 The problem, as Wittgenstein sees it, is that mathematicians, especially foundationalists (e.g., set theorists), have sought to accommodate physical continuity by a theory that ‘describes’ the mathematical continuum (PR §171). When, for example, we think of continuous motion and the (mere) density of the rationals, we reason that if an object moves continuously from A to B, and it travels only the distances marked by “rational points”, then it must skip some distances (intervals, or points) not marked by rational numbers. But if an object in continuous motion travels distances that cannot be commensurately measured by rationals alone, there must be ‘gaps’ between the rationals (PG 460), and so we must fill them, first, with recursive irrationals, and then, because “the set of all recursive irrationals” still leaves gaps, with “lawless irrationals”.