80-110 The Nature of Mathematical Reasoning
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:
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.
- Either the proof that the square root of 2 is irrational, or
- Plato's construction of a square with an area of 8.
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.
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:
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
- What was Heraclitus' view of perceptible things?
- How do mathematical objects differ from Forms and from
perceptible things, according to Plato?
- In what respect are the views of Plato similar to those of the
- How does Plato's views differ from those of the Pythagoreans?
3. Number representations (2 points)
Determine the representation of 1978
- in base 60, and
- in base 2.