The usual objections to infinite numbers, and classes, and series, and the notion that the infinite as such is self-contradictory, may... be dismissed as groundless. There remains, however, a very grave difficulty, connected with the contradiction [of the class of all classes not members of themselves]. This difficulty does not concern the infinite as such, but only certain very large infinite classes.

The ArgumentRecent studies in the philosophy of mathematics have increasingly stressed the social and historical dimensions of mathematical practice. Although this new emphasis has fathered interesting new perspectives, it has also blurred the distinction between mathematics and other scientific fields. This distinction can be clarified by examining the special interaction of thebodyandimagesof mathematics.Mathematics has an objective, ever-expanding hard core, the growth of which is conditioned by socially and historically determined images of mathematics. Mathematics also has reflexive capacities unlike those of (...) any other exact science. In no other exact science can the standard methodological framework usedwithinthe discipline also be used to study the nature of the discipline itself.Although it has always been present in mathematical research, reflexive thinking has become increasingly central to mathematics over the past century. Many of the images of the discipline have been dictated by the increase in reflexive thinking which has also determined a great portion of the contemporary philosophy and historiography of mathematics. (shrink)