Put another way the cardinal notion of a prime (in quantitative terms) can have no strict meaning in the absence of its natural number ordinal members (from a qualitative perspective).

Equally, in reverse terms, the natural number ordinal members can have no strict meaning in the absence of the cardinal notion of the prime.

So for example 3 as a prime has no strict meaning in the absence of its 1

^{st}, 2

^{nd}and 3

^{rd}members; likewise these 1

^{st}, 2

^{nd}and 3

^{rd}members in qualitative terms, have no strict meaning in the absence of the quantitative notion of 3.

Therefore, properly understood, each prime must be understood in a dynamic bi-directional manner entailing both cardinal and ordinal aspects (which are, relatively, quantitative and qualitative with respect to each other).

We saw in yesterday’s blog entry that if s is a prime, that the sth position (with respect to s) always reduces to standard cardinal interpretation and that this represents the default treatment of ordinal numbers in conventional mathematical terms. This in turn indirectly equates in quantitative terms, with the fact that in every case, one of the s roots of 1 = 1.

We then saw that there exists a Zeta 2 function that complements the well known Zeta 1 (i.e. Riemann) zeta function, where its zeros provide an indirect quantitative means of giving unique expression to all the non-trivial ordinal positions associated with each prime.

So the crucial function of the zeros - when appreciated from this dynamic (Type 3) perspective - is that they provide the ready means of indirectly converting from a qualitative to quantitative type interpretation.

So for example in the case of the prime number 3, the Zeta 2 function is given as,

1 + s + s

^{2 }= 0.

Therefore the two solutions to this equation provide unique quantitative conversion of the qualitative notion of 1

^{st}and 2

^{nd}respectively (in the context of 3).

And by extension the Zeta 2 function can thereby be used to provide unique quantitative conversions for (non-trivial) ordinal positions associated with every prime!

All this provides an important basis for appreciating the corresponding external bi-directional relationship as between the primes and the number system as a whole.

Now, when we look externally at this relationship we find that all natural numbers in quantitative terms are uniquely composed of a combination of one or more primes.

So for the primes themselves only one factor is involved (excluding the “trivial” factor of 1 from consideration).

Then for a composite number such as “6” two factors are involved.

So 6 = 2 * 3 (which is the unique prime factor combination for this number).

And of course prime factors can be repeated.

So 12 for example has 3 prime factors where 2 is repeated twice (i.e. 12 = 2 * 2 * 3).

Now this is all well and good insofar as it goes, but unfortunately as we shall see, completely one-sided.

So in standard mathematical interpretation, the primes represent the unique independent “building blocks” of the natural numbers in a merely quantitative manner.

However what is crucially overlooked in such conventional mathematical interpretation is that - quite literally - a corresponding qualitative dimension arises whenever the multiplication of numbers takes place.

This fact - which I have often recited - hit me forcibly at the age of 10 when studying simple concrete problems involving the areas of fields.

So, for example, if one imagines a large field with length 3 km and width 2 km, the corresponding area will be given in square (i.e. 2-dimensional) units.

Though from a quantitative perspective the answer is indeed 6, a qualitative transformation in the nature of the units has thereby taken place through the very process of multiplication.

There is another simple way also of coming to appreciate this qualitative connection.

Imagine there are 2 rows - say of cars - with 3 cars in each row.

Now from the perspective of addition, one would treat all the items as independent.

Therefore one could count up the 3 items in one row (= 3) and then proceed to the second row again counting up the 3 items (= 3) and then add the two rows.

So we are here treating the items in each row as independent (and indeed the rows themselves as independent).

In this way the total no. of cars = 3 + 3 = 6.

However what is vital to carry out multiplication, is the corresponding recognition of the interdependence of each row (which thereby assumes a common similarity as between the 3 items in each row).

So the very reason one can now relate the operator 2 with 3, though multiplication, is because of the recognition of the common similarity (with respect to the cars in each row).

However such similarity applies to the interdependence of the items with each other (which is qualitative in nature).

Therefore, properly understood, multiplication necessarily entails notions of both quantitative independence and qualitative interdependence respectively, which can then only be properly appreciated in a dynamic relative manner.

So the conventional interpretation of multiplication reduces its nature to that of addition (where the quantitative independence of unit members is solely maintained).

Therefore in conventional terms, in our example, we start with the quantitative “building blocks” i.e. 2 and 3 which are defined in linear (1-dimensional terms) i.e. on the number line.

However when we multiply these two numbers, even though their dimensional nature has changed, the result 6 is still represented on the same number line!

So the crucial necessary qualitative nature of multiplication is thereby missed.

Thus if one is to appreciate the true significance of the Zeta 1 (i.e. Riemann) zeros, it is vital to recognise clearly the qualitative nature of multiplication.

Whereas from a quantitative perspective, the identity of the composite natural numbers is indeed based on the primes, strictly speaking this all reversed in qualitative terms, whereby the identity of the primes is now based on their unique relationship with the natural numbers!

