Handout #27
CS103A
Mathematical Proofs
Department of Computer Science
Stanford University
1. Why write proofs?
According to
Webster's Unabridged Dictionary,
the word
prove
comes from the Latin verb
probare
which means
to try
or
to test
. Two Webster's dictionary definitions are
•
To try or to ascertain by an experiment, or by a test or standard .
...
•
To evince, establish, or ascertain, as truth, reality, or fact, by argument .
..
These two definitions reflect a distinction developed by philosophers, namely, the difference
between analytic
and synthetic
statements. Putting it briefly, some statements are best confirmed
by experiment and other statements are best confirmed by argument. If I were to tell you, while
sitting in a room with no windows, that it is raining outside right now, then there is no amount of
argument that would be as convincing as stepping outside to see for yourself. The statement “it is
raining'' is not an analytical statement about the relationship between concepts, but a synthetic
proposition about the world that might or might not be true at any given time. In contrast,
mathematical statements, such as
2200
x
2200
y(x+y = y+x) are analytical statements that are better
proved by argument than by experiment.
Analytic and synthetic statements
Aristotle and his followers for hundreds of years believed that objects fall at a speed proportional
to their weight. This belief is a belief about the world around us.
When Galileo wanted to prove
that objects of different size fall at the same rate, for example, he conducted a series of careful
experiments, timing pendulums and rolling balls of different size down an inclined plane. Many
other basic laws of physics can also be confirmed by experiment.
In terminology used by
philosophers, a statement is synthetic
if its truth or falsity depends upon the way the world is.
Generally speaking, synthetic statements must be verified by direct observation of the world, or
by deduction from statements that have been verified by direct observation.
To see how mathematical properties are different, we can think about Pythagoras' rule for right
triangles: the square of the length of the hypotenuse is equal to the sum of the squares of the
other two sides.
If you want to convince a skeptical friend that this is true, you could try doing
some experiments.
If you draw four or five right triangles and measure their sides, you can build
up some evidence for this rule.
But since the rule is meant to apply to all of the infinitely many
triangles we might draw, experiment is not the most convincing method.
The accepted standard