Cet article se propose d’étudier le concept de « mathématique universelle », apparue chez des philosophes comme Aristote, Jamblique et Proclus, dans son rapport à la mathématique. On essaye notamment de montrer qu’il ne se réduit ni à une interprétation extérieure à la donnée mathématique, ni à une pure et simple référence à une théorie, mais s’appuie sur un problème, celui de l’universalité en mathématiques, qu’il s’agit de reconstituer.

The author shows how Proclo is a precursor of 'Mathesis universalis' concept, without admiting the aporetic method of mathematics which is in Plato, Aristotle and Euclides thought. Today, his paradigm is rejected but it is a decisive factor to understand the sources of western thought. This study deals with the works of Brisson, Cleary, Trudeau, Beierwaltes and Schmitz.

Fondée sous les auspices du père de notre modernité philosophique Descartes, puis consolidée par des penseurs aussi importants que Leibniz, Bolzano ou Husserl, la mathesis universalis paraît représenter à elle seule l'ambitieux programme du « rationalisme classique ». Des philosophes tels que Husserl, Russell, Heidegger ou Cassirer ont pu s'accorder en ce point. Le développement de la « science moderne » aurait porté ce grand « rêve dogmatique » pour mener vers son terme le destin de la métaphysique occidentale. (...) Pourtant les recherches historiques récentes ont montré que l'idée de « mathématique universelle » existait bien avant Descartes, que ce dernier ne revendiquait d'ailleurs aucune rupture sur ce point et que sa réflexion se situait même assez clairement dans l'héritage des Anciens. Comment dès lors justifier que les Anciens, avec lesquels le programme des Classiques était censé rompre, aient pu déjà se préoccuper de « mathématique universelle »? Plus simplement encore, de quoi se préoccupaient donc ces philosophes sous ce concept? Le regain d'intérêt pour la mathesis universalis à la fin du XIXe siècle n'avait-il pas conduit paradoxalement à la perte de son sens comme problème? Cette étude a pour but de suivre ces questions jusqu'à leur origine et de montrer leur importance dans le dialogue entre mathématique et philosophie. Pages de début Remerciements Introduction La constitution de la « mathématique universelle » comme problème philosophique I. Aristote II. « Mathématique universelle » et théories mathématiques : Aristote, Euclide, Epinomis III. Le moment néo-platonicien Vers la science de l'ordre et de la mesure IV. La renaissance de la mathématique universelle V. La mathesis universalis cartésienne Conclusion Annexe I. La quaestio de scientia mathematica communi Annexe II. Essai bibliographique sur la mathesis universalis chez Descartes etLeibniz Bibliographie Index nominum Pages de fin. (shrink)

The article explores the evolution of the idea of correlating numbers and meanings, from ancient numerological systems to modern models of natural language processing based on vector representations and neural networks. The authors demonstrate that the aspiration to uncover hidden properties of objects by associating them with numbers and performing operations on these numbers has been a common thread across various cultures for millennia. The article traces the stages in the formation of the concept of mathesis universalis (universal mathematics), (...) starting from Aristotle and Proclus’s reflections on general mathematics and culminating in R. Descartes’s ideas about the existence of a universally applicable science of order and measure. Particular attention is devoted to G.W. Leibniz’s ambitious project of creating a universal characteristic – a formalized language in which all knowledge could be precisely expressed and logical inference could be carried out through calculations. It is noted that, despite inherent limitations, some of Leibniz’s intuitions have found embodiment in natural language processing technologies based on vector representations of words. However, the simple correlation of concepts with vectors is insufficient for the full realization of the idea of a calculus of reasoning, as it lacks a mechanism for deriving true statements from others. The development of the transformer architecture, which employs the mechanism of self-attention to construct context-dependent vector representations, has been a significant stride towards the automation of reasoning. Nevertheless, modern language models are based on the principle of maximizing likelihood rather than rigorous logical rules. The authors analyze conceptual solutions proposed for creating a comprehensive calculus of knowledge. In conclusion, it is emphasized that, while the idea of reducing reasoning to mathematical operations is undeniably attractive, one should be cautious about its maximalist interpretations. Cognition is constructive in nature, and even the most perfect model of formalization of reasoning cannot substitute for the practical wisdom and freedom of human thought. (shrink)

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)

This volume chronicles the development of philosophical conceptions of space from early antiquity through the medieval period to the early modern era, ending with Kant. The chapters describe the interactions at different moments in history between philosophy and various other disciplines, especially geometry, optics, and natural science more generally. Central figures from the history of mathematics, science and philosophy are discussed, including Euclid, Plato, Aristotle, Proclus, Ibn al-Haytham, Nicole Oresme, Kepler, Descartes, Newton, Leibniz, Berkeley, and Kant.

