L’article met en lumière la continuité intellectuelle de l’Académie à propos d’une question précise, les rapports entre philosophie et géométrie. On soutient d’abord que, dans les livres VI-VII de la République, Platon ne cherche pas à réformer les pratiques des géomètres mais identifie les contraintes incontournables de leurs raisonnements (constructions, hypothèses), qui constituent et limitent leur objectivité. On montre ensuite que cette analyse constitue le cadre des réflexions académiciennes ultérieures sur la géométrie. Speusippe reprend et développe l’analyse platonicienne des constructions (...) géométriques, qui est appliquée (de Speusippe à Crantor) à la cosmologie et même à la doctrine des principes, afin d’expliquer comment elles saisissent leurs objets éternels depuis le devenir, de manière indirecte. Si la Nouvelle Académie d’Arcésilas et de Carnéade critique les mathématiques, on fait l’hypothèse qu’il s’agissait pour elle – dans le contexte des débats philosophiques hellénistiques sur les usages des mathématiques –, de rappeler les limites de leur objectivité contre toute idéalisation dogmatique de la méthode géométrique. (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)

This is an in-depth study of J.G. Fichte’s philosophy of mathematics and theory of geometry. It investigates both the external formal and internal cognitive parallels between the axioms, intuitions and constructions of geometry and the scientific methodology of the Fichtean system of philosophy. In contrast to “ordinary” Euclidean geometry, in his Erlanger Logik of 1805 Fichte posits a model of an “ursprüngliche” or original geometry – that is to say, a synthetic and constructivistic conception grounded in ideal archetypal elements that (...) are grasped through geometrical or intelligible intuition. -/- Accordingly, this study classifies Fichte’s philosophy of mathematics as a whole as a species of mathematical Platonism or neo-Platonism, and concludes that the Wissenschaftslehre itself may be read as an attempt at a new philosophical mathesis, or “mathesis of the mind.” -/- “This work testifies to the author's exact and extensive knowledge of the Fichtean texts, as well as of the philosophical, scientific and historical contexts. Wood has opened up completely new paths for Fichte research, and examines with clarity and precision a domain that up to now has hardly been researched.” Professor Dr. Marco Ivaldo -/- “This study, written in a language distinguished by its limpidity and precision, and constantly supported by a close reading of the Fichtean texts and secondary literature, furnishes highly detailed and convincing demonstrations. In directly confronting the difficult historical relationship between the Wissenschaftslehre and mathematics, the author has broken new ground that is at once stimulating, decidedly innovative, and elegantly audacious.” Professor Dr. Emmanuel Cattin. (shrink)

Tim Maudlin sets out a completely new method for describing the geometrical structure of spaces, and thus a better mathematical tool for describing and understanding space-time. He presents a historical review of the development of geometry and topology, and then his original Theory of Linear Structures.

Historians of modern philosophy have been paying increasing attention to contemporaneous scientific developments. Isaac Newton's Principia is of course crucial to any discussion of the influence of scientific advances on the philosophical currents of the modern period, and two philosophers who have been linked especially closely to Newton are John Locke and Immanuel Kant. My dissertation aims to shed new light on the ties each shared with Newtonian science by treating Newton, Locke, and Kant simultaneously. I adopt Newton's philosophy of (...) geometry as the starting point of investigation, for here I believe we have a constructive means by which to assess Locke and Kant's relationship to Newton, In particular, I defend the thesis that the justification Newton, Locke and Kant offer for applying geometrical principles to nature is central to understanding their respective ties to a Newtonian science characterized by the intermingling of mathematics and experiment, Although little is said by Locke in regard to a mathematical approach to nature, I hope to show that his interpretation of the origins of our geometrical ideas has a close affinity to Newton's own characterization of geometry, leading us to reexamine the extent of Locke's 'Newtonianism.' Kant famously attempts to bridge the gap between geometry and the empirical world by establishing space as a "pure form of intuition." My discussion of Kant's Newtonian approach to nature centers on the imagination, and I argue that the mediating work completed by this faculty in geometrical construction and experience in general is equally important to understanding Kant's application of geometry to the empirical realm, In the end, I hope my treatment of the strategies employed by Newton, Locke, and Kant to account for a mathematical-experimental method of natural philosophy sheds further light on the importance of Newton to the progress of modern philosophy. (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 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)

For over half a century I have been interested in the role of intuitive spatial reasoning in mathematics. My Oxford DPhil Thesis (1962) was an attempt to defend Kant's philosophy of mathematics, especially his claim that mathematical proofs extend our knowledge (so the knowledge is "synthetic", not "analytic") and that the discoveries are not empirical, or contingent, but are in an important sense "a priori" (which does not imply "innate") and also necessarily true. -/- I had made my views clear (...) in courses on philosophy of science and mathematics when teaching at Sussex University (from 1964) which was why one of our former students, Mary Pardoe (then Mary Ensor) who had become a mathematics teacher informed me, while visiting the university, that she had found a new diagrammatic proof of the triangle sum theorem. I reported her proof in some papers and presentations on methods of representation and reasoning, e.g. here, but neither she nor I has encountered anyone else who knew about the proof. (shrink)

