Complementary Connections Explained (1)

In yesterday’s blog entry, I illustrated the relationship as between each individual term (in the product over primes expression) with the collective nature of terms (in the corresponding sum over natural numbers expression).

And the crucial point here is that from the appropriate dynamic perspective that a complementary relationship exists so that what is interpreted in an quantitative manner with respect to each individual term (related to a specific prime), is then interpreted in a qualitative manner with respect to the collection of terms (where this specific prime operates as a shared common factor).

However as always with such dynamic complementary relationships reference frames can be switched, so that equally each individual term (with respect to the product over primes expression) can be viewed in a qualitative manner. Then when this is the case the corresponding interpretation with respect to the collection of natural numbers where the prime operates as a factor is in an quantitative manner.

So let me now illustrate this point at greater length with respect to the example yesterday of ζ(2), where we focussed on the first individual term in the RHS product over primes expression i.e. 4/3, which is based on the prime number “2”.

Now as we have seen in other entries, 2 as a prime can be given both analytic (quantitative) and holistic (qualitative) interpretations, which in the dynamics of understanding keep interchanging with each other.

Thus when we define 2 in cardinal terms as composed of independent homogeneous units (that are thereby devoid of qualitative distinction) i.e. 2 = 1 + 1, we thereby interpret this prime in an analytic (quantitative) manner.

However when we equally recognise that 2 is necessarily composed of both a 1^st and 2^nd unit, which are interchangeable with each other i.e. where each unit now is given a distinct qualitative identity in ordinal terms, then we interpret this prime in a corresponding holistic (qualitative) manner.

Once again, from the dynamic interactive perspective, both of these interpretations are relative, so that the analytic (quantitative) aspect of “2” cannot have any strict meaning in the absence of the corresponding holistic (qualitative) aspect; equally the holistic (qualitative) aspect of “2” cannot have any strict meaning in the absence of the corresponding analytic (quantitative) aspect. 

And of course this dynamic interactive nature equally applies to all other primes, where both analytic (quantitative) and holistic (qualitative) aspects are necessarily related to each other in a relative fashion.

So again referring to the RHS product over primes expression, we can start by defining each prime with respect to its holistic (qualitative) aspect where component units are understood as interdependent (and fully interchangeable) with each other. 

So from this perspective the prime number “2” (to which the first individual term relates) can be given a holistic (qualitative) interpretation.

Then with respect to the corresponding sum over the integers expression, 2 as a factor of all even terms is given a complementary analytic (quantitative) interpretation.

Clearly therefore, though in the earlier case “2” is viewed in holistic terms as a common factor (of all even numbers) it equally enjoys a relatively independent identity (where it can be viewed as separate from other factors). And it is this latter quantitative interpretation of a factor that is now emphasised in this context.

So overall each individual term (with respect to the product over primes expression) related to a specific prime, can be given both an analytic (quantitative) and holistic (qualitative) interpretation.

Likewise the collection of terms (with respect to the sum over integers expression) which includes a specific prime as factor, can be given both analytic (quantitative) and holistic (qualitative) interpretations.

But once again the key requirement in correct dynamic interactive terms is that the relationship between both expressions is of a complementary nature (with analytic complementing holistic and holistic complementing analytic interpretations respectively).

And in this latter case where each individual term (with respect to the product over primes expression) is given a holistic (qualitative) interpretation, then the collective product of these terms is now given a complementary analytic (quantitative) interpretation and vice versa so that when the terms (with respect to the sum over integers expression) that share a specific prime as factor is given a holistic (qualitative) interpretation, then the additive sum of terms is - relatively - of an analytic (quantitative) nature.

So in dynamic interactive terms, as we move from collective (whole) to an individual (part) identity or in reverse from an individual (part) to a collective (whole) identity, the very nature of number likewise keeps switching as between both its analytic (quantitative) and holistic (qualitative) aspects, in a relative complementary manner.

Thus the notion that number somehow can be properly interpreted in an absolute - merely quantitative - manner is utterly misleading and only valid as an important special case in a restricted limited context.

And it is this misleading notion of number - which ultimately is not fit for purpose - that has dominated the development of Mathematics over several millennia.