Ancient Greek and Roman Philosophy of Mathematics

We provide a new interpretation of Zeno’s Paradox of Measure that begins by giving a substantive account, drawn from Aristotle’s text, of the fact that points lack magnitude. The main elements of this account are the Axiom of Archimedes which states that there are no infinitesimal magnitudes, and the principle that all assignments of magnitude, or lack thereof, must be grounded in the magnitude of line segments, the primary objects to which the notion of linear magnitude applies. Armed with this (...) account, we are ineluctably driven to introduce a highly constructive notion of measure based exclusively on the total magnitude of potentially infinite collections of line segments. The Paradox of Measure then consists in the proof that every finite or potentially infinite collection of points lacks magnitude with respect to this notion of measure. We observe that the Paradox of Measure, thus understood, troubled analysts into the 1880’s, despite their knowledge that the linear continuum is uncountable. The Paradox was ultimately resolved by Borel in his thesis of 1893, as a corollary to his celebrated result that every countable open cover of a closed line segment has a finite sub-cover, a result he later called the “First Fundamental Theorem of Measure Theory.” This achievement of Borel has not been sufficiently appreciated. We conclude with a metamathematical analysis of the resolution of the paradox made possible by recent results in reverse mathematics. (shrink)

A fascinating epistemology of science book, to be read alongside contemporary accounts.

Relying upon a very close reading of all of the definitions given in Euclid’s Elements, I argue that this mathematical treatise contains a philosophical treatment of mathematical objects. Specifically, I show that Euclid draws elaborate metaphysical distinctions between substances and non-substantial attributes of substances, different kinds of substance, and different kinds of non-substance. While the general metaphysical theory adopted in the Elements resembles that of Aristotle in many respects, Euclid does not employ Aristotle’s terminology, or indeed, any philosophical terminology at (...) all. Instead, Euclid systematically uses different types of definition to distinguish between metaphysically different kinds of mathematical object. (shrink)

The Greco-Roman mathematician Claudius Ptolemy is one of the most significant figures in the history of science. He is remembered today for his astronomy, but his philosophy is almost entirely lost to history. This groundbreaking book is the first to reconstruct Ptolemy’s general philosophical system—including his metaphysics, epistemology, and ethics—and to explore its relationship to astronomy, harmonics, element theory, astrology, cosmology, psychology, and theology. -/- In this stimulating intellectual history, Jacqueline Feke uncovers references to a complex and sophisticated philosophical agenda (...) scattered among Ptolemy’s technical studies in the physical and mathematical sciences. She shows how he developed a philosophy that was radical and even subversive, appropriating ideas and turning them against the very philosophers from whom he drew influence. Feke reveals how Ptolemy’s unique system is at once a critique of prevailing philosophical trends and a conception of the world in which mathematics reigns supreme. -/- A compelling work of scholarship, Ptolemy’s Philosophy demonstrates how Ptolemy situated mathematics at the very foundation of all philosophy—theoretical and practical—and advanced the mathematical way of life as the true path to human perfection. (shrink)

Bringing the meta-mathematics of Hero of Alexandria and Claudius Ptolemy into conversation for the first time, I argue that they employ identical rhetorical strategies in the introductions to Hero’s Belopoeica, Pneumatica, Metrica and Ptolemy’s Almagest. They each adopt a paradigmatic argument, in which they criticize the discourses of philosophers and declare epistemological supremacy for mathematics by asserting that geometrical demonstration is indisputable. The rarity of this claim—in conjunction with the paradigmatic argument—indicates that Hero and Ptolemy participated in a single meta-mathematical (...) tradition, which made available to them rhetoric designed to introduce, justify, and bolster the value of mathematics. (shrink)

The Neoplatonist Proclus, in his commentary on Euclid's Elements, appears to have been the first to systematically cut imagination's exclusive ties with the sensible realm. According to Proclus, in geometry discursive thinking makes use of innate concepts that are projected on imagination as on a mirror. Despite the crucial role of Proclus' text in early modern epistemology, the concept of a productive imagination seems almost not have been received. It was generally either transplanted into an Aristotelian account of mathematics or (...) simply ignored. In this respect, Johannes Kepler is a remarkable exception. By rejecting the traditional meta-mathematical framework, Kepler was the first to incorporate the productive side of imagination within an early modern philosophy of mathematics. Moreover, by securing imagination's sensory input, he transformed Proclean imagination into a tool for cosmic self-knowledge. -/- . (shrink)

Greek mathematics is usually seen as having reached its height in a “golden age” around 300 b.c., after which it declined, reaching a rather sad stage in late antiquity. In this latter period Pappus of Alexandria stands out as one of the last competent mathematicians, although even his Mathematical Collection has been valued by historians mainly for its wealth of information on earlier mathematical achievements. In her readable book, Serafina Cuomo sets out to correct the conventional view of mathematics in (...) late antiquity: her general goal is “to show that the mathematics of late antiquity deserves a place both in the history of science and in the history of antiquity” . To that end she focuses on Pappus's Collection, for which she attempts “to produce a historical analysis … and at the same time to explore its wider cultural contexts” . Thus her text is not intended primarily as a discussion of the mathematics contained in the Collection; instead Cuomo looks into the historical circumstances in which Pappus produced this work and the image he wanted to convey both of mathematics and of himself.The first chapter presents some general background on the public's perception of mathematics and its practitioners in late antiquity. Emphasizing evidence “apart from what we find in the books of the famous mathematicians” , Cuomo looks at astrological treatises such as the Mathesis by Julius Firmicus Maternus and considers the social and fiscal status of various “technical” professions, such as those of land surveyor, architect, and public administrator. She uses Diocletian's edict on wages and salaries and examines the role of mathematics in education to show that, “far from being invisible or confined to ivory towers” , mathematics was perceived as an integral part of life and regarded positively. All this material is very interesting, but how relevant is it? Cuomo's broadening of the perspective to include more than just the works of famous mathematicians is certainly commendable. But should one equate the mere skill of calculating with numbers, or even that of applying some “higher” mathematics, with theoretical, philosophical, mathematical thought? Of course, Cuomo is aware of this problem. Taking it to be irrelevant for her purpose, she makes a good point: claiming affiliation with mathematics to enhance one's status, as those professional practitioners would do, presupposes a positive image of mathematics among the public . Nevertheless, such evidence would probably not have swayed previous historians' opinion of the decline of mathematics in late antiquity.Chapters 2–4 are devoted to Book 5 of the Collection, which covers isoperimetric problems and Platonic solids; Book 8, on mechanics; Book 3, on cube duplication; and Book 4, on special curves and their classification. In Chapter 2 Cuomo argues that Pappus used the widely known problems discussed in Book 5 to address a general audience so as to promote mathematics and show that “mathematicians are better qualified than philosophers … to talk about isoperimetry and the five Platonic bodies” . At the same time, pointing out the complementarity of mathematics and mechanics , Pappus stresses the usefulness of mathematics in educational and material matters. In Chapter 4 Cuomo examines how Pappus, addressing experts, “marks out his territory within the mathematical field itself” .In the final chapter Cuomo tries “to get a clearer grasp of Pappus' mathematical agenda” . She concludes that Pappus is not just a compiler admiring the past; rather, he selected his subjects and manipulated the mathematical tradition to serve his own immediate goals in the present.This book contains an abundance of interesting historical material, although, regrettably, the mathematical examples are only roughly sketched. Cuomo shows that we can learn much from looking at Pappus's Collection in its own right. Her book deserves careful study. (shrink)