On Definitions

Extract

Present-day logic regards definitions as syntactical devices of abbreviation. This view, summed up in the terse adage: “To define is to eliminate,” obscures the vital part played by definitions in deductive sciences. Abstract calculi, to become noetic tools, must be interpreted; so the richer the interpretation, the more efficient the tool. But most calculi are based on a restricted number of primitives; they thus embrace, at the outset, a restricted number of concepts and can be appraised through and only through the definitions they eventually yield. In a word, the richness of a calculus lies in its defining virtualities. The whole of PRINCIPIA MATHEMATICA would be a futile game of chess, did it not embody all our mathematical concepts; none of these being assumed as primitive, PRINCIPIA MATHEMATICA owes its momentous significance to its definitions and to its definitions only.