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I know that plenty of theorems have been published assuming the Riemann hypothesis to be true. I understand that the main goal of such research is to have a theory ready when someone finally proves the Riemann hypothesis. A secondary goal seems to be to have a chance of spotting a contradiction, thus proving the conjecture false. This must be a secondary goal, since most mathematicians believe the conjecture to be true.

I wonder if it would be a good idea to do the opposite. It is widely believed that $e+\pi\not\in\mathbb Q,$ however no one seems to have any idea how to prove it. I wonder if it would be a good idea to try to build a theory on the statement that $e+\pi\in\mathbb Q$. I don't mean just trying to prove the conjecture by contradiction. I mean really, frowardly assuming we live in a universe in which $e+\pi\in\mathbb Q$ and trying to do maths in this universe. I'm not sure if the distinction is clear, but I hope it is. The ultimate goals mentioned above would switch places in this approach. Now the primary goal would be to spot a contradiction, since the theorems proved would all be very likely to be vacuous. The conjecture is considered very difficult to prove so perhaps it wouldn't be bad to admit our blindness and just move a bit at random and hope we're moving ahead. The theorems proved would of course be likely to be vacuous so it could seem as too focused an approach: it may only serve to prove a single statement, that $e+\pi\not\in\mathbb Q.$ However, I think the techniques developed could still turn out useful in proving other things.

The question whether it's a good idea is probably not a good question on this site, so it it is not my main question. What I would like to know is examples of such an approach being employed. I assume it hasn't been employed often because I have never heard of it except of one case, so I would also like to know why it hasn't. (The one case is the work on the parallel postulate, which was thought by many to follow from other axioms of Euclidean geometry, and which was later shown not to when considering the alternatives resulted in finding consistent geometries violating this axiom only.)

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This is a really cool idea. We should found a journal commited to publishing the consequences of the negation of the Reimann hypothesis. – goblin Apr 21 '13 at 9:48
up vote 3 down vote accepted

I think one of the examples is Fermat's Last Theorem. Frey noted that if there exists a counterexample to Fermat's Last Theorem, i.e. if there exists non-trivial solution to $(a,b,c)$ to $x^p+y^p=z^p$ where $p>2$, then the elliptic curve $y^2 = x (x − a^p)(x + b^p)$ cannot be modular. And this would violate the Taniyama–Shimura conjecture. And Wiles proved Fermat's Last Theorem by proving a weaker version of Taniyama-Shimura conjecture. More can be found in here.

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There's been quite a bit of set theory done under the assumption that the Axiom of Choice is false. In the beginning this might have been in the hope of reaching a contradiction, but it has continued after AC was shown to be independent of ZF, as an object of study in its own right.

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For those interested, The Axiom of Determinacy is a specific example of what Henning Makholm is talking about. – Dave L. Renfro Jun 22 '12 at 21:39
@Dave, if you understand what that is about, would you consider going over to Wikipedia and editing the article there such that it contains an complete statement of the axiom? Currently it just speaks about "certain two-person games" without ever reaching a declarative sentence that specifies "certain". – Henning Makholm Jun 22 '12 at 23:57
I don't know very much about this topic, but I happened to find a Wikipedia page for Determinacy by following the link associated with "winning strategy" near the top of the Axiom of determinacy web page, and I think what you're looking for can be found there. In my opinion, however, I think more of what is at "Determinacy" should be at "Axiom of determinacy", or at least the latter web page should be more clearly labeled (in the former web page) as a useful link to follow. – Dave L. Renfro Jun 25 '12 at 18:08
Entirely by accident (I was looking up something else), I came across what seems to be a very useful historical overview of the Axiom of determinacy: A Brief History of Determinacy by Paul B. Larson, which was published in The Cabal Seminar, Volume IV (2010), and perhaps also in Handbook of the History of Logic, Volume 6 (2012). – Dave L. Renfro Jul 2 '12 at 18:10

I'm not sure most mathematicians believe the Riemann Hypothesis is true. Anyway, there are lots of theorems published assuming RH to be false. One very famous example concerns the class number of imaginary quadratic fields. The first proof that the class number goes to infinity was obtained by showing that it followed from both RH and from the negation of RH.

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That is so cool, conditioning on an unsolved problem like that! I love that, eventually, we'll be able to shorten the proof by dropping one side of it; and that we don't know, in advance, which side will be dropped. – goblin Sep 19 '15 at 7:22
@goblin, that was the first proof. I think there are now proofs that don't make any reference to RH. – Gerry Myerson Sep 19 '15 at 8:15

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