Is Mathematical Truth Time-Dependent?

(c) H' is not Riemann integrable on any closed interval [a, b], for assume that it is. Then H' is continuous a.e. on [a, b]. But it is clear that H'(t) = 0 if H' is continuous at t ,and so H' = 0 a.e. on [a, b]. It follows from (b) that H is a constant on [a, b]-a palpable contradiction. (d) H ' is of Baire class one, being the pointwise limit of the continuous functions H,(x) = n[H(x + lln)-H(x)] ,and so the set of points at which H' is continuous is residual; i.e., its complement is of first category. (e) WriteA = {x: Hf(x) > O) a n d B = {x: H1(x) < O). Thus A n I and B n I both have positive Lebesgue measure for every interval I. In fact, assuming that there exists some interval I = [a, b] such that B n I has measure zero, it follows that H' 2 0 a.e. on I. Therefore, since for all x E I , we conclude that H is nondecreasing on I-a contradiction. Similarly, if A n I had measure zero, then H would be nonincreasing on I. 1. Introduction. Is mathematical truth time-dependent? Our immediate impulse is to answer no. To be sure, we acknowledge that standards of truth in the natural sciences have undergone change; there was a Copernican revolution in astronomy, a Darwinian revolution in biology, an Einsteinian revolution in physics. But do scientific revolutions like these occur in mathematics? Mathematicians have most often answered this question as did the nineteenth-century mathematician Hermann Hankel, who said, "In most sciences, one generation tears down what another has builf, and what one has established, the next undoes. In mathematics alone, each generation builds a new story to the old structure." [20, p. 25.1 Hankel's view is not, however, completely valid. There have been several major upheavals in mathematics. For example, consider the axiomatization of' geometry in ancient Greece, which transformed mathematics from an experimental science into a wholly intellectual one. Again, consider the discovery of non-Euclidean geometries