Available from UMI in association with The British Library. ;Plato's philosophy of mathematics must be a philosophy of 4th century B.C. Greek mathematics, and cannot be understood if one is not aware that the notions involved in this mathematics differ radically from our own notions; particularly, the notion of arithmos is quite different from our notion of number. The development of the post-Renaissance notion of number brought with it a different conception of what mathematics (...) is, and we must be able to see the differences before we can begin to appreciate Plato's views, which must never be seen as applying to a revolutionary notion of mathematics . For this reason the first part of this dissertation must comprise a detailed analysis of these differences, along with some criticisms of those of Plato's modern interpreters who have not taken this point. Much of what will be found here can be found in Jacob Klein's Greek Mathematics and the Origin of Algebra, which will be extensively quoted and interpreted. This book is often quoted, but very seldom understood; or at least, those lessons which should be taken from it are rarely applied to Platonic exegesis. In the second part I attempt a better explanation of Plato's mathematical ontology, beginning with the most extensive passage in the dialogues, the triple image of Sun, Line and Cave in Republic. This reveals a relation between sensibles and forms which is at odds with the traditional view that mathematical objects are never precisely accurate instances of their forms--but this view can be refuted. Plato's view of the nature of mathematical objects will be seen to be a metaphorical way of saying what Aristotle says logically, and is entirely appropriate to the aims and methods of his contemporary mathematicians. Aristotle's criticism of Plato's views is examined and found to make sense and to be appropriate to views which Plato could consistently have held. (shrink)

A straightforward presentation of Plato's views on the nature of mathematics, with special attention to the status of mathematical objects and to the method of mathematical thinking. Mr. Wedberg has summarized his interpretations of Platonic doctrines in a clear and well-organized fashion, devoting one chapter to Plato's views on geometry, one to his views on arithmetic; he then supports these interpretations by a close examination of the relevant passages, not only in Plato's Dialogues, but in Aristotle as well. A (...) comprehensive and useful study.--V. C. C. (shrink)

By the end of his life Plato had rearranged the theory of ideas into his teaching about ideal numbers, but no written records have been left. The Ideal mathematics of Plato is present in all his dialogues. It can be clearly grasped in relation to the effective use of mathematical modelling. Many problems of mathematical modelling were laid in the foundation of the method by cutting the three-level idealism of Plato to the single-level “ideism” of Aristotle. For a long (...) time, the real, ideal numbers of Plato’s Ideal mathematics eliminates many mathematical problems, extends the capabilities of modelling, and improves mathematics. (shrink)

In this article I analyze the issue of many levels of reality that are studied by natural sciences. Particularly interesting is the level of mathematics and the question of the relationship between mathematics and the structure of the real world. The mathematical nature of the world has been considered since ancient times and is the subject of ongoing research for philosophers of science to this day. One of the viewpoints in this field is mathematical Platonism. In contemporary (...) class='Hi'>philosophy it is widely accepted that according to Plato mathematics is the domain of ideal beings that are eternal and unalterable and exist independently from the subject’s beliefs and decisions. Two issues seem to be important here. The first issue concerns the question: was Plato really a proponent of present-day mathematical Platonism? The second one is of greater importance: how mathematics influences our understanding of the nature of the world on its many ontological levels? In the article I consider three issues: the Platonic theory of “two worlds”, the method of building a mathematical structure, and the ontology of mathematics. (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)

It is well known that the largest philosophers differently explain the origin of mathematics. This question was investigated in antiquity, a substantial and decisive role in this respect was played by the Platonic doctrine. Therefore, discussing this issue the problem of interaction of philosophy and mathematics in the teachings of Plato should be taken into consideration. Many mathematicians believe that abstract mathematical objects belong in a certain sense to the world of ideas and that consistency of objects (...) and theories really describes mathematical reality, as Plato quite clearly expressed his views on math, according to which mathematical concepts objectively exist as distinct entities between the world of ideas and the world of material things. In the context of foundations of mathematics, so called ‘Gödel’s Platonism‘ is of particular interest. It is shown in the article how Platonic objectification of mathematical concepts contributes to the development of modern mathematics by revealing philosophical understanding of the nature of abstraction. To substantiate his point of view, the author draws the works of contemporary experts in the field of philosophy of mathematics. (shrink)

