Again in conventional (absolute Type 1) terms,

ζ(4) = 1/1

^{4}+ 1/2

^{4 }+ 1/3

^{4 }+

^{ ... }=

^{ }1/1

^{ }+ 1/16 + 1/81 + ... = π

^{4}/90.

However, when we combine both (linear) Type 1 and (circular) Type 2 aspects in a relative interactive fashion,

ζ(4) = (1/1

^{1 }* 1

^{4}) + (1/16

^{1 }* 1

^{4}) + (1/81

^{1 }* 1

^{4}) + ... = π

^{4}/90.

So the quantitative aspect is expressed through the Type 1 aspect (where varying numbers as base are raised to the fixed dimensional power of 1).

However in inverse fashion, the qualitative aspect is expressed through the Type 2 aspect (where 1 is raised to varying dimensional powers, in this case = 4).

Thus, whereas the Type 1 aspect relates to number as having a relatively independent (quantitative) identity, the Type 2 aspect by contrast relates to the complementary notion of number as possessing a relatively interdependent (qualitative) identity.,

So in particular, 1

^{4 }thereby represents the qualitative notion of 4 as interdependent i.e. where the 4 members of a number group are considered as potentially interchangeable in ordinal fashion with each other.
In other words, here, each individual member can be potentially viewed as 1st, 2nd, 3rd or 4th (depending on context).

Then when x

So the ordinal notion of number - which inherently is of a qualitative nature - is directly related in this context to the Type 2 aspect.

The huge unaddressed problem therefore with the conventional mathematical approach to the Riemann zeta function, is that it confuses the ordinal with the corresponding cardinal notion of number, treating both absolutely in a merely reduced quantitative manner.

However the true nature of number is dynamically interactive in a relative fashion, with complementary aspects that are quantitative and qualitative (and qualitative and quantitative) with respect to each other.

Just as the 4-dimensional appreciation of reality is greatly important in accepted physical analytic terms (based on reduced quantitative notions), the corresponding 4-dimensional holistic appreciation (based on Type 2 qualitative mathematical notions) - though largely unrecognised - is I believe potentially of more fundamental importance.

Therefore in holistic mathematical terms, all reality is governed by 4 "dimensions" where however a dimension now carries a new dynamic interactive meaning (that is equally applicable in both physical and psychological terms).

Thus we can express these dimensions (as with the 4 roots of 1) as relating to two real poles that are positive and negative with respect to each other and equally two imaginary poles that are positive and negative with respect to each other.

The real poles constitute with what psychologically relates to conscious understanding, whereas the imaginary relate to unconscious understanding (that indirectly is conveyed in a conscious manner). In scientific and mathematical terms, the real relate to analytic, whereas the imaginary relate to holistic interpretation respectively.

Positive and negative directions relate to the fact that all understanding is conditioned by external (objective) and internal (subjective) aspects that dynamically interact in a complementary fashion. So we cannot strictly have "objective truth" in the absence of "subjective interpretation". Thus what is objectively held as true always - whether explicitly recognised or not - represents a particular mental interpretation. And this of course fundamentally applies to mathematical truth.

So when we change the nature of interpretation employed - as I am doing here with respect to the Riemann zeta function - we thereby change the nature of the objective truth arising!

Real and imaginary aspects relate to the nature of wholes and parts, which again condition all phenomena.

The most important problem in both science and mathematics is the tendency to reduce - in any context - the whole in terms of its parts corresponding to a merely quantitative approach.

So "reality" is thereby defined literally in terms of what is "real" in this reduced quantitative sense.

However the true relationship as between wholes and parts (and parts and wholes) is as real to imaginary (and imaginary to real). and this is precisely the approach that I have been at pains to illustrate in this blog.

So properly understood all "reality" is complex with both real and imaginary aspects (in both quantitative and qualitative terms).

Thus these four "dimensions" operate like primary colours from which all other dimensional numbers are derived.

I fact I have frequently drawn attention to the importance of 24 in this context.

