Sunday, October 24, 2010

Finite and Infinite

Ultimately the true nature of the Riemann Hypothesis pertains to the key relationship as between finite and infinite (not just in Mathematics but with respect to all living experience).

Indeed the various ways in which I have already expressed this relationship represent the finite/infinite connection.

For example we can formulate this as the essential relationship of quantitative and qualitative, linear and circular, discrete and continuous, classical and quantum mechanical, order and chaos etc.

However conventional mathematical understanding is properly geared solely for (rational) finite interpretation. Though infinite notions are of course recognised they are treated in a strictly reduced manner (i.e. that is amenable to rational type analysis).

In other words the true meaning of the infinite is thereby lost in Mathematics.
This remains the - largely unrecognised - elephant in the room as its key overriding central problem.

To understand what is involved here we have to once again recognise that actual mathematical understanding is never strictly rational (though explicitly in formal terms it is indeed represented as rational!)

Rather, such understanding always involves a dynamic interaction of both rational (conscious) and intuitive (unconscious) elements.

Put quite simply, whereas the rational element of this interaction pertains directly to appreciation of the finite, by contrast the intuitive aspect pertains directly to the infinite.

Though implicitly mathematicians may indeed recognise the importance of intuition - especially in the generation of creative insights - explicitly however Conventional Mathematics is interpreted in a merely reduced rational manner (where in effect the holistic notion of the infinite is reduced in a finite manner).

Thus if we are to properly incorporate the infinite with finite notions then we must include a distinctive qualitative type of mathematical understanding (which I call Holistic Mathematics).

However it is important to recognise that Holistic Mathematics operates in terms of a distinctive logical system (which is circular in nature).

It is only through the appropriate circular use of logic that intuition - pertaining directly to appreciation of the infinite in experience - can be indirectly translated in rational terms.

Once again I will briefly express the key difference here as between the two logical systems.

With linear logic - which defines conventional mathematical experience - the key polarities of experience are treated as separate and independent.

For example internal and external - which necessarily underpin all phenomenal experience - represents one important example of such key polarities.

So in Mathematics the external (objective) aspect is treated as independent of the internal (subjective) aspect. This thereby creates the impression of a strict objective validity to mathematical truths (which ultimately is unwarranted).

Likewise in Mathematics whole notions are treated as independent of parts so that we can study either aspect in isolation from each other. In fact perhaps the most characteristic feature of the linear approach is the manner in which wholes are reduced to parts. In this way both aspects can be treated in a merely quantitative manner (where the whole is literally seen as the sum of the parts).
Thus the key qualitative distinction as between wholes and parts is thereby lost through such interpretation!

With circular logic they key polarities are treated as interdependent (rather than separate). Ultimately of course this means that with full interdependence any notion of polarities as phenomenally separate disappears. We are then left in experience with pure formless appreciation (which is the very nature of intuitive awareness).

However we can characterise the nature of such awareness in an indirect rational manner through the complementarity of opposites. This leads to the recognition - like left and right turns on a road - that opposites have a merely arbitrary definition in any context (depending on the polar frame of reference). So for example what is considered as (quantitatively) whole in one context could be considered as part in another (and vice versa). Now the mysterious dynamic enabling this switching of reference poles is of a qualitative (intuitive) nature and not thereby confused with rational interpretation in isolated contexts.

Put another way - deliberately using holistic mathematical language - there are two key tasks in all understanding (which thereby includes Mathematics). First we must successfully differentiate phenomenal symbols (as independent); secondly we must successfully integrate those same symbols (as interdependent).

In actual mathematical experience, the first of these tasks (of differentiation) is properly achieved through (linear) reason using either/or logic; however the second equally important task (of integration) is properly achieved though intuition that is indirectly represented in a paradoxical manner through circular reason (using both/and logic).

Once again there is a huge unrecognised problem with conventional mathematical interpretation in that is based solely on (linear) reason. Therefore it can only deal with the qualitative task of integral interpretation in a reduced manner.

The relevance of all this for the Riemann Hypothesis is that - properly understood - it actually points to the fundamental condition for the successful harmonisation of both (analytic) quantitative and (holistic) qualitative type understanding.

Imagine in geometrical terms a straight line diameter = 1 unit circumscribed by its circular circumference. Now the midpoint which is common to both circle and line occurs at the midpoint of the line (i.e. at 1/2).

In a nutshell this is what the Riemann Hypothesis is all about i.e. the central condition that is necessary to reconcile both the quantitative (linear) and qualitative (circular) aspects of mathematical understanding.

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