Section 0.2 Definitions, Theorems, and Conjectures
If we think of math like a building, then definitions are the foundation, theorems are the bricks, and logic is the mortar that holds everything together.
Definitions: Definitions introduce new terms in math. They tell us exactly what an object is and what properties it has, so everyone understands and uses the terms in the same way.
Theorems: Theorems are statements about the mathematical objects we’ve defined. A theorem is true because it can be proven. A proof is a step-by-step explanation showing why the theorem must be true. Proofs rely on definitions and on other theorems that have already been proven.
Proving Theorems: You will not be expected to invent new theorems or prove major known theorems on your own. However, most theorems we introduce will include a proof. This helps show how each result follows logically from the definitions and earlier results. Sometimes the proof comes right after the theorem (usually starting with Proof.), and sometimes the reasoning appears before the statement.
Conjectures: A conjecture is a mathematical statement that people think is true, but no one has proven it yet. Conjectures remain unproven until a valid proof is found.
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, ...
https://xkcd.com/263).We also note 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 express in the language of mathematics. They must also fit together so that we obtain a structurally sound mathematical framework. The logical consistency and precise nature of the definitions we use, and the theorems we can prove from them, give us the certainty unique to mathematics. 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 0.2.2.
