So even though number - depending on context - continually switches as between both a particle (analytic) and wave (holistic) identity, no recognition of this takes place in the conventional approach.

Now the basic issue - as I have continually reiterated in my blog entries - relates to the distinction as between the notions of number independence and number interdependence respectively.

Therefore, once again, when a prime such as 3 is employed in conventional mathematical terms, it is interpreted with respect to its independent quantitative status in a cardinal manner, i.e. where its sub-units 1 + 1 + 1 are viewed in a homogeneous impersonal fashion as - literally - devoid of qualitative characteristics.

This treatment then implicitly concurs with the number raised to the default dimension of 1, i.e. as a number defined on the real number line.

This likewise concurs with a linear rational mode of interpretation (based on the making of one-way unambiguous sequential logical connections). And again this is what I define as the Type 1 aspect of the number system.

So in Type 1 terms, 3 is defined as 3

^{1}.

However as well as independence, we should equally recognise the interdependence of all numbers which - strictly - is of a qualitative (rather than quantitative) nature.

So if we return to our example of the prime number "3", there is equally a sense in which its sub-units can be understood as fully interdependent with each other. Now in direct terms, such recognition occurs in an intuitive holistic manner. However indirectly such holistic recognition (of number interdependence) enables one to then make ordinal distinctions at the linear level of understanding.

So effectively, the true intuitive holistic recognition of number is inevitably reduced in a merely quantitative rational manner at the conventional level of mathematical understanding.

And unfortunately such gross reductionism then pervades the understanding of every mathematical relationship in conventional (Type 1) terms.

Therefore, the proper recognition of the uniquely qualitative nature of number requires a distinctive complementary Type 2 approach (where it is defined in a circular manner).

Here every number is inversely defined with respect to a standard base of 1, which then can be raised to all the natural numbers (as powers or exponents representing dimensions).

So in Type 2 terms, 3 is defined as 1

^{3}. Now strictly this refers to the intuitive - rather than the rational - recognition of 3, as the quality of "threeness" (where the interdependence of its sub-units with each other are recognised). Of course as all numbers have both analytic and holistic interpretations, 3 as a number representing a dimensional power i.e. 1

^{3}, equally can be given an analytic meaning i.e. as a cube with common side of 1 unit! However it is the holistic meaning of "3" that I am referring to here in this context.

We can then indirectly express such holistic understanding in a rational manner (i.e. in 1-dimensional terms) by obtaining the 3 roots of 1 (i.e. 1

^{1/3}, 1

^{2/3}and 1

^{3/3 }respectively) where they now appear as circular or paradoxical to rational understanding.

The fundamental significance of these number conversions - which are completely missed from the conventional mathematical perspective - is that they then provide a unique means of indirectly converting the ordinal notions of 1st, 2nd and 3rd (in the context of 3 members) in a quantitative manner.

Well, strictly in fact one of these results, i.e. 1

^{3/3 }is not unique and reduces to 1! And this is always the case in relation to the last of the n roots of any number. And it is this reduced notion of ordinal meaning that thereby defines the conventional mathematical approach.

So in cardinal terms 3 = 1 + 1 + 1. Then in ordinal terms 3 = 1st + 2nd + 3rd.

However conventionally, 1st, 2nd and 3rd are implicitly identified with the last roots of 1(1

^{1/1}) of 2 (1

^{2/2}) and 3 (1

^{3/3}) respectively where they reduce down to 1 in each case.

Therefore from this perspective 1st + 2nd + 3rd = 1 + 1 + 1 (so that ordinal meaning is thereby successfully reduced in the standard quantitative cardinal manner!).

However when one recognises the true distinctive nature of the Type 2 aspect of the number system, the mathematical world of number is turned completely on its head.

Thus, from the Type 1 perspective, 3, as a prime, is unambiguously viewed as an independent "building block" of the natural number system (in quantitative terms).

However from the Type 2 perspective, 3 as a prime is already uniquely defined by its distinctive ordinal natural number members (1st, 2nd and 3rd) in qualitative terms. And these members are fully interdependent with each other in a merely relative manner (depending on context).

So again from the quantitative internal perspective, each prime is viewed in quantitative terms as an individual "unit of linear independence" (with respect to the natural number system).

Then from the corresponding qualitative perspective, each prime is now viewed as a collective "group of circular interdependence" with respect to the natural number system.

Therefore in order to embrace these complementary features of the number system, we must now conceive of the prime in dynamic relative terms based on the two-way interaction of both its quantitative (cardinal) and qualitative (ordinal) aspects. And this equally applies externally to the relationship as between the primes and the natural number system as a whole

Ultimately the number system is characterised by an incredible two-way synchronicity with respect to both its quantitative and qualitative aspects (internally and externally). And the true appreciation of this fundamental fact requires the very refined marriage of both analytic and holistic type understanding (where neither aspect is reduced in terms of the other).

Clearly however, the standard approach of interpreting the number system in an absolute quantitative manner greatly distorts proper understanding of its true nature.

