JoshuaZ comments on Edward Nelson claims proof of inconsistency in Peano Arithmetic - Less Wrong

13 Post author: JoshuaZ 27 September 2011 12:46PM

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Comment author: JoshuaZ 28 September 2011 02:55:37AM 3 points [-]

Whenever there's a paradox that mentions infinity, delete the infinity and see if the paradox still exists. Odds are very high it won't. A lot of diseased mathematical thinking is the result of not being clear with how you take limits, and so when I find a "paradox" that disappears when you get rid of infinity, that's enough for me to drop the problem.

This seems like an unproductive attitude. If everyone took this attitude they would never have hammered out the problems in calculus in the 19th century. And physicists would probably not have every discovered renormalization in the 20th century.

A better approach is to try to define rigorously what is happening with the infinities. When you try that, either it becomes impossible (that is there's something that seems intuitively definable that isn't definable) or one approach turns out to be correct, or you discover a hidden ambiguity in the problem. In any of those cases one learns a lot more than simply saying that there's an infinity so one can ignore the problem.

Comment author: Vaniver 28 September 2011 03:57:26AM -1 points [-]

This seems like an unproductive attitude. If everyone took this attitude they would never have hammered out the problems in calculus in the 19th century. And physicists would probably not have every discovered renormalization in the 20th century.

I will make sure to not spread that opinion in the event that I travel back in time.

A better approach is to try to define rigorously what is happening with the infinities. When you try that, either it becomes impossible (that is there's something that seems intuitively definable that isn't definable) or one approach turns out to be correct, or you discover a hidden ambiguity in the problem. In any of those cases one learns a lot more than simply saying that there's an infinity so one can ignore the problem.

I agree with you that this is a better approach. However, the problem in question is "find the error" not "how conservation of momentum works," and so as soon as you realize "hm, they're not treating this infinity as a limit" then the error is found, the problem is solved, and your curiosity should have annihilated itself.

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