From: Ken.Pledger@vuw.ac.nz (Ken Pledger) Subject: Re: Greek mathematics and history Date: Wed, 14 Feb 2001 15:58:49 +1300 Newsgroups: sci.math Summary: Why did Greek mathematics stagnate? (Zeuthen explanations) In article , I wrote: > In article <96834h$fcb$1@nnrp1.deja.com>, danfux@my-deja.com wrote: > > > Can someone please explain why the Greek mathematics that was so > > dominant in the early history of mathematics mainly > > from 300 BC to 200 AD almost disappeared later?.... > > > That's a deep question!.... It really is. Quick and easy answers just won't do. Your dates for the really active period of mathematics in the Greek-speaking world could be improved to, say, 450 to 200 B.C. The best mathematicians after that were just isolated individuals such as Menelaus, Ptolemy and Pappus. The decline after 200 B.C. can't be explained by any major social or political upheaval or library-burning at that time. There wasn't any. I've already referred to van der Waerden, "Science Awakening" Vol.I. On pp.264-5 he points out that lapses in royal patronage etc. affected astronomy just as much as mathematics; yet astronomy continued to move forward from time to time, whereas mathematics languished and was inherited by the Arabs and early modern Europeans as a dead subject. Here's his summary of all this: "Thus, in astronomy we have a progressive development, even though not uninterrupted, but in mathematics a long-continued decline, followed by a new growth on a quite different foundation. What is the cause of this? "Obviously, general political and economic factors do not adequately explain the decline, since these causes should have had the same effect in astronomy. There must be inner grounds for the decay of antique mathematics. "It was Zeuthen who uncovered these inner causes most clearly..." He goes on to set out Zeuthen's two main ideas. The first is that the Greek geometrical algebra had reached the limits of its power. Around 200 B.C. Apollonius had used it brilliantly in his theory of conics, but it's probably fair to say that everyone else (ancient and modern) finds his work very difficult to follow. As we know, around 1600 the replacement of the ancient techniques by modern symbolic algebra was of immense value in the progress of mathematics ever since. The second internal difficulty was the need for an oral tradition to sustain the geometrical-style Greek mathematics. Fingers on diagrams explain Euclidean proofs far better than the written versions. After explaining this much better than I have, van der Waerden says: "As long as there was no interruption, as long as each generation could hand over its method to the next, everything went well and the science flourished. But as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became extremely difficult to assimilate the work of the great precursors and next to impossible to pass beyond it." Those words constantly haunt me. Think of the present state of geometry. If the teaching of it dies down too much, and there are too few people who know how to put fingers on coloured diagrams, future generations may find it very difficult to recover any good command of geometrical thinking. That's one of the main reasons why I teach geometry rather passionately. (Please don't mock this paragraph. I usually avoid getting so emotional in news groups.) Anyway, the above summary doesn't at all do justice to van der Waerden's discussion. If you're interested, *please* look at pp.264-6 of his book. Ken Pledger.