The great French mathematician Henri Poincaréis also well known, in philosophical circles, as the father of geometrical conventionalism. In particular, the logical positivists appealed especially to Poincaré in articulating and defending their own conception of the conventionality of geometry. As a matter of fact, the logical positivists appealed both to Poincaré and to Einstein here, for they believed that Poincaré's philosophical insight had been realized in Einstein's physical theories. They then used both–Poincaré's insight and Einstein's theories–to support and to illustrate their conventionalism. They thus viewed the combination of Poincaré's geometrical conventionalism and Einstein's theory of relativity as a single unified whole.

How, then, do the logical positivists understand Poincaré's argument? They concentrate on the example Poincaré (1902/1905) presents in the fourth chapter of Science and Hypothesis: the example, namely, of a world endowed with a peculiar temperature field. According to this example, we can interpret the same empirical facts in two different ways. On the one hand, we can imagine, in the given circumstances, that we live in an infinite, non– Euclidean world–in a space of constant negative curvature. On the other hand, we can equally well imagine, in the same empirical circumstances, that we live in the interior of a finite, Euclidean sphere in which there also exists a special temperature field. This field affects all bodies in the same way and thereby produces a contraction, according to which all bodies–and, in particular, our measuring rods–become continuously smaller as they approach the limiting spherical surface.