Fittingly, as the spirit of this book owes much to the spirit of the Pythagoreans, I propose in this chapter to give a multidimensional interpretation of the famous Pythagorean Theorem. In other words I will briefly suggest how this theorem should be interpreted within the various (vertical) paradigms outlined in the previous chapter.
 
 
 

POSITIVE REAL RATIONAL PARADIGM

 
This is the conventional paradigm employed in mathematics. The Pythagorean Theorem states that in a right angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. There would be (almost) universal agreement regarding this statement. However on closer examination it conceals a number of crucial problems.

This paradigm in its operation involves a series of reductionist assumptions.

Firstly, it only considers the positive (i.e. objective) direction of understanding. There is a strong belief among mathematicians that their propositions enjoy an absolute objective truth. However in actual experience, all understanding of mathematical propositions involves a dynamic interplay that is both objective and subjective. Thus the Pythagorean Theorem involves the dual interplay of objective mathematical symbols (in relation to the mind) and the mind (in relation to objective mathematical symbols).

However in this paradigm by recognising solely the positive direction, the subjective aspect of understanding involved is ignored (or more subtly reduced to the objective).

Secondly it only considers the real (i.e. cognitive) mode of understanding. This raises another problem. All symbols - including those used in mathematics - ultimately must be sensibly verified if communication is to take place. However this sensible verification involves the affective mode which is qualitatively different from the cognitive. Thus again the actual understanding of the Pythagorean Theorem involves a dynamic interplay of both cognitive and affective modes. However, once again in this paradigm the affective mode is simply ignored (or more correctly reduced to the cognitive).

Thirdly in being rational it only considers one logical approach (based on the separation of opposites). It thereby ignores the alternative logical approach (based on the complementarity of opposites) which relates to intuition. Thus again in actual experience the Pythagorean Theorem involves a dynamic interplay of reason and intuition involving both explicit (sequential) and implicit (simultaneous) recognition. However, once again in this paradigm the (unconscious) intuitive process is ignored and reduced to the conscious rational process.

Thus to conclude, the positive real rational paradigm involves a series of reductions. Though actual understanding is inherently dynamic involving the interplay of two directions (i.e. objective and subjective), two modes (i.e. cognitive and affective), and two processes (i.e. conscious and unconscious), yet in formal terms understanding of mathematical theorems - such as the Pythagorean Theorem - is interpreted in absolute static terms.
 
 
 
 

NEGATIVE REAL RATIONAL PARADIGM

This paradigm, which is developed during the transition from linear to circular levels, addresses first of these reductions. One begins to realise that there is in fact a dual interpretation of reality. Thus if what is observed (in relation to observer) constitutes one direction of experience then the observer (in relation to what is observed) constitutes an alternative reverse view.

Thus there exists a mirror image interpretation of every mathematical theorem (which is the reverse of the original theorem). This mirror image interpretation is the negative paradigm.

Thus the Pythagorean Theorem has a mirror image alternative or anti-theory which is equally true. In mathematics if we concentrate on the absolute value of a number then positive and negative signs can be ignored. Thus the absolute value of + 1 and - 1 is the same. In like manner if we concentrate on the absolute interpretation of a theorem, likewise positive and negative directions can be ignored. In normal static terms, both a theory and its anti-theory are identical. Thus the Pythagorean Theorem and its mirror theorem are in static absolute terms identical.

However when these two directions are related dynamically - which is the manner in which actual understanding takes place - then they are opposite to each other.

Therefore in dynamic relative terms, the mirror theorem (of the Pythagorean Theorem) has an opposite truth value to the original.

In the dynamics of experience, in order to switch from the objective to the subjective direction of experience, there has to be a degree of negation of (positive) objective phenomena. Likewise to switch back again from the subjective to the objective direction, a corresponding degree of undoing of (negative) subjective phenomena is required. The positive and negative directions of experience are therefore dynamically opposed to each other.

However in static absolute terms this dynamic switching is ignored and positive and negative directions are considered to directly correspond to each other.

Thus to sum up, in static absolute terms the negative paradigm is identical to the positive. The mirror theorem of Pythagoras in such terms is therefore identical to the original.

In dynamic relative terms the negative paradigm is opposed to the positive. Therefore the truth of the mirror theorem (i.e. the mind in relation to the Pythagorean Theorem) is opposite to the truth of the original theorem (i.e. the Pythagorean Theorem in relation to the mind). Both theorems now enjoy a merely partial relative validity.
 
 
 
 

REAL IRRATIONAL PARADIGM: POSITIVE

This involves the attempted harmonisation of the two directions of experience and develops during the circular level. Ultimately this understanding is intuitive. However

it can be given a secondary rational explanation in terms of the principle of the complementarity of opposites. Because this logic is totally opposite to conventional logic and appears deeply paradoxical, it can appear irrational.

