The rational stages of development represent specialisation of conscious understanding. This is extremely useful for fragmenting and analysing reality and is the unquestioned basis of conventional science.

However the (real positive) rational paradigm is very seductive in that it creates the illusion that reality has an independent existence (which mental constructs represent in separate fashion).

However just as the set of real positive rational numbers represents just one limited subset of the total set of number quantities, the real positive rational paradigm represents just one limited example of the total set of number qualities (i.e. paradigms).

Many other more subtle (number) paradigms exist. Here physical and psychological reality are inextricably linked in relative complementary fashion.

We will examine these presently.

The concrete rational stages are more content based where one learns to analyse the objects of experience in a logical manner.

The external phase has both affective and cognitive aspects. The affective aspect involves increased ability to analyse sense experience whereas the cognitive aspect involves a more detached ability to analyse mental perceptions.

The internal phase also has both affective and cognitive aspects. In the former case one learns to analyse emotional feelings and in the latter case one uses (moral) judgement in terms of practical decisions.

The formal rational stages are more holistic (i.e. concerned with qualitative form) where one now learns to analyse pure mental objects logically.

The external phase again has both affective and cognitive aspects with the former related to a more refined form of sense awareness and the latter to a purer mental ability dealing with abstract form.

The internal phase involves the affective aspect of more refined emotional feeling and the cognitive aspect of abstract mental judgement.

There is a crucial difference as between the concrete and formal stages which is completely confused in conventional rational understanding. The concrete (i.e. part) is qualitatively different from form (i.e. whole). In mathematical terms they are horizontal (rational) and vertical (intuitive) with respect to each other.

However in conventional understanding intuition is simply reduced to rational explanation. The whole is thereby simply reduced to the parts and interpreted quantitatively as the sum of the parts.

The starting point for holistic mathematics is the recognition that the whole is qualitatively different from the part bearing a relative complementary relationship to each other. Thus we have two dynamic number systems representing quantitative (parts) and qualitative (wholes) respectively.

Rational (conscious) development is readily suited to the differentiation and analysis of the "parts". It is not suited to the complementary task of integration and synthesis of the "whole" of experience. This is directly based on intuitive (unconscious) development which due to reductionism is largely screened out of conventional development.

This poses a major problem in terms of meaningful integration of mental and body structures which relates directly to spiritual development.

However we have a mathematical way of characterising this central spiritual development. As we have seen the rational is horizontal (where understanding takes place extensively within a fixed linear dimension or paradigm); the intuitive is vertical (where understanding takes place intensively within a variety of paradigms); the spiritual which involves the balanced use of reason and intuition is diagonal (taking place dynamically extensively and intensively within each and every paradigm).