Rationality Quotes September 2012

Jayson_Virissimo

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Please post all quotes separately, so that they can be voted up/down separately. (If they are strongly related, reply to your own comments. If strongly ordered, then go ahead and post them together.)
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Here's the new thread for posting quotes, with the usual rules:

Please post all quotes separately, so that they can be voted up/down separately. (If they are strongly related, reply to your own comments. If strongly ordered, then go ahead and post them together.)
Do not quote yourself
Do not quote comments/posts on LW/OB
No more than 5 quotes per person per monthly thread, please.

The quote says,

could mathematicians afford to use this literary device? How would a reader be able to tell the difference in intent between what I have just written and the following superficially similar passage?

This seems to me to mean: the two cases are different; the first is appropriately handled by serious proof-by-contradiction, while the second is appropriately handled by irony. But readers may not be able to tell the difference, because the two texts are similar and irony is hard to identify reliably. So mathematicians should not use irony.

Whereas I would say: the two cases are the same, and irony or seriousness are equally appropriate to both. If readers could reliably identify irony, they would correctly deduce that the author treated the two cases differently, which is in fact a wrong approach. So readers are better served by treating both texts as serious.

I'm not saying mathematicians should / can effectively use irony; I'm saying the example is flawed so that it doesn't demonstrate the problems with irony.

You're drawing the parallel differently from the quote's author. The second example requires assuming the existence of complex numbers to resolve the contradiction. The first example requires assuming, not the existence of irrational numbers (we already know about those, or we wouldn't be asking the question!), but the existence of integers which are both even and odd. As far as I know, there are no completely satisfactory ways of resolving the latter situation.

4gwern14y

The difference is that mathematicians apply modus tollens and reject sqrt2 being rational, but apply modus ponens and accept the existence of i; why? Because apparently the resultant extensions of theories justify this choice - and this is the irony, the reason one's beliefs are in discordance with one's words/proof and the reader is expected to appreciate this discrepancy. But what one regards as a useful enough extension to justify a modus tollens move is something other may not appreciate or will differ from field to field, and this is a barrier to understanding.