## 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.

- What is the meaning of `a priori'? (Look it up in a dictionary).
- What distinguishes mathematical from
transcendental proofs, according to Kant?
- Why does Kant allow proofs by
contradiction in mathematics, but not in philosophy?
- 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*.

- Describe the
*Socratic method*.
- Study question 1, on page 15.
- 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.