Dear James: I've recently heard a number of talks by physicists to laymen re quantum mechanics, and although I was able to follow them, I'm not sure that most laymen could. The problem is that the physicists were so intent on conveying the _differences_ between quantum mechanics and the layman's world, that they failed to first convey the similarities. I.e., first set up a relationship with the audience and a shared vocabulary prior to taking them out of the familiar world. The biggest problem I usually face is "if we can't even deal with very large numbers, why should we care about even larger numbers?" E.g., if the universe is perhaps 13.7 billion years old, why should I care about 100 billion years? Or if my computer currently has 2 Gigabytes of RAM and 200 Gigabytes of disk, why should I care about petabytes? Or if Avagadro's number is ~6x10^23, why should I care about numbers as big as 10^100 or bigger? The solution of infinity is one of the greatest achievements of the human race -- by considering some calculations to go on far _beyond any bound_, the solution often becomes much simpler! Thus, the calculations of the infinitesimal calculus produce much nicer results when the infinitesimal actually goes to zero, than when it remains finite. Isn't cos(x) much prettier than (sin(x+dx)-sin(x))/dx ? Isn't the single number pi prettier than the various rational approximations to it? Although no perfect circles or perfect spheres exist, we can calculate properties of these "ideal" objects much more easily than with any real approximation to them. I personally would use even numbers instead of square numbers, at least to start with. Many people can't easily square numbers in their heads. Or perhaps n -> 10*n (add zero at the end). Don't get sucked into the "is infinity a number?" debate. If you say it is, then you have to show how to compute with it & the whole conversation gets side-tracked. Countable infinity is simply the name of the "size" of certain sets. Or perhaps you might even start with _real numbers_ rather than whole numbers. First convince them that there are a countably infinite number of points, and if you have time, go on to show that there are even more. E.g., if x /= y, then (x+y)/2 is between x & y, and is different from both x & y. You can also easily set up the correspondence between n <-> 1/n. So there are lots & lots of rational numbers. For certain coordinate calculations in computer graphics/robotics/MRI scans/CAT scans/GPS/etc., it is convenient to utilize homogeneous coordinates which effectively define oo=x/0. In some sense, it doesn't really matter whether you call "oo" a number or not -- the calculations are more uniform and require fewer "special cases" when using homogeneous coordinates. This use of homogeneous coordinates is entirely analogous to calculations using negative or imaginary numbers -- it doesn't matter whether you "believe" in negative or imaginary numbers or not -- the calculations work out more easily. [There is a "logical" rationale for an introduction to infinity using real (actually algebraic) numbers instead of integers: the theory of "real closed fields" is decidable, whereas the theory of integers is undecidable. Counting actually introduces more complexity than rational/algebraic numbers.] There is a real problem in my approach to infinity: "if infinity is so important, how come it doesn't show up in computer languages?" It actually does -- e.g., the homogeneous coordinate example or IEEE-754 "arithmetic" -- but most computer scientists would agree that computer languages have not come up with convenient ways to express infinity. Even symbolic languages such as Macsyma utilize infinity only in certain limited contexts, and this "infinity" is by no means a first-class citizen. Re books: perhaps "One Two Three ... Infinity" by Gamow? At 01:20 PM 10/25/2007, James Propp wrote:

I've been invited to speak on a college morning radio program next week, on the topic of mathematical proof and infinity. It'll be a conversation with two interviewers (no call-ins).

I have no experience with this kind of public speaking, but when the opportunity came my way, it seemed like it might be a fun thing to try. I asked the producer what sort of people listen to the show, and he replied

...

Audience is primarily adults both UML staff and faculty as well as residents of the Merrimack Valley.

The show seems to appeal to people with an interest in a variety of subjects and in "bright" conversations.

Anyone out there have any suggestions for interesting analogies, points worth making, etc.?

When I try to walk through a conversation about infinity in my mind with non-mathematicians, it usually doesn't go very well. There are a lot of ways an intelligent and well-educated person is likely to misunderstand the mathematical enterprise of making up rules about infinity and seeing where they lead us. In fact, the more intelligent a layperson is, the more objections he or she is likely to have to the very first steps of trying to talk about infinity as a well-defined mathematical notion!

(To give just one example of how my inner conversations go awry: If I try to prove that the number of whole numbers has the same size as the number of perfect squares, people are apt to notice and fixate on the fact that one of the sets is a subset of the other, and so "must" be smaller. And even if I can convey the idea that we're using a new notion of measuring size, based on pairing elements, and that we have to relinquish all our intuitions that are based on finite sets until we can justify them in the new setting as consequences of our definition, the fact that the perfect squares have dual citizenship as both whole numbers and perfect squares makes the idea of the pairing confusing.)

Note that you can't draw pictures over the radio, so you can't make a table showing a one-to-one correspondence between two infinite sets.

Part of what I'm missing is a kit of good strategy for evading common pedagogical problems by cleverly choosing a plan of approach that prevents the issue from arising in the first place. For instance, if I use the Hilbert hotel picture, and talk about moving the person in room n to room n^2, then I can argue that there are just as many *rooms* of one kind as the other because the two sets can accomodate the same set of *people*, and the whole "dual-citizenship" thing doesn't arise.

(I'm pretty good at solving pedagogical problems like this, but usually not in real time! Maybe the only way to get good at talking about math on the radio is to get lots of practice and make lots of mistakes; kind of like the way to get good at doing math...)

I also feel that part of what I'm missing is a sales-pitch for a whole style of thought, namely "The Mind at Play", and good, friendly ways of encouraging people to relax, try out ideas, and not be afraid of being wrong.

Also: Are there any books in particular that you think I should plug ("If you enjoyed listening to this conversation, then you should read X")?

Jim

=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!"