Once again my basic contention is that – properly understood – there are two distinct aspects to the number system.
1) An analytic aspect where the fundamental poles of understanding (i.e. internal and internal and quantitative and qualitative are clearly separated). This indeed is what makes such Mathematics linear (i.e. 1-dimensional) in nature as it literally interprets its symbols exclusively – in any context - in terms of the dominance of just one pole of understanding.
So relationships are considered absolutely in terms of the objective nature of symbols (where the external pole dominates); equally relationships are considered absolutely in terms of the quantitative nature of symbols (where the qualitative is thereby reduced to the quantitative).
(It is important to appreciate that when I use the word analytic, typically I use it is this general context of absolute uni-polar interpretation of symbols and not in the narrower more specialised conventional mathematical sense relating to infinite series).
I refer to this aspect of Mathematics as Type 1.
2) A holistic aspect where the same fundamental poles of understanding are now considered as dynamically related. In this new distinctive interpretation the objective nature of mathematical symbols strictly has no meaning in the absence of their corresponding means of (subjective) mental interpretation.
And as a wide variety of partially valid interpretations can be employed the objective nature of such symbols is thereby necessarily of a merely relative nature.
Likewise the quantitative interpretation of symbols strictly has no means in the absence of qualitative appreciation (of a distinctive holistic variety).
To be more accurate the holistic aspect properly refers to the dynamic interaction of poles that are (a) external and internal and (b) quantitative and qualitative with respect to each other.
Likewise the analytic aspect refers to the extreme situation where such interaction is ignored through reducing one entirely in terms of the other. And again in Conventional Mathematics the internal aspect is reduced in terms of the external and the qualitative in terms of the quantitative!
However for convenience I continually contrast analytic and holistic (as shorthand for there respectively different ways of treating such polar interaction)!
I refer to this aspect of Mathematics as Type 2.
3) The actual experience of Mathematics entails the mutual dynamic interaction of both Type 1 and Type 2 understanding. Though customary objective quantitative distinctions of a valid nature can indeed be made, as the polar reference frames to which they relate continually change they are now understood in a merely relative manner.
I refer to this most comprehensive aspect of Mathematics as Type 3.
So properly understood (i.e. in a comprehensive manner) all mathematical relationships are necessarily understood in merely relative terms. The customary fixation with absolute type relationships in Conventional Mathematics reflects its reduced – and thereby limited – (1-dimensional) manner of interpretation.
The distinction as between the Type 1 and Type 2 aspects Mathematics leads to a corresponding distinction with respect to the number system.
For example the natural numbers in Type 1 terms are understood as representing cardinal whole units. So 2 in cardinal terms represent a collective whole unit (i.e. an integer).
The problem with this quantitative approach however is that when one attempts to describe the individual members of 2, one must treat them in homogenous terms as without qualitative distinction. So 2 = 1 + 1.
However this leaves one with the considerable problem of having no means of making a meaningful distinction as between the 1st and 2nd members of 2 (in ordinal terms).
The fact that this fundamental issue is completely glossed over in Conventional Mathematics clearly reflects its reduced nature i.e. where the qualitative (ordinal) nature of 1st and 2nd is misleadingly assumed to be implied by the quantitative (cardinal) understanding of 1 and 2!
So the Type 2 understanding of number resembles – as it were - a sub-atomic approach where each of the individual members of a number group are now given a unique individual identity in relative terms.
So once again from a Type 1 approach, 2 is treated from a cardinal perspective as a collective whole unit (in quantitative terms).
From the complementary Type 2 approach, 2 is treated from an ordinal perspective as comprising unique individual units i.e. 1st and 2nd (in qualitative terms).
Now Type 1 is – literally - defined in 1-dimensional terms, where each natural number is implicitly defined with respect to 1 as exponent (i.e. dimensional number).
So the natural numbers 1, 2, 3, 4,…. are defined in Type 1 terms as
1^1, 2^1, 3^ 1. 4^1,….
The 1-dimensional nature of this approach is clearly illustrated by the fact that any number initially raised to another exponent (dimensional number) will be given a reduced linear interpretation i.e. where its ultimate value is expressed in reduced quantitative terms.
So 2^2 in Type 1 = 4^1!
The Type 1 approach is properly geared to explain the fundamental nature of addition where quantitative change occurs (without qualitative transformation)!
So 2 + 3 in this additive system = 2^ 1 + 3^ 1 = 5^1.
In the Type 2 aspect of the number system, 2 is by contrast is defined with respect to 1 (representing a default base number quantity).
So the natural number 1, 2, 3, 4,….. are defined from a Type 2 perspective as
1^1, 1^2, 1^3, 1^4,…..
Just as the Type 1 approach is geared to the pure nature of addition (where no qualitative transformation in the variables takes place), The Type 2 approach is then geared to the pure nature of multiplication (where no quantitative transformation in the variables takes place).
So 2 + 3 in this system represents 1^2 * 1^ 3 = 1 ^ (2 + 3) = 1^5.
So what appears as multiplication from the Type 1 perspective represents addition from the corresponding Type 2 perspective.
However to properly express the nature of the Type 2 aspect we need to move to a circular number system (based on successive roots of unity).
The key to understanding the nature of this system is that a two-way interaction is necessarily entailed as between variables given both an analytic and holistic interpretation respectively.