So the 24 personality types that I have delineated, represent 24 special configurations with respect to the manner in which the four primary colours (i.e. dimensions) are balanced with each other. And this equally applies to "impersonality types" as the basic ingredients of physical reality (as highlighted recently in the various string and superstring theories).

However all the other integers equally represent unique configurations with respect to the manner that external and internal poles and real and imaginary aspects operate (which geometrically can be represented as equidistant points on the unit circle in the complex plane).

Thus the true significance of the dimensional number i.e. s, with respect to the Riemann zeta function is that - when properly understood in a dynamic relative manner - it defines the ordinal nature of the various members of the group s.

And then the cardinal (quantitative) order with respect to the natural numbers and the ordinal (qualitative) order with respect to s, can be harmonised with each other, as the perfect balance of both linear and circular aspects (both objectively in quantitative terms and subjectively in a qualitative manner).

And this is deeply symbolised by the result (for even values of s) where π is raised to the corresponding value of s.

So when one truly understands in the appropriate 4-dimensional manner (implying both real and imaginary aspects in positive and negative directions) then the quantitative result relating to π

So what we have here then is an indirect rational way - using the most famous of all transcendental numbers - of expressing the intuitive nature of such experience.

Now a corresponding qualitative order applies to the denominator of the expression (where the distinct prime numbers, constituting the factors of this number are ordered in a perfect systematic fashion).

In fact in relation to all the denominators of numerical results for all positive even values of s (in the Riemann zeta function) we can make the following observations (which seem to apply universally).

Where s is a power of 2, the denominator will contain all primes (as factors) from 2 to s + 1 (and only these primes).

Where s is not a power of 2, the denominator will contain all primes (as factors) from 3 to s + 1 (and only these primes).

Thus is this case as s = 4 represents a power of 2, then the denominator 90 will contain all primes from 2 to 5 (and only those primes)

And 90 = 2 * 3

What is surprising here is that the product over primes expression for ζ(s) contains all the primes. And yet the result (where s is even) shows this amazing order with respect to a limited number of primes (defined by the value of s).

So from one perspective (in looking at the power of π), we saw how the quantitative independence of all the natural numbers (in the sum over all natural numbers expression) was then balanced in the resulting sum arising, by the qualitative order i.e. ordinal nature, of all natural numbers (defined by s).

Then from the other perspective (in looking at the denominator of the power of π result), we can likewise see how the initial quantitative independence of all the prime factors (in the product over primes expression) is then balanced in the result by the qualitative order of all prime factors (defined by s).

^{4 }= 1, each of the 4 roots of 1, i.e. i, – 1, – i and 1 , (where x thereby becomes expressed in a 1-dimensional manner), represent the (fixed) notions of 1st, 2nd, 3rd and 4th in any given actual context.So the ordinal notion of number - which inherently is of a qualitative nature - is directly related in this context to the Type 2 aspect.

The huge unaddressed problem therefore with the conventional mathematical approach to the Riemann zeta function, is that it confuses the ordinal with the corresponding cardinal notion of number, treating both absolutely in a merely reduced quantitative manner.

However the true nature of number is dynamically interactive in a relative fashion, with complementary aspects that are quantitative and qualitative (and qualitative and quantitative) with respect to each other.

Just as the 4-dimensional appreciation of reality is greatly important in accepted physical analytic terms (based on reduced quantitative notions), the corresponding 4-dimensional holistic appreciation (based on Type 2 qualitative mathematical notions) - though largely unrecognised - is I believe potentially of more fundamental importance.

Therefore in holistic mathematical terms, all reality is governed by 4 "dimensions" where however a dimension now carries a new dynamic interactive meaning (that is equally applicable in both physical and psychological terms).

Thus we can express these dimensions (as with the 4 roots of 1) as relating to two real poles that are positive and negative with respect to each other and equally two imaginary poles that are positive and negative with respect to each other.

The real poles constitute with what psychologically relates to conscious understanding, whereas the imaginary relate to unconscious understanding (that indirectly is conveyed in a conscious manner). In scientific and mathematical terms, the real relate to analytic, whereas the imaginary relate to holistic interpretation respectively.

