I keep returning to this central point.
The mystery of the relationship of addition and multiplication, with respect to the number system, relates simply to the corresponding mysterious relationship as between its cardinal and ordinal aspects.
Cardinal relates to the quantitative aspect of numbers (considered as independent entities); ordinal - by contrast - relates to the corresponding qualitative aspect (where interdependent relationships as between numbers occur).
Properly understood therefore, the very nature of number is inherently dynamic representing the continual interaction of both its quantitative (cardinal) and qualitative (ordinal) aspects.
Remarkably however for several millennia, we have sought to interpret number - and by extension all mathematical activity - in a highly reduced manner (i.e. solely with respect to its quantitative aspect).
Though this has indeed enabled remarkable progress with respect to the specialisation of one valid aspect, overall it has left us with a severely limited - and thereby gravely distorted - appreciation of the true nature of number (and indeed of the true nature of Mathematics).
As this simple truth regarding the nature of our number system is of the most fundamental nature possible, it has the capacity - I firmly believe - to ultimately lead to the greatest revolution yet in Mathematics (and by extension all the sciences).
Though the inherently dynamic nature of number can be explained without reference to the Riemann Hypothesis, proper appreciation of the Hypothesis however does require such dynamic understanding.
Indeed the Riemann Hypothesis can be validly expressed as the central condition necessary for achieving the ultimate identity of both the cardinal and ordinal notions of number. Looked at in an equivalent manner it serves as the key requirement for the full reconciliation of quantitative and qualitative meaning (with respect to all created phenomena).
The significance of the primes and natural numbers in this context is that the key relationship of quantitative to qualitative is mediated through the two-way relationship as between both sets of numbers.
From the cardinal (quantitative) perspective i.e. Type 1, the prime numbers appear unambiguously as the building blocks of the natural number system. However from the equally valid - though totally neglected ordinal (qualitative) aspect i.e. Type 2 - the natural numbers equally appear in unambiguous terms as the building blocks of each prime number!
Then in the balanced and refined experiential interaction of both cardinal and ordinal aspects i.e. Type 3, both the prime numbers and natural numbers are understood in a transparent two-way complementary manner approaching simultaneity, as perfect mirrors of each other in their common identity (which is of an absolute ineffable nature).
Indeed one could validly maintain without hyperbole that the essential secret underlying all creation relates to this original identity of both the prime and natural numbers (and natural numbers and primes) which inevitably is separated through the process of phenomenal evolution.
One could equally maintain that the very purpose of such evolution is to finally understand (in an ultimately ineffable manner) its original greatest secret.
Again, when viewed from this perspective this true nature of number could hardly be more important as it is thereby inherently encoded in all created phenomena (both physical and psychological) as the original source of their very nature.
Thus in this sense everything in creation represents but a veiled expression of the dynamic interaction of number unfolding in relative space and time (whose ultimate identity however is ineffable).
So the critical fact that we have committed ourselves for so long to a reduced and distorted interpretation of number represents an issue of the very first magnitude. As many of the present great problems facing our world ultimately have their roots in this problem they cannot be solved without a radical new appreciation of its inherent dynamic nature.
I will briefly highlight here once more the essential nature of addition and multiplication.
To do this a number must be defined with respect to both base (or ground) and dimensional aspects.
From the Type 1 perspective the base aspect of number varies while its dimensional aspect remains fixed as 1. By contrast from the Type 2 perspective, the base aspect remains fixed as 1 while the dimensional aspect varies.
So 2 from the Type 1 aspect is represented as 2^1. By contrast from the Type 2 perspective 2 is represented as 1^2.
Thus the Type 1 and Type 2 aspects of number are direct inverses of each other with respect to their base and dimensional aspects respectively.
Properly appreciated the actual experience of number keeps switching as between its Type 1 and Type 2 aspects serving as the very means by which both cardinal (quantitative) and ordinal (qualitative) interpretation arises.
So every number e.g. 2, has a dynamic dual identity whereby meaning keeps switching as between its cardinal and ordinal identities respectively both of which correspond to distinct modes of interpretation. By extension every mathematical notion with an established quantitative (analytic) can be equally given an - as yet unrecognised - qualitative (holistic) interpretation. So properly understood, comprehensive mathematical interpretation represents the balanced interaction of both types of meaning.
Once again the Type 1 aspect of number relates to the quantitative interpretation of number in cardinal terms as comprising a whole unit.
Conventional Mathematics in formal terms operates within a reduced Type 1 interpretation of number that relates solely to its cardinal (quantitative) aspect.
From the Type 1 perspective 2 = 1 + 1 (where each of its units is viewed as homogeneous and thereby identical in quantitative terms). In other words each unit is thereby viewed as without qualitative distinction!
Thus when fully represented in Type 1 terms 2^1 = 1^1 + 1^1. So the cardinal notion of 2 is expressed in terms of the addition of its two constituent units (in quantitative terms).
However as repeatedly stated in my blogs, when we define number merely in this reduced quantitative manner, we lack the means therefore of enabling a qualitative distinction of the units to take place (in ordinal terms).
