The progress of mathematics may be viewed as a movement from the infinite to the finite. At the start, the possibilities of a theory, for example the theory of enumeration, appear boundless. Rules for the enumeration of sets subject to various conditions appear to obey an indefinite variety of recursions and seem to lead to a bounty of generating functions. We are naively led to conjecture that the class of enumerable objects is infinite and unclassifiable.Gian-Carlo Rota
As cases pile upon cases, however, patterns begin to emerge. Freakish instances are quietly disregarded; impossible problems are recognized as such, and what is left gets organized along a few general criteria. We will do all we can to boil these criteria down to one, but we will probably have to be satisfied with a small finite number.
Friday, September 29, 2006
On progress
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