The understanding of the Pythagorean theorem now involves both directions of experience. In other words both the theorem and its anti-theorem are dynamically inter-related.

The understanding of the theorem is now more subtle. When one says that in a right angled triangle, the square on the hypotenuse is equal to the squares on the other two sides one interprets it somewhat differently. For example the right angled triangle is understood in relative rather than absolute terms. In other words its recognition depends in experience on the dynamic interaction of the poles of both mind and matter. Therefore to try and give it an independent objective existence now appears meaningless.

Actually, far from hair splitting this interpretation has a very practical application. The understanding of any proposition such as the Pythagorean Theorem obviously varies from person to person. There is never any ultimate guarantee in communication that another person shares the same understanding. For example - out of strong conviction - I am seriously questioning the conventional view that such a theorem has an absolute validity. Quite clearly it is not absolutely true for me. When one realises that the very symbols and concepts we use in such a theorem arise out of a dynamic mind-matter interaction, then it cannot be absolutely true for anyone else either. In fact, in dynamic terms, mathematical truth represents no more than - an admittedly highly useful - form of social consensus consistently interpreted - or more correctly misinterpreted - in reduced absolute terms. My purpose here is to challenge the nature of this consensus.
 
 
 
 

REAL IRRATIONAL PARADIGM: NEGATIVE

The positive version of this paradigm states that the Pythagorean Theorem must be understood in terms of the complementarity of opposites. In other words the understanding of such a proposition inevitably involves both positive and negative directions of experience in dynamic interaction. All symbols and concepts used in understanding the proposition therefore have only a relative validity.

However a fresh problem now arises. The statement that all things are relative itself is an absolute statement. Thus we now must have the development of a mirror image complement even for this subtle interpretation of the theorem.

Thus all statements based on the complementarity of opposites, have mirror image alternatives or anti-theories.

What happens in practice is that though the truth of the complementarity of opposites is ultimately purely intuitive, it tends to get interpreted - in a secondary manner - in unduly rigid rational terms, where the subjective direction of experience is ignored.
 
 

Thus the anti-theory is necessary to redress this imbalance so that one can move to a more purely intuitive understanding of propositions..
 
 
 
 

IMAGINARY RATIONAL PARADIGM: POSITIVE
 
 

The process by which thought concepts are formulated in consciousness is very different from that by which sense impressions arise.

In direct terms the cognitive mode of control is a conscious act (and indirectly unconscious); however in direct terms the complementary affective mode of response is an unconscious act (and indirectly conscious).

What happens in science and mathematics is that the affective mode of sense impressions is quickly reduced to the cognitive. Therefore there now appears to be a direct correspondence as between these sense perceptions and thought concepts, both of which are now interpreted as "real".

However, in direct terms if the cognitive mode of reason is "real", then the affective mode of sense - in mathematical terms - is "imaginary".

Thus any theorem such as the Pythagorean inevitably involves a dynamic interaction of rational concepts which are "real" and sense perceptions which are "imaginary" with respect to each other. Of course if we take the perceptions as "real", then in relative terms, the concepts are "imaginary".

Again therefore, we can say that just as every "positive" theory has a corresponding "negative" theory or "anti-theory every "real" theory has a corresponding "imaginary" or virtual theory. In static terms these are formally identical; it is in dynamic interaction that the "real" and "imaginary" aspects become apparent.

Remarkably this implies that the truth of a proposition in the general case (i.e. the rational proof) does not apply to the specific case (i.e. an actual concrete example).

Thus if I say - referring to the Pythagorean Theorem - that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides, this being directly conceptual, has a "real" truth value. However strictly speaking this does not establish its real truth in any concrete case. In relative terms, this concrete case has an "imaginary" truth value. In dynamic terms, the Pythagorean Theorem is now a complex rather than a real proposition.

Conventionally we think of the real world as existing "out there" and objective. During this transition from circular to point level, there is a complete inversion where the world is now understood as a projection of one’s inner unconscious. In contrast to former experience, this is "imaginary".

Understanding of all theorems therefore require in part this ability to project from the unconscious (i.e. sense impressions). Conventionally, of course this "imaginary" aspect is reduced to the "real".
 
 
 
 

IMAGINARY RATIONAL PARADIGM: NEGATIVE

Just as we have two directions in relation to the "real" paradigm, we also have two directions in relation to the "imaginary" paradigm.

If the phenomena (in relation to the mind) constitutes the positive direction of experience, the mind (in relation to the phenomena) constitutes the negative direction.

In the dynamics of experience, these two directions are very closely associated.
 
 
 
 

REAL TRANSCENDENTAL PARADIGM

We have seen that at the linear level, the objective direction of experience is separated from the negative. Here the Pythagorean Theorem has a static absolute validity.

