Tuesday, May 22, 2012

The Central Issue for Mathematics

The implications of the Riemann Hypothesis can be stated in a number of different ways.

Ultimately however it relates to the fundamental nature of the number system - which strange as it it might appear to many - is grossly misrepresented in conventional mathematical terms.

The critical basic issue relates to the relationship as between independence and interdependence. Now this applies to all relationships (including of course mathematical).

If we are to understand this relationship in a meaningful fashion then the two aspects independence and interdependence must be understood in a dynamic relative fashion.

This therefore implies that the number system itself must be understood in a dynamic interactive fashion where numbers - from one valid perspective - possees a relative independent existence; yet - from an equally valid perspective - they can be understood as - relatively - interdependent with other numbers.

Expressed in an equivalent manner, numbers thereby possess both quantitative (independent) and qualitative (interdependent) aspects; alternatively we can say that numbers possess both a cardinal and ordinal identity.

Now putting it quite simply the prevailing intellectual paradigm that governs Mathematics (as conventionally understood) is fundamentally unsuited for a coherent dynamic treatment of the number system.

As I have continually stated this paradigm is characterised by its linear logical approach.

Now it may be of value to clarify once again what this precisely means!

All phenomenal experience is governed by the interaction of fundamental poles (that operate as opposites).

At the most basic level these can be reduced to two key sets.

On the one hand we have, in any experiential context, external (objective) and internal (subjective) aspects; also we have whole (collective) and part (individual) aspects.

So for example we cannot have knowledge of a number "object" such as "2" without a corresponding internal (mental) perception of this "object". So properly speaking, what we always have in mathematical experience is the dynamic interaction of two aspects of understanding (external and internal) that are - relatively - positive and negative with respect to each other.

Likewise we cannot have knowledge of a specific number - again such as "2" - without a corresponding collective number concept (that holistically contains this number).
Thus again, mathematical experience properly entails the interaction of both part and whole notions that are positive and negative with respect to each other.

Now, like the four key directions on a compass we could illustrate these four aspects as four equidistant points on the circle of unit radius (in the complex plane). So we position external and internal as opposites on the real axis and whole and part as corresponding opposites on the imaginary axis.

When one stops for a moment to reflect on this, the very manner in which the fundamental poles (dynamically underlying all experience) interact, gives rise to a circular number system that corresponds (from a qualitative perspective) with the roots of 1 (in quantitative terms).

So the four roots of 1 are 1, - 1, i and - i (understood as separate co-ordinates). Likewise in the context of 4, the four dimensions qualitatively are understood in terms of the same set of co-ordinates (this time with opposites interpreted in a complementary manner). Thus the relationship of external to internal can be modelled in terms of + 1 and - 1 (as real complementary opposites). Likewise the relationship of whole to part can be modelled in terms of i and - i (as imaginary complementary opposites).

In conventional terms the relationship as between whole and part is understood in a highly reduced - merely quantitative - fashion. Here objects are understood as holons (part-wholes) with every whole part of a larger whole (in a quantitative manner).

However the unreduced) nature of wholes and parts requires appreciating them in true holistic fashion as archetypes of what is universal. So in this refined intuitive fashion, from one perspective, parts are qualitatively contained in the collective whole (that preserves a qualitative distinction from the parts). Equally from the reverse perspective, the whole is uniquely reflected in each part (again in a manner that preserves the qualitative distinction of each part from the whole).

So in the context of numbers, each individual number (as a quantitative part of the number system) is contained in the collective number concept (that is understood as qualitatively distinct from the specific number); likewise the collective number concept is uniquely reflected through each specific number that now attains a distinctive qualitative nature (which is not confused with the quantitative).

And this is the very meaning of the notion of the imaginary in a qualitative mathematical sense i.e. where a symbol such as number reflects a holistic qualitative meaning (as distinct from its real quantitative interpretation)!

Therefore to properly understand the number system, not alone do we need to allow for real and imaginary aspects (as quantitatively understood). Equally, we need to allow for real and imaginary interpretation (as qualitatively appreciated).

So the relatively independent aspect (as quantitative) comes from the real aspect of understanding (relating to reason). The relatively interdependent aspect (as qualitative) comes from the imaginary aspect (relating to holistic intuition).

However, interpretation corresponding to such (imaginary) intuition, emanating from the unconscious, can indirectly be expressed in a rational manner through the circular use of logic.

Now the simplest form of such circular logic relates to the complementarity of just two opposite poles i.e. external and internal, which constitutes 2-dimensional understanding.

Here, mathematical interpretation is inherently of a dynamic interactive nature.
For example with respect to the understanding of the number "2", we recognise that this number has an external (objective) status as relatively independent in understanding; equally we recognise that it has an internal (subjective) status i.e. as a mental perception that also attains a relative independence. So the actual experience of number necessarily entails both of these poles (as relatively independent) in rational terms and also a recognition of their combined interdependence (in a directly intuitive manner).

So the key point to recognise is that both a quantitative recognition of independent identity and also the qualitative recognition of interdependent identity are necessarily involved in all number experience.

In this manner we are enabled to understand number with respect to both its (quantitative) cardinal and (qualitative) ordinal nature.

However with conventional mathematical understanding, gross reductionism operates at every turn.

Because such mathematics is formally based on mere rational understanding, only the independent quantitative aspect is recognised. This, misleadingly leads to the notion of a number as having an absolute objective identity (independent of subjective interpretation). Then when focus is placed on the mental theoretical side of recognition, again this is given an absolute identity (independent of objective circumstances).

When the independent aspect is treated as absolute this leaves no proper role for the interdependent aspect of recognition (arising from the dynamic interaction of opposite poles) which is inherently of a qualitative - rather than quantitative - nature.

So quite simply no distinct recognition of the ordinal (qualitative) nature of number can exist in Conventional Mathematics. It is thereby grossly reduced in a merely quantitative manner (i.e. as the rankings of cardinal numbers).

Even as a child of 10, I could vaguely see something fundamentally wrong with the manner in which the number system is conventionally understood.

In a sense, my subsequent intellectual journey has involved an on-going attempt to rectify this problem (which I see as central to the very nature of Mathematics, Science and indeed of life generally).

Frankly I am amazed that such little apparent questioning with respect to such a fundamental issue takes place for certainly from my perspective I see clearly the basic structure of Mathematics as deeply flawed and in urgent need of the most radical revision.

In fact the nature of the number system cannot be properly addressed within Conventional Mathematics. For the fundamental paradigm that defines such interpretation, misrepresents its nature in the most fundamental manner possible!

So to sum up! Mathematical experience is of a dynamic interactive nature entailing distinctive quantitative and qualitative aspects.

The true nature therefore of the number system is likewise of a dynamic interactive nature with both quantitative and qualitative aspects.

The Riemann Zeta Function - when correctly decoded - provides a wonderful map of the relationship as between the quantitative and qualitative aspects of the number system.

However, crucially, the one dimensional value for which the Riemann Function is undefined is where s = 1.

As in qualitative terms, Conventional Mathematics is defined by its 1-dimensional rational paradigm (based on isolated poles of recognition) the Riemann Function cannot be coherently interpreted from this standpoint.

Once again when appropriately understood, the Riemann Zeta Function establishes the dynamic complementary relationship as between both quantitative and qualitative aspects of the number system (with the Riemann Hypothesis the central condition for attaining the mutual identity of both aspects).

So clearly the Function (and of course the Riemann Hypothesis) cannot be properly appreciated in conventional mathematical terms, as it defines number in a merely quantitative manner with no formal recognition whatsoever of its equally important qualitative aspect!