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Section 0.2 Definitions, Theorems, and Conjectures

If we think of mathematics as a building, then definitions provide the foundation, theorems are the bricks, and logic is the mortar that connects them and holds them together.

Definitions introduce terminology to define mathematical objects and properties.

Theorems are statements about defined objects. A theorem uses defined terms and is derived from a sequence of logical arguments using definitions and other, previously proven theorems. To prove a theorem is to construct a sequence of logical arguments that make it a true statement (there can be more than one such sequence). The sequence of logical arguments used to derive the theorem is called a proof of the theorem.

In this course we do not expect you to come up with new theorems or to be able to prove known theorems. Nevertheless we will prove most theorems in these notes, if only to show you that everything follows from the definitions in a sequence of logical steps. Proofs of theorems are either given after the theorems (they start with Proof.) or the argument for the correctness is given before the statement.

Although it is possible to give definitions of the integers and their arithmetic and to prove their properties, we will assume familiarity with them.

We will also encounter statements that are believed to be true, but nobody has been able to find a proof yet. These statements are called conjectures.

A door seen from a hallway, with "Teachers' Lounge" on the glass. Inside, two teachers are talking. Teacher 1: My students drew me into another political argument. Teacher 2: Eh; it happens. Teacher 1: Lately, political debates bother me. They just show how good smart people are at rationalizing. The two teachers continue talking. A third one is seen reading a book on a sofa. Teacher 1: The world is so complicated - the more I learn, the less clear anything gets. There are too many ideas and arguments to pick and choose from. How can I trust myself to know the truth about anything? And if everything I know is so shaky, what on Earth am I doing teaching? Teacher 2: I guess you just do your best. No one can impart perfect universal truths to their students. Teacher 3: 'ahem' Teacher 2: ...Except math teachers. Teacher 3: Thank you. \(\mathsf{a\cdot(b+c)=(a\cdot b)+(a\cdot c)}\text{.}\) Politicize that, ...

\(\mathsf{a\cdot(b+c)=(a\cdot b)+(a\cdot c)}\text{.}\) Politicize that, ...

Figure 0.2.0.1. Certainty by Randall Munroe (https://xkcd.com/263).

We would also remark that all the definitions presented here are man-made and, to some extent, arbitrary. We use these particular definitions because they work and help us solve problems that we can formulate in the language of mathematics. It remains a constraint, of course, that the definitions have to work together so that we obtain a structurally-sound mathematics building. The logical consistency and the precise nature of the definitions we choose to use and the theorems that we can prove starting with them give us the certainty that is unique to the discipline of mathematics as referred to in Figure 0.2.0.1.