Therefore from one perspective, numbers possess a certain separate identity (which we customarily identify with its quantitative aspect).
However equally, numbers possess an overall relational identity as interdependent which is properly related to a distinct qualitative identity. What remains completely overlooked in conventional mathematical understanding is that this aspect cannot be successfully reduced in a mere quantitative manner.
This relationship as between analytic and holistic identities (quantitative and qualitative aspects) is especially important in the context of the overall relationship of the primes with the natural numbers (and the natural numbers with the primes).
Indeed this relationship is frequently explained in terms of musical analogies. So for example if the Riemann Hypothesis is true this suggests that the various separate notes of the great number symphony of the natural numbers i.e. the primes, while maintaining their unique individual identities (as separate) yet are related to each other in a perfectly synchronised manner (so that not one note is out of place).
However because of the exclusive quantitative bias of Conventional Mathematics, a persistent failure remains in recognising that this remarkable collective behaviour of the primes is necessarily of a (qualitative) holistic rather than a (quantitative) analytic nature.
A mathematician would readily accept, I imagine that while music can indeed be analysed in a quantitative manner (e.g. with respect to individual notes and chords), that it would be ludicrous to deny its overall qualitative nature, in the manner in which these are combined with each other.
So while the quantitative nature of music relates to the analysis of its individual components, its qualitative nature relates to the overall manner in which all these individual components are successively combined in a holistic manner!
However when we come to the nature of the natural number system - which in truth requires that the most perfect degree of synchronicity be maintained as between its various prime number notes - we still remain totally blind to the fact that this requires a distinct qualitative - as opposed to quantitative - mathematical dimension!
And the reason why we remain so blind as to a totally obvious realisation, is that we have committed ourselves for several millennia now to a distorted - and thereby false - view of the number system (that is defined in a merely quantitative manner).
The reduced quantitative view of the number system has created the erroneous impression that the world of number relates to an abstract set of absolute fixed entities operating independently of natural physical and psychological processes.
However nothing could be further from the truth, with number inherent in all such processes as the deepest encoding of their intrinsic nature (which is ultimately ineffable).
Therefore the reason why a holistic qualitative aspect applies to the dynamic nature of number is because this inherently applies to all natural processes.
Of course because our notions of physical science are so heavily influenced by the quantitative nature of Conventional Mathematics, there is likewise a deep refusal to recognise the obvious implications of - now - accepted physical behaviour at a quantum level.
For example it has been demonstrated through several carefully conducted experiments that subatomic particles (such as light) can communicate with each other, even at a great distance.
Now this clearly relates to a qualitative holistic capability of matter (akin to what might be recognised as a form of para-natural telepathy!)
Indeed if physicists were then to properly accepted such findings, they would be forced to accept that our normal macro physical world essentially operates on a para-natural basis (dictated by its underlying quantum processes).
Of course they deeply resist making such obvious connections precisely because this would call directly into question the reduced paradigm on which accepted notions are based.
However what I am attempting to convey clearly here is that the mysterious quantum behaviour of matter is itself rooted in the prior nature of the number system (when correctly interpreted in a dynamic interactive manner).
The two great issues that therefore need to be faced with respect to the nature of the number system (and indeed all mathematical notions) are as follows:
(i) that we cannot hope to divorce the external objective nature of mathematical relationships from the relative internal subjective mental means of their interpretation.
A deep problem relates to standard mathematical interpretation in attempting to reduce this relationship in an absolute manner (i.e. 1-dimensional interpretation).
This leads to the mistaken view that just one valid system of Mathematics exists.
In fact, properly understood - in dynamic interactive terms - an unlimited number of possible mathematical systems exist (each with an arbitrary partial validity).
So number here in its true dimensional sense is associated with a unique means of interpreting mathematical symbols. The use of 1 as dimension, refers therefore to the limited special case where internal and external polarities are reduced in terms of each other (in an absolute manner).
(ii) that we cannot hope to divorce the quantitative aspect of interpretation from its related qualitative aspect.
Again a deep problem relates to standard interpretation in again attempting to reduce this relationship in an absolute manner (in accordance with 1-dimensional interpretation).
This leads to the deeply mistaken view that number can be consistently understood with respect to its mere quantitative aspect. In effect this leads to a profound confusion with respect to the ordinal (qualitative) nature of number which is effectively reduced in a cardinal manner.
As I have repeatedly stated in my blog entries, the true nature of the number system is that it provides, through the non-trivial zeta zeros (Zeta 1 and Zeta 2), the means by which both its quantitative (cardinal) and qualitative (ordinal) aspects can be consistently related with each other.
In fact the zeta zeros represent the true indispensable holistic basis of the number system, which necessarily underlines our customary analytic understanding of this system.