Abstract: Jacob Klein raises two important questions about Aristotle's account of number: (1) How does the intellect come to grasp a sensible as an intelligible unit? (2) What makes a collection of these intelligible units into one number? His answer to both questions is “abstraction.” First, we abstract (or, better, disregard) a thing's sensible characteristics to grasp it as a noetic unit. Second, after counting like things, we again disregard their other characteristics and grasp the group as a noetic entity composed of “pure” units. As Klein explains them, Aristotle's numbers are each “heaps” of counted units; in contrast, each of Plato's numbers is one. This paper argues that Klein is right to understand a noetic unit existing in the sensible entity, but that his answer to the second question is not consonant with Aristotle's insistence that Plato cannot account for the unity of a number, whereas he can. Slightly modifying Klein's analysis, I show that Aristotle's numbers are each one.

Keywords: unity of units in a number; abstraction; Jacob Klein; counting; Aristotle's account of number; Plato's account of number.

Let me begin with a story. The year was 1976, and I was writing a dissertation on Aristotle's Metaphysics. For reasons I can no longer recall, I decided to spend the summer in London working at the British Library, then still located in the British Museum. After some weeks on my own, I began to look around for people who were working in my area.

Few scientists, or philosophers, have patience for a priori science. It is widely supposed that modern science owes its progress to subjecting hypotheses to experimental tests, and that nature is simply too intricate and surprising to determine without empirical investigation. Philosophers who have tried to study issues of substantial scientific doctrine or theory are regarded as embarrassments, and recent philosophers of science have narrowed their vision to scientific method. Probably no philosopher is more embarrassing than Hegel because he couples a priori science with a dialectical method that purports to derive concepts from each other in ways that bear no connection with either experience or material processes. Some contemporary scholars emphasize the empirical elements in his text, hoping, perhaps, to make his philosophy of nature more palatable against the long tide of philosophers who quickly dismiss his philosophy of nature. In my view, the current antipathy toward a priori science is misplaced: many great scientific achievements came from thinking through the implications of concepts through so-called thought experiments and other modes of nonempirical or, at least, not wholly empirical inference. Be that as it may, my concern here is Hegel's account of mechanics and, in particular, his criticism of Newtonian mechanics. I argue that Hegel not only discovered a contradiction in Newton or, rather, in Newton plausibly interpreted, but proposed a solution that carried the day in its tenor if not in its substance. Whether the solution was accepted because of Hegel is an historical question that I cannot address here.