We will now return to the subject matter of the past few blog entries in an attempt to provide a richer appreciation of the true dynamics involved.

I had earlier attempted to do this in "Another Interesting Relationship" and "Further Investigation" on my companion "Spectrum of Mathematics" blog.

I then recognised subsequently that the π relationship I suggested with respect to the circular number system (based on the roots of 1) was in error.

So just to recap! We have now established (with respect to the Type 1 aspect of the number system) that the probability a number chosen at random will not contain s or more prime factors is 1/ζ(s).

So in the important case where s = 2, the probability that such a number will not contain 2 or more factors = 1/ζ2) = 6/π

^{2}.

This equally can be expressed as the probability that a number chosen at random is square-free or alternatively the probability that 2 numbers chosen at random will contain no common (proper) factor!

With respect to the Type 2 circular system, the n roots of every number comprise a distinct part of the overall system. Now again the primes are unique in this respect in that the n roots of 1 for every prime (apart from the default root of 1) cannot be repeated for any other prime.

Thus as I expressed it in the earlier entry, the fractions expressing the roots for such prime numbers are irreducible.

However with the roots of the composite numbers, some fractions will be reducible and other factors non-reducible.

So my contention was that that this would be governed by the same relationship (as in Type 1 terms).

Therefore we can now state, the probability that the fractional value chosen at random (associated with the roots of 1 in Type 2 terms) is irreducible, is the same as the probability that a number chosen at random (in Type 1 terms) will not contain 2 or more factors.

So once again this probability is given as 1/ζ2) = 6/π

^{2}.

In fact, given that we have already established that this equally expresses the probability that 2 numbers chosen at random (in Type 1 terms) contain no proper factors, the result must necessarily hold.

The reason for this is as follows. Any two numbers chosen at random can be expressed as the numerator and denominator respectively with respect to a fractional value in the Type 2 system.

For example let us choose two numbers (in Type 1 terms) ar random from 1 to 100. Imagine we obtain 16 and 60! Then 16/60 can be expressed in Type 2 terms as the fractional value associated with the 16th root of the 60 roots of 1.

So the issue in Type 1 terms as to whether these contain common factors, amounts in Type 2 terms as to whether both the numerator and denominator of the fraction involved contain common factors.

And as 16 and 60 contain common factors, then equally the numerator and denominator of 16/60 contain common factors!.

However though we are confirming the same numerical result here in both cases, crucially it relates to two different interpretations of number!

In other words if the Type 1 result relates to the particle aspect of number interpretation - then - relatively - The Type 2 results relates to the wave aspect!

In fact this is very revealing in another context.

As we have seen the quantitative result (in both cases) = 6/π

^{2}.

Now when one looks at number in its proper dynamic (Type 3 context), which incorporates Type 1 and Type 2 aspects as interacting partners, this entails a linear frame of reference for the Type 1 aspect and a circular frame of reference for the Type 2 aspect of the number system respectively.

Now the important symbol π, expresses in quantitative terms the (pure) relationship as between a (circular) circumference and its (line) diameter.

Equally in holistic terms, it expresses the pure relationship as between intuition (i.e. that appears circular or paradoxical when indirectly expressed in a conventional manner) and (linear) rational type understanding.

Therefore the deeper holistic (i.e. qualitative) appreciation as to why these two relationships involve a simple relationship (involving π) is because the proper understanding of both relationships entails the coherent incorporation of both Type 1 (linear) analytic and Type 2 (circular) holistic understanding, entailing notions of quantitative independence and qualitative interdependence respectively.

In fact we can even provide a rationale as to why the square of π is involved in the relationship.

From the Type 1 perspective, 6/π

^{2}expresses the probability that the product of two numbers (chosen at random) are square-free.

Then in Type 2 terms it expresses the probability that the product of the two numbers (representing numerator and denominator respectively) are likewise square-free. As we have now switched from a linear (Type 1) to a circular (Type 2) frame of reference, the dynamic explanation therefore entails the integrated relationship of circular and linear type aspects of the number system!

Therefore the richer interpretation of these simple results based on the Riemann zeta function, entails that both Type 1 (quantitative) and Type 2 (qualitative) aspects of the number system be incorporated with each other in a dynamic complementary manner.

Expressed more generally, complete understanding of the Riemann zeta functions entails at least 4 separate stages (two of which are only presently recognised in conventional mathematical terms).

1. The analytical understanding of the function that is confined to real quantitative values that are positive, which corresponds with the Euler zeta function. (Now I am of course aware of the more specialised mathematical interpretation of "analytic" in the context of the Riemann function. However I am using analytical here in a more general sense as relating to the conventional quantitative interpretation of number, which thereby includes the narrower specialised mathematical meaning of "analytic").

2. We then extend analytical understanding to complex quantitative interpretation (both positive and negative) which corresponds with present understanding of the Riemann zeta function.

However, though this does indeed enable significant advances to be made with respect to the quantitative relationship of the primes to the natural numbers, it is still crucially limited (and indeed distorted) by the attempt to understand number in a merely quantitative absolute manner.

