Infinitesimals: Finally, we have seen how to tackle the paradoxes using the resources of mathematics as developed in the Nineteenth century. For a long time it was considered one of the great virtues of this system that it finally showed that infinitesimal quantities, smaller than any finite number but larger than zero, are unnecessary. (Newton’s calculus for instance effectively made use of such numbers, treating them sometimes as zero and sometimes as finite; the problem with such an approach is that how to treat the numbers is a matter of intuition not rigor.) However, in the Twentieth century Robinson showed how to introduce infinitesimal numbers into mathematics: this is the system of ‘non-standard analysis’ (the familiar system of real numbers, given a rigorous foundation by Dedekind, is by contrast just ‘analysis’). Analogously, Bell (1988) explains how infinitesimal line segments can be introduced into geometry, and comments on their relation to Zeno. Moreover, McLaughlin (1992, 1994) shows how Zeno’s paradoxes can be resolved in non-standard analysis; they are no more argument against non-standard analysis than against the standard mathematics we have assumed here. It should be emphasized however that—contrary to McLaughlin’s suggestions—there is no need for non-standard analysis to solve the paradoxes: either system is equally successful. (Reeder, 2015, argues that non-standard analysis is unsatisfactory regarding the arrow, and offers an alternative account using a different conception of infinitesimals.) The construction of non-standard analysis does however raise a further question about the applicability of analysis to physical space and time: it seems plausible that all physical theories can be formulated in either terms, and so as far as our experience extends both seem equally confirmed. But they cannot both be true of space and time: either space has infinitesimal parts or it doesn’t.