Convention is an invention that plays a distinctive role in Poincaré’s philosophy of science. In terms of how they contribute to the framework of science, conventions are not empirical. They are presupposed in certain empirical tests, so they are (relatively) isolated from doubt. Yet they are not pure stipulations, or analytic, since conventional choices are guided by, and modified in the light of, experience. Finally they have a different character from genuine mathematical intuitions, which provide a fixed, a priori synthetic foundation for mathematics. Conventions are thus distinct from the synthetic a posteriori (empirical), the synthetic a priori and the analytic a priori.
The importance of Poincaré’s invention lies in the recognition of a new category of proposition and its centrality in scientific judgment. This is more important than the special place Poincaré gives Euclidean geometry. Nevertheless, it’s possible to accommodate some of what he says about the priority of Euclidean geometry with the use of non-Euclidean geometry in science, including the inapplicability of any geometry of constant curvature in physical theories of global space. Poincaré’s insistence on Euclidean geometry is based on criteria of simplicity and convenience. But these criteria surely entail that if giving up Euclidean geometry somehow results in an overall gain in simplicity then that would be condoned by conventionalism.
The a priori conditions on geometry – in particular the group concept, and the hypothesis of rigid body motion it encourages – might seem a lingering obstacle to a more flexible attitude towards applied geometry, or an empirical approach to physical space. However, just as the apriority of the intuitive continuum does not restrict physical theories to the continuous; so the apriority of the group concept does not mean that all possible theories of space must allow free mobility. This, too, can be “corrected”, or overruled, by new theories and new data, just as, Poincaré comes to admit, the new quantum theory might overrule our intuitive assumption that nature is continuous. That is, he acknowledges that reality might actually be discontinuous – despite the apriority of the intuitive continuum.