The Axiomatic MethodOur AxiomsProofs in PracticeProving an ImplicationMethod #1Method #2 - Prove the ContrapositiveA Bogus Technique: Reasoning BackwardWhy Reasoning Backward Is BadReasoning Backward as ScratchworkProving an ``If and Only If''Method #1: Prove Each Statement Implies the OtherMethod #2: Construct a Chain of IffsHow to Write Good ProofsProof by ContradictionMethodPotential PitfallDoes All This Really Work?6.042/18.062J Mathematics for Computer Science February 3, 2005 Srini Devadas and Eric Lehman Lecture Notes Proofs Why do you believe that 3 + 3 = 6? Is it because your second-grade teacher, Miss Dalrymple, told you so? She might have been lying, you know. Or are you trusting life experience? If you have three coconuts and someone gives you three more coconuts, then you have— aha!— six coconuts. But if that is the true basis for your belief, then why do you also believe that 3, 000, 000, 000 + 3, 000, 000, 000 = 6, 000, 000, 000? Surely you’ve never actually counted six billion of anything! Maybe 3 + 3 = 6 is just “intuitively obvious”, and we shouldn’t talk about it anymore. Hey, here is a game! I secretly put one or more dollar bills into an envelope and then put twice as many dollar bills into a second envelope: $n $2n I seal both envelopes, mix them up, and present them to you. You can pick one and look inside to see how much money it contains. Then you can either take the money in that envelope or take the unknown amount of money in the other envelope. Those are the rules. For example, suppose we play this game and you find $8 in the envelope you initially selected. $8 ? What is your most profitable course? Keep the $8? Or take the unknown amount in the other envelope? Are both options equally good? This situation is hardly more com-plicated than having three coconuts and being given three more. Yet now the correct conclusion is far from “intuitively obvious”. Seemingly plausible arguments about this problem degenerate to absurdities and contradictions. So you may want to dismiss 3 + 3 = 6 and hurry along, but eventually we do have to confront the underlying question: how can we know anything in mathematics? When intuition falters as a guide to truth, how can we distinguish valid mathematics from crack-pot ravings? These are show-stopper questions. Without clear answers, all the number crunching and variable juggling in mathematics is just so much nonsense.2 Proofs The Two-Envelopes Game Here are some “solutions”: • You picked an envelope at random, so you were just as likely to pick the one with more dollars as the one with fewer. Therefore, if you see $8, then the other envelope is equally likely to contain either $4 or $16. On average, the other envelope contains (4 + 16)/2 = 10 dollars, which is more than the $8 you see. So you should clearly switch to the unopened envelope. However, a similar argument applies no matter what amount you see initially. So you should always switch, regardless of the amount in the first envelope. Thus, the best strategy is to pick one envelope and then, without even bothering to look inside, take the amount of money in the other. But that’s absurd! • You were just as likely to pick the envelope with more dollars as with fewer. So, on average, the amount of money in the envelope you picked is the same as the amount in the unopened envelope. So staying or switching makes no difference. But what if you saw $1? In that case, you could be certain that the other enve-lope contained $2. So your strategy would make a difference! • Look at the problem from my perspective. Clearly, I should not put $1 in either envelope; that would give you a big advantage. If you opened the envelope with $1, then you would know to switch. But then if you see $2 in an envelope, you know that the other envelope must contain $4 since we just ruled out $1. So you also have a big advantage in this situation. My only choice is to never put $2 in an envelope either. And I can’t use $3; if you saw that, you’d know the other envelope held $6. And if you see $4, you know the other envelope must contain $8. Apparently, I can’t run the game at all! We’re revisit this problem later in the term when we study probability.Proofs 3 1 The Axiomatic Method The standard procedure for establishing truth in mathematics was invented by Euclid, a mathematician working in Alexadria, Egypt around 300 BC. His idea was to begin with five assumptions about geometry, which seemed undeniable based on direct experience. (For example, “There is a straight line segment between every pair of points.) Proposi-tions like these that are simply accepted as true are called axioms. Starting from these axioms, Euclid established the truth of many additional proposi-tions by providing “proofs”. A proof is a sequence of logical deductions from axioms and previously-proved statements that concludes with the proposition in question. You probably wrote many proofs in high school geometry class, and you’ll see a lot more in this course. There are several common terms for a proposition that has been proved. The different terms hint at the role of the proposition within a larger body of work. • Important propositions are called theorems. • A lemma is a preliminary proposition useful for proving later propositions. • A corollary is an afterthought, a proposition that follows in just a few logical steps from a theorem. The definitions are not precise. In fact, sometimes a good lemma turns out to be far more important than the theorem it was originally used to prove. Euclid’s axiom-and-proof approach, now called the axiomatic method, is the founda-tion for mathematics today. Amazingly, essentially all mathematics can be derived from just a handful of axioms called ZFC together with a few logical principles. This does not completely settle the question of truth in mathematics, but it greatly focuses the issue. You can still deny a mathematical theorem, but only if you reject one of the elementary ZFC axioms or find a logical error in the proof. 1.1 Our Axioms For our purposes, the ZFC axioms are too primitive— by one reckoning, proving that 2 + 2 = 4 requires more than 20,000 steps! So instead of starting with ZFC, we’re going to take a huge set of axioms as our foundation: we’ll accept all familiar facts from high school math! This will give us a quick launch, but you will
View Full Document
Unlocking...