Number Types and Stages of Development

The are various number types (as quantities) recognised within Mathematics.

For example we can start with the original - and most fundamental - numbers 0 and 1.

Then we have 2 and the other odd prime numbers e.g. 2, 3, 5, 7,.....

These then lead on through multiplication to the natural number system 1, 2, 3, 4,...

Then we have the integers allowing - including 0 - which allow for for both positive and negative natural numbers.

Next come the rational numbers which can be expressed as fractions i.e. the ratios of the integers

Then we have the (algebraic) irrationals such as the square root of 2 and phi which arise as solutions to polynomial equations using integer coefficients.

We also have (transcendental) irrationals such as pi and e (which do not arise as solutions to the aforementioned polynomial equations.

We also have imaginary as well as real numbers and the combination of both as complex numbers.

And finally we can define transfinite numbers representing infinite classes (as derived in a linear manner).


However a key realisation for me (following on the attempt to resolve the Pythagorean Dilemma) was that all these number types could equally be defined in a coherent qualitative manner.

Furthermore the qualitative interpretation of each number type defines a unique scientific paradigm (or more properly metaparadigm) with which to interpret reality.


Right away this suggests how limited is the accepted rational paradigm which defines the conventional scientific - and indeed mathematical - approach to reality.


Though clearly very important in their own right, the rational numbers (as quantities) represent just one subset with respect to the overall set of numbers.

In like manner the rational paradigm likewise represents just one important form of scientific interpretation (from a varied range of possible interpretations).


Looked at from another perspective each qualitative number type closely corresponds with the understanding that unfolds at a particular stage of development.

Rational understanding - that defines conventional scientific interpretation - conforms to a narrow band towards the centre of the psychological spectrum. However there are several "higher" and "lower" bands that potentially define equally important modes of scientific interpretation (that are all but unrecognised).


My own interest over the years has largely been devoted to the "higher" stages of development.

In the spiritual traditions these generally are associated with contemplative type development requiring an ever more refined type of intuitive awareness.

However what has been greatly missing from such accounts is any proper clarification as to to the implications of such awareness for scientific (and mathematical) appreciation.

So recognising this deficit I decided to try and fill in the gaps to best of my own limited abilities. and it is through these endeavours that Holistic Mathematics (and associated Holistic Sciences) emerged.

Indeed nearly 20 years ago I wrote two online books "Transforming Voyage" and "The Number Paradigms" summarising my progress to that point.
So I basically was at pains to trace the nature of each of the major stages of development before then showing how a distinctive number paradigm was closely associated with each main stage.


In the present context i.e. resolution of the Riemann Hypothesis, two number types of special importance are the prime numbers and the imaginary numbers.


Once again though these are conventionally understood merely as quantities, they equally possess an important qualitative aspect.


So in the next contribution we will look firstly at the important holistic (i.e. qualitative) nature of prime numbers which will help to illuminate their true inherent nature (which in turn is vital in terms of obtaining proper appreciation of the fundamental significance of the Riemann Hypothesis).