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- The purpose of this book has been to demonstrate the complete complementarity
of mathematics and (transpersonal) psychology. This appreciation requires
a special approach which I have referred to as transrational.

This involves making explicit the dynamic relationship as between reason and intuition.

Now in actual understanding, these are always so related. However conventionally
intuition is largely screened out of formal interpretations so that this
amazing marriage of mathematical and psychological reality is rarely recognised.

The implications of this marriage are of immense importance, opening the way for a true TOE, incorporating both physical and psychological reality.

The basis of all this is ultimately very simple. Mathematical understanding
- at whatever level - of necessity always involves the dynamic interaction
of mind and matter. In other words mathematical understanding equally incorporates
complementary psychological understanding. This can be stated in more formal
terms as follows.

Every mathematical relationship (interpreted rationally in quantitative terms) has a complementary relationship (interpreted intuitively in qualitative terms).

Because physical and psychological reality are themselves complementary, this means that all mathematical relationships have a direct relationship to both aspects of this (psycho-physical) reality.

While being strongly critical of conventional reductionist approaches
to mathematics, paradoxically, I believe that the power of mathematics
is greatly under-estimated. Because it is interpreted within just one restricted
level of experience (i.e. positive real rational approach of the linear
level), its applications to all other levels of reality are continually
overlooked. Thus, for example, the new physics, increasingly throws up
experimental results which do not correspond to our conventional intuitions
of the world. This is due to the fact that our intuitions are largely confined
to just one level of reality, whereas these experimental results relate
to different levels. Therefore, when we understand reality at the correct
level, our intuitions will be appropriate to the level involved.

Let me briefly illustrate this important point.

Mathematical dimensions are given by the power or exponent of a number. Now such dimensions (mathematically) can be positive or negative, whole numbers or fractions. Indeed they do not have to be rational numbers. We could express exponents (or dimensions) as irrational (algebraic) or transcendental numbers. Also they could be expressed by imaginary rather than real numbers. Finally we could even use transfinite rather than finite numbers.

Now all these dimensions - which are mathematically possible - have corresponding interpretations in terms of both physical and psychological reality.

We can now see how limited is the conventional world-view which is confined to positive real rational dimensions that are whole numbers (i.e. three dimensions of space and one of time).

Properly understood however - depending on the level of experience involved - both physical and psychological reality behave dynamically in terms of positive and negative, whole and fractional, rational and irrational, real and imaginary, finite and transfinite dimensions respectively.

There are many different bands and levels of reality. Conventionally the understanding of science and mathematics is confined to just one band within one level (i.e. the positive real rational approach of the linear level). However there are many other possible bands and levels where reality (physically and psychologically) behaves in a different manner.

In this dynamic world-view, physics and psychology are inextricably
linked. The observations of physical reality subtly change when the psychological
manner of observing reality changes. In this dynamic approach, each (new)
level of psychological reality unveiled is the very means by which the
complementary (new) level of physical reality is interpreted.

We have built up a comprehensive map of the various levels and the various bands or stages within each level.

Looked at from one perspective, these represent different psychological models of reality. However from the complementary perspective, they represent different physical models of reality. Both are indeed inextricably linked. If we change the band or level of one perspective, we must in complementary manner change the band and level of the other.

The purpose of this book has been to translate all the bands and levels of the comprehensive map into mathematical terms. We have found that the number system - with its own well recognised bands and levels, provides a remarkably precise means of achieving this. From this dynamic perspective, every relationship has both a rational (quantitative) and intuitive (qualitative) interpretation. Reality is now not so much written in the language of mathematics. Rather the language of mathematics now dynamically writes (i.e. creates) reality. The possibilities inherent in this reality are the possibilities inherent in mathematics itself.

The view therefore that we live in four dimensional space-time is the product of just one band of one level (i.e. the rational linear level). Viewed from the perspective of the circular level, the dimensions of space-time are experienced as (mathematically) irrational. From the appropriate bands of the point level they are imaginary, and from the radial level as transfinite.

Correctly interpreted, what is mathematically possible - in terms of dimensions - is exactly replicated both physically and psychologically (in terms of the appropriate band and level of experience).

In this book I have concentrated largely on the psycho-mathematical significance of number. However this significance - interpreted from the transrational perspective - can be extended to all mathematical relationships. Correctly understood, - because the process of understanding itself is dynamic and living - there is no such thing as abstract mathematics. For every mathematical relationship there is a corresponding interpretation which has both complementary physical and psychological aspects.

For example, the physical significance of Einstein’s famous formula
E=MC^{2}_{, }has been amply demonstrated. Matter in fact
is just another form of (physical) energy. However, this formula equally
has profound psychological significance where matter can be seen as just
a condensed form of (spiritual energy). Indeed it could be validly said
that the very goal of mystical transformation is the conversion of psychic
matter (i.e. the phenomena of experience) into pure spiritual energy.

Thus to unleash "the fire in the equations", we need this dynamic transrational
approach.

THE DOUBLE BINARY SYSTEM

Using computer language, the transrational approach I have been outlining
in this book can be expressed in terms of a double binary system. I will
conclude by briefly outlining this system.

We have two logical systems which are themselves complementary.

The first is the traditional rational system of differentiation, where opposites are clearly separated. This system can be represented of course in terms of the binary digits 1 and 0. The miracles of computer technology and our modern information system have been made possible through this system.

The complementary binary intuitive system is of integration, where opposites are now closely complementary (and ultimately identical). This intuitive logic is in fact the principle of all systems of transformation (e.g. personality transformation).

Putting it another way the rational binary system is a horizontal, whereas the intuitive binary system - in relative terms - is a vertical approach to reality.

A comprehensive method clearly involves both the horizontal and vertical binary systems in what I refer to as a double binary approach.

This double binary system involves both separation and complementarity.

1 representing an on state vertically (intuition), equally represents an off state horizontally (reason), where it is 0. In complementary terms 1 representing an on state horizontally (reason), equally represents an off state vertically (intuition), where again it is 0. Thus in terms of both horizontal and vertical scales (considered separately), a state is either on or off. However when these scales are related, states are both on and off simultaneously.

Reality - at all levels - in fact can be seen as a dynamic double binary system of control and response incorporating both information and transformation. This is true of both physical and psychological aspects. (Indeed reason is essentially a system of control and intuition one of response.)

Just as quantitatively, all information can be translated in binary terms, also qualitatively, all information can equally be translated in binary terms. However, whereas the interpretation of binary information is understood in finite (rational) terms, the corresponding qualitative transformation must be understood (directly) in transfinite (intuitive) terms.

This double binary system is not just an artificial attempt at translating reality. In the deepest sense it is the way reality actually functions i.e. as a complex computer network, incorporating information and transformation processes simultaneously.

The great advances in computer information technology relate to one binary system (based on reason). However this system will never (directly) meet the complementary need for transformation in society which relates to a second binary system (based on intuition).

My dream is that future generations will understand the need for the
balanced use of both systems.