Timeline for An example of a beautiful proof that would be accessible at the high school level?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 17, 2011 at 16:23 | comment | added | Douglas Zare | Thanks for the clarification/correction about the oracles vs sets. | |
| Sep 17, 2011 at 15:55 | comment | added | Andreas Blass | I agree with Douglas Zare that this topic would not be appropriate for the intended audience. But I disagree with him abut the axiom of choice "really saying, that there exist oracles for situations where we don't have oracles." The axiom of choice is about sets, not oracles (unless you take "oracle" to mean simply set), and it certainly isn't about our "having" anything. After all, we don't even "have" all the natural numbers. It seems to me that part of the problem with the hats puzzle is that the solution pretends that sets (obtained by AC) can be used as oracles. | |
| Sep 17, 2011 at 0:26 | comment | added | Douglas Zare | Also, I disagree that the Axiom of Choice is intuitive. I think it's only intuitive if you overlook what it is really saying, that there exist oracles for situations where we don't have oracles. (The reals can be well-ordered? Ok, tell me how to compare them, or the least element of this set.) You are asking students to accept the step of memorizing the output of an oracle they don't have, but which somehow "exists." This would have turned me off from mathematics. There are much better things to show middle school and high school students than paradoxes which follow from questionable axioms. | |
| Sep 17, 2011 at 0:18 | comment | added | Douglas Zare | The positive integers are not the only infinite set involved in this example. If it were, you wouldn't need to use the Axiom of Choice. You are using the Axiom of Choice on equivalence classes of subsets of the natural numbers. I really think this example would be damaging. Normally, we start proving things we think are obvious, then prove things we believe, then use proofs to establish the truths of things which are uncertain. These students might or might not have hit the first stage and you want to jump to proving paradoxes with questionable axioms. Students will reject math. | |
| Sep 16, 2011 at 23:32 | comment | added | Mark | The axiom of choice seems intuitive to most of us, as are the positive integers (which is the only thing infinite here). I agree that the presentation needs a lot of careful planning, but if the object is to make high school students interested in mathematics, I think that this will achieve this (of course, everyone has a different idea about what mathematics is...). | |
| Sep 16, 2011 at 23:25 | comment | added | Douglas Zare | I disagree with, "You don't need to introduce any complicated concept." Do you think the students are already familiar with the Axiom of Choice? They aren't even comfortable with infinite sets. Presenting counterintuitive results which they can't otherwise verify and which are not connected to anything they have seen may be damaging to students new to mathematics. They may well reject mathematics (and at least the AOC) as not sensible, and they may be right unless you are extremely careful in the presentation, much more so than the blog you link, which was not aimed at high school students. | |
| Sep 16, 2011 at 17:17 | history | answered | Mark | CC BY-SA 3.0 |