As I have repeated often in these blogs, the true nature of number (as indeed all mathematical activity) is of an inherently dynamic interactive nature. Unfortunately Conventional Mathematics provides but a reduced and thereby distorted interpretation of number.
Firstly number inherently has both external (objective) and internal (subjective) aspects.
We cannot externally envisage a physical number “object” in the absence of the corresponding psychological mental perception of the number. So properly understood these two aspects necessarily continually interact in a relative manner with respect to experience.
Put another way, experience necessarily entails the interaction of two aspects of number that are physical and psychological with respect to each other.
Once again Conventional Mathematics gives but a reduced interpretation of this interaction.
Now a professional mathematician if sufficiently pressed might eventually concede that we cannot form knowledge of the number “object” in the absence of its corresponding mental perception. However the necessary interaction thereby involved is then completely ignored with interpretation taking place in a misleading absolute fashion. Thus the erroneous notion of numbers as abstract objective entities still dominates conventional thinking.
Because of its linear 1-dimensional nature Conventional Mathematics can only handle such dynamic interactions in a reduced manner whereby the subjective mental aspect is identified with the objective (which is predominant) or alternatively the objective aspect wirh its mental perception. In either case we then get an absolute rather than - more correctly - a truly relative interpretation of the nature of number.
Therefore to repeat once more the true dynamic nature of number necessarily entails twin interacting elements that are external (physical) and internal (psychological) with respect to each other.
This means in effect that once we identify for example – as recently with the (Type 1) non-trivial zeros – their physical similarity to certain quantum chaotic processes, this automatically entails that they must necessarily also have an equally important significance in complementary psychological terms.
However because Conventional Mathematics is completely lacking in dynamic interpretation it thereby places no emphasis on such complementary type relationships.
Therefore the extremely important psycho spiritual significance of the non-trivial zeros still remains completely unrecognised by the conventional mathematical community!
The other key distinction is with respect to the quantitative and qualitative aspects of number!
If we take the number “3” to illustrate we cannot experientially identify the cardinal nature of this number without implicitly recognising that it necessarily contains a 1st, a 2nd and 3rd member in ordinal terms. Thus the quantitative recognition of “3” implies corresponding ordinal recognition of its 1st, 2nd and 3rd members in a corresponding qualitative manner. And in reverse manner we cannot form knowledge of the ordinal members of a group without implicitly recognising its cardinal (quantitative) identity.
So all mathematical experience is fundamentally conditioned by the dynamic interaction as between opposite sets of polarities.
Chief among these are the external/internal that operate in a horizontal manner and the quantitative/qualitative that operates in a corresponding vertical manner.
In fact it may help to initially recognise the relationship as between them as like a compass with the four directions East and West along the horizontal axis and North and South along the vertical axis respectively.
However we can give a firmer mathematical rationale to these locations (in terms of the Type 2 aspect of the number system) by recognising these four equidistant points as corresponding to the four roots of 1. So the external and internal polarities are – relatively – complementary in a real manner (with directions that are + 1 and – 1 with respect to each other).
The qualitative therefore has likewise two directions that are + i and – i with respect to each other.
Quantitative and qualitative polarities are thereby real and imaginary with respect to each other.
What this means in effect is that the individual members of a number set have a unique qualitative meaning i.e. in their ordinal identity. However the overall set – which we initially identified in quantitative terms as cardinal likewise has an ordinal identity when related to other numbers.
So for example 2 and 3 are prime numbers (in a cardinal manner). However they equally enjoy a qualitative ordinal identity as the 1st and 2nd prime numbers respectively.
So the individual members of a cardinal prime number such as 3 enjoy a unique qualitative identity (in terms of its 1st, 2nd and 3rd members).
Thus the cardinal prime is ordinally defined in terms of its natural number members (in a corresponding qualitative manner).
However the same prime number 3 enjoys a unique collective qualitative identity as the 2nd prime in the natural number system. And all natural numbers represent a unique combination of these prime number factors.
Thus when properly understood, the all important relationship as between the primes and the natural numbers represents the fundamental manner by which their quantitative and qualitative aspects are related.
And as always – in dynamic interactive terms – there are two complementary ways in which this relationship can be understood:
1) whereby the natural number system as a whole is collectively defined through unique combinations of its prime number members.
2) whereby each prime number is uniquely defined through a collection of its natural number members.
Putting it bluntly therefore the conventional mathematical attempt to define the relationship as between the primes and the natural numbers misses the crucial point that this relationship entails a dynamic two-way complementarity as between its quantitative and qualitative aspects.
From one perspective, it is quite extraordinary how we have remained blind to this key relationship for so long!
We have tried to convince ourselves that Mathematics is solely concerned with the quantitative aspect of number. However strictly speaking we cannot even begin to identify the quantitative aspect without implicit recognition of its corresponding qualitative aspect.
Thus Mathematics is properly – in dynamic terms – as for example here with number, concerned with the relationship as between its quantitative and qualitative aspects.
Likewise – in relation to the other polarity set - we cannot form an objective knowledge of number (as external) without a corresponding mental interpretation (that is - relatively - internal).
Thus again in dynamic terms, Mathematics is properly about the relationship of objective type results to the corresponding interpretations (through which they are viewed). And from this perspective there is not just one absolute type interpretation that is valid but potentially an unlimited number (each enjoying a partial relative validity).
Once again the relationship as between the quantitative (analytic) and qualitative (holistic) aspects of number fundamentally points to the corresponding relationship as between the primes and natural numbers (and natural numbers and the primes).
And mediating this relationship are two important sets of zeta zeros (corresponding to the Type 1 and Type 2 number systems respectively).
As I have stated on a number of occasions, these zeros essentially can be viewed as the shadow of our one-sided quantitative view of number (i.e. where the qualitative aspect is directly confused with its quantitative expression).
So we can fruitfully view both sets of zeros as a means of giving two distinctive expressions (in a related complementary fashion) to the long unrecognised qualitative aspect of the number system.