Edward Halper

In the first lines of Metaphysics 3, Aristotle argues that any progress in this discipline hinges on carefully working through the problems peculiar to it, the metaphysical aporiai; and he devotes all of book 3 to drawing up these problems. Despite this warning, book 3 and its doublet, book 11.1–2, have received relatively little attention. Many of the problems Aristotle sets out here are not addressed explicitly elsewhere in the Metaphysics, their discussion in book 3 is inconclusive, and most seem today of little intrinsic interest on their own. Book 3 has often been read as a merely historical record of that stage in his development when Aristotle personally struggled with Platonism, other philosophies, or his own earlier philosophy. Since there is no detailed and thorough study of book 3 in the recent literature, Madigan’s contribution is sorely needed and a worthy addition to the Clarendon Aristotle series. Like other volumes in the series, this one includes a literal translation followed by a line by line commentary that aims to explore different construals of Aristotle’s arguments and a glossary. Madigan also provides a few pages of textual notes that, unlike the rest of the book, presuppose some knowledge of Greek. A lengthy introduction does a good job of connecting book 3 to the rest of the Metaphysics. Book 11 receives relatively little attention—three pages in the introduction, six in the commentary. Madigan divides the text by aporia, he works through each argument that falls under each aporia, emphasizing in the process Aristotle’s frequent shifts in focus and assumption; Madigan tries to trace the origin of the various assumptions; and he concludes his discussion of each aporia by explaining where and how Aristotle solves it. Additionally, Madigan groups some of book 3’s aporiai and provides, either for the group or each nongrouped aporia, an introduction and an epilogue that lists the premises mentioned explicitly in each aporia along with a statement of whether or not Aristotle continued to accept each or how his view changed. Madigan counts 124 such premises in the whole of book 3. His conclusion is that “working through the aporiai of book 3 would heighten one’s awareness of the difficulties of the issues, but book 3 does not state in so many words what the ‘knots’ in our thinking are, much less how to untie them”.

ABSTRACT In Greek Mathematical Thought and the Origin of Algebra, Jacob Klein contrasts ancient Greek philosophy's direct engagement with things through arithmetic with the ancient science of numeric calculation, logistic. By chronicling the later development of logistic, by means of increasing symbolization, ultimately into algebra, he argues that logistic has come to displace arithmetic and, thereby, to submerge the ontological issues at the center of Greek thought. This article argues, first, that Klein's target is Ernst Cassirer's notion of number as a “symbolic function”—what Cassirer would later call “symbolic form.” In Substance and Function Cassirer argues that number constitutes a conceptual structure progressively extended by thought so as to include the results of different sorts of calculation (e.g., negative numbers, irrational roots, and infinite numbers) within itself and, thereby, to be a closed system. Drawing upon Dedekind, Cassirer claims that no individual number can be understood apart from the entire conceptual structure. This article argues, second, that Cassirer recognizes, as it were, a different set of things, namely, relations, and that it is possible to engage these “things” directly by recognizing the ontological issues they raise.