Discrete Mathematics 55

Mathematical Habits

Recognize when you have a proof and when you do not. This is an important, diffi-

cult step in one’s mathematical development. Unfortunately, beginners are sometimes

satisfied by an argument that experienced mathematicians will say is nonsense. Per-

haps the attempted argument makes unwarranted assumptions, or misuses terms, or

does not logically establish the full conclusion, or perhaps it makes any number of

other mathematical errors, without the beginner realizing this. Therefore, be honest

with yourself in your work; be skeptical about your argument; avoid talking nonsense.

If you are not sure whether you have a proof, then you probably do not. Verify that

your arguments logically establish their conclusions. Do not offer hand-waving argu-

ments that avoid difficult but essential details. Never offer arguments that you do not

understand.

Use metaphor. Express your mathematical issues metaphorically in terms of a famil-

iar human experience, if doing so makes them easier to understand. Find evocative

terminology that represents your mathematical quantities or relationships in familiar

terms, if doing so supports the mathematical analysis.

Make conjectures. Use your mathematical insight, based on examples or suggestive

reasoning, to guess the answer to a mathematical question or the mathematical fact that

would explain a given mathematical phenomenon. Test your conjecture by checking

whether it is consistent with known facts or examples. Try to prove your conjecture.

Exercises

5.1 Suppose that a finite group of people has some pattern of pointing at each other, with each

person pointing at some or all or none of the others or themselves. Prove that if there is a

person who is more often pointed at than pointing, then there is another person who is less

often pointed at than pointing.

5.2 Suppose that you could control who follows whom on Twitter. Could you arrange it so that

every person has more followers than people they follow? For example, some extremely

famous people currently have many millions of followers, and one might hope to reassign

most of those followers in such a way that everyone will be more followed than following.

5.3 Show that if there are infinitely many people, then it could be possible for every person to be

more pointed at than pointing. Indeed, can you arrange infinitely many people, such that each

person points at only one person but is pointed at by infinitely many people? How does this

situation interact with the money-making third proof of theorem 34?