The Pythagorean Dilemma

Recently there has been an increased interest in the philosophical vision of the Pythagoreans.

Partly this is due to the recent proof of Fermat’s Last Theorem (which had become the best-known unsolved problem in Mathematics).

The roots of Fermat’s Last Theorem lie deep in the famous Pythagorean Theorem.

Indeed this theorem (i.e. Pythagorean) is perhaps the most famous in mathematics. It states that in any right-angled triangle the square on the hypotenuse (diagonal line) is equal to the sum of the squares on the other two sides (opposite and adjacent). Thus in the well-known example when opposite and adjacent sides measure 4 and 3 units respectively, the hypotenuse is 5.

However the Pythagorean Theorem gives rise directly to another intriguing problem of a qualitatively different order which I refer to as the Pythagorean Dilemma. Indeed the failure to solve this problem led to the break-up of the Pythagorean School and subsequently exercised a profound influence on the development of Western Mathematics.

To appreciate the nature of the Pythagorean Dilemma we have to understand the nature of their approach to mathematics.

Nowadays mathematics is conventionally understood as a merely quantitative (analytical) discipline and is clearly separated from (qualitative) philosophical investigations. However this has not always been the case.

The Pythagorean approach to mathematics represented a dynamic balance as between two aspects (which were understood as mutually complementary).

It had of course a quantitative (analytical) aspect. Indeed besides the famous triangle (which bears their name), the Pythagoreans made many interesting contributions to number theory and applied science.

Equally however it had a qualitative (holistic) aspect.

The scientific worldview of the Pythagoreans was based on - what we understand as - the rational paradigm.

They then looked out a world that they believed corresponded in quantitative terms with this qualitative paradigm. Thus they believed that all numerical quantities could be expressed in rational terms (i.e. as whole numbers or fractions).

I think that it is vital to grasp this essential point. Mathematical activity (in quantitative rational terms) served as a continual confirmation of their qualitative rational paradigm. Equally the rational paradigm provided the correct basis for their quantitative investigations.

Indeed they believed that the interaction of these aspects served as an ideal basis for spiritual contemplation (which was the true end of mathematical activity).

Initially all seemed well with this worldview. Gradually however they became aware of a very disturbing problem which seemingly had no resolution. Ironically this was directly associated with their most famous discovery (i.e. the Pythagorean triangle).

In the simplest possible case of the right-angled triangle both the opposite and adjacent sides are both 1 unit (in length). Since the square of 1 is 1, the square on the hypotenuse is (1 + 1 = 2 units). Therefore the length of the hypotenuse is the square root of 2.

However this number cannot be expressed in whole or fractional terms (as a rational number). Indeed it is the best known example of what is termed in mathematics an (algebraic) irrational number.

The real reason for the disturbance caused by the discovery of the square root of 2 was the fact that the Pythagoreans were unable to offer a qualitative holistic explanation for its existence.

As we have seen they employed – in qualitative terms – the rational paradigm.

This was consistent with the existence of matching rational quantities in nature. However it was not consistent with these new irrational quantities.

Thus with the discovery of the square root of 2, the balance as between quantitative and qualitative appreciation of nature was broken. As they were not interested in pursuing a merely quantitative approach to mathematics, their School quickly disintegrated.

Since the Pythagoreans, a merely (reduced) quantitative interpretation of mathematical symbols is given by Western Mathematics. Many other number types besides (algebraic) irrational have been discovered. We also have (transcendental) irrational numbers. (The famous mathematical symbol pi is a transcendental number). We have imaginary numbers, complex numbers and transfinite numbers.

However the holistic qualitative appreciation of the nature of these numbers has been lost (with a merely reduced rational interpretation remaining). By extension this reduced interpretation applies to all mathematical symbols.

Now there are many different levels of understanding on the Spectrum of Consciousness).

The great difficulty however is that scientific understanding of reality is invariably associated with just one of these levels (i.e. L0 – the rational linear level).

Now L0 has three distinct sub-levels. These are commonly referred to as conop (concrete operational thinking), formop (formal operational thinking) and vision-logic (which is more intuitive and represents the dynamic interaction of conop and formop).

Now though the Pythagoreans were rightly concerned with the goal of spiritual enlightenment, their scientific paradigm did not formally extend beyond that of L0. Indeed I have made the same very same criticism of Ken Wilber (whose scientific paradigm is clearly based on the "highest" of the rational stages i.e. vision-logic). Properly understood this is suitable as a multi-differentiated approach. However it does not provide the correct basis for a truly integrated paradigm.

