35 Mathematical Proofs

35 Mathematical Proofs - Handout #35 Feb. 15, 2008 CS103A...

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CS103A Feb. 15, 2008 Robert Plummer 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 x 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 in mathematics is that statements must be proved by a form of argument that conforms to rigorous standards. In geometry, the Pythagorean Theorem is proved from accepted principles by a sequence of deductive steps that will convince anyone familiar with mathematics or logic that the theorem is true for all triangles. In terminology used by philosophers, a statement is
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This note was uploaded on 10/01/2011 for the course CS 103A taught by Professor Plummer,r during the Winter '07 term at Stanford.

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35 Mathematical Proofs - Handout #35 Feb. 15, 2008 CS103A...

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