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)

Proclus, head of the Philosophy School at Athens for fifty years, was one of the leading philosophical figures in Late Antiquity. Lucas Siorvanes here introduces Proclus to English-language readers, discussing his metaphysics and theory of knowledge and focusing in particular on his Neo-Platonism. Proclus lived in the turbulent fifth century A.D., a time of struggles among Christians, Jews, and pagans, the invasion of Attila the Hun, the fall of the Western Roman Empire, and the rise of the Eastern Roman Empire (...) in Byzantium. While Late Antiquity has been regarded as a time of superstition and forbiddingly complex philosophies, recent scholarship has shown it to be full of cultural and intellectual vigor. During Proclus's tenure as head of the Philosophy School, he systematized Neo-Platonism as the summit of ancient Greek thought, brought it to its peak of influence, and became responsible for the form in which it was transmitted to the Byzantine, Western European, and Islamic civilizations. Siorvanes's extensive original research presents Proclus as much more than just a metaphysician. He surveys all of Proclus's philosophical interests—including religion, physics, astronomy, mathematics, and poetry—revealing the philosopher's central concern with the problems of being and knowledge and relating the ideas of Proclus to those of such other major thinkers as Plato, Aristotle, and Ptolemy. To help the newcomer, Siorvanes supplies more than 200 quotations from Proclus's works. He also traces the impact of Proclus's Neo-Platonism across cultures, religions, and centuries, in such diverse areas as Christian and Islamic theologies, Renaissance art, Kepler's astronomy, Romantic poetry, Emerson's thought, modern philosophy and science, and current popular phrases. (shrink)

In a fragment entitled Elementa Nova Matheseos Universalis Leibniz writes “the mathesis [...] shall deliver the method through which things that are conceivable can be exactly determined”; in another fragment he takes the mathesis to be “the science of all things that are conceivable.” Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our (...) imagination, i.e. that is possible at least in principle. As a general science of forms the mathesis investigates possible relations between “arbitrary objects”. It is an abstract theory of combinations and relations among objects whatsoever. In 1810 the mathematician and philosopher Bernard Bolzano published a booklet entitled Contributions to a Better-Grounded Presentation of Mathematics. There is, according to him, a certain objective connection among the truths that are germane to a certain homogeneous field of objects: some truths are the “reasons” of others, and the latter are “consequences” of the former. The reason-consequence relation seems to be the counterpart of causality at the level of a relation between true propositions. Arigorous proof is characterized in this context as a proof that shows the reason of the proposition that is to be proven. Requirements imposed on rigorous proofs seem to anticipate normalization results in current proof theory. The contributors of Mathesis Universalis, Computability and Proof, leading experts in the fields of computer science, mathematics, logic and philosophy, show the evolution of these and related ideas exploring topics in proof theory, computability theory, intuitionistic logic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification. (shrink)

I examine the question why in Proclus' view genetic processes provide demonstrative explanations, in light of the interpretation of Aristotle's theory of demonstration in late antiquity. I show that in this interpretation mathematics is not an explanatory science in the strict sense because its objects, being immaterial, do not admit causal explanation. Placing Proclus' account of demonstrative explanation in this context, I argue that this account is aimed at answering the question whether mathematical proofs provide causal explanation as opposed to (...) grounds. I show further that Proclus can answer this question in the affirmative due to his realist view of mathematical objects and the priority he ascribes to causal relations over logical relations. (shrink)

Book review of John Schuster: Descartes-agonistes: Physico-mathematics, method and corpuscular-mechanism 1618-33. (Studies in History and Philosophy of Science, Volume 27.) Dordrecht: Springer, 2013, xix + 631pp. Descartes-Agonistes is the magnum opus of John Schuster, formerly of the University of New South Wales, honorary fellow at the University of Sydney. Its roots go back to the dissertation he wrote 35 years ago under Thomas Kuhn at Princeton University. As Schuster correctly remarks, some regard his dissertation as an underground classic. I count (...) myself among them: Schuster’s work has been directional in my work on the history of early modern science. Schuster himself prefers to regard his dissertation as ‘a vestige’ of a history of science of the 1970s that he has since gone beyond. In the 1980s, he elaborated and published on some central insights in a series of seminal articles, but his work on Descartes really got into its stride again around the turn of the millennium (Schuster 1980, 1990; Gaukroger et al. 2000). This book is the acme of this work, and grand it is: clocking 600 pages (excluding appendices) it extends, deepens and reinforces his analysis of Descartes into a dense and .. (shrink)

Adriaan van Roomen published an outline of what he called a Mathesis Universalis in 1597. This earned him a well-deserved place in the history of early modern ideas about a universal mathematics which was intended to encompass both geometry and arithmetic and to provide general rules valid for operations involving numbers, geometrical magnitudes, and all other quantities amenable to measurement and calculation. ‘Mathesis Universalis’ (MU) became the most common (though not the only) term for mathematical theories developed with (...) that aim. At some time around 1600 van Roomen composed a new version of his MU, considerably different from the earlier one. This second version was never effectively published and it has not been discussed in detail in the secondary literature before. The text has, however, survived and the two versions are presented and compared in the present article. Sections 1–6 are about the first version of van Roomen’s MU the occasion of its publication (a controversy about Archimedes’ treatise on the circle, Sect. 2), its conceptual context (Sect. 3), its structure (with an overview of its definitions, axioms, and theorems) and its dependence on Clavius’ use of numbers in dealing with both rational and irrational ratios (Sect. 4), the geometrical interpretation of arithmetical operations multiplication and division (Sect. 5), and an analysis of its content in modern terms. In his second version of a MU van Roomen took algebra into account, inspired by Viète’s early treatises; he planned to publish it as part of a new edition of Al-Khwarizmi’s treatise on algebra (Sect. 7). Section 8 describes the conceptual background and the difficulties involved in the merging of algebra and geometry; Sect. 9 summarizes and analyzes the definitions, axioms and theorems of the second version, noting the differences with the first version and tracing the influence of Viète. Section 10 deals with the influence of van Roomen on later discussions of MU, and briefly sketches Descartes’ ideas about MU as expressed in the latter’s Regulae. (shrink)

