Min-Max Math, ep. 02: Definitions, theorems, proofs

The scientific method versus mathematical thinking

What we’ve seen in the previous lesson is mathematics at a high level.

To be able to communicate mathematical concepts and results, we need to move beyond the ad hoc descriptions such as “a set is a collection of things”. (As we’ll see, this particular description of a set goes horribly wrong really fast.)

For this purpose, mathematics is recorded in a definition-theorem-proof structure. In this chapter, we’ll learn about this. This is essential to understand; otherwise, you won’t be able to read and write about math.

Consider the previously introduced notion of graphs. We described them as “nodes and their connections”, but this is not a precise definition.

Why? Take a look at these two graphs.

Are these graphs different?

Are these two graphs really different? No. They are two different representations of the same graph. Check out the following figure, where I numbered the nodes to help you match the two representations.

These graphs are the same

Yet, this is not obvious from the “definition” nodes and their connections.

The reason why graphs as nodes and connections are imprecise is that we haven’t told what nodes and connections exactly are. (Don’t worry about this just yet, we’ll talk about graphs later in great detail.) The point is, mathematical definitions (and theorems) are arranged in a hierarchy, building on top of each other. To illustrate the definition-theorem-proof structure, let’s put graphs aside for a while, and talk about something much simpler: numbers.

Definitions...

Without further ado, here's our first definition.