Given its brevity, Plato's Meno covers an astonishingly wide array of topics: politics, education, virtue, definition, philosophical method, mathematics, the nature and acquisition of knowledge and immortality. Its treatment of these, though profound, is tantalisingly short, leaving the reader with many unresolved questions. This book confronts the dialogue's many enigmas and attempts to solve them in a way that is both lucid and sympathetic to Plato's philosophy. Reading the dialogue as a whole, it explains how different arguments are (...) related to one another and how the interplay between characters is connected to the philosophical content of the work. In a new departure, this book's exploration focuses primarily on the content and coherence of the dialogue in its own right and not merely in the context of other dialogues, making it required reading for all students of Plato, be they from the world of classics or philosophy. (shrink)

Mathematics plays an inordinate role in the work of many of famous Western philosophers, from the time of Plato, through Husserl and Wittgenstein, and even to the present. Why? This paper points to the experience of learning or making mathematics, with an emphasis on proof. It distinguishes two sources of the perennial impact of mathematics on philosophy. They are classified as Ancient and Enlightenment. Plato is emblematic of the former, and Kant of the latter. The Ancient (...) fascination arises from the sense that mathematics explores something ‘out there’. This is illustrated by recent discussions by distinguished contemporary mathematicians. The Enlightenment strand often uses Kant's argot: ‘absolute necessity’, ‘apodictic certainty’ and ‘a priori’ judgement or knowledge. The experience of being compelled by proof, the sense that something must be true, that a result is certain, generates the philosophy. It also creates the illusion that mathematics is certain. Kant's leading question, ‘How is pure mathematics possible?’, is easily misunderstood because the modern distinction between pure and applied is an artefact of the 19th century. As Russell put it, the issue is to explain ‘the apparent power of anticipating facts about things of which we have no experience’. More generally the question is, how is it that pure mathematics is so rich in applications? Some six types of application are distinguished, each of which engenders its own philosophical problems which are descendants of the Enlightenment, and which differ from those descended from the Ancient strand. (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 chapter advances a reformulation of the classical problem of the nature of mathematical objects (if any), here called “Plato’s problem,” in line with the program of a philosophy of mathematical practice. It then provides a sketch of a platonist solution, following the same perspective. This solution disregards as nonsensical the question of the existence of abstract, and specifically mathematical, objects, by rather focusing on the modalities of our access to them: objects (in general, both concrete and abstract) (...) are regarded as individual contents that we have (or can have) a de re epistemic access to. The question of the existence of mathematical objects is then replaced by that of the modalities of our de re epistemic access to individual mathematical contents. (shrink)

