Reflections on Time and Politics

2. Point, Line, Curve

2 Point, Line, Curve modern set theory rejects the horror infiniti of earlier thought. An infinite like Aristotle’s, linked to endless division of finite magnitudes, is a potentiality that cannot become actual; point and line are incompatible because in- finity remains a never completed process of approximation. Conversely, an infinite set, immediately and fully given, offers a way for a continuum to be composed of unlimited indivisibles. The assertion of actual infinities in pure mathematics remains controversial.1 Nevertheless, its introduction was crucial to the arithmetization of analysis and the grounding of calculus and geometry in number theory. In Our Knowledge of the External World, Bertrand Russell employs the developments of set theory to answer philosophical quandaries concerning spatiotemporal continuity and the infinite. The apparent opposition between discrete points and instants and the continuums they compose results from ‘‘a failure to realize imaginatively, as well as abstractly, the nature of continuous series as they appear in mathematics’’ (Russell 1926, 136). Without necessarily conceding the existence of mathematical points and instants, one can still define mathematical series that are isomorphic with the continuums of time and space: ‘‘I do not see any reason to suppose that the points and instants which mathematicians introduce in dealing with space and time are actual physically existing entities, but I do see reason to suppose that the continuity of actual space and time may be more or less analogous to mathematical continuity’’ (137).2 This leaves no reason to fol1 . On intuitionist mathematics’ rejection of actual infinites, see George and Velleman (2002, chapters 4–5). 2. Also: ‘‘although the particles, points, and instants with which physics operates are not themselves given in experience, and are very likely not actually existing things, yet, out of the materials provided in sensation, together with other particulars structurally similar to these materials, it is possible to make logical constructions having the mathematical properties 23 Point, Line, Curve low Bergson’s ‘‘heroic methods’’ (143), which banish mathematics and number from concrete accounts of lived experience.3 Continuity is conceivable in terms of an infinite series because infinity is not found primarily in endless divisibility. While the ability to divide a finite distance into smaller distances ‘‘must be admitted’’ (141), ‘‘it is a mistake to define the infinity of a series as ‘impossibility of completion by successive synthesis.’ The notion of infinity . . . is primarily a property of classes, and only derivatively applicable to series; classes which are infinite are given all at once by the defining property of their members, so that there is no question of ‘completion’ or of ‘successive synthesis’’’ (160). Set theory defines an infinite class intensionally rather than extensionally, according to its members’ properties rather than by enumeration. The inability to reach indivisibles through successive division does not mean that a continuum cannot be composed of them: ‘‘But just as an infinite class can be given all at once by its defining concept, though it cannot be reached by successive enumeration, so an infinite set of points can be given all at once as making up a line or area or volume, though they can never be reached by the process of successive division. Thus the infinite divisibility of space gives no ground for denying that space is composed of points’’ (163). Given this actual infinity, continuity is defined by the ‘‘compactness’’ of an ordered series. ‘‘Continuity, in mathematics, is a property only possible to a series of terms, i.e. to terms arranged in an order, so that we can say of any two that one comes before the other . . . it does not belong to a set of terms in themselves, but only to a set in a certain order’’ (Russell 1926, 137–38). A series is ‘‘called ‘compact’ when no two terms are consecutive, but between any two there are others’’ (138). In the series of rational numbers , for example, ‘‘given any two fractions, however near together, there are other fractions greater than the one and smaller than the other, and therefore no two fractions are consecutive’’ (138). This ensures both the continuity of change and the discreteness of the points and instants through which change occurs. With a true infinity of indivisibles, motion is smooth rather than jerky: ‘‘The moving body never jumps from one position to another, but always passes by a gradual transition through an infinite number...