Pythagoras Revisited

Special Numbers

There is a very interesting example, in the history of mathematics of an explicit holistic approach to mathematics. It is associated with the Pythagoreans who flourished in Greece (circa 600 BC).

Their discoveries in number theory illustrate the approach very well.

For example there is the special class of numbers (associated with them) called perfect numbers.

A perfect number is one that is equal to the sum of its proper factors. 6 is the first of these numbers. Its proper factors are 1, 2 and 3 and the sum of these is 6. The next three perfect numbers are 28, 496 and 8128 respectively. (In all more than 30 are known).

Perfect numbers have several unique properties and were frequently used, in earlier times, as numerical symbols of psychological perfection.

I wish here to comment on one interesting point. All (even) perfect numbers are derived from the product of a Mersenne prime (which we have already met) and a power of 2, e.g. 6 = (22 - 1) * 21

Now Mersenne primes (which have no factors) are a particularily "independent" type of number, representing the "masculine" extreme. Powers of 2, on the other hand, being highly composite (with the maximum factors possible for the size of number), represent the "feminine" extreme. Thus, there is a complementarity of opposites apparent in the derivation of perfect numbers, always being a combination of two numbers, one of which represents the masculine, and the other the feminine pole.

In Jungian terms, psychological perfection can be represented as combining the two poles of the personality masculine and feminine (both having initially being highly differentiated). So indeed, there is a complementary underlying structure both to perfect numbers and psychological perfection.

Another type of number attributed to the Pythagoreans is the set of amicable (friendly) numbers. The simplest type involves pairs of numbers so that the sum of the proper factors of each number is equal to the other number. This immediately suggests comparison with friendship in psychological terms based on two people sharing complementary personality characteristics. The best known amicable pair comprises 220 and 284. The sum of this pair = 504, which is the product of three consecutive numbers (6 *7 * 8). Also, if we take the reverses of 220 and 284 (i.e. 022 and 482) and add we again get 504.

504 is also divisible by 9. This is a feature shared by the vast majority of sums of amicable pairs, triples etc.

9 in turn again has a strong psycho-mathematical importance serving as a symbol of integration.

The Enneagram, for example, which is a widely used system for classifying personality types, is based on the number 9.

Pythagorean Triangle

Returning to the Pythagoreans, perhaps their most important psycho-mathematical symbol is the triangle named after them.

This of course is the right angled triangle where the square on the hypotenuse (the diagonal line) is equal to the sum of squares on the other two sides (i.e horizontal and vertical lines).

Now the Pythagoreans were not the first to discover this mathematical result. However its significance for them extended way beyond mathematics and really summed up their whole philosophy of life.

They did not see mathematics - in marked contrast to the modern approach - solely as an independent objective discipline. Rather they saw the quantitative order of the physical universe as complementing an inner qualitative psychological order. In other words rather than a one sided mathematical approach, they adopted a two sided approach (i.e. with complementary mathematical and psychological aspects).

Furthermore, the road to spiritual contemplation was understood as the combination of these two aspects.

Now the diagonal in the triangle can be viewed as relating to the central spiritual aspect, whereas the horizontal and vertical lines represent the mathematical and psychological aspects respectively. It is fascinating, in this context that any right angled triangle can be fitted within a circle with the diagonal line (i.e. the hypotenuse) always being the diameter (i.e. the central line of the circle).

The mathematical result (i.e. that the square on the hypotenuse equals the squares on the other two sides), really symbolises the deeper reality that in this two sided holistic mathematical approach, spiritual wisdom (i.e. diagonal understanding) is to be achieved by balancing the (horizontal) quantitative order of physical reality with the (vertical) qualitative order of the corresponding philosophical paradigm.

Initially, this school believed that all numbers were rational. Their underlying philosophical approach in turn was based on the rational paradigm. Therefore - given the holistic approach adopted - they firmly believed that the rational mental order (qualitative) was matched by a corresponding physical order (quantitative).

However in the most basic right angled triangle, where both horizontal and vertical lines are 1, the diagonal line is not a rational number. In other words the square root of 2 cannot be expressed as a fraction and is thus irrational. (Paradoxically this very discovery of irrational numbers came from the use of their famed mathematical theorem)!

This was why the discovery of irrational numbers was so devastating. In short it undermined the universal validity of the rational paradigm. Where complementarity of mental and physical orders is of the essence, then irrational quantities cannot be successfully incorporated within a (qualitative) paradigm that is rational.

Subsequently this problem has been dealt with in mathematics through a process of reductionism where the authentic philosophical understanding of the meaning of irrational numbers has been lost. A (reduced) quantitative rather than a qualitative appreciation is now all that remains.

However from a holistic mathematical perspective, this approach simply is inadequate. From a holistic perspective irrational (mathematical) quantities must be explained by a corresponding irrational (psychological) paradigm. I will return to this important point soon.