Spiritual Illumination

Again with reference to Marcus du Sautoy's "The Music of the Primes", I found an interesting quote on P.297 attributed to Andre Weil.

"Every mathematician worthy of the name has experienced ... the state of lucid exaltation in which one thought succeeds another as if miraculously...this feeling may last for hours at a time, even for days. Once you have experienced it you are eager to repeat it but unable to do it at will, unless perhaps by dogged work..."

What Weil is desribing here is in fact - what spiritual writers refer to as - "illumination" where for a while one has a peak experience of holistic intuitive insight. It is in such moments the truly great mathematical insights are obtained and those decisive creative breakthroughs where for a brief moment one is able to "see" certain important relationships - perhaps for the first time - in an enhanced manner.

In fact quite clearly such moments relate directly to the qualitative - rather than quantitative - aspect of mathematical appreciation. Though the intuitive insights obtained may indeed be later expressed in a (reduced) rational manner that wins the acceptance of the mathematical community, the initial intuitive realisation properly remains of a qualitative nature.

Remarkably however, the qualitative aspect of Mathematics is given no formal recognition at all in conventional terms.

In other words though every mathematical symbol, relationship, hypothesis etc. has a distinctive (holistic) qualitative as well as (analytical) quantitative interpretation in Type 1 Mathematics only the the latter is recognised.

Thus in a comprehesive treatment we should have both Type 1 (quantitative) and Type 2 (qualitative) aspects that initially are developed in relative independence from each other.

Then when both of these aspects have achieved appropriate degrees of specialisation they can be fruitfully combined with each other in the most advanced form of Mathematics (i.e. Type 3).