TITLE: Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false
QUESTION [50 upvotes]: What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false?
One interesting example, and the impetus for this question, is work in number theory based on the assumption that Siegel zeros exist. If there were such things, then the Generalized Riemann Hypothesis would be false, which it presumably isn't. So it's unlikely that there are Siegel zeros. Still, lots of effort has gone into exploring the consequences of their existence, which have turned out to be numerous, interesting, surprising and so far self-consistent. The phenomena generated by the Siegel zero hypothesis are sometimes referred to as an "illusory world" or "parallel universe" sitting alongside that of ordinary number theory. (There's some further MO discussion e.g. here and here.)
I'd like to hear about other examples like this. I'd be particularly grateful for references, especially those that discuss the motivations behind and benefits of undertaking such studies. I should clarify that I'm mainly interested in "illusory worlds" built on hypotheses that were believed to be false all along, rather than those which were originally believed true or plausible and only came to be disbelieved after the theory-building was done.
Further context: I'm a philosopher interested in counterfactual reasoning in mathematics. I'd like to better understand how, when and why mathematicians engage with counterfactual scenarios, especially those that are taken seriously for research purposes and whose study is viewed as useful and interesting. But I'd like to think this question might be stimulating for the MO broader community.

REPLY [5 votes]: This is more of a theoretical/mathematical physics example, but it can happen in mathematical physics that a lot of stuff is built around hypotheses which are essentially known to be false from the beginning or objects which are known not to exist.  Some examples:
1.) I know that you asked for hypotheses which were always known to be false, but Tait originally started to develop knot theory because Kelvin had hypothesised that atoms can be obtained as knots in the ether.  The concept of the ether was invalidated by experiments of Michelson and Morley which obviously also invalidated the hypothesis of Kelvin regarding the physical basis of knot theory.  The mathematics of knots has survived and flourished up to the present day although this application did not work.
2.) A huge amount of theoretical and mathematical papers have been published on magnetic monopoles (including Dirac and 't Hooft-Polyakov monopoles) although the consensus looks to be that magnetic monopoles likely do not exist in this Universe which we live in.  I am not sure if Dirac regarded the hypothesis of existence of monopoles to be false ''all along'', but this is certainly possible, as magnetic monopoles are forbidden by the mathematical equations of classical electromagnetism and Dirac was considering what happened theoretically if you increased the amount of symmetry which the equations have.