Leibniz in Paris: A Discussion Concerning the Infinite Number of All Units

Abstract

In this paper, we analyze the arguments that Leibniz develops against the concept of infinite number in his first Parisian text on the mathematics of the infinite, the Accessio ad arithmeticam infinitorum. With this goal, we approach this problem from two angles. The first, rather philosophical or axiomatic, argues against the number of all numbers appealing to a reductio ad absurdum, showing that the acceptance of the infinite number goes against the principle of the whole and the part, which is analytically demonstrated. So, discussing the ideas of Galileo, Leibniz concludes that the infinite number equals 0. Moreover, Leibniz seems to arrive at the same conclusion through his rule for adding the infinite series resulting from the harmonic triangle. Although he acknowledges the conjectural character of this conclusion, he seems to consider it to be a reinforcement of his first argument. Moreover, in reconstructing the justification of the given rule, we try to show that Leibniz does not appeal to the application of infinitesimal quantities, but rather to a treatment of the infinite series in terms of totalities.