PYTHAGORAS REVISITED

PSYCHO-MATHEMATICAL APPROACH

Special Numbers

There is a very interesting example, in the history of mathematics of an explicit psycho-mathematical method. It is associated with the Pythagoreans who flourished in Greece (circa 600 BC).

Their many discoveries in number theory help to illustrate this approach.

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 slightly 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 particularly "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.

Reverse numbers and Palindromes

I now wish to illustrate this psycho-mathematical approach further now by combining reverse numbers and the number 9 in a simple demonstration involving whole numbers.

The difference of any number and its reverse is always divisible by 9. We conventionally operate in a base 10 number system and the significance of 9 is that it is one less. Relative to 10, it is -1. Now this is just a particular example of a more general pattern. In any number base n, the difference of a given number and its reverse, is always divisible by n - 1.

Now a reverse number is in fact a mirror number (where the direction of sequence of digits is opposite to the given number).

Thus (given) numbers and reverse numbers bear obvious comparison with psychological structures and mirror structures (where the direction of understanding is reversed).

Combining a given number with its reverse can therefore be viewed as a complementary numerical activity to the combining of structures and mirror structures.

(Again as structures and mirror structures are positive and negative relative to each other, it is the difference rather than the sum of numbers that is involved).

In our base system the number (obtained from the difference of a given number and its reverse) is as we have seen always divisible by 9.

A remarkable fact about the result, is that it shows very distinctive palindromic tendencies. Frequently an actual palindrome will emerge. In many other cases, one digit or at most two have to be altered by 1 to render a palindrome. (A number palindrome is the same when read from either direction. Alternatively if a given number equals its reverse it is palindromic).

To illustrate let us take a 5 digit number 14117 (which I have just obtained from a random number generator). Its reverse is 71141 and the difference of this number and the given number = 57024.

This is divisible by 9 with the result = 6336 which is a palindrome.

There is a complementary psychological significance to this palindromic tendency which is fascinating. With a palindrome there is no distinction as between the positive direction of the number and the negative direction. Both have been harmonised eliminating all distinct polar tendencies. It is likewise with personality integration. When the positive and negative directions of conscious understanding are successfully combined with the unconscious, we get a new form without any distinct polarities (i.e. spiritual intuition). Thus, when viewed in this light, we can see that there is a remarkable complementarity as between these basic mathematical and psychological processes.

The given number and its reverse complement the positive and negative directions of understanding. The difference of the numbers reflects the result from combining the positive and negative directions of digits, which complements the psychological switching of direction e.g. external to internal understanding. 9 which is -1 in terms of the base number has a holistic unconscious symbolism. Thus the division of the resulting difference by 9 matches the attempt psychologically to balance the positive and negative polarities of (conscious) understanding with the unconscious. Finally the palindromic tendency in the numerical result in turn complements the successful elimination of conscious polarities in the conscious.

There is also - as we might expect - a special significance to the number immediately to the right of the base number. This will be 11 in any base system, which relative to the base number is +1 (i.e. n + 1).

Now whereas 9 (n - 1) has an unconscious significance, 11 (i.e. n + 1) has a corresponding conscious significance with one-sided polar tendencies.

This is remarkably demonstrated in number behaviour.

If the total of digits in a (given) number is odd, then the difference of this number and its reverse will always be divisible by 11.

However if the total of digits in a given number is even, then the sum of this number and its reverse will always be divisible by 11.

55512 is a randomly generated number with an odd total of digits.

Its reverse is 21555 and the difference of this and given number is 33957. This is divisible by 11 = 3087. (Note there is no palindromic tendency here).

877275 is a random generated number with an even total of digits.

Its reverse is 572778 and the sum of this and given number = 1450053. This is divisible by 11 = 131823.

This also implies that the difference of any number and its reverse (with an odd total of digits) in any base n is divisible by (n + 1) * (n - 1).

Thus 55512 - 21555 = 33957 is divisible by 99 = 343 (Again there is a distinct palindromic quality to the result).

Incidentally, all perfect numbers on division by 9, leave a remainder of 1.

Indeed we can even see a distinctive "personality" in these various kinds of number behaviour complementing our three personality groupings.

The vertical type (based on unconscious specialisation of personality) complements the difference (i.e. of given number and reverse) which is always divisible by 9 (i.e. in base 10).

The horizontal type (based on conscious specialisation of personality) complements the sum (i.e. of given number and reverse with even total of digits) which is always divisible by 11.

The diagonal type (based on both conscious and unconscious specialisation of personality) complements the difference (i.e. of given number and odd total of digits) which is always divisible by 99.

Significance of 9

The significance of 9 can be readily appreciated by expressing it in the binary system (which is especially suited to psycho-mathematical investigation).

9 in binary form = 1001. This of course is palindromic with two 1"s and two 0"s.

However if we rearrange the digits to form two numbers at polar extremes (i.e. in ascending and descending order of magnitude ) we get 1100 and 0011. The difference of these two numbers = 1001.

Thus 1001 which is palindromic represents the combination of two polar extremes. This complements personality integration (i.e. where rigid polarities are eliminated) through the successful combination of fully differentiated opposites.

The special significance of 1001 (i.e. 9 in denary terms) is that it is the first and only symmetrical number (i.e. with equal number of 1"s and 0"s) in the binary system to possess this unique property.

Psycho-mathematically therefore, 9 serves as a unique symbol of integration.

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 psycho-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 psycho-mathematical approach adopted - to preserve overall harmony, the rational mental order needed to be matched by a corresponding rational physical order.

However in the most basic right angled triangle, where both horizontal and vertical lines are 1, the diagonal line (Ö 2) 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. It was not so much the mathematical significance, but rather that the discovery 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 psycho-mathematical perspective, this approach simply is inadequate. Irrational quantities must be explained by a corresponding irrational (qualitative) paradigm. It is to this task that I now return.