Monday, January 24, 2011

A Category-Theory Dream

Had a dream the other day in which Category-theoretic operations had somehow become three-dimensionalized and actualized into a huge machine of sorts, a wooden box hiding its bells and whistles inside, with an opening on top into which I (or anyone else) could jump inside.  It was assumed in the dream, it seems, that there was an opening on the bottom of the huge box that would give results and answers.  It was assumed in the dream again, that the results and answers would come from the instantaneous churning that this box would provide, and specific details regarding algebraic systems would come to be known, even though what entered the machine could be very general data.  In went the general data (myself) and out came the details, plus new information regarding the general data, their relationships, and structures that were unknown prior to their going into the box, something like a "synthetic a priori" of sorts.  It was supposedly the “Categories” box of functors and morphisms.  I somehow thought of Saunders MacLane in the dream, and about how dreamlike his (co-)invention was. 

Notwithstanding the obvious symbolism in the dream with regard to Categories, the dream reveals something about the nature of formal systems in general in a “Turing machine,” theoretical-machine sort of way.  Formal systems reveal nested hierarchies, the structure of which can be simply generalized into two parts, viz., the combinatoric and the analytic.  This duality of formal systems can be viewed respectively in terms of discrete and continuous, or analog and digital, and so on, and even in terms of hardware and software.  There is also a nested hierarchy of number sets (and their algebras (N, , 0): number set, operation(s), unit element, respectively from the one-generator free monoid of natural numbers to the field of Reals, and finally to the field of Complex numbers) which has its transitions in gradations from discrete to smoother, to smoothest.  The same holds true of methods over these exemplary objects; perhaps it could be said that the strictly combinatoric ends with the set and ring of Rational numbers.      

The dream symbolizes the extrapolation of analytic information from the combinatoric, and I suppose the implication of the dream’s symbolism is that this works both ways, e.g., extrapolating intuitive, Boolean truisms from the Jordan Curve Theorem, extrapolating power sets of transfinite Alephs (א) from the cardinality of Real numbers, and so on.