Tuesday, May 7, 2013

Mathematics at the Crossroads

In the last blog entry, I highlighted once again how in our experience of number, two complementary interpretations are necessarily intertwined, which are quantitative (cardinal) and qualitative (ordinal) with respect to each other.

So when properly interpreted, the inherent nature of number is dynamic and interactive. Thus, from the quantitative perspective numbers have a relatively distinct independent identity (whereby they can be clearly separated from other numbers). However from the equally important qualitative perspective, numbers have an overlapping interdependent identity, whereby they can be coherently placed in an ordered relationship with each other.

In this dynamic interactive sense number behaviour is necessarily conditioned by polar opposites i.e. quantitative and qualitative of a complementary nature.

Equally, number behaviour is dynamically conditioned by external and internal polarities that are - relatively - objective and subjective with respect to each other.

In other words the external individual "object" of  an  number has no meaning in the absence of a corresponding mental perception that is - relatively - of an internal nature. Likewise the general "object" class of number as external, has no meaning in the absence of the corresponding mental concept of number as - relatively - internal.  

So once again in a dynamic interactive sense we cannot divorce the external reality of number  from corresponding mental interpretation . Number behaviour therefore has no absolute basis. (This arises from the fallacy of attempting to view such behaviour as somehow independent of interpretation)!
Therefore all number behaviour is dynamically conditioned by two key sets of polar opposites.
Likewise - by extension - every mathematical notion is conditioned by these same two sets.

Thus we can no longer attempt to view mathematical relationships in absolute terms as solely quantitative i.e. analytic in nature. 
All mathematical relationships possess a qualitative i.e. holistic meaning that is of equal importance. Therefore comprehensive mathematical understanding requires the balanced interaction of both aspects. 

Likewise we can no longer view the external (objective) aspect of mathematical relationships as somehow separate from corresponding mental interpretation of a - relatively - internal (subjective) nature. For we now have not just one default absolute interpretation but a wide range of possible relative interpretations of mathematical reality. And in a very valid sense each of these differing interpretations now corresponds to a distinctive external reality! So when we change our mental interpretation, the objective mathematical world to which it relates likewise changes!

The very manner in which we look at the prime numbers subtly changes from the dynamic perspective.

The conventional mathematical perspective is strongly built on the notion of the prime numbers as absolute number quantities with inherent objective properties (as somehow independent of our interaction with them). However this is strictly meaningless as we cannot understand numbers without such mental interaction!

Likewise it is strictly meaningless to attempt to view the prime numbers as independent of the natural numbers.

So from the dynamic perspective we no longer attempt to understand the prime numbers (as independent) but rather as the relationship of the primes to the natural numbers.

And right away from the dynamic perspective, this is seen to have two complementary directions i.e. the relationship of the primes to the natural numbers and the corresponding inverse relationship of the relationship of the natural numbers with the primes.

So from the former perspective each natural number is seen in cardinal terms, through multiplication, as the unique expression of prime factors (as building blocks).
However from the complementary perspective, each prime number is seen in ordinal terms (indirectly) through addition, as the unique expression of a succession of natural numbers (as building blocks).

Therefore the notion of any prior causation in the origin of the primes and natural numbers loses its meaning from this dynamic standpoint. Both mutually give rise to each other in an ineffable manner (with the appearance of some prior causation arising from their phenomenal unfolding in space and time). 

Likewise the (external) physical aspect of the primes and natural numbers cannot be divorced from the (internal) psychological means of their interpretation. The deeper significance of this relates to the key fact that all physical and psychological processes in nature are encoded in number (as the original secret of their inherent nature).

 So from this perspective nothing indeed is more important than number! 

Once again I mentioned in the last blog that without explicitly incorporating a qualitative (holistic) aspect to Mathematics that we have no adequate means of establishing relationships between numbers.
Because - quite misleadingly - in Conventional Mathematics, numbers are viewed in absolute terms as independent entities, subsequent attempts to establish relationships as between such numbers (which  implies interdependence) must necessarily be of a reduced - and thereby distorted - nature.

This equally of course applies to the relationship as between the primes and the natural numbers. Here the attempt to view it from a merely quantitative perspective blinds us to its fundamental nature.

Though I have repeated this analogy on countless occasions before on these blogs, it remains of the utmost importance as appreciation of its significance highlights the key overriding limitation of the present mathematical approach.

When we fix the frame of reference e.g. with one travelling North on a road and encountering a crossroads, a left turn can be given an unambiguous meaning.

Then we fix the frame of reference in terms of the opposite pole of reference i.e. with one travelling South on the same road encountering the same crossroads, again a left turn can be given an unambiguous meaning. 

In each case here we are operating within independent frames of reference!

However though the designation of a left turn is unambiguous from each independent frame considered separately, in terms of both frames as interdependent,  it is rendered paradoxical. So clearly in this simultaneous context, a left turn implies its opposite pole i.e. right turn and a right turn likewise implies its opposite (i.e. a left turn).

It is exactly analogous in terms of the attempted study of prime numbers.

When we adopt an independent frame of reference (regarding the individual behaviour of primes with respect to the natural numbers) we can unambiguously identify numerical results of a quantitative nature.
Then when we adopt the opposite independent frame of reference (of the general behaviour of the primes with respect to the natural numbers) again we can unambiguously identify numerical results of a quantitative nature.

However when we attempt to relate both frames as interdependent (with respect to both their individual and general behaviour) numerical results of a merely quantitative nature are rendered paradoxical.
So just as a left turn in this context implies its opposite pole (in a right) and a right its opposite in a left turn, here with respect to the relationship between the primes and the natural numbers, quantitative interpretation implies qualitative and likewise qualitative interpretation implies quantitative respectively.

In other words it is strictly futile attempting to deal with the two-way interdependence of the primes and the natural numbers in a merely quantitative manner!

This two-way interdependence is indeed now recognised in Conventional Mathematics, with the non-trivial zeros understood as encoding the behaviour of the primes and the primes in turn encoding the behaviour of the non-trivial zeros. 
However what is not yet appreciated is that this interdependence necessarily implies qualitative (holistic) as well as quantitative (analytic) interpretation.

Put another way the relationship between the primes and natural numbers implies both quantitative (cardinal) and qualitative (ordinal) aspects.

However because these aspects must be properly considered in a dynamic relative manner, the cardinal has a qualitative while the ordinal also has a quantitative aspect.

So whereas the analytic (quantitative) appreciation is indeed of an unambiguous linear nature (when considered within either separate frame of reference), the holistic (qualitative) appreciation is of a paradoxical circular nature (when both frames are considered as interdependent).

We can appreciate this instinctively in the context of road directions at a crossroads. However we have yet to realise that the same appreciation intimately applies to the very nature of Mathematics.

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