Tuesday, June 3, 2014

More on Randomness and Order (5)

We have seen that randomness and order are complementary notions that can only be appropriately understood in a dynamic interactive manner. So they are not capable of any absolute definition in isolation, but are necessarily of a relative approximate nature (that implicitly imply each other).

We also saw that when approached analytically (i.e. through single independent frames of reference) from the cardinal and ordinal perspective respectively, what is random and what is ordered in each case are the reverse of each other.

So in cardinal terms, the behaviour of each individual prime appears highly random, whereas the behaviour of the overall collection of primes (withing the number system) appears highly ordered; then in ordinal terms, the behaviour of each natural number member (of a prime group) appears highly ordered, whereas the overall collection of primes (representing groups of individual members) appears highly random.

Therefore when we bring both reference frames together (i.e. cardinal and ordinal) holistically through the simultaneous recognition of interdependent reference frames, the very notions of randomness and order are rendered paradoxical in terms of each other.

This holistic recognition is then vital for the direct appreciation of the true nature of the zeta zeros (Zeta 1 and Zeta 2).

Thus from both perspectives, the zeta zeros imply the - seemingly paradoxical - situation where the notions of randomness and order approximate perfect identity (in relative terms) with respect to the number system.

In the case of the Riemann (Zeta 1) zeros , in terms of the number system as a whole (uniquely derived as the product of primes) we have an unlimited series of points (through 1/2 ) on an imaginary number scale (with matching positive and negative values) where each point  represents approximation to  a state that is equally random and ordered in number terms.

Then in the case of the Zeta 2 zeros, within each prime, we have an arrangement on natural number members in ordinal terms, where each individual member can be quantitatively chosen at random, while the overall qualitative relationship between the various members maintains a perfect order.

Thus the double paradox as between notion of randomness and order in this case arise from the two-way relationship as between the corresponding fundamental polarities of whole and part in both their individual and collective manifestations.

So there are a number of ways of expressing this holistic interdependence of the number system (expressed through the zeta zeros).

As we have seen the zeros represent the two-way approximate identity of the notion of randomness and order in dynamic relative terms.
Equally we could express this as the two-way identity of independent and interdependent, of cardinal and ordinal, of analytic and holistic, of quantitative and qualitative notions with respect to the number system.

There is also another complementary aspect involved in this relationship that is of equal importance.

As well as necessarily involving the dynamic two-way relationship as between whole and part, the zeta zeros equally entail the two-way relationship as between external and internal polarities.

In other words our experience of numbers in an objective external manner necessarily implies corresponding mental constructs that are - relatively - of an internal nature.

Thus strictly speaking the absolute notion of number - as some timeless entity existing in abstract mathematical space - is without foundation, for the very assertion of this position is not possible in the absence of corresponding psychological constructs of an internal nature.

Therefore all mathematical understanding necessarily takes place in a dynamic interactive context, entailing both objective notions (as external) and subjective mental notions (as - relatively - internal).

We cannot therefore have objective knowledge of number in the absence of corresponding mental interpretation (both of which are necessarily of a dynamic relative nature).

Crucially therefore, the very belief in an abstract unchanging world of number, reflects an absolute type  interpretation that is strictly untenable in terms of the actual dynamics of mathematical understanding.

Of course there is considerable value in exploration of the quantitative extreme where number relationships correspond well to such rigid assumptions. But rather like Newtonian Physics this represents an approximation that is invalidated at a deeper level of mathematical experience.
And just as Quantum Mechanics shows up the shortcomings of so many Newtonian assumptions, likewise the zeta zeros shows up the severe shortcomings of the the very paradigm that underlines Conventional Mathematics (as we know it).

Quite simply, proper appreciation of the zeta zeros will require a greatly enlarged mathematical paradigm.

This will entail (at a minimum) three distinct areas.

1) Conventional (Type 1) Mathematics. This represents the merely linear (1-dimensional) quantitative approach in analytic terms to mathematical relationships, based on single independent frames of reference.

2) Holistic (Type 2) Mathematics. This represents the greatly unrecognised circular qualitative approach in holistic terms to mathematical relationships based on multiple frames of reference that are simultaneously appreciated. Though this qualitative aspects is as equally important as the quantitative, we have not even begun yet to form an appreciation of its enormous potential in opening up in a completely new manner, vast fields of unexplored mathematical territory. Indeed its very existence is still resolutely denied at the formal level of accepted mathematical inquiry!

3) Comprehensive (Type 3) Mathematics. This entails the balanced dynamic interpenetration of  both Conventional (Type 1) and Holistic (Type 2) aspects of mathematical understanding.

In the most preliminary manner, I have been attempting to give some consistent indication of what this Type 3 approach might entail in all my blog entries on "The Riemann Hypothesis".

So I say again with considerable conviction that we are fast reaching by far the greatest watershed in our intellectual history, where the very nature of Mathematics will undergo profound change and with it, appreciation of all the sciences, the natural environment and our social relationships with each other.

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