In my mapping of the Spectrum there are three higher integrated levels (i.e. HL1, HL2, HL3) and finally Radial Reality (which represents the most mature understanding at both a differentiated and integrated level). When translated in appropriate holistic mathematical terms, the first of these levels i.e. HL1 which is the circular level (or subtle realm) provides the solution to the Pythagorean Dilemma.

The understanding of L0 (the rational linear level) is associated with - what I have referred to - as one-directional (or one-dimensional) understanding. This is a form of understanding where movement can only be appreciated in positive terms (i.e. as moving forward). Thus for example in terms of L0, time has a strictly positive interpretation (i.e. time moves in a solely forward direction).

I have identified the crucial transition from L0 to HL1 as the development of mirror understanding (i.e. the negative direction). This leads initially to a more subtle appreciation of dynamic interaction of horizontal polarities (e.g. internal and external). Thus from this dynamic relative perspective if time is moving forward for the world (relative to the self), then it is moving backwards for the self (relative to the world). The very appreciation of this negative backward movement (i.e. mirror understanding) comes from a profound existential crisis where the customary interpretations of external reality are largely eroded.

Thus in holistic mathematical terms, the rational understanding of L0 is positive (i.e. where interpretations of relationships are one-directional).

Likewise in holistic mathematical terms the rational understanding of the transition from L0 to HL1 is negative. So here the former rational interpretation is greatly eroded (in horizontal terms) with the consequent development of intuitive mirror understanding.

Now with the development of HL1 a new type of understanding unfolds which can be described as the complementarity of polar opposites (and is bi-directional in horizontal terms).

Thus at HL1, horizontal opposites (e.g. external-internal, exterior-interior, object-subject) are now understood in bi-directional fashion (with both positive and negative interpretations).

This understanding by its very nature represents a subtle fusion of reason and intuition.

In rational terms it leads to a paradoxical form of understanding (in terms of former rational understanding).

Conventional rational understanding is characterised by an either/or logic (requiring the separation of poles). For example a proposition is either true or false in absolute terms.

However the "new" paradoxical logic is characterised by a both/and logic (requiring the complementarity of both poles). Thus a proposition is both true and false in terms of this logic (i.e. has a merely relative validity).

Thus in terms of former quantitative rational understanding this new paradoxical logic could be accurately described as "irrational" (as it confounds the very basis of rational logic).

However it is very important to realise that these quantitative paradoxes have only an indirect rational interpretation. Their direct qualitative interpretation is itself intuitive in origin.

Thus the very role of rational paradox (i.e. irrational understanding) at HL1 is to prepare the mind for a transformation through pure intuitive understanding (which is of a qualitatively different order).

This intuition then leads readily to (reduced) rational interpretations that are paradoxical (in quantitative terms). The very dynamic of experience at this level is this ceaseless interaction where phenomena are transformed (intuitively) and reduced (rationally).

Unfortunately in practice the intuitive qualitative dimension is often lost. Hegelian philosophy is a fine example of irrational (i.e. paradoxical) understanding in horizontal terms. Thus Hegel saw reality in dynamic terms as. the complementarity of polar opposites (i.e. thesis and antithesis). However though Hegel’s philosophy necessarily started with the qualitative intuitive aspect (of refined spiritual understanding) this aspect became progressively lost as his work developed. In other words his philosophy became a paradoxical and largely reduced rational interpretation of reality.

Now surprisingly the behaviour of irrational number quantities (such as the square root of 2) exactly complement this holistic qualitative interpretation.

The square root of 2 in fact has two solutions. In one solution the answer is positive. In the other solution the answer is negative.

So in qualitative terms irrational understanding results from expressing what is inherently intuitive and two dimensional, in reduced one-dimensional rational format. Understanding then is expressed as the complementarity of opposite poles and is both positive and negative.

In quantitative terms, irrational numbers (such as the square root of 2) result from attempting to express a two-dimensional quantity in reduced one-dimensional format. The result then expresses itself as two opposite poles (i.e. the same number with opposite signs) that are clearly separated.

The connections are even deeper than this.

Again in holistic terms, irrational understanding has both rational (quantitative) and intuitive (qualitative) aspects.

Now the rational aspect gives a directly finite interpretation of reality; the intuitive aspect by contrast gives a directly infinite interpretation.

The same behaviour is replicated in the behaviour of (algebraic) irrational numbers.

The square root of 2 has a finite (reduced) quantitative interpretation. Thus its value can be approximated as a rational number. Thus correct to four decimal places its value is 1.4142.

However the square root of 2 also has an infinite qualitative interpretation. Its decimal sequence extends indefinitely (in random fashion) and can never be exactly determined.

Thus this qualitative infinite aspect is inherent in its very behaviour. This is what troubled the Pythagoreans so greatly. They had not got a paradigm to explain its behaviour.

