On Tuesday, February 18, 2014 10:03:18 PM UTC+2, Ken Pledger wrote:
KP: Surely he was. (Remember his theory of levers, and his calculations of centres of gravity.) But he was also an outstanding pure mathematician.
JG: There was never any question about that.
KP: For example, Riemann sums just express in modern notation the way that Archimedes handled integrals.
JG: False. The method of exhaustion is not remotely similar to anything Riemann had to say. Before I discuss it, let me state that in fact, proposition 3 and 4 On Spirals are actually the Archimedean property disguised. Stated correctly (by me only!), it says:
If x is any magnitude (commensurable or incommensurable), then there exist commensurable magnitudes (or Natural numbers) m and n such that m < x < n.
You have just seen the property stated correctly for the first time in your life (unless you've read it before in one of my many articles). Archie used the knowledge of this property to calculate pi, other areas and volumes. The only numbers he knew about were the rational numbers. Archie rejected irrational numbers, not non-existent *infinitesimals* as most ignorant academics think, but irrational numbers (or real numbers). The property is incorrectly stated in all other sources.
The method of exhaustion (moe) works because of the Archimedean Property. In the following article, I explain how he and Cavalieri thought:
Dedekind and Riemann tried to use the property to define real numbers and evaluate integrals respectively. Dedekind in the sense of cuts on the number line and Riemann with upper and lower sums.
The moe is in fact the same as the method of indivisibles (moi). As for Riemann sums and the Riemann integral, it works because it is actually the product of two averages, not the sum of infinitely many rectangular areas - that is just ignorant nonsense. My article RiemannFaux proves these facts.
So, I think you need to clear the confusion in your mind surrounding these concepts and facts.
KP: If you read Archimedes, you won't always find it easy, but you will realize that you are in the presence of an impeccable mathematician.
JG: It's never easy.
KP: I don't know who "they" are.
JG: Sure you do. Look in the mirror.
KP: Text-book authors often make simplistic remarks like that in historical footnotes, which are uncritically copied from one book to another.
JG: And that's why publications should always be taken with a pinch of salt. I can say that Thomas Heath is not one of those guilty.
KQ: In fact most areas of thought seem to have built up gradually through the work of many people.
JG: And the reason why there are so many incompetent and truly ignorant academics today.