Easy to understand philosophy papers in all areas. Table of contents: Three Short Philosophy Papers on Human Freedom The Paradox of Religions Institutions Different Perspectives on Religious Belief: O’Reilly v. Dawkins. v. James v. Clifford Schopenhauer on Suicide Schopenhauer’s Fractal Conception of Reality Theodore Roszak’s Views on Bicameral Consciousness Philosophy Exam Questions and Answers Locke, Aristotle and Kant on Virtue Logic Lecture for Erika Kant’s Ethics Van Cleve on Epistemic Circularity Plato’s Theory of Forms Can we (...) trust our senses? Yes we can Descartes on What He Believes Himself to Be The Role of Values in Science Modern Science Kant’s Moral Philosophy Plato’s Republic as Pol Potist Bureaucracy Schopenhauer on Human Suffering Bertrand Russell on the Value of Philosophy The Philosophical Value of Uncertainty Logic Homework: Theorems and Models Searle vs. Turing on the Imitation Game Hume, Frankfurt, and Holbach on Personal Freedom Manifesto of the University of Wisconsin, Madison Secular Society Michael’s Analysis of the Limits of Civil Protections Bentham and Mill on Different Types of Pleasure Set Theory Homework Aristotle on Virtue Nagel On the Hard Problem Wittgenstein on Language and Thought Camus and Schopenhauer on the Meaning of Life Camus’ Hero as Rebel without a Cause My Little Finger: Camus’ Absurdism Illustrated Are Late-term Abortions Ethical? Does Mathematics Assume the Truth of Platonism? The Self-defeating Nature of Utilitarianism and Consequentialism Generally What is The Good Life? Bentham and Mill regarding types of pleasures Kant’s Moral Philosophy Five Short Papers on Mind-body Dualism Tracy Latimer’s Father had the Right to Kill Her: Towards a doctrine of generalized self-defense Arguments Concerning God and Morality Goldman, Rousseau and von Hayek on the Ideal State J.S Mill on Liberty and Personal Freedom A Kantian Analysis of a Borderline Date-rape Situation Living Well as Flourishing: Aristotle’s Conception of the Good Life Three Essays on Medical Ethics: Answers to Exam Questions on Elective Amputation, Vaccination, and Informed Consent Hobbes, Marx, Rousseau, Nietzsche: Their Central Themes De Tocqueville on Egoism Mill vs. Hobbes on Liberty Exam-Essays on the Moral Systems of Mill, Bentham, and Kant Kant’s Moral System Aristotle on Virtue Plato’s Cave Allegory An Ethical Quandary Superorganisms The Tuskegee Experiment A Rawlsian Analysis Why Moore’s Proof of an External World Fails A Defense of Nagel’s Argument Against Materialism A Utilitarian Analysis of a Case of Theft The Paradox of the Self-aware Wretch: An Analysis of Pascal’s Moral Philosophy Jean-Paul Sartre: Decline and Fall of a Marxist Sell-out A One Page Proof of Plato’s Theory of Forms Plato’s Republic as Pol Potist Bureaucracy The problem of the one and the many Four Short Essays on Truth and Knowledge What is ‘the Good Life?’ The Ontological Argument Different Political Philosophies: Plato, Locke, Madison, Rousseau, Hayek, and Mill on the State What do I know with certainty? Skepticism about skepticism Neuroscience and Freewill Operant Conditioning What makes us special? Are Late-term Abortions Ethical? No Two Papers on Epistemology: Gettier and Bostrom Examination Nietzsche on Punishment God’s Foreknowledge and Moral Responsibility . (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS 113 phers); nevertheless, I feel that the book would have been more effective pedagogically had they devoted more attention to them than they have. As a specific recommendation, I would suggest several short introductions at strategic places in the text devoted to a brief resum~ of the historical setting in which the philosophers to be discussed found themselves. Once