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While most of the world is fine with proofs performed by contradicting the thesis, direct proofs are sometimes considered more elegant than indirect ones. Those who prefer intuitionism or structuralism as their philosophy of mathematics may even consider indirect proofs as strictly invalid.

Therefore while proofs by contradiction are enough to convince others that a thesis is valid, there is a tendency to search also for a direct proof, as such are considered more valuable.

This makes me wonder whether there are theorems that have been proven only by contradiction, and a direct proof is not known.

Maybe there are none, and we can prove everything directly. Maybe there are some, which are they? Or maybe this is quite a common case, and there is a lot of such theorems - if that is the case, I am wondering how would the math (in general) look like if we removed these theorems, considering their proofs as invalid, and therefore treating them as hypotheses.

Mikhail Katz
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Rafał Cieślak
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1 Answers1

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The question is certainly on-topic here. There are in fact numerous theorems that cannot be proved without arguing by contradiction. A nice example is the extreme value theorem (EVT). One cannot prove this theorem without an argument by contradiction, whose main ingredient is the Law of Excluded Middle (LEM). There are alternatives to classical logic where the LEM is not part of the package. Such logics are generally known as intuitionistic logics. It turns out that the extreme value theorem is actually false in one such setting. From this it follows that the EVT cannot be proved without LEM. A description of a counterexample can be found in the book by Troelstra and van Dalen on intuitionism.

A detailed discussion of the Extreme Value Theorem in the context of intuitionistic logics can be found in this forthcoming article in Logica Universalis.

Mikhail Katz
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