The Sense of Arithmetic

Abstract

In the preceding chapters we were concerned with some basic phenomenological problems relating to the constitution of number. Following Husserl’s lead, we attempted to bring to light the fundamental intentional structures at work in our dealings with number. We sought to make clear how numbers are authentically presented, how they are given in the most original manner. We also stressed that numbers can be intended ‘emptily,’ and indeed that they can be somehow ‘there’ for us even when we do not actually intend them at all. Numbers, we suggested, are identities in certain modes of presence and absence. In Husserl’s view, such investigations as these form an essential part of the philosophy of arithmetic. They are not, however, the whole of it. We saw in Chapter I of this study that Husserl’s ultimate goal was to make a contribution that ‘desideratum of centuries,’ the ‘true philosophy of the calculus.’¹ His aim was not merely to clarify the concept of number for its own sake, but rather to shed light on the puzzling problem of the nature of mathematical analysis. He sought to clarify the ‘true sense’ of this discipline and to give a theoretical justification for it and the broadened number concept that is essential to it. This twin project of clarification and justification is therefore the central theme of the remainder of this study.