trying to be a mile wide AND a mile deep
I had the good fortune as an undergraduate to have a “bridge class” in my math curriculum. We learned basic logic, set manipulation, formal functions and relations, proof structure and induction. In that class our instructor had us read an article by Reuben Hirsch called “Math Lingo vs. Plain English: Double Entendre” (published in the […]
Theorem: Suppose a, b, c are natural numbers. If a divides b and b divides c, then a divides c. Bad Proof: Suppose the hypothesis. We can substitute a times something for b in the equation b times something equals c. The two somethings together multiply a to c so we’re done.[way too vague; might […]
Proof traits explicit/specific (non-vague) logically sound, including complete lacking irrelevant statements understandable to the reader self-contained (may assume basic things; anything else needs explicit reference to previous work or must be written out in the proof) Proof kinds Direct proof: Assume hypothesis and march to conclusion. Contradiction: If proving a single clause, assume its negation […]
1. Willie Nelson’s song “I’d Have to be Crazy” contains the line I’d have to be crazy to fall out of love with you. Based on that hypothesis, he asserts The place where I hold you is true, so I know I’m all right. Is this a logically valid argument? That is, is still being […]