Abstract: By examining Klein's discussion of the difference between Plato and Aristotle regarding the ontology of number, this article aims to spells out the significance of that debate both in itself and for the development of the later mathematical sciences. This is accomplished by explicating and expanding Klein's account of the differences that exist in the understanding of number presented by these two thinkers. It is ultimately argued that Klein's analysis can be used to show that the transition from the ancient to the modern number concept has some roots in this disagreement between Plato and Aristotle regarding number. This, in turn, sets up that dispute as an essential part of the background to the more general break between ancient and modern conceptuality, the uncovering of which is Klein's main concern.

Keywords: Jacob Klein; Plato; Aristotle; Diophantus; theoretical logistic; number ontology; conceptuality.

Jacob Klein's Greek Mathematical Thought and the Origin of Algebra is an immensely rich book, touching on a wide range of topics. Its main goal is to uncover the nature of modern symbolic mathematics, which lies at the heart of modern science. The most important result of Klein's analysis is that the ancients and the moderns relate their concepts to the world in different fashions. Klein shows that the conceptuality (Begrifflichkeit) of ancient science was first intentional, meaning that the concepts of this science always directly intend or refer to a definite object, whether it be sensible or intelligible. Modern conceptuality, on the other hand, is symbolic in that it has conflated second intentional concepts with first intentional ones, whereby its concepts only intend a definite object indirectly.