Introduction

Mathematicians use a large number of methods to discover new results--trial and error, computation of special cases, inspired guessing, pulling results from thin air. The difference in this and an astrologer, for example, is that we have an accepted method, called the axiomatic method, for proving that these results are correct. Proofs give us assurance that results are correct. In many cases they also give more general results. For example, the Egyptians and Hindus knew by experiment that if a triangle has sides of lengths 3, 4, and 5, it is then a right triangle. But the Greeks proved that if a triangle has sides of lengths a, b, and c, and if , then the triangle is a right triangle. There is no amount of checking by experiment that could give this general result. Proofs give us insight into relationships among different things that we are studying, forcing us to organize our thoughts in a coherent way. If you gain nothing else from the course than this, you have still gained the greatest gift that mathematics has to offer.

I wish to persuade you that a certain statement is true or false by pure reasoning. I could do this by showing you that the statement follows logically from some other statement that you may already believe. I may have to convince you that that statement is also true, and follows from another statement. This process may continue until I reach a statement which you are willing to believe, one which does not need justification. That statement plays the role of an axiom. If no such statement exists, then I will be caught in an infinite regress, giving one proof after another ad infinitum. There are three requirements that must be met before we can agree that a proof is correct.

Requirement 1

There must be mutual understanding of the words and symbols used in the discourse.

Requirement 2

There must be acceptance of certain statements called axioms without justification.

Requirement 3

There must be agreement on how and when one statement follows logically from another, i.e., agreement on certain rules of reasoning.

 

There should be no problem in reaching mutual understanding so long as we use terms familiar to both and use them consistently. If I use an unfamiliar term, you have the right to demand a definition of this term. Definitions cannot be given arbitrarily; they are subject to the rules of reasoning referred to in Requirement 2. Also, we cannot define every term that we use. In order to define one term we must use other terms, and to define these terms we must use still other terms, and so on. If we were not allowed to leave some terms undefined, we would get involved in infinite regress.

Let us begin with this.