A Theory of Mathematical Knowledge: Can Rules Account for Creative Behavior?

This paper describes some elements of an emerging theory of mathematical knowledge. It is proposed that the mathematical behavior any given (idealized) S is potentially capable of at any given stage of learning can be accounted for precisely in terms of a finite set of rules. In particular, it is hypothesized that rule characterizations of this sort can account for creative, as well as more routine, behavior. The main argument centers on the notion of allowing rules to operate on other rules (i.e., to act as higher order rules) and thus allowing rule sets to “grow.” (Newly generated rules are assumed to be available from then on.) Attention is first centered on the behaviors involved in knowing mathematical systems (semantic knowledge). Then, the same ideas are extended to logical interrelationships between properties of mathematical systems, e.g., proving theorems.