 So suppose that you are just given the number of points in a line and that their lengths are all zero; how would you determine the length? Do we need a new definition, one that extends Cauchy’s to uncountably infinite sums? It turns out that that would not help, because Cauchy further showed that any segment, of any length whatsoever (and indeed an entire infinite line) have exactly the same number of points as our unit segment. So knowing the number of points won’t determine the length of the line, and so nothing like familiar addition—in which the whole is determined by the parts—is possible. Instead we must think of the distance properties of a line as logically posterior to its point composition: first we have a set of points (ordered in a certain way, so that there is some fact, for example, about which of any three is between the others) then we define a function of pairs of points which specifies how far apart they are (satisfying such conditions as that the distance between \(A\) and \(B\) plus the distance between \(B\) and \(C\) equals the distance between \(A\) and \(C\)—if \(B\) is between \(A\) and \(C)\). Thus we answer Zeno as follows: the argument assumed that the size of the body was a sum of the sizes of point parts, but that is not the case; according to modern mathematics, a geometric line segment is an uncountable infinity of points plus a distance function. (Note that Grünbaum used the fact that the point composition fails to determine a length to support his ‘conventionalist’ view that a line has no determinate length at all, independent of a standard of measurement.)