again I should emphasize that, despite the criticisms (...) I have made, I think the book to be an excellent piece of work, which should prove quite successful as an undergraduate text in the history of philosophy. OLIVER JOHNSON University of California, Riverside Plato's Theory of Knowledge. By Norman Gulley. (London: Methuen & Co. Ltd.; New York: Barnes & Noble Inc. Pp. viii ~ 203.) Mr. Gulley gives an account of the development of Plato's theory of knowledge. The impetus to this development he finds in an inconsistency that underlies the attempt to combine an a priori method of analysis with a theory of knowledge based on the doctrines of recollection and the archetype-copy relation of Forms and sensibles. Only in the last chapter, "Mathematical Knowledge," does he suggest a theory of the relation of a priori to empirical knowledge which he thinks might have satisfied Plato. The major part of the work, the first three chapters, concerns the conflicts and readjustments occasioned by the underlying inconsistency. I shall discuss these chapters in terms of a few central points. In chapter I, GuUey attempts to get at the basis of the theoretical difficulties. He points out that Plato first considers philosophic method as independent of the theory of Forms and the theory of recollection (pp. 40 f). In the Meno it is a method of analysis, which is a forerunner of the Phaedo's method of hypothesis, of which the dialectical method is a further development. It is an a priori method. But the theories, which are first hypotheses examined by the method (Phaedo), become later "presuppositions for the effective practice of the method," as, for example, in the Republic. In "recollection" there is an intimate connection between Forms and sensiblcs, but in knowledge there is complete separation. This is what Gulley considers to be the source of the major difficulties. Since I do not find these difficulties, I shall concentrate on their exposition. Gulley notes in the Phaedo two meanings of "giving an account": (1) in terms of a method of propositional analysis; (2) in terms of participation of sensibles in Forms. His explanation of the first offers no difficulties, but of the second there are serious problems. His explication of the second is as follows: To "give an account" is "to explain one's possession of a Form by showing that it is necessarily implied by that judgment of the 'deficiency' of sensible characters which arises as soon as they are recognized" (p. 40). His argument for this interpretation is as follows: "... if to know a Form brings with it the ability 'to give an account', and if to know a Form is to be reminded of it by the sensibles which resemble it (74 b-c), the 'giving of an account' must be explicable within the conception of 'knowledge' implied in the use of the term in 74 b-c" (p. 46). He assumes that Plato is at least consistent. Now in 74 b-c Simmias agrees with Socrates that we have knowledge of equality "in and by itself" and that such things as equal logs and stones "lead us to conceive" of it. Only "recollection," not the preexistent knowledge of Forms is in question here. Since it is assumed that knowledge implies the ability to give an account of what is known, then this instance of knowledge, which occurs in conjunction with perception, implies a similar ability. But one cannot infer from this that the discussion of equality as Plato has discussed it, which is a discussion of it as an example of recollection, is the same thing as the giving an account of equality itself, much less that it is "the only type of account possible" within the theory of Forms. That it is the only possible type is in fact contradicted by the assumption of pre... (shrink)