Thus for example the number 2 in type 2 terms = 1^2.
2 here is identified with the two individual unique dimensions of 2 i.e. its 1st and 2nd dimensions (with a holistic interpretation as interdependent).
Now in analytic terms these two dimensions can be given expression as the 1st and 2nd roots of 1 respectively or more correctly the roots of 1^2 and 1^1 respectively.
So the first root of 1 is therefore (1^2) raised to the power ½ = + 1 (i.e. 1^1).
The second root of 1 is (1^1) raised to the power of ½ = – 1.
However in dynamic terms these two separate values (as analytically understood) are in continual relationship with the corresponding interdependent value (as holistically understood).
So the quantitative independent interpretation of 1 and – 1 as separate strictly have no meaning in dynamic relative terms apart from the combined holistic interpretation of 1 and – 1 as interdependent.
Once again I have repeatedly used the example of a crossroads to illustrate this type of understanding. When understood (within independent reference frames) left and right turns at a crossroads can be given a – relatively – separate interpretation which are + 1 and – 1 with respect to each other. (1 refers here to the dimensional unit i.e. representing a direction).
However the appreciation that both left and right ultimately have a purely relative i.e. paradoxical meaning (when both frames are simultaneously combined) represents the true qualitative holistic nature of 2!
And this could be represented quantitatively as 1 – 1 = 0, i.e. as strictly without quantitative significance!
So in the actual understanding of the nature of a direction at a crossroads, both types of understanding (analytic and holistic) are necessarily combined.
Indeed this 2-dimensional type of dynamic understanding is well-recognised in spiritual literature (e.g. Taoism and Buddhism) in philosophy (e.g. Heraclitus and Hegel) in psychology (e.g. Jung) and indeed even indirectly in quantum physics (e.g. wave-particle duality).
What is vital to appreciate however is that it necessarily combines both analytic and holistic aspects of understanding in a complementary fashion.
Now clearly this cannot be appreciated in Type 1 terms (where holistic notions are reduced to analytic in a static absolute manner).
As Conventional Mathematics is fundamentally based on such absolute notions it is severely limited in terms of dealing with the very notion of interdependence.
And as the nature of prime numbers likewise involves such interdependence e.g. in the relationship of the primes - constituting the natural number system in cardinal terms - to the non-trivial zeta zeros, it is fatally flawed in terms of appreciating the true nature of the number system.
Another key point is that this interpretation of the number 2 from a Type 2 perspective represents but the simplest version of Type 2 understanding.
Indeed associated with every number as an exponent (i.e. dimensional number) is a unique means of configuring the dynamic interaction as between its analytic and holistic elements! Or using the shorthand means that I have been employing every number has a unique holistic significance serving as a distinct means of interpretation of the dynamic interaction of polarities (underlying all mathematical understanding).
In this holistic context 1 (as 1-dimensional) has an extreme limiting interpretation where the holistic aspect is directly reduced to analytic interpretation.
And this is what precisely defines the nature of Conventional Mathematics which is based on absolute objective interpretation of its symbols in a merely (reduced) quantitative manner!
However in truth in a more comprehensive vision of Mathematics, an unlimited number of possible interpretations exist (all with a partial relative validity). And in all these other systems (based on a dimensional number ≠ 1).
And if you understand this you will immediately understand why from a holistic perspective, the Riemann zeta function is uniquely undefined for s = 1 (as this is the one value where the inherent dynamic interaction as between the analytic and holistic aspects of the number system is broken).
So once again, putting it bluntly the Riemann zeta function cannot be properly interpreted in conventional mathematical terms. And of course neither can the Riemann Hypothesis (which is fundamentally based on true appropriate interpretation of the Riemann Function).
Finally it is futile therefore attempting to prove the Riemann Hypothesis from a conventional (1-dimensional) perspective!
However fortunately, the key to appreciating the nature of the Riemann Hypothesis can be related to the simplest case of Type 2 understanding (with respect to the number 2).
Once again the most comprehensive form of mathematical understanding involves the interaction of both Type 1 and Type 2 aspects where both analytic (as separate) and holistic appreciation can reach a high level of refinement.
In fact whenever a non-unitary base number is raised to a non-unitary power, both Type 1 and Type 2 aspects of interpretation are both required.
So in the simplest case of 2^2, strictly both a quantitative (analytic) and qualitative (holistic) transformation is entailed the mutual interaction of which entails Type 3 understanding.
Next we find that associated with the Type 1 and Type 2 aspects of the number system are corresponding Type 1 and Type 2 zeta functions.
Now the Riemann zeta function refers strictly to the Type 1 aspect; however proper appreciation of the Riemann zeta function and its associated Riemann Hypothesis requires incorporation of a complementary Type 2 zeta function.
Though the conventional Riemann zeta function can indeed be given an analytic Type 1 formulation (as the Zeta 1 function), strictly this has no proper meaning in the absence of corresponding holistic Type 2 appreciation.
In like manner through the Riemann zeta function can be given an – unrecognised – Type 2 formulation (as the Zeta 2 function), strictly again this has no proper meaning in the absence of corresponding analytic Type 1 formulation.
Finally the most comprehensive understanding – in what I refer to as the Type 3 zeta function - entails the simultaneous interaction of both Type 1 and Type 2 formulations in a manner ultimately approaching purely ineffable understanding!