Internal Nature of Number

However while establishing at University in Dublin a clear philosophical position regarding the limits of the conventional absolute approach, it took considerable further time to properly flesh out the framework for a new dynamic treatment that could throw light on outstanding issues within number theory such as the Riemann Hypothesis.

Much of my early attention focussed on the internal nature of number.

For example, if we take the number 2 (and first prime) as an illustration, in conventional additive terms, it can be expressed as 

2 = 1 + 1.

So the units here (as components of the cardinal number 2) are treated as absolutely independent in a homogeneous quantitative manner. From this perspective therefore, the individual units cannot be distinguished from each other.

However, if the units were indeed independent in such a manner, no means would thereby exist for subsequently relating them to the new whole number identity of 2!

Therefore, the very capacity to combine the unit numbers with each other, requires a distinctive qualitative identity (of relational interdependence), which is expressed through their ordinal nature.

So from this latter qualitative perspective, the key characteristic of the number 2 is that it contains uniquely distinctive members that are potentially 1^st and 2^nd with respect to each other. In this context, it implies that individual members are fully interchangeable so that the same unit can be 1^st or 2^nd, depending on context.

Though the ordinal nature of number is indeed recognised in conventional mathematical terms, it is customarily understood in a fixed actual manner (where positions are non-interchangeable).

In this way, the true qualitative nature of ordinal identity (that expresses the relative interdependence of individual units) is thereby reduced in a cardinal manner.



The significance of this observation cannot be over-stated.

Because of the limited nature of conventional mathematical interpretation, the quantitative identity of each prime is recognised in a rigid cardinal fashion that is absolute.

This then leads to the somewhat misleading notion that the primes are thereby the indivisible “building blocks” of the natural number system.

However, when one properly recognises the qualitative relational aspect of number, each prime is now understood in dynamic interactive terms, as comprising a group of individual unit members, which shares both quantitative aspects (as relatively independent) and qualitative aspects (as relatively interdependent) respectively.

And when the cardinal identity of a prime relates directly to its quantitative, then in complementary fashion, its ordinal identity relates to its qualitative nature.

So 2 (two) for example expresses the quantitative notion of the first prime (where individual independent units cannot be distinguished from each other).

However 2 (“twoness”) expresses the complementary qualitative notion of the same prime (where unique individual units as 1^st and 2^nd respectively, are interchangeable with each other). So it is this unique interdependence with respect to units that directly relates to the qualitative notion of 2 (i.e. as “twoness”)

Thus from a dynamic interactive perspective - which reflects the true nature of mathematical understanding - one cannot experience the quantitative notion of 2 (two) in the absence of its qualitative aspect (as “twoness”); likewise one cannot experience the qualitative notion of 2 (as “twoness”) in the absence of its corresponding quantitative aspect (as two).   


When one views a prime in this dynamic interactive manner, its fundamental nature is thereby completely changed.

Again, from the conventional mathematical perspective, each prime is viewed in cardinal terms as a pre-given “building block” of the natural number system (in an absolute quantitative manner).

However from this new perspective, a prime is viewed in dynamic terms as representing the complementary interaction of both quantitative (cardinal) and qualitative (ordinal) aspects that are relatively independent and interdependent with respect to each other.

And this new perspective reveals the inherent paradoxical nature of each prime.

So again in cardinal terms, a prime appears as an indivisible “building block” of the natural numbers in quantitative terms; however from the complementary ordinal perspective, the same prime is revealed as already composed of a unique series of individual natural number members in a qualitative manner.

Thus from the former cardinal perspective, the natural numbers appear to depend on the individual primes; however from the complementary ordinal perspective, each individual prime already appears to depend on a unique ordered series of natural numbers.

So, in cardinal terms for example, 5 as a prime number, serves as an indivisible “building block” of the natural numbers; however in a complementary ordinal manner, 5 is uniquely composed of its 1^st, 2^nd, 3^rd, 4^th and 5^th natural number members (which are potentially interchangeable with each other).


So, from the dynamic perspective - where both cardinal and ordinal aspects are understood as complementary aspects of number - this paradox points to a fundamental holistic synchronicity, connecting in two-way fashion each prime with a unique set of natural numbers, which is ultimately ineffable.