Oskar Becker - History of Mathematics

History of Mathematics

Becker also made important contributions to the history and interpretation of ancient Greek mathematics. Becker, as did several others, emphasized the "crisis" in Greek mathematics occasioned by the discovery of incommensurability of the side of the pentagon (or in the later, simpler proofs, the triangle) by Hippasus of Metapontum, and the threat of (literally) "irrational" numbers. To German theorists of the "crisis", the Pythagorean diagonal of the square was similar in its impact to Cantor's diagonalization method of generating higher order infinities, and Gödel's diagonalization method in Gödel's proof of incompleteness of formalized arithmetic. Becker, like several earlier historians, suggests that the avoidance of arithmetic statement of geometrical magnitude in Euclid is avoided for ratios and proportions, as a consequence of recoil from the shock of incommensurability. Becker also showed that all the theorems of Euclidean proportion theory could be proved using an earlier alternative to the Eudoxus technique which Becker found stated in Aristotle's Topics, and which Becker attributes to Theatetus. Becker also showed how a constructive logic that denied unrestricted excluded middle could be used to reconstruct most of Euclid's proofs.

More recent revisionist commentators such as Wilbur Knorr and David Fowler have accused historians of early Greek mathematics writing in the early twentieth century, such as Becker, of reading the crisis of their own times illegitimately into the early Greek period. (This “crisis” may include both the crisis of twentieth century set theory and foundations of mathematics, and the general crisis of WWI, the overthrow of the Kaiser, communist uprisings, and the Weimar Republic.)