Spring 2001

Dirk Schlimm

# Homework No.2

Thursday, January 25, 2001
Due Tuesday, January 30, 2001

1. Early Greek proof (5 points)

Write out one of the following as detailed and accurate as you can:

1. Either the proof that the square root of 2 is irrational, or

2.
3. Plato's construction of a square with an area of 8.
In either case, state all your assumptions at the beginning and the conclusion at the end of the proof. Justify every step you make. Also state the definitions that you are using. If you draw a diagram, label the parts that you are using in the text. If you use variables, make sure to state what type they have, e.g., natural numbers, rational numbers.

For 1.:
You may use the following lemmas without proving them:

• Lemma 1: For every natural number a, a2 is even if and only if a is even.
• Lemma 2: For every positive rational number x, there are natural numbers a and b not both even, such that x=a/b.

For 2.:
If you are familiar with the proof that the square root of 2 is irrational, you may want to read the copied pages from Plato's Meno and describe in detail how Plato constructs a square with an area of 8. Explain also why the so-constructed square really has the area of 8. Please answer the following question in a few sentences, too: What is the purpose of Plato's proof?

2. Plato's and Pythagoras' and views of mathematical objects (3 points)

Read the Aristotle's presentation of Plato's and Pythagoras' views on mathematical objects from Metaphysics (987a26--988a5) and answer the following questions:

1. What was Heraclitus' view of perceptible things?
2. How do mathematical objects differ from Forms and from perceptible things, according to Plato?
3. In what respect are the views of Plato similar to those of the Pythagoreans?
4. 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. Number representations (2 points)

Determine the representation of 1978

1. in base 60, and
2. in base 2.