The notion of mathesis universalis appears in many of Edmund Husserl’s works, where it corresponds essentially to “a universal a priori ontology”. This paper has two purposes; one, largely exegetical, of clarifying how Husserl elaborates on Leibniz’ concept of mathesis universalis and associated notions like symbolic thinking and symbolic knowledge filtering them through the lesson of the so called “bohemian Leibniz”, Bernard Bolzano; another, more properly philosophical, of examining the role that the universal mathesis is allowed to (...) play, and the space it occupies in Husserl’s intuition-based epistemology. (shrink)

Francesco Patrizi was a competent Greek scholar, a mathematician, and a Neoplatonic thinker, well known for his sharp critique of Aristotle and the Aristotelian tradition. In this article I shall present, in the first part, the importance of the concept of a three-dimensional space which is regarded as a body, as opposed to the Aristotelian two-dimensional space or interval, in Patrizi’s discussion of physical space. This point, I shall argue, is an essential part of Patrizi’s overall critique of Aristotelian science, (...) in which Epicurean, Stoic, and mainly Neoplatonic elements were brought together, in what seems like an original theory of space and a radical revision of Aristotelian physics. Moreover, I shall try to show Patrizi’s dialectical method of definition, his geometrical argumentation, and trace some of the ideas and terms used by him back to Proclus’ Commentary on Euclid. This text of Proclus, as will be shown in the second part of the article, was also important for Patrizi’s discussion of mathematical space, where Patrizi deals with the status of mathematics and redefines some mathematical concepts such as the point and the line according to his new theory of space.Keywords: Francesco Patrizi; Space; Geometry; Mathematics; Aristotle; Proclus. (shrink)

Lorenzo Valla's rhetorical reform of logic resulted in important changes in sixteenth-century mathematical sciences, and not only in mathematical education and in the use of mathematics in other sciences, but also in mathematical theory itself. Logic came to be identified with dialectic, syllogisms with enthymemes and necessary truth with the limit case of probable truth. Two main ancient authorities mediated between logical and mathematical concerns: Cicero and Proclus. Cicero's 'common notions' were identified with Euclid's axioms, so that mathematics could (...) be viewed as core knowledge shared by all humankind. Proclus' interpretation of Euclid's axioms gave rise to the idea of a universal human natural light of reasoning and of a mathesis universalis as a basic mathematics common to both arithmetic and geometry and as an art of thinking interpretable as algebra. (shrink)

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)

The Pythagorean idea that numbers are the key to understanding reality inspired philosophers in late Antiquity (4th and 5th centuries A.D.) to develop theories in physics and metaphysics based on mathematical models. This book draws on some newly discovered evidence, including fragments of Iamblichus's On Pythagoreanism, to examine these early theories and trace their influence on later Neoplatonists (particularly Proclus and Syrianus) and on medieval and early modern philosophy.

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.

It is said that all philosophy is nothing other than a commentary on Plato.Maybe.But was not Plato himself a commentary on Parmenides, Heraclitus, the Pythagoreans, and the Sophists, not to mention Socrates?And conversely, too, the Commentary on Aristotle composed by St Thomas was not the personal philosophy of Thomas Aquinas? Or then again, do Proclus’ Commentarii in primum Euclidis Elementorum librum not embody a new and original neoplatonic philosophy of mathematics?

John J. Cleary was an internationally recognised authority in ancient Greek philosophy. This volume of penetrating studies of Plato, Aristotle, and Proclus, philosophy of mathematics, and ancient theories of education, display Cleary’s range of expertise and originality of approach.

This volume gathers an international team of renowned scholars in the fields of ancient greek philosophy, in order to explore the continuous but changing dialogue between Platonism and Aristotelianism from the early imperial age to the end of Antiquity. While most chapters concern Platonists (Philo, Plutarch, Plotinus, Syrianus, Proclus, Damascius, Philoponus), and their uses or criticism of Aristotle's doctrines, several chapters are also devoted to Peripatetic authors (Boethius and mostly Alexander of Aphrodisias) and their attitudes towards Plato's positions. Each of (...) the eleven chapters draws the attention to specific polemical contexts and philosophical problems which made Platonists and Aristotelians unite against common adversaries like the Stoics, or split up, not only in the fields of metaphysics and cosmology, but also in epistemology, psychology and ethics. (shrink)

John J. Cleary was an internationally recognised authority in ancient Greek philosophy. This volume of penetrating studies of Plato, Aristotle, and Proclus, philosophy of mathematics, and ancient theories of education, display Cleary’s range of expertise and originality of approach.

