This project explored a number of interconnected philosophical issues that arise in connection with the history of Greek mathematics. Much of the project´s work has been closely connected with contemporary philosophy.
Research
Jonathan Beere and Ben Morison (Princeton, Senior Fellow in Topoi) have been working on a major project on the problem-propositions in Greek mathematics. They argue that these propositions constitute a specifically mathematical form of knowing how to do something. Building on Jason Stanley’s theory of knowing how, they explain how the theorem-propositions and the problem-propositions form a unified deductive system. Stanley’s work explains how the knowledge how to do something might be the knowledge of a proposition, which, like other propositions, can be proven.
Results
Stephen Menn and Jonathan Beere organized a large conference, “Apollonius and the History of Conic Sections” in July 2014. The talks drew together many different perspectives on conic sections, including their connection to parabolic sundials (Elisabeth Rinner), the pre-Apollonian history of conic sections (Stephen Menn), and the reception of Apollonius in the early modern period (Vincenzo di Risi). Talks by Jonathan Beere and Ben Morison, Alison Laywine, Henry Mendell, Ken Saito, Heike Sefrin-Weiss, and Sabetai Unguru provided an overview of Books I through IV of Apollonius’ Conic Sections.
The paper just mentioned by Stephen Menn explain Archytas’ doubling of the cube. His extraordinary construction involves the intersection of a cone, a cylinder, and a torus. Menn reconstructs an analysis which could have led to this construction. Jonathan Beere shows how Archytas could be led to this analysis by trying to extend techniques of analysis in plane geometry to solid geometry using some elementary propositions of spherics, i.e., of the mathematical foundations of astronomy. Although the sphere is not explicitly mentioned in Archytas’ construction, it is crucial to understanding the underlying analysis, and to making the construction seem non-miraculous. Archytas’ work allows us a glimpse into the history of Greek geometry before the discovery of the conic sections (credited to Menaechmus two generations later). This also explains the terminology for problems that we find in later Greek geometry: a “solid problem,” i.e., one solved by “solid methods,” is one solved using conic sections. But this is a narrowing of an earlier practice – of which Archytas is an example – when “solid problems” such as the doubling of the cube were solved using techniques of solid geometry, always using surfaces of revolution and their intersections with each other and with planes to find the locus of the point being sought, but with no restriction to cones or their plane sections.
Jonathan Beere and James Conant (University of Chicago) jointly taught a course on Ludwig Wittgenstein’s Reflections on the Foundations of Mathematics and Philosophical Investigations, exploring the way in which Wittgenstein’s reflections on mathematics led him to think about rules. On their view, Wittgenstein thinks that the practices of deducing and inferring are prior to the explicit rules that might claim to govern those practices. This bears on how we should interpret Greek mathematical texts. We should not start from logic, but rather from the specific language of the text itself in reconstructing the rules of deduction that Greek mathematicians followed. Visiting Scholar Fabio Acerbi (CNRS) will follow up on this work with a book, The Logical Syntax of Greek Mathematics.
Fabio Acerbi and Jonathan Beere taught a course together on the ancient theory of irrational lines reported in Euclid’s Elements X, developing a new synthetic overview of that theory. The theory is a unique instance in ancient Greek geometry of a complete classification of a class of mathematical objects. All of the many strange features of Book X can be explained in terms of the project of showing that classification of the kinds of irrational line is exclusive and exhaustive. The book also raises numerous philosophical questions, particularly about the relationship between construction and definition.
Definition is also the focus of a dissertation by Benjamin Wilck about definition in ancient philosophy and in Greek mathematical practice. Benjamin Wilck is comparing and contrasting the way in which Aristotle used mathematical practice in developing his theory of definition and the way in which Greek mathematicians formulated definitions. In his dialectical treatise Topics, Aristotle presents a series of dialectical arguments to attack several pre-Euclidean definitions of fundamental mathematical terms. It turns out that Aristotle is criticizing the contemporary practice of mathematical definition in the Platonic Academy. Since some of the definitions that Aristotle rejects in the Topics are restated and deductively employed as explanatory premises in mathematical proofs in Euclid’s Elements, the question arises how seriously a mathematician should take Aristotle’s dialectical arguments against particular mathematical definitions.
A further dissertation project, pursued by Jan Gerhold, explores the way in which Plato and, especially later Neo-Platonists (Theon, Iamblichus), considered mathematics to be an ideal propaedeutic to philosophy.
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