Positive and negative directions relate to the fact that all understanding is conditioned by external (objective) and internal (subjective) aspects that dynamically interact in a complementary fashion. So we cannot strictly have "objective truth" in the absence of "subjective interpretation". Thus what is objectively held as true always - whether explicitly recognised or not - represents a particular mental interpretation. And this of course fundamentally applies to mathematical truth.

So when we change the nature of interpretation employed - as I am doing here with respect to the Riemann zeta function - we thereby change the nature of the objective truth arising!

Real and imaginary aspects relate to the nature of wholes and parts, which again condition all phenomena.

The most important problem in both science and mathematics is the tendency to reduce - in any context - the whole in terms of its parts corresponding to a merely quantitative approach.

So "reality" is thereby defined literally in terms of what is "real" in this reduced quantitative sense.

However the true relationship as between wholes and parts (and parts and wholes) is as real to imaginary (and imaginary to real). and this is precisely the approach that I have been at pains to illustrate in this blog.

So properly understood all "reality" is complex with both real and imaginary aspects (in both quantitative and qualitative terms).

Thus these four "dimensions" operate like primary colours from which all other dimensional numbers are derived.

I fact I have frequently drawn attention to the importance of 24 in this context.

So the 24 personality types that I have delineated, represent 24 special configurations with respect to the manner in which the four primary colours (i.e. dimensions) are balanced with each other. And this equally applies to "impersonality types" as the basic ingredients of physical reality (as highlighted recently in the various string and superstring theories).

However all the other integers equally represent unique configurations with respect to the manner that external and internal poles and real and imaginary aspects operate (which geometrically can be represented as equidistant points on the unit circle in the complex plane).

Thus the true significance of the dimensional number i.e. s, with respect to the Riemann zeta function is that - when properly understood in a dynamic relative manner - it defines the ordinal nature of the various members of the group s.

And then the cardinal (quantitative) order with respect to the natural numbers and the ordinal (qualitative) order with respect to s, can be harmonised with each other, as the perfect balance of both linear and circular aspects (both objectively in quantitative terms and subjectively in a qualitative manner).

And this is deeply symbolised by the result (for even values of s) where π is raised to the corresponding value of s.

So when one truly understands in the appropriate 4-dimensional manner (implying both real and imaginary aspects in positive and negative directions) then the quantitative result relating to π

^{4 }exactly matches one's qualitative manner of understanding so that both are perfectly united in an intuitive manner.So what we have here then is an indirect rational way - using the most famous of all transcendental numbers - of expressing the intuitive nature of such experience.

Now a corresponding qualitative order applies to the denominator of the expression (where the distinct prime numbers, constituting the factors of this number are ordered in a perfect systematic fashion).

In fact in relation to all the denominators of numerical results for all positive even values of s (in the Riemann zeta function) we can make the following observations (which seem to apply universally).

Where s is a power of 2, the denominator will contain all primes (as factors) from 2 to s + 1 (and only these primes).

Where s is not a power of 2, the denominator will contain all primes (as factors) from 3 to s + 1 (and only these primes).

Thus is this case as s = 4 represents a power of 2, then the denominator 90 will contain all primes from 2 to 5 (and only those primes)

And 90 = 2 * 3

^{2 }* 5!What is surprising here is that the product over primes expression for ζ(s) contains all the primes. And yet the result (where s is even) shows this amazing order with respect to a limited number of primes (defined by the value of s).

So from one perspective (in looking at the power of π), we saw how the quantitative independence of all the natural numbers (in the sum over all natural numbers expression) was then balanced in the resulting sum arising, by the qualitative order i.e. ordinal nature, of all natural numbers (defined by s).

Then from the other perspective (in looking at the denominator of the power of π result), we can likewise see how the initial quantitative independence of all the prime factors (in the product over primes expression) is then balanced in the result by the qualitative order of all prime factors (defined by s).

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