Philosophers since antiquity have argued the merits of mathematics as a normative aid in ethical decision-making and of the mathematization of ethics a theoretical discipline. Recently, Anagnostopoulos, Annas, Broadie and Hutchinson have probed such issues said to be of interest to Aristotle. Despite their studies, the sense in which Aristotle either opposed or proposed a mathematical ethics in subject-matter and method remains unclear. This paper attempts to clarify the matter. It shows Aristotle’s matrix of exactness and inexactness for ethical subject-matter (...) and ethical method in the Nicomachean Ethics. Then it probes a resultant puzzle from the matrix, namely, the HL model of the happy life without consideration of mathematical justice and the HJL model of the happy life with such consideration. Finally, it examines Aristotle’s twofold rationale for differentiating these two models in his overall moral feedback loop system: differences in the intellectual virtue of good deliberation; the priority of friendship over justice for the happy life. This suggests Aristotle saw no objection either to using mathematics as an aid to ethical decision-making for a happy life, or to mathematizing at least some parts of an ethical theory of eudaimonism. (shrink)

Dominic O'Meara has produced a scholarly and sympathetic account of a most enigmatic subject, namely, the role of mathematics in late Greek Platonic thought. O'Meara traces the path of mathematical philosophy from the Neopythagoreanism of the second and third centuries A.D. through that master of Athenian Neoplatonism, Proclus. Without this study few would recognize the paradigmatic role that mathematics played in Platonic thinkers throughout this period, for mathematics became the model for many forms of philosophical inquiry--not only theology and physics, (...) but even ethics. (shrink)

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current debate on (...) Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. (shrink)

The gods that guard the poles have been assigned the function of assembling the separate and unifying the manifold members of the whole, while those appointed to the axes keep the circuits in everlasting revolution around and around. And if I may add my own conceit, the centers and poles of all the spheres symbolize the wry-necked gods by imitating the mysterious union and synthesis which they effect; the axes represent the connectors of all the cosmic orders … and the (...) very spheres are likenesses of the perfecting divinities … 1 Proclus’s Commentary on the First Book of Euclid’s Elements contains one of the most important discussions of the nature of mathematical objects and preserves a wealth of historical information about mathematics from antiquity. However, large sections of the text have been neglected by scholars since the text’s first publication in the sixteenth century. Proclus expounds at length on correspondences between mathematical objects and the gods. At times he uses the language of “theurgy,” the ritual practices that later Neoplatonists insisted were necessary for the human soul to ascend to the intelligibles. The contention of this article is that the Commentary, taken as a whole, concerns the practice of “inner theurgy,” mental rituals used by advanced students to raise the soul beyond its assigned bounds. This practice can be traced back to Plotinus and the very beginnings of Neoplatonism; however, the particular methods presented by Proclus are his own development. This interpretation of the Commentary is supported by Proclus’s works Commentary on the Parmenides and Platonic Theology. Parts of the Commentary that have been studied carefully by modern scholars, especially the theory of the mathematical phantasia or imagination, will be shown to be crucial elements in Proclus’s theory and practice of inner theurgy. (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)

Dans La Philosophie de l’algèbre (1962), Jules Vuillemin présente sa démarche comme une manière d’instruire « le problème, si important et si négligé aujourd’hui, de la mathesis universalis dans ses rapports à la philosophie ». Il intitule d’ailleurs la seconde partie du traité « mathématique universelle », titre qu’il reprend pour la conclusion. Présentant le projet du second tome, il avance que cette étude devait le conduire « aux questions concrètes de la mathématique universelle ». Pourtant, à aucun moment, (...) on ne voit Vuillemin expliciter la nature de ce problème. Le but de cet article sera de produire une telle explicitation en s’appuyant, en particulier, sur certains éléments avancés dans le manuscrit préparatoire du second tome. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Proclus: Neo-Platonic Philosophy and Science by Lucas SiorvanesP.A. MeijerLucas Siorvanes. Proclus: Neo-Platonic Philosophy and Science. New Haven: Yale University Press, 1996. Pp. xv+ 340. Cloth, $35.00.This book will be welcomed by scholars of Proclus and by readers unfamiliar with Proclus alike. There are not many introductory books on Proclus. And Siorvanes presents in an interesting way the latest developments in scholarship. [End Page 160]Siorvanes gives an account of (...) Proclus’s life and times, his position in the Athenian Neoplatonic school, and his influence and offers a panoramic view of Proclus’s philosophy in the chapters: “General Metaphysics,” “Knowledge and the levels of Being,” “Physics and Metaphysics,” and “Stars and Planets.” He presents not only Proclus’s views on physics and astronomy, but also his poetics (189).Siorvanes deals competently with Proclus’s metaphysics, analyzing it in its own right and as the indispensable background of his scientific views. His summaries are on the whole adequate and instructive. But a scheme or outline of levels of the multi-layered Proclean system of Gods (Henads), such as that offered by H. D. Saffrey and L. G. Westerink in Proclus, Théologie Platonicienne (Paris 1968) for example, would have facilitated understanding.In an impressive description of Proclus’s historical influence, Siorvanes establishes connections between Proclus’s ideas and modern views in metaphysics, mathematics, physics, logic and astronomy. Yet a warning would be appropriate in this context. There is, for instance, the strikingly modern Proclean theory that planets have satellites. Perhaps Siorvanes is too much enamoured with these modern Proclean theories. Though he recognizes that the theory about the satellites is really a consequence of Proclus’s metaphysics (268–271), he fails to emphasize that Proclus’s modern-seeming theories are no less speculative than other ancient theories. Neither Proclus’s metaphysics nor his mathematics can replace evidence as used in modern science. Thus, the rather triumphal tones Sioravanes sounds in this regard are somewhat out of place. The incidental similarities in theoretical structures and patterns that emerge in the course of history are interesting to note, but we should beware of becoming too enthusiastic about their significance. The so-called indirect influence may be due precisely to the structure of problems and the (limited) range of possible solutions.I would raise some objections to Siorvanes’s manner of translating and interpreting the Greek word for “to be” (einai) or “Being” (on). As is often done, especially in the Anglo-Saxon world, Siorvanes alternatively uses “being,” “existence” and “reality,” without being aware of the fact that in ancient philosophy the Greek words “on/ousia” (“Being”) generally do not admit of the translation “existence.” Such translations are anachronistic and wrong. “Reality” is entirely inappropriate as a translation. As an essentially medieval expression, the word “reality” (realitas) as such is nowhere to be found in antiquity. When Proclus presents his multilevel system of Being, there are no degrees of reality in the sense that one transcendent level is more real than another. There are only degrees of Being. The problematic nature of the issue stands out even more when one speaks about not being. “Being” interpreted as “reality” suggests that what is not being, has no reality. But that is not what Proclus meant to say. The translation “existence” is bound to bring about similar misunderstandings. The notion “existence” is seldom in the focus of the interest when an ancient philosopher uses “einai.” Thus, an innocent reader of Siorvanes’s book may despair when he encounters “real existence” as a translation for “ousia” (124). Do the levels of being other than the one called “ousia” exist or do they not exist? If they do not, Proclus’s entire philosophy would be incoherent, and our world would not even exist or possess “reality.” But this is not what Proclus or Siorvanes mean. To avoid misunderstandings, we should use only “Being” (and “to be”) as translations. [End Page 161]There are also problems with some aspects of Siorvanes’s description of participation. In picturing the connection of the entity that is participated (methekton), what is given to the participant (metechomenon) and the participant (metechon), the terms “predication” and... (shrink)