Mr. Brumbaugh gives several accounts in the course of his work of the main purpose of his study, and the emphasis falls now one way and now another. Readers may easily be misled by the opening sentence of the introduction, which suggests that Plato's mathematical illustrations are pointers to "diagrams which Plato had designed, and were intended to accompany and clarify his text." If that is what Mr. Brumbaugh intended, he has failed to make out his case. There is no (...) direct evidence that Plato designed diagrams; and there is some good inferential evidence to the contrary. It would be surprising, for example, if Plato's diagrams had survived that long, that Aristotle should have described the Divided Line in terms of an arithmetical progression and ignored Plato's express indication of proportion between the upper and lower segments. Mr. Brumbaugh himself faces this passage and admits that "no continuous tradition connects the figures of Hellenistic scholars with Plato's original design." But it really does not matter. If we design diagrams in the light of our knowledge of Plato's mathematical imagination, it is most certainly a help to the interpretation of the passages concerned. This, as a matter of fact, seems to be Mr. Brumbaugh's main reason for ascribing the diagrams to Plato, and while the conclusion does not follow, the premiss is correct; and the premiss, not the conclusion, is the main point at issue. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Dialectic and Dialogue: Plato's Practice of Philosophical Inquiry (SPEP Studies in Historical Philosophy)Rosamond Kent SpragueFrancisco J. Gonzalez. Dialectic and Dialogue: Plato's Practice of Philosophical Inquiry (SPEP Studies in Historical Philosophy). Evanston, IL: Northwestern University Press, 1998. Pp. 418. Paper, $29.95.What this rich and independent-minded book asks us to do is to give serious consideration to the question, "What, in Plato's view, are we doing when we (...) philosophize?" (1) If we employ our own conceptions of philosophy as the models for formulating and discussing specific Platonic questions, we may be imposing upon him ways of thinking that are foreign to his thought. To do this is particularly distorting in the case of dialectic.Gonzalez has chosen to confine himself to the early and middle dialogues, so that his book is a direct challenge to the work of Richard Robinson. Of the important views that have more recently arisen, Gonzalez allies himself most nearly with the non-doctrinal approach, but not to the extent of embracing scepticism. There is much philosophical knowledge in the dialogues, but such knowledge is primarily non-propositional. As for the esotericist and the developmentalist, both are misguided in proceeding on the assumption that Plato is attempting to formulate definitions and to arrive at a systematic philosophy; they regard the dialogues as treatises, which they are not. As Gonzalez quite justly inquires, "can there be any form of writing less suited to presenting a systematic philosophy than Plato's dramatic dialogues?" (4)In contrast to Robinson, who relies on the analysis of passages taken out of context, Gonzalez focusses mainly on whole dialogues, taking them to be dramatic portrayals of dialectic at work. He begins with detailed studies of the Laches and Charmides. In both cases, Socrates starts with ordinary knowledge (in the persons of the title characters), but "clearly intends to transcend it" (20). Neither will he accept "the 'sophistic' understanding that attempts to substitute definitions for this experience" (21) on the part of Nicias and Critias. The knowledge sought by Socrates by means of the dialectical method is shown to thave three characteristics: it is "knowledge how," it is "self-knowledge," and it is "non-propositional" (61).The next chapters consider three presuppositions of Plato's dialectic: words (in the Cratylus), arguments (in the Euthydemus), and images (in Republic X). All three are necessary to the dialectical enterprise, but, like ordinary knowledge in the aporetic dialogues, are transcended.Gonzalez' confrontation with Robinson naturally requires that he address the topic [End Page 113] of the method of hypothesis in the Meno, Phaedo, and Republic. This is a method that does yield propositional results, but it remains at the level of poion ti and fails to provide similar knowledge of the ti. Gonzalez' interpretation of the divided line is of particular interest here: as he sees it, dialectic is not intended as simply an extension continuous with the method of hypothesis. To take it this way is, with Robinson, to collapse dialectic into mathematics, making the former as deductive as the latter. The good (which "is not for Plato an exclusively moral term" 216) is "already there before the inquiry as the ground of its possibility" (236); this is what "unhypothesized" means. The movement of thought is circular, but not with a vicious circularity; "reflexivity" is the term preferred.To my mind the most rewarding and most convincing section of the book is the concluding chapter on the philosophical digression in the Seventh Letter. (The issue of authenticity is not significant, since, if the Letter is a forgery, "this forger had a better understanding of Plato than many other scholars, both ancient and contemporary" (246). Although "the subject of this wisdom simply cannot be expressed in words" (252) (and here Gonzalez emphasizes that Plato means spoken as much as written words), there is no contradiction in being able to talk or write about it. If "our only means of attaining true knowledge of a thing have the defect of being unable to express this nature" (263), Plato suggests an up and down movement which transcends this defect. The nature of this movement he describes as tribein, a process... (shrink)

