Inconsistencies in the history of mathematics: the case of infinitesimals. - Van Bendegem, Jean Paul (Inconsistency in Science, 2002)
In this paper I will not confine myself exclusively to historical considerations. Both philosophical and technical matters will be raised, all with the purpose of trying to understand (better) what Newton, Leibniz and the many precursors (might have) meant when they talked about infinitesimals. The technical part will consist of an analysis why apparently infinitesimals have resisted so well to be formally expressed. The philosophical part, actually the most important part of this paper, concerns a discussion that has been going on for some decennia now. After the Kuhnian revolution in philosophy of science, notwithstanding Kuhns own suggestion that mathematics is something quite special, the question was nevertheless asked how mathematics develops. Are there revolutions in mathematics? If so, what do we have to think of? If not, why do they not occur? Is mathematics the so often claimed totally free creation of the human spirit? As usual, there is a continuum of positions, but let me sketch briefly the two extremes: the completists (as I call them) on the one hand, and the contingents (as I call them as well) on the other hand.