The rules of logic specify the meaning of mathematical statements. For instance, these rules
help us understand and reason with statements such as “There exists an integer that is
not the sum of two squares” and “For every positive integer n, the sum of the positive integers
not exceeding n is n(n + 1)/2.” Logic is the basis of all mathematical reasoning, and of all
automated reasoning. It has practical applications to the design of computing machines, to the
specification of systems, to artificial intelligence, to computer programming, to programming
languages, and to other areas of computer science, as well as to many other fields of study.
To understand mathematics, we must understand what makes up a correct mathematical
argument, that is, a proof. Once we prove a mathematical statement is true, we call it a theorem.A
collection of theorems on a topic organize what we knowabout this topic.To learn a mathematical
topic, a person needs to actively construct mathematical arguments on this topic, and not just
read exposition. Moreover, knowing the proof of a theorem often makes it possible to modify
the result to fit new situations.
