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 lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Studies in Plato's Two-Level ModelIan MuellerHolger Thesleff. Studies in Plato's Two-Level Model. Helsinki: Societas Scientarum Fennica, 1999. Pp. vi + 143. N. P.After some 30 years of incisive intervention in Platonic scholarship, Holger Thesleff here offers us "an attempt to elaborate and ground more firmly some basic theses which I have propounded in various contexts before," (1) a rather modest description of what he also describes as an (...) attempt to contribute toward "a more general agreement on how to read and interpret Plato" (5). The richness of this book makes it impossible to describe in any detail, and I shall content myself with indicating some of what I take to be its principal points of view.Thesleff"s concern for agreement arises from his perception of the vastly discrepant schools of Plato interpretation, of which the most important for him might be labeled esotericist, analytic, and "literary." The last, which can hardly be called a school at all, includes a variety of scholars who downplay "literal" interpretation and emphasize such things as "dramatic artistry, the structure of the dialogues, the characterization of settings and persons, shifts of style, allusions, play, and irony" (3). Thesleff's sympathies clearly lie with these people, perhaps to an extent which, as I shall suggest, will limit the effectiveness of his attempt to move toward a consensus.As the title indicates Thesleff builds his reading around what he calls Plato's two-level model. This two-level model is not, as one might suppose, what has been called Plato's two worlds view, the belief in a world of pure being beyond the perceived world we inhabit. Indeed, it is not a doctrine or set of doctrines at all. Rather "it is an intuitive 'vision' which offers only crude basic patterns and frames for the structures of Plato's thinking, feelings, intuitions, and intentions, from his youth to his old age, and for our understanding of them" (123). Clearly one cannot expect a full and precise articulation of this vision. Thesleff proceeds by discussing in terms of their role in the earlier dialogues ten hierarchic pairs which form part of Plato's vision: divine/human, soul/body, leading/being led (particularly important in the conduct of a Socratic discussion), truth/appearance, knowledge/opinion, intellect/senses, defined/undefined, stability/change, one/many, same/different. These hierarchic or asymmetric pairs are contrasted by Thesleff with what he calls polar opposites like large/small and light/dark. He also stresses that the hierarchic pairs of Plato's two-level model involve no separation (chorismos), but rather a communion (koinônia).As the words "from his youth to his old age" in the quotation of the previous paragraph indicate, Thesleff is a "unitarian" in his approach to Plato. His Plato is not a philosopher whose positions changed through time as a result of reasoned argument. What many scholars take to be doctrines later given up, e.g., the view that learning is being reminded (anamnêsis), are "thought experiments," which are not followed up, thought experiments made within the context of the unchanging two-level vision.Although Thesleff is a unitarian in the sense I have indicated, he does believe that a kind of order of composition can be constructed for the dialogues. I say "kind of" because Thesleff's chronology involves assumptions of revisions over time, multiple authorship, and even dialogues "written down" by people other than Plato. The resulting chronology is complex, and I here only indicate it in rough form in a way which [End Page 272] brings out what I take to be its major points. Leaving out of account those Socratic" dialogues which Thesleff takes to be latish and the posthumously completed Laws, we can say that a series of late dialogues starts with the Parmenides and includes the Timaeus, Critias, Sophist, Politicus, and Philebus. All other dialogues, including the Theaetetus, were written prior to the Parmenides with the exception of the Republic, which Thesleff proposes is in its ultimate shape a late work, representing "only and purely Plato's 'own' and personal philosophy" (115). As for the other late dialogues, which have played an important... (shrink)

This chapter contains sections titled: Euclid's Elements First Principles Aspects of Euclid's Plane Geometry Proportionality Greek Arithmetic and its History On the History of Greek Geometry Conclusion Bibliography.

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)

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)

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.

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)