6.042/18.] Mathematics for Computer Science February 3, 2005 Srini devadas and Eric Lehman Lecture notes Proof 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--ahal--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 youve never actually counted six billion of anything Maybe 3+3=6 is just"intuitively obvious", and we shouldnt 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 envelo n n 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 he other envelope? Are both options equally good? This situation is hardly more com licated 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 nonsense6.042/18.062J 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 second­grade 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 show­stopper questions. Without clear answers, all the number crunching and variable juggling in mathematics is just so much nonsense