Tuesday, August 9, 2016

Riemann Hypothesis: New Perspective (11)

I have mentioned before how a future golden age of Mathematics will contain at least three distinctive ways of interpreting mathematical symbols.

1) The conventional rational approach based on the quantitative interpretation of mathematical symbols in a conscious manner.

2) The - largely - unrecognised intuitive approach based on the qualitative interpretation of mathematical symbols in an unconscious manner. Though this approach is indeed directly based on refined intuitive type recognition that cannot be successfully reduced in standard rational terms, indirectly it can be intellectually translated in a (circular) rational manner (entailing paradox from a dualistic perspective).

3) the comprehensive radial approach based on the mutual interpenetration, in a coherent integrated fashion, of mathematical symbols in both a conscious and unconscious manner.

However the great surprise that awaits entails the additional recognition that all mathematical symbols have both cognitive rational and affective sense interpretations.

So comprehensive mathematical appreciation of symbols entails the emotional as well as rational domain!

Indeed ultimately it entails also the volitional domain as the very means for successfully harmonising - relatively - both conscious and unconscious aspects with respect to cognitive and affective aspects is through the volitional aspect (i.e. will).

Now in this context it would be helpful to carefully distinguish both these two aspects of the psychological recognition of mathematical symbols through a simple example.

Imagine one is looking at 3 cars (say parked in a driveway)!

The cognitive recognition here relates to the common collective identity of the cars (as belonging to the same class).

This directly concurs with the cardinal notion of number (where each unit of the number in question enjoys an impersonal homogeneous identity in quantitative terms.
So 3 = 1 + 1 + 1.

However affective (sense) recognition is quite distinct in relating to the unique individual identity of each car (arising from their ordinal relationship with each other.

.This then directly concurs with the ordinal notion of number where each unit now enjoys a distinct personal identity in a qualitative manner.

So from this perspective 3 = 1st + 2nd + 3rd!

In conventional mathematical terms, the latter each interpretation is simply reduced in a cardinal manner. So the personal unique identity of each item - corresponding initially with sense recognition of an affective kind - is thereby lost.

However properly understood - when we recognise the true complementary nature of both aspects of number recognition - cognitive and affective aspects are necessarily involved in the dynamics of all number recognition.

Indeed these dynamics relate directly to the true relation of whole and part (and part and whole).
Once more each prime, from the conventional mathematical perspective is considered in a quantitative whole manner (where all units are considered as homogenous and thereby lacking any qualitative distinction).

So again to illustrate, 3 (as a prime) = 1 + 1 + 1 (where the quantitative units lack any qualitative identity).

This concurs with the standard rational (i.e. cognitive) interpretation of number where each prime is considered as a "building block" of the cardinal natural number system.

However, from the complementary (unrecognised) perspective 3 (as a prime) is uniquely defined by its ordinal members in natural number terms.

So 3 = 1st + 2nd + 3rd.

Now here, in reverse, 3 (as the unique combination of individual ordinal units )  strictly lacks a quantitative identity. So 3, in this context, properly relates to "threeness" (as the qualitative nature of 3).

However this latter qualitative recognition i.e. that number units bear a necessary relationship with each other, pertains directly to sense recognition (of an affective kind).

So when one fails to recognise the necessary interaction of both cognitive and affective recognition with respect to each prime, a fundamentally distorted interpretation of the relationship of whole and parts results.

So in conventional mathematical terms - reflecting the dominance of the merely cognitive (rational) aspect of understanding - the number system is interpreted in a merely reduced quantitative manner (where primes are unambiguously viewed as the "building blocks" of the natural numbers).

However when one properly allows for the corresponding affective (sense) aspect of understanding, the number system is likewise seen in a true qualitative manner (where each prime is defined by its natural number members in an ordinal manner).

So from the customary analytic perspective, the relationship between the primes and natural numbers (and natural numbers and primes) is considered in a one-way unambiguous manner.
So in standard (Type 1) terms each prime serves as a quantitative "building block" of the natural number system a cardinal manner.

Then in corresponding (Type 2) terms each prime is already uniquely defined by its natural number mebers in an ordinal manner.

Then the simultaneous recognition of both Type 1 and Type 2 aspects (i.e. Type 3) requires true holistic understanding in the inherent understanding of the number system in a dynamic interactive manner where both aspects - that appear unambiguous from within each reference frame considered in isolation - now appear as deeply paradoxical (when indirectly conveyed in a circular rational manner).

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