Yesterday, my attention was drawn to a headline in a newspaper relating to a story regarding “the circle of the 60 most influential people in Ireland”.
This set me thinking once again about the fundamental nature of the number system and how in fact it is substantially misrepresented in conventional mathematical terms.
We are accustomed through training – especially in a formal mathematical context – to think of number with respect merely to its quantitative aspect. Indeed the identification of people nowadays by a number (rather than a personal name) has become synonymous with the impersonal nature of modern society.
Now this quantitative impersonal treatment of number correlates well with the linear interpretation that characterises Conventional (Type 1) Mathematics.
At its deepest level the linear approach is characterised by the use of isolated uni-polar reference frames for mathematical interpretation.
All experience of reality (including of course mathematical) is necessarily conditioned by polar opposites which dynamically interact in a relative manner.
So strictly an external object such as a number has no meaning in the absence of a corresponding mental perception that - relatively – is of an internal nature.
Likewise the actual identification of a specific number in quantitative terms has no meaning in the absence of a holistic number concept (potentially applying to “all” numbers) that is - relatively - of a qualitative nature.
However when linear (one-dimensional) polar reference frames are used, numbers are misleadingly given an absolute identity (independent of subjective interpretation); likewise they are given a merely quantitative identity (independent of holistic qualitative considerations).
In other words, though the true nature of number is inherently of a dynamic interactive nature, it is misleadingly portrayed through Conventional Mathematics in a somewhat static absolute manner.
Thus, despite its so-called rigour, the most fundamental issues of mathematical interpretation are glossed over leading to reduced interpretation at every turn.
For if numbers are defined in absolute terms as independent entities, then this excludes – except in a reduced manner – corresponding consideration of their interdependence with other numbers. Thus the very notion of interdependence – which inherently is of a qualitative nature – is thereby inevitably confused in conventional mathematical terms with the quantitative aspect.
And as I have pointed out this for example is the key problem in the failure to recognise the true relationship of the primes to the natural numbers (and the natural numbers to the primes).
So the starting point for the more accurate understanding of number – and indeed all mathematical relationships – is the recognition that they must be defined in a merely relative manner (befitting the dynamic interaction entailed by complementary opposite poles).
Now it is interesting how informally the use of number with respect to its relational aspect of interdependence is characterised by a circular rather than linear reference. So a grouping of a number of friends for example (thereby entailing close knit relationships) will be referred to as a “circle of friends”.
And unfortunately we often have in society “golden circles” where the relationships of power and influence among a small group of people can be too strong and exclusive.
However this informal use of circles to suggest relationship provides a direct clue as to the nature of the qualitative aspect of mathematical understanding.
Rather than being based on the line, numbers are now considered in a circular manner. Now the roots of 1 comprise such a circular system; however we cannot possibly attempt to understand the inherently qualitative nature of this system while still using linear logic!
So the qualitative nature of the circular system of numbers is only revealed when appropriately viewed through a corresponding circular logical approach (based on the dynamic relationship between opposite poles of understanding).
Thus in this new understanding of numbers both the linear and circular aspects of interpretation are closely integrated in a dynamic interactive manner.
Therefore the linear (cardinal) aspect - Type 1 - is initially defined with respect to the relatively independent aspect of number behaviour; the circular (ordinal) aspect - Type 2 - is then defined with respect to the relatively interdependent aspect. However just as in physics, particles have wave aspects and waves particle aspects respectively, likewise it is with number.
So we will see that corresponding to this dynamic understanding, the number system possesses both a cardinal (particle) aspect that is quantitative and a corresponding ordinal (wave) aspect that is qualitative in relative terms.
However the cardinal aspect as quantitative equally has a corresponding aspect as qualitative; likewise the ordinal aspect as qualitative equally has an ordinal aspect as quantitative. And this recognition of the pure interdependence of quantitative and qualitative aspects gives rise - to what I term - Type 3 Mathematics.
Now until Riemann’s pioneering work only the cardinal quantitative aspect of number was properly realised. What Riemann has shown is that underlying this particle notion of number is a harmonic wave system (which is however viewed solely in quantitative terms).
What is still totally missing from this picture is any clear recognition of the corresponding (mirror) qualitative interpretation of number (which itself has both particle and wave aspects). And without this qualitative aspect the quantitative aspect itself cannot be properly appreciated.
So to put it simply, Riemann’s findings regarding an underlying wave harmonic structure to the number system points to its inherent dynamic nature.
However we cannot properly appreciate this dynamic nature while still trying to understand number relationships in absolute terms (with respect to their merely quantitative aspect).
The very dynamism that can now be seen to exist arises directly from the fact that number has both quantitative and qualitative aspects (in continual interaction with each other).
And with this realisation the mystery of the primes can at last be resolved where it is now seen from this dual perspective that both the natural and prime numbers (and prime numbers and natural) are perfect reflections of each other in an ultimate identity that is ineffable!
However once again, we will never appreciate this mutual identity while trying to understand numbers in a merely quantitative manner!