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# 80-110 Nature of Mathematical Reasoning

Spring 2002

Dirk Schlimm

Homework 2

Wednesday, January 23

Due Monday, January 28

1. Babylonian and Roman numerals (2 points)

a) How would the Babylonians have written the number 1978? (Remember that they used a representation with base 60.)
b) How would the Romans write 1978?

2. Plato's and Pythagoras' and views of mathematical objects (2 points)

Read the Aristotle's presentation of Plato's and Pythagoras' views on mathematical objects from Metaphysics (987a26--988a5). The text is below (on the handout).

Answer the following questions:
a) What was Heraclitus' view of perceptible things?
b) According to Plato, how do mathematical objects differ from Forms and from perceptible things?
c) In what respect are the views of Plato similar to those of the Pythagoreans?
d) How does Plato's views differ from those of the Pythagoreans?

You do not need to answer the following question for this homework, but why not think about it on Friday evening with your friends (or Sunday morning for that matter): What do you think mathematical objects, e.g., the numbers, are?

3. Euclid's proof that there are infinitely many primes (6 points)

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

Euclid's Proposition 20 in Book IX says: Prime numbers are more than any assigned multitude of prime numbers.

Explain this proof in your own words--as if you would explain it to a friend. In particular, try to justify every step in the proof. (For this you also have to look at Book VII, Proposition 31 and Definitions 11 and 13. If there are steps that you find unconvincing, say so and state your reasons.)

Here are some hints:

• Euclid actually proves 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.
• If A is measured by B, this means that B divides A without remainder.
• "The least number measured by A, B, and C" is the product of A, B and C.
• The "unit DF" is simply 1.

This problem might seem extremely difficult: try to do the best you can.