Math Department (Thesis/2003-04)

Ancient Greek Mathematics

Under the direction of: Julius Barbanel

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Most mathematical historians agree that the notion of "pure mathematics" began with the ancient Greeks. Students interested in a senior thesis in ancient Greek mathematics will have a number of specific areas from which to choose. The area I know best involves the ancient Greek discovery of irrational numbers. A summary of this area follows. I am also happy to work with students in other areas of ancient Greek mathematics.

A real number is irrational if and only if it cannot be expressed as the ratio of two integers. Ancient Greek mathematicians are credited with the discovery of irrational numbers, but their approach and presentation are very different from anything that you have probably seen before. Their work was geometric. It focussed on the notion of commensurable line segments: two line segments are commensurable if and only if each is a (whole number) multiple of some common line segment. The Greeks showed that there are line segments that are not commensurable. Although it may not be obvious, this does imply that there are irrational numbers.

Among the many areas related to incommensurabilility that students may choose to explore are the following:

• the theory of proportionality that was needed to prove the existence of incommensurables

• various conjectures regarding how the existence of incommensurables was first established

• a fascinating historical puzzle that is presented to us in one of Plato's dialogues, the Theaetetus

• the division of irrationals into various classes based upon certain geometric properties

Among the Greek mathematicians and philosophers whose work we shall consider are Pythagoras, Eudoxus, Plato, Theodorus, Theaetetus, and Euclid. Our main texts will be Books 5 and 10 of Euclid's Elements. We shall also use various modern sources.

Math Department web pages Created: 25 Apr 2003 Last modified: 25 Apr 2003 07:32:16 Comments to: math@union.edu

Thesis Topics - B. Carswell Thesis Topics for 2003-2004 2003-2004 Thesis Letter