Ancient Greek and Roman Philosophy of Mathematics

This book is dedicated to V.A. Yankov’s seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic. The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov’s results and their applications in algebraic logic, the theory (...) of admissible rules and refutation systems is included in the book. In addition, the reader can find the studies on splitting and join-splitting in intermediate propositional logics that are based on Yankov-type formulas which are closely related to canonical formulas, and the study of properties of predicate extensions of non-classical propositional logics. The book also contains an exposition of Yankov’s revolutionary approach to constructive proof theory. The editors also include Yankov’s contributions to history and philosophy of mathematics and foundations of mathematics, as well as an examination of his original interpretation of history of Greek philosophy and mathematics. (shrink)

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 (1) the Axiom of Archimedes which states that there are no infinitesimal magnitudes, and (2) 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 (outer) 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)

The paper examines the main points of Yankov’s hypothesis on the rise of Greek mathematics. The novelty of Yankov’s interpretation is that the rise of mathematics is examined within the context of the rise of ontological theories of the early Greek philosophers, which mark the beginning of rational thinking, as understood in the Western tradition.

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)

There is little agreement about Aristotle’s philosophy of geometry, partly due to the textual evidence and partly part to disagreement over what constitutes a plausible view. I keep separate the questions ‘What is Aristotle’s philosophy of geometry?’ and ‘Is Aristotle right?’, and consider the textual evidence in the context of Greek geometrical practice, and show that, for Aristotle, plane geometry is about properties of certain sensible objects—specifically, dimensional continuity—and certain properties possessed by actual and potential compass-and-straightedge drawings qua quantitative and (...) continuous. For their part, the objects of stereometry are potential sensible three-dimensional figures qua quantitative and continuous. (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)

In his commentary on Euclid, Proclus says both that the first principle of geometry are self-evident and that they are hypotheses received from the single, highest, unhypothetical science, which is probably dialectic. The implication of this seems to be that a geometer both does and does not know geometrical truths. This dilemma only exists if we assume that Proclus follows Aristotle in his understanding of these terms. This paper shows that this is not the case, and explains what Proclus himself (...) means by definition, hypothesis, axiom, postulate, and the self-evident, and how geometry is a science that receives its principles from dialectic. (shrink)

Book 1 of Euclid's Elements opens with a set of unproved assumptions: definitions, postulates, and ‘common notions’. The common notions are general rules validating deductions that involve the relations of equality and congruence. The attested postulates are five in number, even if a part of the manuscript tradition adds a sixth, almost surely spurious, that in some manuscripts features as the ninth, and last, common notion. The postulates are called αἰτήματα both in the manuscripts of the Elements and in the (...) ancient exegetic tradition. It is not said, however, that this denomination is original, as it coincides with the nomen rei actae associated with the verbal form introducing the postulates themselves. Since antiquity it has been recognized that the postulates naturally split into two groups of quite different character: the first three are rules licensing basic constructions, the fourth and the fifth are assertions stating properties of particular geometric objects. I transcribe postulates 1–3 in the text established by Heiberg : ᾐτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖνκαὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖνκαὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράϕεσθαι. (shrink)

Much of Aristotle's thought developed in reaction to Plato's views, and this is certainly true of his philosophy of mathematics. To judge from his dialogue, the Meno, the first thing that struck Plato as an interesting and important feature of mathematics was its epistemology: in this subject we can apparently just “draw knowledge out of ourselves.” Aristotle certainly thinks that Plato was wrong to “separate” the objects of mathematics from the familiar objects that we experience in this world. His main (...) arguments on this point are in Chapter 2 of Book XIII of the Metaphysics. There are three distinct lines of argument: The first concerns the objects of geometry ; the second deals with the Platonist principles which are applied to arithmetic and geometry; the third is about substance as living things, especially animals, and perhaps man in particular. In addition to the above, this article also examines Aristotle's treatment of infinity. (shrink)

Both literalism, the view that mathematical objects simply exist in the empirical world, and fictionalism, the view that mathematical objects do not exist but are rather harmless fictions, have been both ascribed to Aristotle. The ascription of literalism to Aristotle, however, commits Aristotle to the unattractive view that mathematics studies but a small fragment of the physical world; and there is evidence that Aristotle would deny the literalist position that mathematical objects are perceivable. The ascription of fictionalism also faces a (...) difficult challenge: there is evidence that Aristotle would deny the fictionalist position that mathematics is false. I argue that, in Aristotle's view, the fiction of mathematics is not to treat what does not exist as if existing but to treat mathematical objects with an ontological status they lack. This form of fictionalism is consistent with holding that mathematics is true. (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)

The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in Posterior (...) analytics is used to distinguish between conceptions that share the same name but are substantively different: for example the search for a broader genus including all mathematical objects; the search for a common character of different species of mathematical objects; and the effort to treat magnitudes as numbers. (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)

In Proclus' penetrating exposition of Euclid's method's and principles, the only one of its kind extant, we are afforded a unique vantage point for understanding the structure and strenght of the Euclidean system. A primary source for the history and philosophy of mathematics, Proclus' treatise contains much priceless information about the mathematics and mathematicians of the previous seven or eight centuries that has not been preserved elsewhere.

Whether aristotle wrote a work on mathematics as he did on physics is not known, and sources differ. this book attempts to present the main features of aristotle's philosophy of mathematics. methodologically, the presentation is based on aristotle's "posterior analytics", which discusses the nature of scientific knowledge and procedure. concerning aristotle's views on mathematics in particular, they are presented with the support of numerous references to his extant works. his criticism of his predecessors is added at the end.

If we were to name the two greatest mathematicians of antiquity, we would probably choose Archimedes and Euclid. The first excelled in research, the second in synthesis or system. The synthesis or system is closely associated with the theory or philosophy of that subject; and Euclid's Elements, which has been characterized as “one of the noblest monuments of antiquity”, is the best concrete instance of the theory of mathematics according to the ancient Greeks. Now Aristotle had a theory of mathematics, (...) although he was not a practicing mathematician. It is the purpose of this article to show that the mathematical method according to Aristotle is superior to and more modern than that according to Euclid. For references to Aristotle's works we shall use pages and lines according to the standard Bekker's edition. By “a number” we shall mean a positive integer or a whole number. We proceed to a brief outline of Euclid's Elements. (shrink)

The work of Euclid is situated in the Aristotelian rather than the Platonic perspective, although it incorporates the methodological results of both. The Alexandrian mathematician stamped with an Aristotelian seal the whole system of Greek mathematics.