=James Propp I've been invited to speak on a college morning radio program next week, on the topic of mathematical proof and infinity. ... Anyone out there have any suggestions for interesting analogies, points worth making, etc.? ...
I haven't ever attempted anything like this under radio constraints, but I *did* manage to convey some of these ideas successfully to my kids' classes (as an guest "enrichment" speaker) when they were in kindergarten through about third grade, so it's plausibly do-able... perhaps even for adults! Here's a few random comments based on that experience... First, it should be FUN! Concentrate on conveying what interests you. It's not so much about truth, as about motivating the search for truth. Start with what most intrigues you, say exploring some cool things about infinity. People want to be shown wonders, not get into fussy debates. Proceed via a sequence of clear incremental setups and payoffs to a climax. I asked the kids: "Suppose I gave two of you each a handful of jelly beans. Can you tell if I gave you each the same number WITHOUT counting them?" Then we got into the pairing strategy in various cases: Each gets 5 beans. Each gets 8 beans. One gets 5 and one gets 8 beans. Etc. We can answer "Same number of different?" without counting (which is no good for a lot of reasons--what if you spoke different languages? Does "Five" equal "Cinco"?). Finally we note that pairing works even if there were a LOT more beans, even if you don't know what number comes after a jillion, even if the number's so big you can't count up to it... So now we DECIDE it'll be fun to see what comes of pairing with "infinity". More like "let's explore what happens" as opposed to "here's the truth". Soon we're admiring a wonder: You can show there are as many even numbers as there are numbers! (No need for post-kindergarten ideas like squares!) We've discovered something unique about infinite sets that is different from finite ones--merely being way bigger isn't the whole story. So what else is cool? Having learned a magic trick we have a ticket for a tour: Are all infinities the same? Is an infinitely long proof a proof? Who is that fat man in the red costume (a Large Cardinal<;-)? You can take it wherever you want to go. But always front-load with dramatic hooks--"There's actually a proof that there are true things that can't be proved!"--to motivate any exposition. And mix in some of the fun math-lore. I got a lot of mileage talking about kids putting stones in Ramanujan's pants, and they were so motivated hearing the tale about how young Gauss out-foxed his crabby teacher that they really *got* summing 1..100 by pairing... (Your audience demographics may differ<;-) As they say in show biz, "Break a leg!"