3. The next stage in understanding entails the recognition that associated with every analytic (quantitative) interpretation with respect to the Riemann zeta function is a corresponding unrecognised holistic (qualitative) interpretation.

Though recognition of this holistic aspect did at once co-exist with the analytic, especially with the Pythagoreans (although in a somewhat undeveloped manner) subsequently it has been completely discarded - at least in formal terms - from accepted mathematical understanding.

So it would be very difficult, I imagine, for most practising mathematicians to form any realistic conception of what this holistic aspect means.

I will attempt here however to give just one brief example - which if properly followed - will be seen to have immense repercussions for the misguided quest to seek a proof of the Riemann Hypothesis.

As is well known, the one value (in accepted Type 1 terms) for which the Riemann zeta function is undefined is where s (representing a common dimensional power with respect to the terms of the function) = 1.

Now this - using my terminology - represents the accepted analytic interpretation (where number relationships are interpreted in an absolute quantitative manner).

The corresponding holistic explanation (in Type 2 terms) again states that the function is undefined where s = 1. However 1 in this holistic context takes on the qualitative meaning of rational linear interpretation (which in fact defines the conventional mathematical approach).

So, remarkably what this entails, is that strictly it is not possible to properly understand the Riemann zeta function (i.e. it remains undefined) in an absolute 1-dimensional rational manner.

In other words, the true nature of the number system is inherently dynamic and interactive, entailing the two-way relative interaction of both quantitative and qualitative type interpretation.

And we can only give meaning to the mapping of values through the functional equation connecting positive values for s on the RHS with negative values for (1 – s) on the LHS, through recognition of the holistic aspect.

For example the conventional interpretation of the series 1 + 2 + 3 + .... is that the sum continually gets larger and ultimately diverges (to infinity).

However according to the Riemann zeta function (where s = – 1) the sum of this series = – 1/12.

So at the very least, we are faced with a major task of interpretation in explaining how two completely different results can be apparently given for the same series!

And of course the real problem is that whereas number results concur with standard analytic interpretation on the RHS, they then - in relative terms - concur with holistic interpretation on the LHS.

So the real purpose of the Riemann functional equation - when suitably interpreted - is that it shows how analytic can be mapped with holistic (and holistic with analytic) values with respect to both sides of the function. However again this entails understanding number as the two-way interaction of both quantitative (Type 1) and qualitative (Type 2) meaning.

So the very nature of the Riemann Hypothesis - when suitably understood - transcends conventional mathematical interpretation.

The key issue relates to the condition for the mutual identity of both the quantitative (analytic) and qualitative (holistic) type notions of number. And the Riemann Hypothesis amounts to a precise statement of this requirement!

However clearly it is futile attempting to prove this condition when the qualitative (holistic) aspect is not even recognised in formal mathematical terms. Expressed in an equivalent manner, the consistency of qualitative with the quantitative interpretation of mathematical symbols is already assumed in the axioms used to obtain mathematical proof! So these axioms therefore cannot be meaningfully used to obtain the proof (or disproof) of the Riemann Hypothesis.

So the fundamental revolution that is greatly required in Mathematics could eventually be slowly precipitated through the continued failure to prove the Hypothesis, eventually awakening mathematicians to the fact that there are indeed crucially important dimensions to the understanding of number that have hitherto been totally suppressed with respect to conventional interpretation.

4. At the most advanced stage of understanding, the Riemann zeta function - at least in terms of what I can envisage - both Type 1 (analytic) and Type 2 (holistic) aspects are dynamically incorporated in an integrated harmonious fashion. This is what I refer to as Type 3 understanding, or alternatively radial mathematical understanding.

Even within this category, I would refer to three - relatively - distinct subcategories.

With Type 3 (a) the emphasis is primarily on creative analytic type developments (within the context of established holistic appreciation).

With Type 3 (b) the emphasis is primarily on holistic type interpretation (against a background of established analytic appreciation).

I would see my own recent approach - in a most preliminary fashion - as representative of this subtype.

In other words, though I necessarily grew up - even to the extent of studying Mathematics at University - with the accepted analytic approach, my ability is primarily with respect to the (unrecognised) holistic dimension. So in this context, I would see this contribution with respect to the Riemann zeta function as one of radical re-interpretation, where the very nature of the number system is now interpreted in a completely new light.

Type 3 (c) would then represent the pinnacle of mathematical understanding, where one would be equally gifted with respect to both analytic and holistic type appreciation enabling a contribution to Mathematics both highly creative (holistically) and immensely productive (in analytic terms).

So a proper understanding of the Riemann zeta function (and with it the number system) will require eventual progression to stages 3 and 4. Thus the holistic aspect of mathematical understanding firstly will need to be recognised as equally important - though utterly distinct - from the analytic. Then when this neglected aspect undergoes appropriate specialisation, both the analytic and holistic aspects will finally be required to be coherently integrated with each other in a harmonious interactive manner.

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