A survey of Euclid's Elements, this text provides an understanding of the classical Greek conception of mathematics and its similarities to modern views as well as its differences. It focuses on philosophical, foundational, and logical questions — rather than strictly historical and mathematical issues — and features several helpful appendixes.
In the first part of chapter 2 of book II of the Physics Aristotle addresses the issue of the difference between mathematics and physics. In the course of his discussion he says some things about astronomy and the ‘ ‘ more physical branches of mathematics”. In this paper I discuss historical issues concerning the text, translation, and interpretation of the passage, focusing on two cruxes, the first reference to astronomy at 193b25–26 and the reference to the more physical branches at 194a7–8. In (...) section I, I criticize Ross’s interpretation of the passage and point out that his alteration of has no warrant in the Greek manuscripts. In the next three sections I treat three other interpretations, all of which depart from Ross's: in section II that of Simplicius, which I commend; in section III that of Thomas Aquinas, which is importantly influenced by a mistranslation of, and in section IV that of Ibn Rushd, which is based on an Arabic text corresponding to that printed by Ross. In the concluding section of the paper I describe the modern history of the Greek text of our passage and translations of it from the early twelfth century until the appearance of Ross's text in 1936. (shrink)
In his critical study of Speusippus Leonardo Tarán (T.) expounds an interpretation of a considerable part of the controversial books M and N of Aristotle's Metaphysics. In this essay I want to consider three aspects of the interpretation, the account of Plato's ‘ideal numbers’ (section I), the account of Speusippus’ mathematical ontology (section II), and the account of the principles of that ontology (section III). T. builds his interpretation squarely on the work of Harold Cherniss (C.), to whom I will (...) also refer. I concentrate on T. because he has brought the ideas in which I am interested together and given them a concise formulation; he is also meticulous in indicating the secondary sources with which he agrees or disagrees, so that anyone interested in pursuing particular points can do so easily by consulting his book. (shrink)
In this note i argue against harold n. lee's assertion ("mind," october, 1965) that resolution of zeno's paradoxes is closely connected with the modern mathematical distinction between density and continuity. zeno's paradoxes would arise as much if space or time is dense as they do if it is continuous. in fact the paradoxes only arise if one combines a mathematical analysis of space and time with a non-mathematical conception of motion.
Proclus Platonic Academy) is undoubtedly one of the most influential figures in the history of western philosophy; his writings did more to shape pre-twentieth-century understandings of Plato than any other person. But today few students of ancient philosophy would cite Proclus as an authority on Plato, and only a few scholars and certain people whom many would identify as enthusiasts or mystics are likely to have read a whole work of Proclus, even in translation. And although there are some passages (...) which can be read as original philosophical investigations, most notably—but perhaps this reflects my own philosophical interests—Proclus's discussion of the role of imagination in geometrical reasoning, even those passages have to be abstracted from their intellectual context to be made palatable to contemporary academic philosophical taste. The context is an elaborately triadic hierarchical metaphysics ranging between the limits of a One which is neither describable nor apprehensible because it is above being and a matter which is neither describable nor apprehensible because it is below being. But the metaphysics is also a representation of a pagan theology in which all the gods of fifth-century Greco-Roman religion find their place or places, and it is accompanied by a serious belief in practices standardly labeled magical. (shrink)