At the circular level, these two directions are treated as complementary. Here the Pythagorean Theorem has a dynamic relative validity.

During the point level a new paradigm emerges which is very subtle indeed. This is associated with the emergence of advanced superstructures. Since all understanding in practice involves elements of the "rational" linear and "intuitive" circular levels, then a richer paradigm must attempt to combine both. The transcendental paradigm therefore establishes truth as the relationship between the static linear and dynamic relative paradigms. This simply represents the attempt to explicitly combine both reason and intuition in terms of mathematical understanding. In relation to the Pythagorean Theorem we would now maintain that both the static absolute and dynamic relative views have both a partial validity.

It is possible - to a limited extent - to have positive and negative directions even in relation to transcendental understanding. Because of the high subtlety of experience at the point level, these directions however become very closely associated.
 
 
 

IMAGINARY TRANSCENDENTAL PARADIGM
 
 

This again addresses the direct nature of sense symbols which relative to rational concepts are imaginary.

This development is related to the varied stages of development of substructures during the point level.

There is in all understanding a projected element coming from the unconscious which relative to direct real conscious understanding is imaginary.

We now have a very subtle form of understanding where this imaginary understanding combine elements of both the linear and circular levels. Projected symbols have in part an absolute static validity and in part a dynamic relative validity.

Again we can distinguish positive and negative directions, though - like virtual particles in physics - these tend inevitably to be very closely associated in experience
 
 
 

TRANSFINITE PARADIGM: REAL
 
 

This represents a purely intuitive understanding. Though intuition is extremely important at both circular and point levels, it is given an indirect rational interpretation (i.e. irrational and transcendental structures).

When the unconscious is fully differentiated in experience, it is no longer reduced to - and thereby confused with - rational interpretations.

This understanding emerges during the transition from point to radial level.

Putting it another away, this highlights the purely implicit nature of recognition.

In formal terms, mathematical propositions only emphasise the explicit nature of recognition. Thus, if I state that a = b and b= c, therefore a = c, this will in formal terms be interpreted as a valid rational conclusion. In a sense the conclusion is tautological. Its truth is already contained in the opening two statements. In the same way Euclidean geometry is tautological. All of its conclusions are contained in five starting axioms. Now, whereas reason makes explicit what is implied by the opening axioms, without intuitive implicit recognition one would never be able to follow its various steps. In practice this problem is very common especially wherever abstract reasoning is involved. The various steps in a proof may be demonstrated. However one may still be unable to follow these steps to a conclusion, because of lack of implicit recognition.

This paradigm therefore is merely highlighting this vital intuitive nature of understanding which literally provides the mental light to follow explicit reasoning.
 
 
 

TRANSFINTE PARADIGM: IMAGINARY
 
 

Again this focuses on the qualitative difference as between rational concepts and sense symbols. In relative terms they are "real" and "imaginary" with respect to each other.

The intuitive light involved in terms of implicit recognition is similarly different.

Pure intuition underlying understanding of rational concepts is referred to as the contemplation of transcendence. Pure understanding of sense symbols - in relative terms - is referred to as the contemplation of immanence.

By this stage i.e. the transition therefore from point to radial levels, understanding of mathematics - just as the Pythagoreans had advocated has reached the level of pure contemplation.
 
 

FINITE COMPLEX PARADIGM
 
 

We are now into the radial level. Due to the full differentiation of the cognitive rational and affective sense modes, one has now a complex rational rather than a real rational understanding of all mathematical propositions.

Thus the Pythagoreans Theorem - understood from this perspective - has both real and imaginary components.

Again there is a deep practical implication here which is usually completely overlooked.

The Pythagorean Theorem establishes a proof for "any" triangle. Strictly speaking this is not any actual finite triangle but rather a potential infinite triangle. Therefore the general case - applying to the potential triangle, is qualitatively different from the particular concrete case - applying to the actual triangle. The conventional approach which derives the truth in the particular case from truth in the general case involves a confusion of finite and infinite notions. The infinite is essentially treated as an extension of the finite notion, whereas it is qualitatively different. A "real" infinite notion can be only given an indirect finite status through conversion to "imaginary" format.

In other words, the truth of a proposition in any particular case does not follow from the proof of the proposition (established for the general case.)

More correctly if we take the general proof as "real" then this establishes its application to the concrete triangle (which is "imaginary").

Alternatively if the general proof (for the potential triangle) is "imaginary", then this establishes its application in any concrete case to the "real" triangle.

This is why understanding of the relationship between general and particular must now be couched in complex rather than real terms.
 
 
 

TRANSFINITE COMPLEX PARADIGM
 
 

This is the intuitive counterpart underlying rational understanding of the radial level. It involves a mixture of the two types of intuitions (transcendent and immanent) respectively. In transfinite terms these are "real" and "imaginary" with respect to each other. In direct terms they refer to the light (mental energy) giving insight into the connections established through reason.

