Finite Precision Measurement Nullifies Euclid’s Postulates

Following Meyer’s argument [Phys. Rev. Lett. 83, 3751 (1999)] the set of all directions in space is replaced by the dense subset of rational directions. The result conflicts with Euclidean geometry. Meyer’s claim [1] that “finite precision measurement nullifies the KochenSpecker theorem” (that is, makes it irrelevant to physics) and some of its generalizations [2] have caused considerable controversy that lasts until today [3]. Meyer’s proposal was to replace the set of all directions in space by the dense subset of rational directions, arguing that a finite precision measurement cannot decide whether or not a number is rational. Let us apply the same argument to ordinary geometry and consider only points with rational coordinates. Then the line x = y and the unit circle x + y 2 = 1 are both dense but they do not intersect, in contradiction to Euclid’s postulates [4]. This work was supported by the Gerard Swope Fund.