This chapter contains sections titled: Going Beyond Nature in Order to Explain it Technē, epistēmē and alēthēs doxa Mathematics, pure and applied Observation and Experimental Verification Bibliography.

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)

In his Republic Plato considers grasping the unity of mathematics as the ultimate goal of the mathematical studies in which the future philosopher-rulers must engage before they turn to philosophy. How the unity of mathematics is supposed to be understood is not explained, however. This book argues that Plato conceives of the unity of mathematics in terms of the mutually benefiting links between its branches, just as he conceives of the unity of the state outlined in (...) the Republic in terms of the common benefit for all citizens. Evidence for this view is provided by a discussion of his conception of astronomy as a propedeutic to philosophy, which can be best understood as hinting at a historically possible link between fourth-century-BC astronomy and solid geometry. The monograph also includes a detailed discussion of two well-known stories about Plato: not only he motivated Greek mathematicians to solve a difficult problem in solid geometry with his interpretation of a Delphic oracle given to the inhabitants of the island of Delos but he also posed the question which led to the development of the astronomical theory of homocentric spheres. It is argued that these stories are best understood as fictional episodes in Plato's life, constructed from passages in his works. (shrink)

This paper re-constructs Plato's ‘philosophy of geometry’ by arguing that he uses a geometrical method of hypothesis in his account of the cosmos’ generation in the Timaeus. Commentators on Plato's philosophy of mathematics often start from Aristotle's report in the Metaphysics that Plato admitted the existence of mathematical objects in-between ( metaxu) Forms and sensible particulars ( Meta. 1.6, 987b14–18). I argue, however, that Plato's interest in mathematics was centred on its methodological usefulness for philosophical inquiry, (...) rather than on questions of mathematical ontology. My key passage of interest is Timaeus’ account of the generation of the primary bodies in the cosmos, i.e. fire, air, water and earth ( Tim. 48b–c, 53b–56c). Timaeus explains the primary bodies’ origin by hypothesising two right-angled triangles as their starting-point ( arkhê) and describing their individual geometrical constitution. This hypothetical operation recalls the hypothetical method which Socrates introduces in the Meno (86e–87b), as well as the use of hypotheses by mathematicians which is described in the Republic (510b–c). Throughout the passage, Timaeus is focussed on explicating the bodies in terms of their formal structure, without however considering the ontological status of the triangles in relation to the physical world. (shrink)

The late 1960s saw the emergence of new philosophical interest in Kant's philosophy of mathematics, and since then this interest has developed into a major and dynamic field of study. In this state-of-the-art survey of contemporary scholarship on Kant's mathematical thinking, Carl Posy and Ofra Rechter gather leading authors who approach it from multiple perspectives, engaging with topics including geometry, arithmetic, logic, and metaphysics. Their essays offer fine-grained analysis of Kant's philosophy of mathematics in the context (...) of his Critical philosophy, and also show sensitivity to its historical background. The volume will be important for readers seeking a comprehensive picture of the current scholarship about the development of Kant's philosophy of mathematics, its place in his overall philosophy, and the Kantian themes that influenced mathematics and its philosophy after Kant. (shrink)

Wittgenstein's role was vital in establishing mathematics as one of this century's principal areas of philosophic inquiry. In this book, the three phases of Wittgenstein's reflections on mathematics are viewed as a progressive whole, rather than as separate entities. Frascolla builds up a systematic construction of Wittgenstein's representation of the role of arithmetic in the theory of logical operations. He also presents a new interpretation of Wittgenstein's rule-following considerations - the `community view of internal relations'.

Once we realize that the indivisible and infinitely repeatable One of the arithmetic lesson in Republic7 is generated by διάνοια at Parmenides 143a6-9, it becomes possible to revisit the Divided Line’s Second Part and see that Aristotle’s error was not to claim that Plato placed Intermediates between the Ideas and sensible things but to restrict that class to the mathematical objects Socrates used to explain it. All of the One-Over-Many Forms of Republic10 that Aristotle, following Plato, attacked with the Third (...) Man, are equally dependent on Images and above all on the Hypothesis of the One. (shrink)

One of the hardest questions to answer for a (Neo)platonist is to what extent and how the changing and unreliable world of sense perception can itself be an object of scientific knowledge. My dissertation is a study of the answer given to that question by the Neoplatonist Proclus (Athens, 411-485) in his Commentary on Plato’s Timaeus. I present a new explanation of Proclus’ concept of nature and show that philosophy of nature consists of several related subdisciplines matching the (...) ontological stratification of nature. Moreover, I demonstrate that for Proclus philosophy of nature is a science, albeit a hypothetical one, which takes geometry as its methodological paradigm. I also offer an explanation of Proclus’ view of what is later called the mathematization of physics, i.e. the role of the substance of mathematics, as opposed to its method, in explaining the natural world. Finally, I discuss Proclus’ views of the discourse of philosophy of nature and its iconic character. (shrink)

William Tait is one of the most distinguished philosophers of mathematics of the last fifty years. This volume collects his most important published philosophical papers from the 1980's to the present. The articles cover a wide range of issues in the foundations and philosophy of mathematics, including some on historical figures ranging from Plato to Gdel.

