Teaching CPSC 326, Theoretical Foundations of Computer Science always brings the challenge of getting people who deal with making things work, think about proof techniques.
This semester I started that all out with a presentation “Mathematics and Proofs” The presentation includes mostly all CC materials, except for some Jimi Hendrix music and an Abbott and Costello skit on why 13 X 7 = 28. I hope it had some positive effect on my students. We’ll see how that works when I ask them to prove somethings using Math Induction.
Here are a few links that I’ve found useful in this context.
- Mathematical Induction
- About Proofs
“Let’s look at a very famous mathematical problem: are there numbers that are not fractions of integers? Pythagoras had visited Egypt, and returned with a lot of mathematics, and a firm belief that all numbers were rational, i.e., fractions of integers. His precise argument is unknown (nothing he wrote survives: he led a religious group dedicated to an identification of arithmetic and religion), but it was philosophical if not religious, and may have gone something like this: Since all things arise out of the first thing, all whole numbers arise out of repeatedly adding one. From these whole numbers, we obtain fractions that represent ratios of arbitrary refinement. Thus for any real number, we must be able to find a ratio of whole numbers for it.” - Proofs – some good advice on understanding definitions, theorems, and proofs
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