Origen has been always studied as a theologian and too much credit has been given to Eusebius’ implausible hagiography of him. This book explores who Origen really was, by pondering into his philosophical background, which determines his theological exposition implicitly, yet decisively. For this background to come to light, it took a ground-breaking exposition of Anaxagoras’ philosophy and its legacy to Classical and Late Antiquity, assessing critically Aristotle’s distorted representation of Anaxagoras. Origen, formerly a Greek philosopher of note, whom Proclus (...) styled an anti-Platonist, is placed in the history of philosophy for the first time. By drawing on his Anaxagorean background, and being the first to revive the Anaxagorean Theory of Logoi, he paved the way to Nicaea. He was an anti-Platonist because he was an Anaxagorean philosopher with far-reaching influence, also on Neoplatonists such as Porphyry. His theology made an impact not only on the Cappadocians, but also on later Christian authors. His theory of the soul, now expounded in the light of his philosophical background, turns out more orthodox than that of some Christian stars of the Byzantine imperial orthodoxy. (shrink)

이 글은 데카르트의 정신지도를 위한 규칙에서 보편 수리학의 관점에서 비례의 개념을 해명하는 것을 목표로 한다. 보편 수리학이란 데카르트 자신의 설명에 따르면, “순서와 척도에 관해 연구될 수 있는 것을 모두 설명하는 어떤 일반적인 학문”이다. 구체적으로 말하자면 측량할 수 있는 모든 크기를 그 차원이나 존재 방식과 무관하게 어떤 보편적 인식 원리에 따라 규정하고 인식하는 것이 보편적 수리학의 과제인 것이다. 그런데 이런 보편 학문의 이념을 실현하는 것을 불가능하게 만들었던 장벽이 있었는데, 그것이 수와 도형의 통약 불가능성 그리고 같은 도형에서도 차원이 다른 도형들 사이의 연산 (...) 불가능성이다. 그런데 데카르트는 『정신지도를 위한 규칙』에서 수와 도형 사이의 장벽과 도형의 상이한 차원에 따른 연산 불가능성을 비례의 개념을 통하여 극복한다. 그리고 이를 통해 대수와 기하의 장벽을 허물고 보편적 수리과학의 기초를 마련했다. 이 글은 『정신지도를 위한 규칙』에서 비례의 개념이 어떻게 보편적 수리과학의 기초가 되는지를 주요 구절들을 통해 간단 명료하게 보이려는 시도이다. (shrink)

The concept of a Mannigfaltigkeit in Husserl has been given various interpretations, due to its shifting role in his works. Many authors have been misled by this term, placing it in the context of Husserl’s early period in Halle, while writing the Philosophy of Arithmetic, as a friend and colleague of Georg Cantor.Yet at the time, Husserl distanced himself explicitly from Cantor’s definition and rather took Bernhard Riemann as example, having studied and lectured extensively on Riemann’s theories of space. Husserl’s (...) Mannigfaltigkeitslehre would then not be a Cantorian set-theory, but come rather closer to topology. Then, in the Prolegomena, Husserl introduces the idea of a pure Mannigfaltigkeitslehre as a meta-theoretical enterprise which studies the relations among theories, e.g. how to derive or found one upon another. When Husserl announces that in fact the best example of such a pure theory of manifolds is what is actually practiced in mathematics, this sounds slightly misleading. The pure theory of theories cannot simply be the mathematics underlying topology, but should rather be considered as a mathesis universalis. Indeed, while this might not have been fully clear yet in 1900/1901, Husserl will explicitly tie together the notions of pure theory of manifolds and mathesis universalis. The mathesis universalis in this sense is formal, a priori and analytic, as theory of theory in general. It is an analysis of the highest categories of meaning and their correlative categories of objects. In my paper I try to understand the development of the notion of Mannigfaltigkeit in Husserl’s thought from its mathematical beginnings to its later central philosophical role, taking into account the mathematical background and context of Husserl’s own development. (shrink)