Of course by this stage both finite complex and transfinite complex understanding are used in close parallel.

The directions of experience (positive and negative polarities) have been fully differentiated (circular level)

The modes of experience (real and imaginary polarities) have been fully differentiated (point level).

The processes of experience (finite and transfinite polarities) now are fully differentiated (radial level).
 
 
 

SIMPLE PARADIGM
 
 

The final stage involves equally the integration - as well as differentiation - of direction, mode and process.

Because such experience constitutes a dynamic seamless whole, it is no longer capable of differentiation. This is the most mature form of understanding and yet simple.
 
 
 

SUMMARY
 
 

The positive real rational paradigm - which is the conventional approach of mathematics - despite its great achievements is highly distorted. It reduces the negative (subjective) to the positive (objective) direction of experience, the imaginary (affective) mode to the real (cognitive), and the transfinite (intuitive) to the finite (rational) process.
 
 

The negative real rational paradigm, concentrates on the complementary direction of experience. Every theory has a mirror image anti-theory, which in static absolute terms is formally identical with the original theory. However in dynamic relative terms, theory and anti-theory constitute direct opposites.
 
 

The positive real irrational paradigm, is a dynamic relative approach where a theory is treated in terms of the complementarity of polar opposites. Both objective and subjective directions of experience are now inextricably linked in an approach which - in terms of the absolute approach - is deeply paradoxical. What this really means is that the truth of all mathematical theorems represents no more than an especially useful form of social consensus. Where there is no opposition to this consensus, such theorems may appear to have an absolute validity. Indeed there is no essential difference here regarding the status of theorems in science and mathematics. In both cases truth is always relative and approximate. The most interesting situation is when there is a breakdown in mathematical consensus. Then the subjective elements - always present but hitherto unquestioned - come to the fore.
 
 

The negative real irrational paradigm is the mirror image anti-theory to the positive direction. Though more subtle than the rational, the irrational paradigm - of the complementarity of polar opposites - is itself is a static reduced form of expression where the belief that all things are relative, itself becomes absolute. Again in static terms the anti-theory is identical to the original. In dynamic terms, once more they are opposed, leading to a more purely intuitive understanding of truth.
 
 

Conventionally the world is seen as existing "out there", with the mind in various ways reflecting this reality. This is the basis of "real" theories.

However the world equally can be see as existing "in here" in the unconscious and only indirectly expressed in outer phenomenal form. This is the basis of the "imaginary"

approach where reality is seen as the projection of the unconscious mind.

Indeed it is the direct basis of the affective sense mode. Though necessarily involved in mathematical understanding (i.e. in the use of appropriate symbols) the affective sense is invariably reduced to the cognitive mode of reason. In other words the "imaginary" aspect of understanding is lost.

The imaginary paradigm recovers this.

As with real paradigms we can have rational and irrational versions. We can also have positive and negative versions in the formulation of "imaginary" theories and their corresponding anti-theories.
 
 

The transcendental paradigm is even more subtle. The truth of a theorem is understood in neither rational terms (as static absolute truth) nor irrational terms (as dynamic relative truth), but rather as the relationship between these two approaches.

Again such paradigms can have positive and negative directions (theorems and anti-theorems) and real and imaginary modes (theorems and virtual theorems).
 
 

The transfinite paradigm relates to the purely intuitive aspect of understanding involved in all theorems. Mathematics conventionally concentrates on the rational extreme. We are now concentrating on the complementary intuitive extreme, which literally provides the "light" or mental insight to grasp propositions. In physical terms objects and their relationships cannot be seen in the absence of light. In psychological terms it is precisely similar.

We could say that whereas reason is the (direct) means of informed understanding, intuition is the (direct) means of transformed understanding.

The transfinite paradigm itself has real and imaginary aspects. The real "particle" aspect of spiritual light (contemplation of immanence) enables one to (directly) see objects. The imaginary "wave" aspect of spiritual light (contemplation of transcendence) enables one to directly see the dimensions (in which the objects are placed).
 
 

The complex paradigm mirrors the complex number systems. All phenomena have both real and imaginary aspects, with positive and negative directions. In other words both objective and subjective directions and cognitive and affective modes are properly differentiated so that further reductionism does not take place in understanding.

This paradigm has finite and transfinite aspects; the finite aspects relate to the (conscious) phenomena that arise in experience; the transfinite aspects relate to the spiritual light (through which these phenomena are seen).
 
 

The simple paradigm involves the full interpenetration of all aspects (direction, mode and process). This represents a dynamic seamless web, where experience is both fully differentiated and yet fully integrated at the same time. About this no more can be said.