----- Original Message ---- From: James Propp <jpropp@cs.uml.edu> To: math-fun@mailman.xmission.com Sent: Thursday, October 25, 2007 1:20:28 PM Subject: [math-fun] radio math 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 _______________________________________________ 1. You won't have time to cover two topics, i.e. both proofs and infinity, so arrange with the interviewers which one they will start with. My guess is that infinity is the easier of the two. There is so much that can be discussed: the countable and uncountable cardinals, Cantor diagonalization, the ordinals, the asymptotic infinity as in "let x approach infinity". 2. You can't practice within your own head; you must try it out in advance with several typical people. Try to explain the different infinities, see where they get hung up, and modify the explanation until it flows well. 3. I first learned, while still in elementary school, about the countable and uncountable from George Gamow's "One, Two, Three, Infinity". Gamow gave the diagonalization proof that the real numbers are uncountable, and I understood it readily. You could have a look at that book and see if it's helpful. Gamow calls the cardinality of the reals aleph-1; years later I learned that this is not correct, but that mistake is no big deal. That same book was also my first exposure to imaginary numbers, and that might also be a suitable topic for discussion. 4. If you're trying to demonstrate the countability of the rationals, the hard part is embedding the rationals into the integers. Here's a clever trick that I saw somewhere. Write a rational number as a character string as for example "123/456". Now think of "/" as a digit and the character string as a base-11 integer. The same character string trick can show the countability of polynomials with integer (or rational) coefficients, and even polynomials in several variables, and even countably many variables. Likewise, there are only countably many algebraic numbers, or computable numbers. Gene __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com
On 10/25/07, Eugene Salamin <gene_salamin@yahoo.com> wrote:
... 1. You won't have time to cover two topics, i.e. both proofs and infinity, so arrange with the interviewers which one they will start with. My guess is that infinity is the easier of the two. There is so much that can be discussed: the countable and uncountable cardinals, Cantor diagonalization, the ordinals, the asymptotic infinity as in "let x approach infinity".
Also complex infinity, distinct from the last --- although you could argue that "+oo" and "-oo" are shorthand for the direction of approach to the same limit. Some computer systems (CDC 6600 hardware, Maple CAS software) have attempted to introduce infinity as a floating-point value, though with little apparent success. Also the projective prime at infinity (line in plane geometry). Both these have the useful property that they can be re-interpreted as finite objects in some distinct space (see Riemann sphere, vanishing points, etc). WFL
First I would establish the need for infinity. Point out that the alternative is to declare a greatest legitimate number. That implies that the basic axiom that every integer has a unique successor needs to be modified in a complicated way. Also the rules for addition, etc., need to be modified too. Then I would point out the usefulness of infinity, for example using the fact that the value of sqrt(2) can not be written down with complete precision with a finite number of digits. Then mention that certain important values can't even be written in closed form, such as sin(40 degrees) etc. So far as the cardinality of the evens vs. the integers, don't speak of "just as many" or "equal in number", but mention only that they can be put in 1:1 correspondence, as if that were an odd idea used only by mathematicians. That should stop any arguments. Next, and perhaps finally, say that the reals cannot be put 1:1 with the integers, etc. I would not mention anything geometric, given the absence of illustrations, especially the hard-to-visualize point and line at infinity. (I "get it" only by not trying to picture it.) Steve Gray Fred lunnon wrote:
On 10/25/07, Eugene Salamin <gene_salamin@yahoo.com> wrote:
... 1. You won't have time to cover two topics, i.e. both proofs and infinity, so arrange with the interviewers which one they will start with. My guess is that infinity is the easier of the two. There is so much that can be discussed: the countable and uncountable cardinals, Cantor diagonalization, the ordinals, the asymptotic infinity as in "let x approach infinity".
Also complex infinity, distinct from the last --- although you could argue that "+oo" and "-oo" are shorthand for the direction of approach to the same limit. Some computer systems (CDC 6600 hardware, Maple CAS software) have attempted to introduce infinity as a floating-point value, though with little apparent success.
Also the projective prime at infinity (line in plane geometry). Both these have the useful property that they can be re-interpreted as finite objects in some distinct space (see Riemann sphere, vanishing points, etc).
WFL
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I forgot to say that infinity is both absolutely necessary for a useful (or rich?) mathematics, and the source of almost all its problems. This makes for an interesting "paradox" which can be discussed at some length. Mathematicians have been propelled into a world almost against their will, like physicists wondering why the rules of physics have to be so incredibly complicated and almost beyond comprehension. Steve Gray Fred lunnon wrote:
On 10/25/07, Eugene Salamin <gene_salamin@yahoo.com> wrote:
... 1. You won't have time to cover two topics, i.e. both proofs and infinity, so arrange with the interviewers which one they will start with. My guess is that infinity is the easier of the two. There is so much that can be discussed: the countable and uncountable cardinals, Cantor diagonalization, the ordinals, the asymptotic infinity as in "let x approach infinity".
Also complex infinity, distinct from the last --- although you could argue that "+oo" and "-oo" are shorthand for the direction of approach to the same limit. Some computer systems (CDC 6600 hardware, Maple CAS software) have attempted to introduce infinity as a floating-point value, though with little apparent success.
Also the projective prime at infinity (line in plane geometry). Both these have the useful property that they can be re-interpreted as finite objects in some distinct space (see Riemann sphere, vanishing points, etc).
WFL
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There's a new novel called _A Certain Ambiguity_ by Gaurav Suri (I think I have the spelling right). A big chunk of the novel is about the quest for certainty in general, and whether it can be found in mathematics, and talks about how upsetting it could be to realize that noneuclidean geometries are not only equally viable mathematically but might actually be the real geometry of the universe (as in general relativity). Another big chunk talks about infinity and has some really nice conversations aimed at exactly what you're trying to do: clear up the misconceptions. It might be a fun read and it's a good (new, just published a couple months ago) book to plug for people interested in thinking more about the ideas. But emphasis on the fact that it's a novel with some math in it (it's not a math book built around a story). --Joshua Zucker
participants (4)
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Eugene Salamin -
Fred lunnon -
Joshua Zucker -
Steve Gray