Does syllogistic logic have the resources to capture mathematical proof? This volume provides the first unified account of the history of attempts to answer this question, the reasoning behind the different positions taken, and their far-reaching implications. Aristotle had claimed that scientific knowledge, which includes mathematics, is provided by syllogisms of a special sort: 'scientific' ('demonstrative') syllogisms. In ancient Greece and in the Middle Ages, the claim that Euclid's theorems could be recast syllogistically was accepted without further scrutiny. Nevertheless, (...) as early as Galen, the importance of relational reasoning for mathematics had already been recognized. Further critical voices emerged in the Renaissance and the question of whether mathematical proofs could be recast syllogistically attracted more sustained attention over the following three centuries. Supported by more detailed analyses of Euclidean theorems, this led to attempts to extend logical theory to include relational reasoning, and to arguments purporting to reduce relational reasoning to a syllogistic form. Philosophical proposals to the effect that mathematical reasoning is heterogenous with respect to logical proofs were famously defended by Kant, and the implications of the debate about the adequacy of syllogistic logic for mathematics are at the very core of Kant's account of synthetic a priori judgments. While it is now widely accepted that syllogistic logic is not sufficient to account for the logic of mathematical proof, the history and the analysis of this debate, running from Aristotle to de Morgan and beyond, is a fascinating and crucial insight into the relationship between philosophy and mathematics. (shrink)

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)

In Plato’s eponymous dialogue, Timaeus, the main character presents the universe as an (almost) perfect sphere filled by tiny, invisible particles having the form of four regular polyhedrons. At first glance, such a construction may seem close to an atomistic theory. However, one does not find any text in Antiquity that links Timaeus’ cosmology to the atomists, while Aristotle opposes clearly Plato to the latter. Nevertheless, Plato is commonly presented in contemporary literature as some sort of atomist, sometimes as supporting (...) a form of so-called ‘mathematical atomism’. However, the term ‘atomism’ is rarely defined when applied to Plato. Since it covers many different theories, it seems that this term has almost as different meanings as different authors. The purpose of this article is to consider whether it is correct to connect Timaeus’ cosmology to some kind of ‘atomism’, however this term may be understood. Its purpose is double: to obtain a better understanding of the cosmology of the Timaeus, and to consider the different modern ‘atomistic’ interpretations of this cosmology. In short, we would like to show that such a claim, in any form whatsoever, is misleading, an impediment to the understanding of the dialogue, and more generally of Plato’s philosophy. (shrink)

Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are devoted. (...) At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)

This paper seeks to indicate some connections between a major philosophi- cal project of the seventeenth century, the conception of a mathesis universalis, and the practice of baroque poetry. I shall argue that these connections consist in a peculiar view of language and systems of notation which was particularly common in European baroque culture and which provided the necessary conceptual background for both poetry and the mathesis universalis.

The “common notions” prefacing the Elements of Euclid are a very peculiar set of axioms, and their authenticity, as well as their actual role in the demonstrations, have been object of debate. In the first part of this essay, I offer a survey of the evidence for the authenticity of the common notions, and conclude that only three of them are likely to have been in place at the times of Euclid, whereas others were added in Late Antiquity. (...) In the second part of the essay, I consider the meaning and uses of the common notions in Greek mathematics, and argue that they were originally conceived in order to axiomatize a theory of equivalence in geometry. I also claim that two interpolated common notions responded to different epistemic needs and regulated diagrammatic inferences. (shrink)

로저 베이컨(Roger Bacon)의 『상형증가론』은 서유럽 후기중세에서 전개된 다양한 상 형론의 원천을 이루는 작품으로 거론되어 왔다. 그는 로베르투스 그로세테스테(Robertus Grosseteste)로 대표되는 이른바 옥스퍼드학파의 자연철학에 기반한 상형론을 전개한다 는 점에서 차후 페트루스 요한네스 올리비(Petrus Johannes Olivi)에 이르는 중세 프란치 스코회 상형론 전통의 한 주축을 형성한다. 그럼에도 지난 베이컨 연구들은 대부분 근대과 학자의 원형으로서 베이컨을 조명하는 과학사적 접근에만 의존할 뿐, 철학자 베이컨의 초 상을 그리는데 인색했던 것처럼 보인다. 특히 과학사의 관점은 베이컨의 상형론을 중세광 학이라는 제한된 틀 안에서 그것의 일부로 다루었던 까닭에 베이컨의 상형론이 (...) 가지고 있 는 철학적 지반에 대한 분석에 충실하지 못한 일면적 관점만 제공했을 뿐이다. 하지만 우 리가 철학적-문헌학적으로 접근한다면 베이컨의 상형론은 아리스토텔레스철학의 재해석 에서 비롯된 질료의 능동적 가능태 이론의 산물로서 나타난다. 더욱이 베이컨의 상형론은 이처럼 독창적인 질료형상론의 토대로부터 우주와 지구 사이의 자연적 변화작용과 지각하 는 주체의 시각이론 양자 모두를 동시에 설명할 수 있는 일종의 보편수학을 구성한다. 따 라서 우리는 베이컨의 상형론의 원천적 기원과 그것이 도달하는 학문적 결과를 분석함으 로써 기존의 과학사 서술에서 벗어난 철학자 로저 베이컨의 면모들 중 하나를 드러내고자 하며, 나아가 이 분석을 기초로 베이컨의 상형론은 ‘중세광학’보다도 ‘시각학적 자연철학’ 이라 규정될 필요가 있다고 주장하려 한다. (shrink)