Widespread interest in Frege's general philosophical writings is, relatively speaking, a fairly recent phenomenon. But it is only very recently that his philosophy of mathematics has begun to attract the attention it now enjoys. This interest has been elicited by the discovery of the remarkable mathematical properties of Frege's contextual definition of number and of the unique character of his proposals for a theory of the real numbers. This collection of essays addresses three main developments in recent work (...) on Frege's philosophy of mathematics: the emerging interest in the intellectual background to his logicism; the rediscovery of Frege's theorem; and the reevaluation of the mathematical content of The Basic Laws of Arithmetic. Each essay attempts a sympathetic, if not uncritical, reconstruction, evaluation, or extension of a facet of Frege's theory of arithmetic. Together they form an accessible and authoritative introduction to aspects of Frege's thought that have, until now, been largely missed by the philosophical community. (shrink)

A curious collection of snippets from the world of mathematics, mainly of historical significance. Beside Newton there are selections from Chaucer and Dürer, on the one hand, and Plato and Sun-Tsu on the other. The author provides historical and biographical sketches for all fifty-four of the cross-cultural selections.—P. S.

Attention is drawn to the fact that what is alternatively known as Dummett logic, Gödel logic, or Gödel-Dummett logic, was actually introduced by Skolem already in 1913. A related work of 1919 introduces implicative lattices, or Heyting algebras in today's terminology.

I construe mathematical philosophy not in the narrow sense of philosophy of mathematics but in a broad indefinite sense of different manners of giving mathematics a privileged place in the study of philosophy. For example, in one way or another, mathematics plays an important part in the philosophy of Plato, Descartes, Spinoza, Leibniz, and Kant. In contrast, history plays a central role in the philosophy of Vico, Hegel, and Marx. In more recent (...) times, Frege, Husserl, Russell, Ramsey, and Gödel all began as mathematicians. One way of viewing Kant’s system of philosophy might be to stress that he was struck by the synthetic a priori character of mathematical propositions. He went on to offer a remarkable account of this fact and also look for and propose synthetic a priori foundations of physics, morality, and esthetics. (shrink)

In her 2014 monograph, Sarah Broadie argues that Timaeus’s cosmology points to a radical Platonic insight: the full rationality of the cosmos requires the existence of individualized, autonomous, and finite beings like us. Only human life makes the cosmos truly complete. But can Timaeus do full justice to the uniquely human way of being and hence to his own insight? My paper argues that he cannot and that Plato means for us to see that he cannot, by showing how Timaeus (...) treats a famous Platonic theme: eros. Timaeus describes human perfection as assimilation to the mathematical proportions of the cosmos, but by comparing Timaeus with the Symposium I show that, given his deeply mathematized conception of reason, Timaeus cannot provide what Diotima can: a phenomenologically satisfactory account of how we come to identify ourselves with this perfection. Such identification is a transformation in our self-understanding explicable only because of the desirous and reflexive character of the soul. Expressing this character, however, requires combining the mathematical with a poetic, or even mantic, register. Only these sensibilities together grant access to Plato’s cosmology in its fullness. (shrink)

Sebastian Grève interviews Felix Mühlhölzer on his work on the philosophy of mathematics.

Wittgenstein's remarks on mathematics have not received the recogni tion they deserve; they have for the most part been either ignored, or dismissed as unworthy of the author of the Tractatus and the I nvestiga tions. This is unfortunate, I believe, and not at all fair, for these remarks are not only enjoyable reading, as even the harshest critics have con ceded, but also a rich and genuine source of insight into the nature of mathematics. It is perhaps (...) the fact that they are more suggestive than systematic which has put so many people off; there is nothing here of formal derivation and very little attempt even at sustained and organized argumentation. The remarks are fragmentary and often obscure, if one does not recognize the point at which they are directed. Nevertheless, there is much here that is good, and even a fairly system atic and coherent account of mathematics. What I have tried to do in the following pages is to reconstruct the system behind the often rather disconnected commentary, and to show that when the theory emerges, most of the harsh criticism which has been directed against these re marks is seen to be without foundation. This is meant to be a sym pathetic account of Wittgenstein's views on mathematics, and I hope that it will at least contribute to a further reading and reassessment of his contributions to the philosophy of mathematics. (shrink)

