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gwern comments on Rationality Quotes September 2012 - Less Wrong
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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.
I hadn't considered that irony. I was thinking about the explicit irony of the text itself in its proof of sqrt(2) being irrational. The reader is expected to know the punchline, that sqrt(2) is irrational but that irrational numbers are important and useful. So the text that (ironically) appears to dismiss the concept of irrational numbers is in fact wrong in its dismissal, and that is a meta-irony.
...I feel confused by the meta levels of irony. Which strengthens my belief that mathematical proofs should not be ironical if undertaken seriously.
Yes, I feel similarly about this modus stuff; it seems simple and trivial, but the applications become increasingly subtle and challenging, especially when people aren't being explicit about the exact reasoning.
If mathematicians behaved simply as you describe, then those resultant extension theories would never have been developed, because everyone would have applied modus tollens regarding in a not-yet-proven-useful case. (Disclaimer: I know nothing about the actual historical reasons for the first explorations of complex numbers.)
Therefore, it's best for mathematicians to always keep the M-T and M-P cases in mind when using a proof by contradiction. Of course, a lot of time the contradiction arises due to theorems already proven from axioms, and what happens if any one of the axioms in a theory is removed is usually well explored.