The project of this Précis de philosophie de la logique et des mathématiques (vol. 1 under the direction of F. Poggiolesi and P. Wagner, vol. 2 under the direction of A. Arana and M. Panza) aims to offer a rich, systematic and clear introduction to the main contemporary debates in the philosophy of mathematics and logic. The two volumes bring together the contributions of thirty researchers (twelve for the philosophy of logic and eighteen for the philosophy of mathematics), specialists in (...) the history or philosophy of logic or mathematics. Each volume consists of ten chapters; each chapter is about forty pages, is independent from the others and deals with a particular philosophical question about logic or mathematics. The objective is to offer to the French-speaking reader a reference book. On many of the issues addressed, this Précis offers the first clear and thorough synthesis that is available in French. -/- This book is intended for third year / master's students, but also for teachers of philosophy in secondary or higher education, for logicians and mathematicians willing to take a philosophical look at the fundamental concepts of their discipline, and for any reader interested in the basic issues discussed in the philosophy of logic and mathematics. Each chapter is self-contained, although some basic training in logic (or in mathematics, as the case may be) is required. The book should be looked at as a comprehensive framework for contemporary discussions in the philosophy of logic and mathematics, while also giving some guidance on the disciplinary content required by these discussions. -/- The first three chapters of the present volume (on the philosophy of mathematics) are devoted to the history of the philosophy of mathematics: from antiquity to the modern era, and from this period to the foundational crisis of the nineteenth century, to the twentieth century. Next, four chapters deal with the crucial questions for the philosophy of mathematics of the twentieth century: the opposition and/or comparison of set theory and category theory as foundational frameworks for mathematics; mathematical constructivism; the analysis of calculability; and Benaceraff's problem. The two following chapters are focused on the philosophy of mathematical practice, by treating the notion of ideals of proof, in particular explanation and purity, and the notion of informal proof and the use of visual artifacts in mathematical argumentation. Finally the last chapter deals with the applicability of mathematics, including the role of probability. -/- In Chapter 1, Sébastien Maronne and David Rabouin present the history of the philosophy of mathematics from antiquity to the modern era. The goal is to show that the philosophy of mathematics has a history as ancient as mathematics and philosophy itself, a history largely continuous with the latter, often by staying out of sync with respect to the former. In Chapter 2, Sébastien Gandon discusses the question of the foundations of mathematics from Kant to the end of the nineteenth century. By tying this question to Kant, he is able to show that the discussion of the foundations of mathematics that occupied a large part of these two centuries has much older roots. Accent is placed on the differences between the framework proposed by Kant for making sense of classical mathematics and the use made by Frege and then Russell of logical tools for giving a different reading of arithmetic and real analysis. The link between between these different conceptions and the evolution of mathematics itself is also underlined. -/- Chapter 3, written by Hourya Benis-Sinaceur and Mirna Džamonja, deals with the evolution of mathematics in the twentieth century, notably the structuralist turn of Dedekind, E. Noether, and Bourbaki, and certain more recent developments. They devote themselves, among other things, to explaining how these developments can be understood as responses to and extensions of questions posed during the foundational debate of the immediately preceding era. -/- In Chapter 4, Jean-Pierre Marquis and Jean-Jacques Szczeciniarz discuss the different foundational options coming from set theory and category theory. They discuss, among other things, Lawvere's program to replace ZF with an axiomatization of the category of sets, the latter judged better adapted to the needs of contemporary mathematics. Different reactions, as much philosophical as technical, raised by this program are also taken into account. -/- Constructive mathematics are the object of Chapter 5, in which Gerhard Heinzmann and Mark van Atten examine in a critical and comparative manner a variety of constructivisms, including intuitionism, constructive type theory, predicativism, and finitism. -/- In Chapter 6, Guido Gherardi and Maël Pegny present computability theory, stressing its philosophical consequences. They discuss the basic concepts of this theory, for the computability of both the natural numbers and the real numbers. They then tackle the theory of computational complexity. -/- In Chapter 7, Andrea Sereni and Fabrice Pataut present Benacerraf's problem, opposing every possible account of mathematical knowledge to the availability of a theory of truth founded on an abstract ontology of mathematical objects. They give an overview of the original problem and its reformulation by Hartry Field, and reconstruct the debate that this problem has raised, by meeting at once the major opposing positions today of the analytic affiliation of the philosophy of mathematics, as much those on the platonist side as on that of nominalism. -/- In Chapter 8, Valeria Giardino and Yacin Hamami treat certain crucial aspects of the philosophy of mathematical practice. They deal in particular with the notion of an informal proof and on the role of artifacts in mathematical reasoning. They consider how such proofs, making use of these means, can contribute to the understanding of theorems and mathematical theories, thanks also to the role of visualisation and the aid of computer tools. -/- Why do mathematics often give several proofs of the same theorem? This is the question that Andrew Arana raises in Chapter 9, introducing the notion of an epistemic ideal and discuss two such ideals, the explanatoriness and purity of proof. -/- Chapter 10 is devoted to the applicability of mathematics in the study of empirical phenomena. After summarizing the history of this question, going back to Plato and Aristotle and passing by Mill and Kant, Daniele Molinini and Marco Panza reconstruct the contemporary debate, in relation, among other things, to questions raised in the philosophy of science. -/- Two appendices complete the volume. In the first Frédéric Patras treats the French tradition in philosophy of mathematics of the 20th century. In the second, Maria Carla Galavotti discusses the notion of probability and its use in science. (shrink)