I will point to one final striking correlation.

Paradoxical understanding is irrational (in terms of "lower" one dimensional rational understanding).

However when viewed from the "higher" two-dimensional perspective it appears quite rational.

(Indeed Hegel identified reason with this two-dimensional rather than one-dimensional interpretation).

It is quite similar with the square root of 2. This is clearly irrational in terms of its one-dimensional value. However if we now raise this number to two dimensions (by obtaining its square) it will now be rational.

Thus the solution to the Pythagorean Dilemma requires a holistic mathematical understanding of the next "higher" level of understanding (i.e. HL1 – the circular level or subtle realm).

We are then able to appreciate not alone that the quantitative explanation as to why the square root of 2 is irrational but also the deeper qualitative explanation.

I have stated before on this Forum that our notions of what scientific understanding entails are very limited.

Basically we interpret science in quantitative analytical terms which is suitable for a differentiated appreciation of reality. We neglect its equally important holistic aspect which is suitable for an integrated appreciation.

Science in my view has both quantitative (differentiated) and qualitative (integrated) aspects.

Now L0 is by definition ideally suited to the quantitative (differentiated) aspect; however the "higher" levels are required for qualitative (integrated) understanding.

The need for this integrated aspect is very much exemplified by a field such as physics.

Though the brightest minds are engaged in the quantitative mathematical understanding of key areas such as quantum mechanics and superstrings, a huge lack of proper holistic understanding is in evidence. If even a small fraction of the time devoted to quantitative understanding was devoted instead to holistic understanding it would pay rich dividends. However this is not possible in terms of the present paradigm of science.

This is why the Pythagorean Dilemma is so important. The Pythagoreans were not interested in a merely quantitative proof that the square root of 2 is irrational. They equally wanted a convincing qualitative proof.

In other words from this perspective irrational quantities can only be properly explained in terms of irrational qualities (a paradigm which provides a proper intuitive interpretation of its findings).

Modern science is greatly lacking in this holistic intuitive vision. Despite its great success the finding of modern physics increasingly conflict with the intuitions of the rational linear paradigm.

As the Pythagoreans suspected alternative scientific paradigms are necessary that require satisfactory scientific translations of the "higher" levels of understanding.

This is a task on which I have been engaged for some time.

Thus when properly understood all the stages of HL1 correspond to (algebraic) irrational understanding. This level can be subdivided conop, formop and vision-logic sub-levels with a markedly different interpretation from the rational linear level.

This translation not alone provides a satisfactory solution to the Pythagorean Dilemma but it equally provides a fascinating way of understanding many thorny areas (such as quantum physics) where qualitative (intuitive) understanding can be shown to correspond with quantitative (rational) understanding.

In like manner the holistic mathematical translation of imaginary numbers provides the crucial explanation of the important transition as between HL1 and HL2 (i.e. the point level or causal realm).

The Holistic mathematical translation of transcendental (irrational) numbers then provides the appropriate scientific (number) paradigm for HL2. These again can be divided into conop, formop and vision-logic sub-levels which have unique interpretations.

The holistic translation of complex numbers then provides the appropriate explanation of the transition from HL2 to HL3. This culminates in the null level (nondual reality) which has a holistic interpretation in terms of transfinite numbers.

Thus all the "higher" number types (in quantitative terms) have "higher" holistic interpretations (in qualitative terms).

These then provide appropriate number paradigms for the development of integrated science.

Again the rational paradigm is ideally suited to a differentiated interpretation of reality.

However other number paradigms are necessary for a truly integrated understanding. Among these the solution to the Pythagorean Dilemma (i.e. the irrational paradigm) has an important place.

It provides the appropriate model for an integrated scientific understanding of reality (at the level of HL1).

I have stated repeatedly on this Forum that there is an enormous – and still largely uncharted area of integrated science that requires suitable scientific translations of the "higher" spiritual levels.

Many would claim that Ken Wilber is tackling this problem. However I would not accept this viewpoint. Ken does not yet seem to see beyond the vision-logic of L0 as a scientific integrative tool. Again this is really only suitable as a versatile differentiated approach to reality. It is not appropriate as a true integrative tool, and when employed in this manner leads to deep inconsistencies (which I have attempted to demonstrate in many posts).

True integration requires "higher" rational paradigms whose underlying features can be precisely translated – as the Pythagoreans would have wished - in holistic mathematical terms. They lead to a fundamentally different qualitative approach to science which is still very much lacking in contemporary understanding.

If I succeed – even in a general way – in alerting Forum participants to the true nature of scientific integration, I will consider that my efforts have been well worthwhile.