And this can only be properly appreciated in a dynamic interactive context, where analytic and holistic appreciation - representing the independent and interdependent aspects of the number system respectively - are seen as directly complementary with each other.
Looked at from the more psychological perspective, when the natural number system is interpreted in the customary (linear) rational conscious manner, then the Riemann zeros represent the (unrecognised) unconscious appreciation of this system. Now whereas the direct nature of this appreciation is purely intuitive in nature, indirectly it is then objectively conveyed in a (circular) paradoxical rational fashion. And this again is the precise meaning of "imaginary" from a holistic mathematical perspective!
Therefore the very appreciation of the Riemann zeros requires that the (hidden) unconscious aspect of mathematical understanding be brought fully into the conscious light.
Only then can both the conscious and unconscious aspects of mathematical understanding - which in truth are ultimately truly interdependent - be fully integrated with each other!
And of course, here we face again the most fundamental problem possible with accepted mathematical interpretation in that as a formal discipline of thought it remains totally blind to its deep unconscious aspect.
In other words, everywhere in interpretation the unconscious (intuitive) aspect of understanding is reduced in a merely formal (linear) rational conscious manner.
Thus in basic terms, in every context the holistic notion of "interdependence" is reduced in an analytic (i.e. independent) manner.
Therefore, as I have repeatedly stated in these blog entries, nothing less than a total revolution is now required in mathematical thinking representing by far the most radical transformation yet in our intellectual history. Not alone will this have the most fundamental repercussions possible for Mathematics (changing and enlarging its scope in an utterly unparalleled manner) but equally it will have the most profound implications for all the sciences and for society generally.
So at present we have just one recognised form of Mathematics i.e. Type 1 Mathematics which is interpreted in an absolute analytical manner.
However in a future golden age of Mathematics we will have at least three interacting forms.
1) Type 1 Mathematics. This will largely equate with present analytic interpretation, though now understood in a relative rather than absolute manner.
2) Type 2 Mathematics. This will be identified with the qualitative interpretation of mathematical symbols, in what I refer to as "Holistic Mathematics".
3) Type 3 Mathematics. This will represent the most comprehensive type of mathematical understanding as the integrated appreciation of both analytic (Type 1) and holistic (Type 2) aspects.
Ultimately both Type 1 and Type aspects can only obtain their true - relatively - separate status through the perspective of integrated Type 3 understanding. In the past I have referred to this as "Radial Mathematics".
In fact these blog entries on the nature of the number system are designed to represent but the most preliminary excursion into Radial Mathematics.
So far however, we have looked at the holistic nature of the Riemann zeros (from the starting appreciation of the standard analytic interpretation of the number system).
However in dynamic interactive terms, these interchange with each other, so that we equally have a holistic interpretation of the natural numbers and an analytic interpretation of the Riemann zeros respectively.
Now one might ask what the holistic appreciation of the natural number system entails!
Whereas with standard analytic interpretation we address the natural number system in collective terms (i.e. as a collective whole), here in complementary fashion we view each member of this system in individual terms (where each number now represents a unique whole with respect to its unit members).
Once again let us use the number "3" to illustrate!
From the analytic perspective this number is viewed as composed of independent unit members
i.e. 3 = 1 + 1 + 1 (where each unit is homogeneous and independent).
However from the corresponding holistic perspective this number is now composed of interdependent unit members i.e. 3 = 1 + 1 + 1 (where each unit has a unique meaning through interdependence with the other units).
So what happens in actual experience is that one keeps switching - though not formally recognised in standard interpretation - from the quantitative notion of "3" (representing independent units) to the qualitative notion of "3" (representing "threeness" as the interdependence of units).
This also directly equates with the continual switch from cardinal notions of "3" (where component units are homogeneous) to ordinal notions of "3" (where component units are unique as 1st, 2nd and 3rd respectively).
However just as the analytic interpretation of the (cardinal) natural number system is complemented by the holistic interpretation of the Riemann zeros, equally the holistic interpretation of each natural number (i.e. ultimately with respect to the unique ordinal positions of each prime number) is then complemented by the analytic interpretation of the Riemann zeros.
And this very much conforms with the standard appreciation of the zeros, where they can now be used collectively in a quantitative manner to exactly predict the locations of the primes (having corrected with respect to their initial general frequency).
One of the great problems with the current attempt to appreciate the nature of primes is that they are invariably studied with respect to their cardinal nature. So in effect the ordinal aspect of the primes is simply reduced in cardinal terms.
However properly understood both the cardinal and ordinal aspects of primes - as quantitative to qualitative (and qualitative to quantitative respectively) - are ultimately fully complementary with each other (as in experience) in a dynamic interactive manner.
Now when one begins to realise this, one can then appreciate that the relationship between the primes and natural numbers (and natural numbers and primes) is inherently of a paradoxical nature.
In other words, from a dynamic interactive perspective, they mutually depend on each other.
So, for example from the conventional cardinal perspective "3" as a prime is viewed as an independent quantitative "building block" of the natural number system.
However from the complementary ordinal perspective "3" as a prime is already uniquely defined and thereby depends on its 1st, 2nd and 3rd members.
So from the cardinal perspective, each (composite) natural number system appears to depend on individual primes (as building blocks). However from the ordinal perspective, each individual prime already depends on a unique collection of natural number members.
Thus there are two complementary forces at work with respect to the relationship between the primes and the natural numbers.
In Type 1 terms the number 6 (as quantitative aspect) is expressed as (2 * 3)1
However in Type 2 terms the number 6 (as - relatively - qualitative aspect) is expressed as 1(2 * 3)
So we see that both base and dimensional numbers switch in complementary fashion as between the two definitions.
This simply implies that whereas the natural numbers appear to be quantitatively generated from the primes in Type 1 terms that it is the reverse from a Type 2 perspective (i.e. the primes are qualitatively generated - i.e. assume a unique quality of interdependence - through their relationship with the natural numbers).
And in both cases - with respect to both the Type 1 and Type interpretations of the natural number system, we have a (shadow) set of Riemann zeros, which enables the consistent interface of both Type 1 and Type meanings.
So the zeros enable the conversion (1) from quantitative to qualitative meaning and (2) from qualitative to quantitative meaning.
Putting it more simply, underlying our conventional (conscious) understanding of both the cardinal and ordinal nature of number is a hidden (unconscious) system (Riemann zeros) that enables the successful transition as between both aspects.
So the great task in terms of a coherent understanding of the number system is to bring this unconscious system fully into the conscious light.