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The Babylonians were wizards at computation, they had at least one of the two ways of generating what we now call Pythagorean triples (as Otto Neugebauer brilliantly demonstrated from the Plimpton cuneiform tablets), they could approximate the square root of two to as many sexagesimal places as desired, but the question of the rationality or irrationality of the square root of two simply would have made no sense to them, because they had no numbers in their conceptual framework. This is not surprising, because numbers did not yet exist in those days. Numbers were invented (or revealed, as believers would maintain) by Pythagoras.
But Eudoxus eliminated the dualism of number and magnitude. His idea was this: rather than say what the ratio of two magnitudes is, it suffices to define a notion of two such (possibly nonexistent) ratios being equal, and this he did by a subtle quantification over all the Pythagorean numbers. Despite the fact that it was in the Elements for all to read, this construction of the real number system was not understood – not even by Galilei. It was just last century that the notion was re-invented by Richard Dedekind. I know of no parallel to this in the history of human thought.
This article emphasizes Archimedes’ original contribution to the principle or lemma of Archimedes (in modern terms, given two homogeneous magnitudes a, b, there exists an integer n so that na > b), which tends to be attributed to Eudoxus before him. According to the author, Archimedes had to develop the lemma, because he dealt with more complex problems including the length of curved lines.
There was only a real arithmetical theoretical field: music, and there we can find the roots of the arithmetical theory of proportions (Szabo). Eudoxos' theory was tailored to face geometrical 'logoi' to be employed in astronomy or in exhaustion methods. In Aristotle we read the awareness of possible common proofs, but never in classic times it became a general theory. ...The Greeks understood with Eudoxus the problem of multiplying numbers and magnitudes, but never the problem of attaching numbers to magnitudes, except for trade.
Originally posted by ZeuZZ
The problem is that today, some knowledge still feels too dangerous.
Because our times are not so different to Cantor or Boltzmann or Gödel’s time.
We too feel things we thought were solid, being challenged, feel our certainties slipping away.
And so, as then, we still desperately want to cling to belief in certainty.
It makes us feel safe.
At the end of this journey the question, I think we are left with, is actually the same as it was in Cantor and Boltzmann’s time.
Are we grown up enough to live with uncertainties?
Or will we repeat the mistakes of the twentieth century and pledge blind allegiance to yet another certainty?
Thanks for readingedit on 7-7-2012 by ZeuZZ because: (no reason given)