In other words with respect to the composites, the primes have a new interdependent relationship as factors. Therefore though the primes are indeed independent in a quantitative sense (as separate “building blocks”), through relationship with each other as the factors of composite numbers, they likewise share a qualitative interdependence with each other.

Now I have made this simple point before though in truth it is extremely subtle to understanding, that the very nature of number keeps switching, as it were, between both a particle and wave identity (without ever being noticed in conventional terms).

Thus once again the standard quantitative definition of number is based on the independence of each of its unit members.

So 3 = 1 + 1 + 1 (with each homogeneous unit independent in a quantitative sense).

However if we now say for example that a number - say 30 - has 3 factors the very nature of 3 has now changed (from a dynamic interactive perspective).

The very point about each factor is that its identtity implies that it is related to a composite number. So in this context, 2, 3 and 5 assume a new qualitative identity as factors through their common relationship to the number 30!

And likewise when we now say that 30 has 3 factors, 3 = 1 + 1 + 1 (but now - relatively - in an interdependent rather than independent sense).

So once again this is akin to nature of left and right turns at a crossroads, where what is deemed left and right is merely relative depending on a reference frame based on the direction of approach (which can keep switching).

So just as we saw how the Zeta 2 zeros can be used to deal with the very important internal question of how ordinal qualitative notions (with respect to the members of each prime) can be indirectly converted in a consistent quantitative manner, we now have the parallel problem of how the Zeta 1 (Riemann) zeros can equally be used externally to indirectly convert the qualitative nature of the primes (as factors of natural numbers) in a quantitative manner.

And I have shown before how the frequency of the Riemann zeros is intimately linked to the factor frequency of the natural numbers.

Therefore once again if one accumulates the frequency of the proper factors of the natural numbers (up to n), the total will bear a remarkably close relationship with the corresponding frequency of the non-trivial Riemann zeros (up to t) where n = t/2π.

Therefore the key importance of the Riemann zeros (from this perspective) is that they provide an indirect means of converting the qualitative nature of the primes (through their relationship with the natural numbers as factors) in a quantitative manner.

And both the Zeta 1 and Zeta 2 functions are themselves complementary, relating to the bi-directional relationship of the primes and natural numbers (externally and internally) in both quantitative and qualitative terms.

So the Zeta 1 (Riemann) function can be expressed:

1

^{– s }+ 2

^{– s }+ 3

^{– s }+ 4

^{– s }+ …… = 0.

The corresponding Zeta 2 function can then be expressed as

1

^{ }+ s

^{1 }+ s

^{2 }+ s

^{ 3 }+ …… = 0 (initially where s is prime).

Notice the complementarity! Whereas the natural numbers represent the base values with respect to the Zeta 1, they represent the dimensional values with respect to the Zeta 2; and whereas the unknown (s) represents the dimensional values with respect to the Zeta 1, they represent the base values with respect to the Zeta 2.

Also whereas the dimensional values are negative in Zeta 1, they are positive in Zeta 2; finally whereas Zeta 1 represents an infinite, Zeta 1 represents a finite series respectively.

Of course ultimately both internal and external aspects of this bi-directional relationship between the primes and natural numbers are themselves fully interdependent.

From the quantitative perspective, it does indeed appear that the natural numbers are derived from the primes; however equally from the qualitative perspective the primes appear to now obtain their positions (expressing their relationship with each other) through their identity as unique factors of the natural numbers!

And if one thinks about this for a moment, without knowledge of the gaps between the primes, it would not be possible to list the primes (as quantitative "building blocks" of the natural number system); likewise without knowledge of their quantitative value it would not be possible to establish the gaps between the primes (that express their qualitative relationship with each other).

So rather than an absolute quantitative relationship connecting the primes with the natural numbers in a one-way static manner, rather we have a dynamic two-way interactive relationship that operates relatively in both quantitative and qualitative terms.

This implies that an incredible dynamic synchronicity characterises the relationship between the primes and natural numbers , where they ultimately approach total identity with each other in an ineffable and utterly mysterious manner.

The very ability to - literally - "see" in a pure intuitive manner this remarkable synchronicity, where number as form becomes dynamically inseparable from number as energy, represents the holistic extreme of mathematical understanding (where quantitative can no longer be separated from qualitative appreciation).

At the other extreme we have the totally abstract rational understanding of number representing absolutely fixed forms where quantitative is totally separated (in formal terms) from qualitative appreciation.

The fundamental requirement for Mathematics is then the consistent integration of both types of understanding. However this will require that equal emphasis is given to both analytic and holistic aspects.

And without any hint of exaggeration, Mathematics at present, despite its enormous advances, is hugely unbalanced (through a complete neglect of its vitally important holistic aspect).

Nothing less than a total revolution in perspective will rectify this problem!

Indeed this may be slowly initiated through the inevitable continued failure to prove the Riemann Hypothesis, which not alone cannot be proven (or disproven) but much more importantly cannot be properly appreciated in conventional mathematical terms.