In other words defining 2 = 1 + 1 provides us with no means of distinguishing the 1st from the 2nd unit in ordinal terms (as this entails a qualitative ranking of units).
Therefore the ordinal nature of the individual units of 2 relate to a distinctive Type 2 interpretation - relating directly to multiplication - where 2 = 1 * 1.
Now to appreciate this properly, we must again define number fully with respect to both base and dimensional values.
So from this perspective, 1^2 = 1^1 * 1^1.
We could also of course write 1^2 = 1^(1 + 1).
So interestingly what represents (pure) multiplication with respect to the Type 1 system i.e. 1 * 1 represents (pure) addition with respect to the Type 2 i.e. 1 + 1.
This points to the very essence of addition and multiplication, with these operations inherently relating to the critical distinction as between the cardinal (quantitative) and ordinal (qualitative) interpretation of number.
It also implies that quantitative and qualitative have a merely relative arbitrary distinction. Thus when we fix the frame of reference with the base number as quantity, the dimensional value is then qualitative (in this relative context). However within its own frame of reference, the dimensional value can likewise be given a quantitative interpretation, whereby the base number is thereby - relatively - qualitative in nature!
To properly appreciate this ordinal nature of 2, we must indirectly represent it in a circular manner through obtaining the corresponding two roots of 1.
So in ordinal terms the two (qualitatively) distinct members of 2 relate to its 1st and 2nd members respectively.
Therefore, to indirectly represent these distinctive members in a quantitative circular manner, we must obtain the roots of 1^1 and 1^2 respectively.
So the square root of 1^2 = + 1 and the square root of 1^1 = - 1 respectively.
Therefore the 1st and 2nd members of 2 can be indirectly represented in a circular manner (as equidistant points on the circle of unit radius in the complex plane) as + 1 and - 1 respectively.
It must be appreciated that understanding here is inherently of a dynamic relative nature!
Thus the very means by which we are enabled to distinguish 1st and 2nd, implicitly entails a continual process of (conscious) positing and (unconscious) negating in experience.
In other words to consciously recognise the 2nd member as distinct from the 1st, we must unconsciously negate in experience exclusive identification with the 1st member.
By this means we literally are enabled to place the two numbers in relation to each other (which is what the qualitative aspect directly implies).
The important implication of this is that the ordinal appreciation of number is necessarily of a dynamic relative nature in experience (entailing unconscious recognition).
And as the cardinal is inextricably linked with its ordinal aspect, this means likewise that cardinal appreciation of number is necessarily also of a merely relative nature.
In other words, properly understood, natural numbers have both a cardinal identity (as relatively independent) and an ordinal identity (as relatively interdependent) respectively.
Thus the accepted absolute appreciation of numbers in rational objective terms as abstract independent entities represents a grave distortion of their true nature.
And it is this distorted interpretation that has dominated Western thought now for several millennia!
With this in mind let us go back again to the fundamental relationship as between the primes and natural numbers (and the natural numbers and the primes).
From a Type 1 perspective, the primes are seen as the building blocks of the (cardinal) natural numbers (in quantitative terms).
So from this perspective the number 6 for example is represented uniquely as the product of its two prime factors i.e. 6 = 2 * 3.
When we represent this more accurately with respect to both base and dimensional aspects,
6^1 = (2*3)^1.
However from a Type 2 perspective, the natural numbers are seen as the ordinal building blocks of the primes (in a qualitative manner).
So again from this perspective the prime number 3 for example is uniquely composed in natural number ordinal terms of a 1st, 2nd and 3rd member!
Thus from this alternative perspective, the number 6 is represented uniquely as
1^6 = 1^(2*3).
So to indirectly represent this dimensional notion of 6 we obtain the 6th root of 1!
Thus the uniqueness of this 6th root of 1 (indirectly representing the ordinal nature of 6) is that it lies as a distinctive point on the circle of unit radius (in the complex plane).
Therefore we have now two complementary notions of the prime numbers as building blocks.
From the first quantitative perspective each natural number in cardinal terms is uniquely built from the product of prime factors (representing base values).
From the 2nd qualitative perspective - indirectly represented in a quantitative manner - each natural number in ordinal terms is likewise uniquely built as the product of prime factors (representing dimensional values).
However the relationship of primes to natural numbers (and natural numbers to primes) is diametrically opposite in both cases.
Therefore when we combine these approaches (in Type 3 understanding) both the quantitative (cardinal) and qualitative (ordinal) aspects of the number system simultaneously arise with the primes and natural numbers ultimately seen as entirely interdependent with each other (in an ineffable manner).
And this is the true mysterious nature of our number system where prior causation
(in terms of either primes or natural numbers) has no strict meaning.
Once again properly understood the Riemann Zeta Function (especially through its Functional Equation) represents the dynamic relationship as between the cardinal and ordinal aspects of number. The Riemann Hypothesis then represents the condition for the ultimate identity of both aspects.
As the very nature of this identity greatly transcends conventional mathematical interpretation there is no way of course of proving (or disproving) the Hypothesis (within its axioms).
The very fact that this is not yet clearly recognised indicates how limited in fact is our current appreciation of the number system!