Aristotelian Axiomatics and Geometrical Axiomatics
Professor Szabó deserves credit for calling our attention to the interplay of philosophical and mathematical influences in the development of Greek axiomatics. It is this interplay that lends a special flavor to much of the early as well as some of the later history of the axiomatic method. I believe, however, that in the last analysis the total picture of the early development of axiomatics will turn out to be quite different from the one Szabó paints. My reasons for this belief are nevertheless subtler than one might first expect. Professor Szabó finds the true ancestors of the central mathematical methodology of the Greeks, including the axiomatic method, in the Eleatic dialectic. In so doing, Szabó prima facie misses a large part of the interdisciplinary interplay with which he is dealing. Most other historians of the axiomatic method would give the pride of place on the philosophical side of the fence to Aristotle, who is sometimes called the first great theoretician of the axiomatic method and whose ideal of a science was by any account explicitly and self-consciously axiomatic. Szabó admittedly discusses Aristotle, but gives the Stagirite short shrift, dismissing him as having played no real part in the development of the axiomatic methods actually used in mathematics.
KeywordsMathematical Practice Indirect Proof Axiomatic Theory Axiomatic Method Greek Mathematic
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