring veils though which number seeks to hide its most intimate secrets.