80-110   The Nature of Mathematical Reasoning

Spring 2001

Dirk Schlimm

Homework No.4

Thursday, February 8, 2001
Due Tuesday, February 13, 2001

1. On proofs by contradiction (3 points)

The following questions refer to Handout #9, Proofs by contradiction from Kant to present. Answer each of them with a sentence or two.

  1. What is the meaning of `a priori'? (Look it up in a dictionary).
  2. What distinguishes mathematical from transcendental proofs, according to Kant?
  3. Why does Kant allow proofs by contradiction in mathematics, but not in philosophy?
  4. What is Bolzano's and Frege's view on apagogical proofs (i.e. by contradiction) in mathematics?

2. Kinds of proofs (3 points)

a) List the proofs you've seen in class so far and classify them as direct, negation introduction, or reductio ad absurdum (RAA).

b) Discuss proofs by contradiction: are they `on the same level' as direct proofs? What are the advantages and disadvantages? If you could prove a theorem directly or using an indirect argument, which one would you choose, and why? (Write a paragraph or two).

3. Socratic method and Euclid's definitions (2 points)

The following questions refer to Handout #10, Proofs.

  1. Describe the Socratic method.
  2. Study question 1, on page 15.
  3. Study question 2, on page 15.

4. Definitions (2 points)

Find a definition in any textbook you like, copy it and deteremine its primitive terms. Discuss in a few sentences whether you think your example is a good definition or not.