# Has anyone ever tried to develop a theory based on a negation of a commonly believed conjecture?

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. – user18921 Apr 21 at 9:48

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.