80-110   The Nature of Mathematical Reasoning

Spring 2001

Dirk Schlimm

Homework No.1

Thursday, January 18, 2001
Due Tuesday, January 23, 2001

1. Greek philosophers and mathematicians (5 points)

An excellent place to find the relevant information is the website of the MacTutor History of Mathematics Archive http://www-groups.dcs.st-and.ac.uk/~history/index.html ). There is also a link to it from the course website.

2. Euclid's proof that there are infinitely many primes (5 points)

Find a copy of  Euclid's Elements, either a book or see the website http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

Write down Euclid's proof of Proposition 20 in Book IX in your own words --- as if you would explain it to a friend. In particular, justify every step in the proof. For this you also have to look at Book VII, Proposition 31 and the Definitions 11 and 13. If there are steps that you find unconvincing, say so and explain your reasons.

Here are some hints: Euclid actually seems to prove the fact that there are at least four prime numbers. Therefore, he assumes at the beginning that there are only three of them, A, B, and C and derives a contradiction. "The least number measured by A, B, and C" is the product of A, B and C. The "unit DF" is simply 1.

Don't worry if this problem appears extremely difficult: try to do the best you can, we will go over this proof in class.