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I loved loved loved learning the diagonal proof in college. I've done it for friends occasionally, as a demonstration of the idea that math can be awesome. The

I loved loved loved learning the diagonal proof in college. I've done it for friends occasionally, as a demonstration of the idea that math can be awesome. There was one guy in the class who simply wouldn't accept it. He kept saying, "But so then you add that new number to the list!" I didn't know that he was representative of a whole larger set of people. I just posted this to his Facebook page.

bmm6o 17y ago
> He kept saying, "But so then you add that new number to the list!" I think if you get down to it, this reveals an interesting point. I think it suggests that the logic behind the proof wasn't made clear enough, that it's not a constructive proof, it's one of the first sophisticated proofs by contradiction that a student sees. Since you aren't disproving a concrete thing - you are showing that some hypothetical number doesn't appear on some hypothetical list - it can seem very unsatisfying. Also, why is it so ridiculous to try to add a missing number to the list? There's an intuition that you're fighting with a proof like this: 0: Create an empty list 1: Find a real number not on your list 2: Add it to the list 3: goto 1 Obviously this process never terminates, but I think you have to think pretty hard before that bothers you. After all, what is fundamentally different if I replace "real" with "integer" or "rational"? (rhetorical) Finally, students are comfortable enough talking about the "set of reals". Showing that there's no such thing as the "list of reals" reveals that there must be some difference between a set and a list - but at the time this proof is presented, has that distinction been made sufficiently clear?
- gabrielroth 17y ago
  That's a good sketch of some of the complexities involved. I remember that some months after I learned the proof, your rhetorical question 'What is fundamentally different if I replace "real" with "integer" or "rational"?' occurred to me, and I had to spend a few weeks thinking about it before the answer came to me. In other words, I got to have an experience of actual mathematical thinking without learning much higher math. This is why I love the diagonal proof, and the guy who taught it to me: Paul Lockhart, author of the essay 'A Mathematician's Lament.' (http://www.maa.org/devlin/devlin_03_08.html http://www.maa.org/devlin/devlin_03_08.html)

bmm6o 17y ago

> He kept saying, "But so then you add that new number to the list!" I think if you get down to it, this reveals an interesting point. I think it suggests that the logic behind the proof wasn't made clear enough, that it's not a constructive proof, it's one of the first sophisticated proofs by contradiction that a student sees. Since you aren't disproving a concrete thing - you are showing that some hypothetical number doesn't appear on some hypothetical list - it can seem very unsatisfying. Also, why is it so ridiculous to try to add a missing number to the list? There's an intuition that you're fighting with a proof like this: 0: Create an empty list 1: Find a real number not on your list 2: Add it to the list 3: goto 1 Obviously this process never terminates, but I think you have to think pretty hard before that bothers you. After all, what is fundamentally different if I replace "real" with "integer" or "rational"? (rhetorical) Finally, students are comfortable enough talking about the "set of reals". Showing that there's no such thing as the "list of reals" reveals that there must be some difference between a set and a list - but at the time this proof is presented, has that distinction been made sufficiently clear?

gabrielroth 17y ago
That's a good sketch of some of the complexities involved. I remember that some months after I learned the proof, your rhetorical question 'What is fundamentally different if I replace "real" with "integer" or "rational"?' occurred to me, and I had to spend a few weeks thinking about it before the answer came to me. In other words, I got to have an experience of actual mathematical thinking without learning much higher math. This is why I love the diagonal proof, and the guy who taught it to me: Paul Lockhart, author of the essay 'A Mathematician's Lament.' (http://www.maa.org/devlin/devlin_03_08.html http://www.maa.org/devlin/devlin_03_08.html)