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. (shrink)

Plato’s philosophy is important to Badiou for a number of reasons, chief among which is that Badiou considered Plato to have recognised that mathematics provides the only sound or adequate basis for ontology. The mathematical basis of ontology is central to Badiou’s philosophy, and his engagement with Plato is instrumental in determining how he positions his philosophy in relation to those approaches to the philosophy of mathematics that endorse an orthodox Platonic realism, i.e. (...) the independent existence of a realm of mathematical objects. The Platonism that Badiou makes claim to bears little resemblance to this orthodoxy. Like Plato, Badiou insists on the primacy of the eternal and immu- table abstraction of the mathematico-ontological Idea; however, Badiou’s reconstructed Platonism champions the mathematics of post-Cantorian set theory, which itself af rms the irreducible multiplicity of being. Badiou in this way recon gures the Platonic notion of the relation between the one and the multiple in terms of the multiple-without-one as represented in the axiom of the void or empty set. Rather than engage with the Plato that is gured in the ontological realism of the orthodox Platonic approach to the philosophy of mathematics, Badiou is intent on characterising the Plato that responds to the demands of a post-Cantorian set theory, and he considers Plato’s philosophy to provide a response to such a challenge. In effect, Badiou reorients mathematical Platonism from an epistemological to an ontological problematic, a move that relies on the plausibility of rejecting the empiricist ontology underlying orthodox mathematical Platonism. To draw a connec- tion between these two approaches to Platonism and to determine what sets them radically apart, this paper focuses on the use that they each make of model theory to further their respective arguments. (shrink)

For several decades, Carnap's philosophy of mathematics used to be either dismissed or ignored. It was perceived as a form of linguistic conventionalism and thus taken to rely on the bankrupt notion of truth by convention. However, recent scholarship has revealed a more subtle picture. It has been forcefully argued that Carnap is not a linguistic conventionalist in any straightforward sense, and that supposedly decisive objections against his position target a straw man. This raises two questions. First, how (...) exactly should we characterise Carnap's actual philosophy of mathematics? Secondly, is his position an attractive alternative to established views? I will tackle these issues by looking at Carnap's response to the incompleteness theorems. Drawing on arguments put forward by Gödel and Beth, I show that some crucial aspects of Carnap's positive account have remained underdeveloped. Suggestions on what a full evaluation of Carnap's position requires are made. (shrink)

In general, scholars have viewed the mathematical detail of Plato’s Divided Line discussion in Republic VI-VII as irrelevant to the substance of his epistemology.Against this stance this essay argues that this detail serves a serious and instructive purpose and makes manifest some central features of Plato’s account of human knowledge.

The article considers Cassirer’s philosophy of mathematics in opposition to empiricist theories, Frege’s logicism, and its realism, Hilbert’s formalism and its nominalism, and Brouwer’s intuitionism grounding mathematics in the intuition of time. For Cassirer mathematical objects are purely relational structures and not abstractions of certain characteristics, as is the case with empiricists and Frege. In opposition to logicists, Cassirer argues for the synthetic nature of mathematics. Contrary to Brouwer, he does not ground this in intuition but (...) ascribes to mathematics a purely logical nature and relates it to the purely rational faculty of the spirit. He thus turns away from Hilbert, who takes sensuous objects to be the object of mathematics; he also cannot agree with his formalism, which reduces all progress in mathematics to a mere combination of signs, thus stripping it of all meaning. The article considers these differences in view of the opposition between the cardinal and ordinal theories of number and the introduction of ideal elements. (shrink)

This chapter contains sections titled: Introduction Plato's Divided Line and Mathematical Cognition The Cartesians and the Problem of Pure Thought Descartes on the Role of the Imagination in Forming a Distinct Idea of Corporeal Nature Malebranche on the Role of the Imagination in Mathematical Cognition Conclusion.