This confusion arises directly from attempting to deal with - what inherently represent - dynamic interactive relationships (possessing a mere relative validity) in a misleading static absolute fashion.
This is especially true - in the most fundamental sense - with respect to the true nature of the number system. Rightly interpreted, this entails in experience the continual interaction of both quantitative (analytic) and qualitative (holistic) aspects (with an arbitrary relative validity). However for several millennia now we have committed ourselves to a reduced interpretation (in absolute terms) with sole recognition given to the quantitative aspect.
Now I would always recognise the important validity of this approach (in a limited partial i.e. quantitative sense). However the huge mistake is in elevating it as being synonymous with all valid Mathematics.
My criticism could not be more fundamental for it embraces all accepted present mathematical activity.
So it cannot be countered through more contemporary developments such as p-adic number systems for these too are interpreted within the same reduced quantitative approach.
The precise nature of the conventional mathematical approach can be accurately described as 1-dimensional.
To understand this more clearly let me once more place this observation in context.
All phenomenal experience - including of course mathematical - is conditioned by the interaction of important polarity sets (two of which are especially important).
So all external (objective) recognition necessarily entails a corresponding internal mental means of (subjective) interpretation.
Thus external and internal polarities necessarily dynamically interact in a continual manner in experience.
Secondly all quantitative (analytic) recognition necessarily entails a corresponding qualitative (holistic) aspect with both of these aspects again continually interacting in experience.
Now a whole new type of mathematical appreciation - operating according to uniquely distinctive means of circular type interpretation - opens up when we consider mathematical symbols and relationships explicitly in such a dynamic context.
So we here recognise in a much more refined rational manner how interpretation keeps switching as between opposite polar reference frames (in a dynamic interactive manner).
Thus for example in the simplest dynamic situation we must use two complementary reference frames. And such higher-dimensional interpretation (where each number represents a certain dynamic configuration with respect to opposite polarities) defines the qualitative (holistic) approach.
Now Conventional Mathematics avoids all such dynamic considerations by simply defining interpretation with respect to just one pole of reference. So typically we portray number as possessing an objective identity (without reference to internal interpretation). Equally we portray numbers as possessing an independent quantitative identity without reference to their overall relationship with other numbers (which is properly of a qualitative nature).
Now it should be immediately obvious that it is not in fact possible to give a cardinal (quantitative) identity to a number without an implied ordinal (qualitative) identity and vice versa.
However the quantitative confusion runs so deep in Conventional Mathematics that we have somehow convinced ourselves that the ordinal nature of number likewise refers to mere quantitative identity!
In other words the reduced nature of our interpretation is so entrenched that we no longer even recognise the huge confusion involved!
So the vital key to appreciating the dynamic nature of number is that in experience, we always have the interaction of both quantitative and qualitative aspects (operating according to two uniquely distinct modes of interpretation)
So in actual experience we necessarily switch as between cardinal and ordinal modes in a dynamic manner.
However Conventional Mathematics completely misrepresents the nature of such interaction by attempting to interpret number with respect to its mere quantitative aspect.
And because it is necessarily defined in a 1-dimensional (quantitative) manner, furthermore it is entirely lacking any means of properly incorporating its neglected qualitative aspect.
This is precisely why an entirely new paradigm is now urgently needed in Mathematics.
Let us now see how understanding of the relationship as between the primes and the natural numbers is transformed through 2-dimensional (rather than 1-dimensional) interpretation.
In the conventional mathematical approach, the primes are seen in cardinal terms as the quantitative building blocks of the natural number system (the atoms as it were of the system).
However this approach is utterly misleading.
If we take the prime number 3 for example, in cardinal terms it is given a collective quantitative identity. If we were to attempt to break this number down in quantitative terms through its constituent units, it would be represented as 1 + 1 + 1.
But there remains a crucial problem here in that this quantitative definition (in terms of homogeneous units) leaves us with no means of making an ordinal (qualitative) distinction as between the units.
In other words in a number group of 3 members we thereby have no means of uniquely distinguishing in any context its 1st, 2nd and 3rd members!
So the ordinal (qualitative) recognition of 1st 2nd and 3rd (which depends on the relational context of each member within the group) cannot be derived from the cardinal (quantitative) definition of number (viewed in collective whole terms as independent).
Though this issue is truly of the first magnitude it is completely overlooked in conventional terms.
So it needs an utterly distinctive (Type 2) holistic interpretation of number to deal with the ordinal aspect! So, as I have outlined repeatedly in various blog entries we can indirectly express in quantitative terms the ordinal nature of n units (with n a prime number). This is done in a circular - rather than linear manner - as points on the circle of unit radius, through obtaining the corresponding n roots of 1. Thus the relative independence of each member of the number group in an indirect quantitative manner is given through each root (in isolation). The overall qualitative interdependence - again of a relative nature - of the n members is then expressed through the sum of roots = 0.
Thus when we adopt both the Type 1 (cardinal) and the Type 2 (ordinal) approaches to number, the relationship between the primes and natural numbers operates in two diametrically opposite ways.
From the Type 1 approach the primes appear to serve as unique quantitative building blocks of the natural numbers (in a cardinal manner).
However from the Type 2 approach the natural numbers appear to serve - in reverse manner - as the unique qualitative building blocks of the primes (in an ordinal fashion). So again from this perspective the prime number 3 comprises 1st, 2nd and 3rd members (which as we have seen can then be indirectly given quantitative expression in a circular numerical fashion).
So the quantitative (cardinal) perspective we see the natural numbers in terms of the primes; then from the qualitative (ordinal) perspective we see the primes in terms of the natural numbers!
Now the holistic (2-dimensional) approach essentially consists in wedding these two complementary perspectives (which in dualistic terms are paradoxical in terms of each other) in a single nondual intuitive type realisation. And the very ability to “see” in a unified manner through such intuition, is intimately related to the corresponding ability to identify opposite characteristics accurately with respect to the two frames of reference!
Therefore the holistic (qualitative) is utterly distinct from the corresponding analytic (quantitative) approach.
he latter is directly rational in a linear manner; the former however is directly intuitive (which is then indirectly expressed rationally in a circular paradoxical fashion).
So when we are enabled to switch smoothly in an increasingly dynamic manner as between both reference frames, we see the primes and natural numbers ever more closely as perfect mirrors of each other (with the ultimate pure spiritual experience of this relationship ineffable in nature).