Monday, April 12, 2010

Monstrous Moonshine - role of the unconscious

As is well known the term "Monstrous Moonshine" was coined by John Conway with respect to an unsuspected mathematical connection demonstrated as between the Monster Group and the j-function (an important part of number theory related to modular functions).

The Monster Group itself appeared somewhat like the final piece in a massive jigsaw puzzle.

The whole area of Group Theory - that has now assumed great importance in Mathematics - arose out of the attempt to classify the different types of symmetrical objects that can arise.
A simple geometrical example of a symmetrical object is provided by the 3-dimensional cube which has six similar faces. Now there are many different ways of rotating the cube (48 in all) so that its symmetry remains unchanged. So the quest that arose in the 19th century was to classify the "atoms of symmetry" or basic building blocks as it were for all symmetrical objects, in what were termed finite simple groups.

In some respects this parallelled the similar quest to find the basic building blocks of the natural number system (i.e. the prime numbers).

Thus a great number of symmetrical objects are based on the different rotations possible with respect to various prime number permutations.

However other simple groups exist where the number of indivisible permutations do not correspond to prime numbers!

In the classification of the various simple groups that subsequently arose, most conveniently fell into four coherent families. However like separated islands, many exceptions remained (26 in all) that were lumped into a Sporadic Groups category, the largest of which is the Monster.

The total number of symmetries in this group - which cannot be decomposed into smaller sub-groups - is truly enormous and greater than all the quarks in the Universe. The lowest number of (non-trivial) dimensions in which they exist is 196,883 (47 * 59 * 71). It is not surprising therefore that this has been termed the Monster Group.

Then by surprise it was discovered that the number 196,883 and other larger dimensional numbers associated with the Monster bore a close relationship with coefficients of the j-function (a seemingly unrelated area of Mathematics).

It is in this connection that the term "Monstrous Moonshine" was coined by Conway.

"The stuff we were getting (i.e. the unexpected connections) was not supported by logical argument. It had the feeling of mysterious moonbeams lighting up dancing Irish leprechauns"

From my own holistic mathematical perspective as an "Irish leprechaun", I find this comment very interesting as it actually points to a deep unrecognised limitation of Conventional Mathematics.

The moon is commonly used to signify unconscious life. Indeed one remnant of this association is in the term lunacy (derived from the Latin word for the moon). So lunacy or madness would reflect a deep disorder with respect to the unconscious.

Then in the spiritual life "dim contemplation" is often used to refer to the faint passive light (like reflected moonlight) that greatly facilitates understanding of a universal holistic quality based on pure unconscious intuition.

As I have continually maintained there are in fact two equally important aspects in balanced (i.e. psychologically symmetrical) understanding of Mathematics.
One is the quantitative rational aspect which obtains its specialised expression in Conventional Mathematics. However the other - largely unrecognised - is the qualitative intuitive aspect which obtains its specialised expression in what I refer to as Holistic Mathematics.

As we will once again see the mathematical notion of Euclidean dimensions (on which the classification of the finite simple groups is based) is but a reduced rational notion based on merely linear logical concepts.

Indeed there is a strong paradox in this mathematical approach to symmetry based - as it is - on classification of irreducible building blocks (i.e. the finite simple groups).

Symmetry in any context entails the holistic relationship as between the various part aspects involved. However such holistic appreciation (literally of the whole context) should not be confused with successful analysis of the various parts. Though both aspects (whole and parts) are necessarily interrelated they should not be identified with each other. However, because of sole recognition of the quantitative aspect (suitable for partial analysis), in any relevant context Conventional Mathematics can only proceed by attempting to reduce the whole to the parts.

So the point I am making - which will be developed in further posts - is that true holistic understanding is needed to qualitatively appreciate what is entailed by "Monstrous Moonshine".

Another obvious paradox about the mathematical drive to understanding symmetry comes through recognition of the extremely important role played by many of its leading proponents who display classic symptoms of Asperger's Syndrome. For example John Conway, Simon Norton and Richard Borcherds would seem to fit readily into this category.

Indeed one could go further and argue that success in the highly specialised abstract task of finding the unique building blocks for all symmetrical objects has been greatly facilitated in this context through Asperger's Syndrome.

And I do not wish by this to question the merit of this achievement which has been truly outstanding; rather I am suggesting that from an overall comprehensive mathematical perspective it is quite unbalanced. Thus a much more rounded appreciation (requiring qualitative rather than quantitative mathematical notions) likewise needs to be provided.