THE WHOLE PROBLEM

COMBINED STAGE: INTEGRATION

There is an attempt in Western psychology, to represent integration of all specialised structures of the linear level, in a final combined stage, sometimes referred to as the centaur level. This, as I have argued is only possible in a very limited sense. The linear level by definition is not geared towards true integration. It is heavily based on the conscious process which favours differentiation of structures.

True integration involves considerable specialisation of unconscious understanding which requires going beyond the linear level.

Now, when we confine numbers to the rational set, there is no great difficulty in achieving integration of the number system. All numbers thus encountered, prime, natural, negative integers and fractions (positive and negative) are all members of the same rational system. Just as psychologically, the rational paradigm is the crowning achievement of the linear level of understanding, likewise, in mathematical terms, the real number system embracing (only) the rational numbers is the complementary equivalent.

In psycho-mathematical terms, if no other number types existed, then the rational paradigm, would be appropriate as the sole paradigm. Indeed the Pythagoreans were keen to ensure there would be then full complementarity as between both the qualitative method of understanding and the nature of quantitative data investigated. Thus they wanted a rational paradigm to conform to a (solely) rational number system.

However, as is well known there are many other classes of numbers. There are irrational numbers (which in turn can be subdivided into algebraic and transcendental).

Then there is an imaginary number system, which in so many respects complements the real. Finally, there are the transfinite numbers, which potentially raise the most serious difficulties in terms of the conventional rational system.

The approach to all these new "finds" has been somewhat similar. Initially, due to their alien appearance they seemed somewhat threatening, and there was consequently great reluctance to consider them as "valid" members of the number club. However, eventually, when made to abide by the rules applying to the existing members, they were reluctantly granted membership. In other words, they have been all simply been given a reduced interpretation so as to fit in with the rational paradigm. However, in many respects they still rest uneasily with the rational numbers.