Logic

Often when dealing with problems in mathematics, we make use of certain facts. For example, when dealing with geometry problems on a flat surface, we often make use of the Pythagorean Theorem and the Angle Bisector Theorem. When dealing with quadratic equations, we can use the Quadratic Formula to calculate its roots. The Fundamental Theorem of Calculus gives us a way to relate integrals and derivatives.

But how do we know that the Pythagorean Theorem, Angle Bisector Theorem, or the Fundamental Theorem of Calculus are true? How do we know that the Quadratic Formula gives us the correct roots, and not some mere approximation, or worse outright lies about the roots? Can we trust that these theorems, or the myriad of other theorems, accurately describe reality?

The answer is that in order to establish these theorems as fact, they must be grounded in logic. If we use a system of logic to derive these theorems, then we can be confident in their application. While the study of logic is a deep philosophical rabbit hole, we can at least get by with the basics. Over the course of this chapter, we will become acquainted with the basic tools of logic. Even with the basic tools, a rich mathematical tapestry can be artfully weaved!