OElig;he thesis is sustained that the definitions of fundamental geometric entities which open Euclids Elements actually are excerpts from the Definitions by Heron of Alexandria, interpolated in late antiquity into Euclids treatise. As a consequence, one of the main bases of the traditional Platonist interpretation of Euclid is refuted. Arguments about the constructivist nature of Euclids mathematical philosophy are given.

In this paper, we examine a fundamental problem that appears in Greek philosophy: the paradoxes of self-reference of the type of “Third Man” that appears first in Plato’s 'Parmenides', and is further discussed in Aristotle and the Peripatetic commentators and Proclus. We show that the various versions are analysed using different language, reflecting different understandings by Plato and the Platonists, such as Proclus, on the one hand, and the Peripatetics (Aristotle, Alexander, Eudemus), on the other hand. We show that the (...) Peripatetic commentators do not focus on Plato’s solution but primarily on the formulation of the “Third Man” paradox. On the contrary, Proclus seems to be convinced that Plato suggests a sound solution to the paradox by defining the predicate of similarity (homogeneity) that demarcates two types of homogeneous entities – the eide and the participants in them in a way that their confusion would be inadmissible. We claim that Plato’s solution follows a sound line of reasoning that is formalisable in a language of Frege-Russell type; hence there exists a model in which Plato’s reasoning is valid. Furthermore, we notice that Plato’s definition of the second-order predicate of similarity is attained by resorting to first-order entities. In this sense, Plato’s definition is comparable to Eudoxus’ definition of ratio, which is also attained by resorting to first-order objects. Consequently, Plato seems to follow a logical practice established by the mathematicians of the 5th century, notably Eudoxus, in his solution to the paradox. (shrink)

This book is not a conventional history of mathematics as such, a museum of documents and scientific curiosities. Instead, it identifies this vital science with the thought of those who constructed it and in its relation to the changing cultural context in which it evolved. Particular emphasis is placed on the philosophic and logical systems, from Aristotle onward, that provide the basis for the fusion of mathematics and logic in contemporary thought. Ettore Carruccio covers the evolution of mathematics from the (...) most ancient times to our own day. In simple and non-technical language, he observes the changes that have taken place in the conception of rational theory, until we reach the lively, delicate and often disconcerting problems of modern logical analysis. The book contains an unusual wealth of detail (including specimen demonstrations) on such subjects as the critique of Euclid's fifth postulate, the rise of non-Euclidean geometry, the introduction of theories of infinite sets, the construction of abstract geometry, and-in a notably intelligible discussion-the development of modern symbolic logic and meta-mathematics. Scientific problems in general and mathematical problems in particular show their full meaning only when they are considered in the light of their own history. This book accordingly takes the reader to the heart of mathematical questions, in a way that teacher, student and layman alike will find absorbing and illuminating. The history of mathematics is a field that continues to fascinate people interested in the course of creativity, and logical inference quite part and in addition to those with direct mathematical interests. Ettore Carruccio, who until his retirement was professor of philosophy at the University of Turin. He has made many contributions to mathematical and logical theory as well as to the history of the science. Isabel Quigly was the literary editor of The Tablet for many years. (shrink)

By a thorough study of the Posterior Analytics and related Aristotelian texts, Richard McKirahan reconstructs Aristotle's theory of episteme--science. The Posterior Analytics contains the first extensive treatment of the nature and structure of science in the history of philosophy, and McKirahan's aim is to interpret it sympathetically, following the lead of the text, rather than imposing contemporary frameworks on it. In addition to treating the theory as a whole, the author uses textual and philological as well as philosophical material to (...) interpret many important but difficult individual passages. A number of issues left obscure by the Aristotelian material are settled by reference to Euclid's geometrical practice in the Elements. To justify this use of Euclid, McKirahan makes a comparative analysis of fundamental features of Euclidian geometry with the corresponding elements of Aristotle's theory. Emerging from that discussion is a more precise and more complex picture of the relation between Aristotle's theory and Greek mathematics--a picture of mutual, rather than one-way, dependence. Originally published in 1992. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)