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.
Best Answer
Girolamo Saccheri in his Euclides Vindicatus (1733) essentially discovered Hyperbolic Geometry, by building around the hypothesis that the angles of a triangle add up less than 180°. This was widely believed to be always impossible, since people at that time were convinced of the absolute nature of Eucliden Geometry.