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When studying mathematics some fifty years ago I was amazed of how few inference rules that actually was used to prove theorems: modus ponens, reductio ad absurdum and tertium non datur, as far as I remember. Kurt Gödel seemed to be of the same opinion:

The class of provable formulas is defined to be the smallest class of formulas that contains the axioms and is closed under the relation ’immediate consequence’, i.e. formula c of a and b defined as an immediate consequence in terms of modus ponens or substitution.

(Kurt Gödel in: Jean van Heijenoort, 1976, "From Frege to Gödel: A SourceBook in Mathematical Logic, 1879-1931", p. 601, Harvard University Press.)

But due to Wikipedia there are theorems being proved by the derivation rule $$((R\to T)\wedge(\neg R\to T))\to T$$ where $R$ is the Riemann hypothesis and $T$ is the theorem.

Is this unique for the Riemann hypothesis or the mathematics developed aound it, to catalyze proofs of some theorems, or do there exist other non trivial examples of conjecture based derivation rules suitable for capture proofs of certain theorems?

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    $\begingroup$ This form of proof is ubiquitous. Any proof that uses cases implicitly uses this rule. $\endgroup$
    – Paul
    Commented Jan 30, 2020 at 15:05

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The most elementary one (and probably the famous one) is the following.

Question. Are there irrational numbers $a,b>0$ such that $a^b$ is rational?

Answer. Consider the proposition $R$: $$\sqrt{2}^{\sqrt{2}}\mbox{ is rational}$$ and the proposition $T:$ $$\mbox{there irrational numbers $a,b>0$ such that $a^b$ is rational}$$ Clearly $R\Rightarrow T$. Now if $\neg R$, then $\sqrt{2}^{\sqrt{2}}$ is irrational and in this case $$a^b = (\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = \sqrt{2}^{\sqrt{2}\cdot \sqrt{2}} = \sqrt{2}^2 = 2$$ is rational for irrational $a= \sqrt{2}^{\sqrt{2}}$ and irrational $b = \sqrt{2}$. Thus $\neg R\Rightarrow T$. Now using the inference rule $$\left(\left(R\Rightarrow T\right)\wedge \left(\neg R\Rightarrow T\right)\right)\Rightarrow T$$ we deduce that $T$ holds.

Remark.

Main feature of constructivism in mathematics is that it does not use the inference rule $$\left(\left(R\Rightarrow T\right)\wedge \left(\neg R\Rightarrow T\right)\right)\Rightarrow T$$ So basically what you are asking for is a difference between classical and constructive mathematics. People wrote monographs on this subject (see link above).

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    $\begingroup$ @Paul let me quote wikpedia: "Much constructive mathematics uses intuitionistic logic, which is essentially classical logic without the law of the excluded middle". Note that the inference rule we are talking about is precisely the law of excluded middle in disguise. $\endgroup$
    – Slup
    Commented Jan 30, 2020 at 15:20
  • $\begingroup$ You're right--deleted my comment. $\endgroup$
    – Paul
    Commented Jan 30, 2020 at 15:25
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    $\begingroup$ @Paul You anticipated me. I was about to quote Russel's and Whitehead's "Principia Mathematica" :). $\endgroup$
    – Slup
    Commented Jan 30, 2020 at 15:27
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    $\begingroup$ In number theory, I think the main practical distinction is between effective (computable in principle) and ineffective (depending on the law of excluded middle, so not computable until we know which hypothesis holds) constants. There is a very famous folklore "theorem" that I learned from one of Iwaniec lectures which shows this clearly: Theorem (folklore) - there is a number $T_0>0$ s.t if the RH holds for all the non trivial zeros with imaginary part between $0$ and $T_0$ it then holds. Proof: RH true let $T_0=1$, otherwise $T_0=T_1+1$, $T_1$ is the lowest imaginary part counterexample $\endgroup$
    – Conrad
    Commented Jan 31, 2020 at 3:13

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