Brief Interlude

It may perhaps help to place some of my recent blogs on the intimate complementarity of the Zeta 1 and Zeta 2 Functions in perspective.

My overriding purpose in these blogs is simple.

Over the past 50 years or so I have reached the firm conclusion that – strictly speaking - Mathematics, as we know it, is simply not fit for purpose!

So in this context I am using the Riemann Hypothesis to illustrate the nature of this dilemma.

The true significance of the Riemann Hypothesis could hardly be more significant as it relates directly to the fundamental nature of our number system.

Now the inherent nature of this system - as indeed all mathematical activity - is truly dynamic with twin analytic (quantitative) and holistic (qualitative) aspects that continually interact in a complementary fashion with respect to each other.

Now as Conventionally Mathematics is formally interpreted in a merely reduced manner (that gives sole recognition to its quantitative aspect) not alone is the Riemann Hypothesis incapable of proof from within this perspective, more importantly it cannot even be properly understood in this manner!


Right at the heart of the conventional understanding exists a basic form of reductionism where the ordinal (i.e. qualitative) interpretation of number is assumed to be implied directly by its corresponding cardinal (i.e. quantitative) interpretation!

So we tend to look at the cardinal numbers as independent units in quantitative terms. However momentary reflection on the matter would suggest that corresponding ordinal interpretation entails a relationship between numbers (which is necessarily of a qualitative nature).


In actual experience the notion of number (such as 2 in this example) continually alternates as between its cardinal (quantitative) and ordinal (qualitative) meanings.

However far from realising the significance of this relative dynamic interaction, a highly reduced - and thereby utterly distorted - interpretation of the nature of number has come to dominate Western culture (and indeed other cultures) whereby number is misleadingly interpreted in an absolute static manner (with respect to its quantitative attributes).

Thus we mistakenly believe that the ordinal notion of the natural numbers for example such as 1st, 2nd, 3rd, 4th and so on are directly implied by the corresponding cardinal notions of 1, 2, 3, 4!

Indeed very often cardinal numbers are used directly to refer to ordinal rankings. So Tiger Woods is now once again no. 1 (i.e. 1st) in the PGA professional golf rankings, illustrating how the cardinal notion of 1 as a quantitative number is readily interchanged with the corresponding ordinal notion of 1st, as its relational qualitative counterpart!


However this fallacy i.e. of the ordinal being directly implied by its cardinal aspect, is immediately exposed when we attempt to explain the relationship of the primes to the natural numbers (from the conventional mathematical perspective).

Here the primes are viewed as the cardinal building blocks of the natural numbers so that each natural number represents a unique combination of prime factors.

However if we reflect on it for a moment a prime number strictly has no meaning in the absence of its ordinal natural number counterpart.

So for example the very recognition of 2 and 3 and 5 as prime numbers implies a natural number ordinal ranking among the primes of 1st 2nd and 3rd respectively. And if we assume that the ordinal is implied by its cardinal aspect, then this means that we must already assume the pre-existent identity of the cardinal numbers before we can even begin to explain their derivation from the primes!


This observation in fact implies a related circular notion of number (the significance of which is effectively unrecognised in conventional terms) that represents the appropriate (unreduced) appreciation of ordinal meaning.

Again even a little reflection on the matter will reveal how ambiguous is the ordinal notion of number! Clearly the meaning of 1st, 2nd, 3rd etc is of a relative nature depending on the size of the number group to which it refers.

So for example the 2nd of a group of 2 has a relatively distinct meaning from the second of a group of 3, 4, 5 etc. with potentially an unlimited range applying.


Therefore all natural numbers (in ordinal terms) have an unlimited range of potential meanings (depending on the size of the finite group to which they belong).

A little further reflection would clearly indicate that since cardinal and ordinal meaning are thereby in dynamic terms interconnected, that indirectly the cardinal number system is also of a merely relative nature!

In other words, in dynamic terms, all numbers have two complementary aspects that are - relatively - independent and also - relatively - interdependent with each other.

The static notion of numbers as abstract entities is therefore just an illusion (based on a reduced - merely quantitative – interpretation).


In my own dynamic treatment of number, I am therefore at pains to demonstrate that the number system is composed of two interacting aspects - which I refer to as Type 1 and Type 2 respectively.

The Type 1 is directly associated with the quantitative aspect (though indirectly through interdependence equally possessing a qualitative aspect).

The Type 2 is then directly associated with the qualitative aspect (though again through interdependence) indirectly possessing a quantitative aspect.

Then I refer to the full combined interaction of both aspects (Type 1 and Type 2) as Type 3.


Basically the Type 1 aspect treats numbers as homogeneous collective units in quantitative terms (thus allowing for no qualitative distinction as between individual units). So 3 as a cardinal number from this perspective = 1 + 1 + 1 (in quantitative terms).

From the Type 1 cardinal aspect the prime numbers represent the basic building blocks of the natural number system. All natural numbers (except 1) from this perspective represent a unique combination of prime factors.

The Type 2 aspect by contrast is based the notion of a number group as composed of unique individual members (in qualitative terms). Thus from this perspective 3 relates to its distinctive 1st, 2nd and 3rd individual members as ordinally defined.

From the Type 2 aspect the natural numbers represent the basic building blocks of each prime number (in ordinal terms). So every prime number (p) from an ordinal perspective is composed of a unique set of natural numbers 1st, 2nd, 3rd,...pth!


Then using these distinctions, I go on to define both Zeta 1 and Zeta 2 Functions with respect to uncovering the true nature of the Riemann Hypothesis. This relates to the ultimate identity of both the quantitative and qualitative aspects of number.


From this enlarged perspective we now have two complementary sets of (non-trivial) zeta zeros.

Indeed perhaps the best explanation of these incredibly significant sets of numbers can be given as follows.

The Type 1 zeta zeros indirectly represent the holistic aspect of the cardinal number system, whereas the Type 2 zeta zeros directly represent its corresponding holistic ordinal aspect.

Ultimately - which can be approximated in Type 3 terms where both systems simultaneously interact - both Type 1 and Type 2 zeros can be seen as fully complementary and indeed identical in an ineffable manner with the natural number system (in both cardinal and ordinal terms).

The very manner by which the quantitative and qualitative aspects of the number system interact is though the two-way interaction of the primes with the natural numbers (and the natural numbers with the primes), both of which are mediated through the two sets of zeta zeros respectively.


This leads to the remarkable conclusion that the hidden holistic activity of both sets of zeta zeros necessarily underlies all human experience in an innate fashion in the seemingly obvious identification of cardinal with ordinal meaning (i.e. 1st with 1, 2nd with 2, 3rd with 3 etc.)

Indeed even more remarkably this innate activity necessarily underlies all natural processes as the very means by which they acquire a phenomenal identity!

So what seems most accessible at a conscious level (appearing totally obvious) therefore remains least accessible in corresponding unconscious terms.


Mathematics is now in urgent need of addressing its great shadow i.e. the unrecognised holistic aspect of interpretation. As such understanding intimately affects all of the sciences - now greatly in need of a more integrated understanding - our very civilisation may now well depend on such holistic realisation occurring rapidly in the very near future.