Wednesday, July 4, 2007

Mathematics and culture

William Byers' How Mathematicians Think (Princeton, 2007) states that the deepest motivation for mathematical practice rests on the fact that "There exists that which cannot be expressed yet we must express it." (p. 120; italics in original.) He amplifies this as "putting the ineffable into words, describing the indescribable, or expressing the inexpressible." Exploring the example of the "infinite," he finds that historically two clusters of characerizations are found: on the one hand the infinite is boundless, endless, immeasurable; and on the other hand it is complete, whole, perfect, and absolute. The first cluster belongs more to the context of expression, the second to the intuition of the wholeness that lies beyond expression. In my terms, the indescribable infinite belongs to World Two, and the attempt to say the infinite in terms such as "immeasurable" to World Three, the realm of culture and specifically mathematical culture. Godel's incompleteness theorem is a paradigmatic example of mathematical culture because it shows that "within the context of logical thought one can deduce limitations on that very thought." (Byers, p. 282). This is quite similar to the argument in the Samkhya Karika that leads the highest embodied intelligence (sattvic buddhi) to see that "I am not, I have nothing, there is no self in me" (naham na me nasmi, SK 64).

The self referentiality of the Samkhya text closely parallels that of Godel. This seems to be a general feature of (genuine) culture. There is always self-reflection in the service of self-transcendence. We are bound into the world of duhkha but that suffering life contains within it the possibility--indeed the necessity--of a beyond. Great ideas in mathematics parallel the great realizations of figures like the Buddha and Patanjali (of the Yoga Sutra) that show the limitations and thereby the potentialities for release in our World One of constriction and pain.