||The birth of mathematics in Greece
||1. Philosophy, science and mathematics in Greece. Plato and Aristotle drew the distinction between primary and secondary (or necessary) causes. Although dealing mainly with necessary causes, Greek mathematics are by no way empty of philosphical interests. 2. The “scientific”, the “pre-scientific” and the “extra-scientific”. Elements of knowledge appearing in a period previous to the shaping of the science can be designed “pre-scientific”, provided we give a constitutive, not temporal sense to this word. Some elements remain permanently outside the range of science (p. e. astrology); we could term them “extra-scientific”. The criterium of scientificity will be, beyond the somewhat ambiguous character of systematicity, the historical fecundity of a given element. 3. Demonstration in Greek mathematics. Practical mathematics were widely employed before the Greeks. A Mesopotamic tablet from the Yale collection shows a systematic use of the theorem of Pythagoras. Nevertheless, the novelty of the Greek matematicians was the demonstration of such practical formules. On the other hand, it is not proved that Thales was the first thinker in proposing a deductive method in geometry. 4. Demonstration and deduction. Proves by rule and compass and the equivalent method of epharmozein do not constitute scientific mathematics. In Euclides' Elements, such proves are found up to I 34. But at I 35 appears a theorem that cannot be proved by the practical method, therefore requiring deductive demonstration. Mathematical deduction can be adscribed first to Theodorus of Cirene. 5. Deduction and axiomatic foundation. The establishment of evident propositions (axioms) as basis for the necessity of a deductive argument was a procedure elaborated probably in the Academy; it was exposed in Aristotle's Analytica Posteriors 2 and 10. “As for “historical fecundity”, the method of deductive demonstration appears as the most decisive instrument of scientific progress in mathematics.
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||Enrahonar : quaderns de filosofia, N. 21 (1993) p. 7-26, ISSN 0211-402X