This book discusses in dialogue form the basic principles of mathematics and its applications including the question: What is mathematics? What does its specific method consist of? What is its relation to the sciences and humanities? What can it offer to specialists in different fields? How can it be applied in practice and in discovering the laws of nature? Dramatized by the dialogue form and shown in the historical movements in which they originated, these questions are discussed in their (...) full complexity, yet are easily comprehended. The first dialogue, whose chief actor is Socrates, leads the reader to the source of modern mathematics in Athens in the 5th Century BC. The second dialogue, featuring Archimedes, takes place during the siege of Syracuse in 212 BC and shows the birth of applied mathematics. The third dialogue occurs in the year 1633 in Rome, its chief character being Galileo Galilei who fully realized the central importance of the mathematical method in discovering the laws of nature. Intended as supplemental reading for philosophy of mathematics courses at the high school or college level it will be of interest to both specialists and non-specialists in mathematics. Alfréd Rényi was born in Budapest Hungary in 1921. He studied mathematics and physics at the University of Budapest and received his Ph. D. from the University of Szaged in 1945. Since 1950 he has been Director of the Mathematical Research Institute of the Hungarian Academy of Sciences and since 1952 a professor at the University of Budapest. Dr. Renyi was a visiting professor at Michigan State University in 1961, at the University of Michigan in 1964 and at Stanford University in 1966. His main fields of research are probability theory, mathematical statistics and information theory, and he has also worked in analytic number theory as well as in various branches of analysis, combinatorial analysis and geometry. (shrink)

Au début du xixe siècle, la formation des officiers de l’armée des États-Unis s’effectue à l’Académie militaire de West Point. Défaillante en de nombreux points, y compris sur le terrain de l’enseignement mathématique, elle est transformée par Sylvanus Thayer en 1817, alors qu’il revient d’un séjour en Europe lors duquel les établissements militaires français ont fait l’objet de scrupuleuses observations. La supériorité des méthodes françaises – l’articulation mathématico-ingéniérique qui structure les curricula, le rôle de la géométrie descriptive et l’analyse dans (...) l’appréhension de l’art de la guerre, l’usage des manuels – semble chose donnée pour acquise du côté américain, si bien que, dans le sillage de West Point, d’autres académies militaires « à la française » ouvrent avant-guerre. Soutenant l’idée d’une « infusion française » plus ou moins passive, l’historiographie alimente bien souvent un récit qui mythifie et cristallise le rôle de la France dans la formation mathématique des militaires américains de la première moitié du xixe siècle, un récit qui minore, de fait, par son caractère univoque, l’importance des espaces intermédiaires – scientifiques, pédagogiques, institutionnels et éditoriaux – où entrent en contact les territoires émetteurs et récepteurs. Cet article montre comment les acteurs qui œuvrent à la formation mathématique dans les écoles militaires aux États-Unis – superintendants, professeurs et auteurs de manuels – procèdent par décomposition et réassemblage des savoirs et méthodes français, dont le « transfert » depuis l’autre côté de l’Atlantique et au sein du territoire américain emprunte des circuits moins prétendument polarisés et hiérarchisés, et très fortement soumis à une forme de rationalisation locale. (shrink)

Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact of (...) model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Heirs of Plato: A Study of the Old Academy (347–274 BC)Carlos SteelJohn M. Dillon. The Heirs of Plato: A Study of the Old Academy (347–274 BC). Oxford: Clarendon Press, 2003. Pp. x + 252. Cloth, $65.00.When Plato died, in 347 BC, he left behind not only the collection of philosophical dialogues we still read with admiration, but also a remarkable organization, the "Academy," wherein his students continued (...) the discussion he had initiated. Dillon's latest book is an attempt to recreate the intellectual climate in the Academy during the first generations after Plato's death, the "Old Academy," called thus to distinguish it from the "New Academy," which started with Arcesilaus in the mid 270's, who abandoned doctrinal speculations for a sceptical stance. Given the elusive nature of the dialogues, the interpretation of Plato's philosophy has been controversial from the beginning. Therefore, scholars always have been eager to know in which direction Plato's first successors—Speusippus, Xenocrates, Polemo—took the Academy; thus they hoped to find some justification for their own interpretation of Plato. Unfortunately, our information on the doctrines of Plato's first successors is scarce. Browsing through the fragments collected by scholars, one may be tempted to conclude that it is not such a great disaster that most texts of the Old Academy have been lost. For it seems that the Academy after Plato was leaning towards a rather esoteric metaphysical speculation about the first principles, which makes us lament on what has become of the philosophical heritage of Plato, a genius fallen into the hands of epigones. One gets the impression that the original and refreshing philosophical contributions came from outside the Academy, from the Lyceum founded by Aristotle and continued by Theophrastus, and later from Epicurus and Zeno and their new schools, which will dominate the Hellenistic period. This, however, is not the view of Dillon: "It will be the thesis of this book that Speusippus and Xenocrates set the agenda for what was to become, over the succeeding centuries, the intellectual tradition which we call Platonism" (v). Even early Stoicism is, in Dillon's view, a "development and formalization of contemporary Platonism" (235).In an introductory chapter Dillon discusses the location and juridical structure of the Academy and gives a general survey of the main doctrines of the school. The following three chapters present the lives, work and thought of the first heads of the Academy; a final chapter is devoted to some minor members of the early Academy. The book concludes with a short epilogue on Arcesilaus and the reasons for his turn to scepticism. Dillon is an [End Page 204] excellent writer who knows how to tease out and expound the most abstruse philosophical doctrines in a captivating way, mixing, as he does here, biographical anecdotes and witty comments with a sharp analysis of arguments and a speculative reconstruction of lost doctrines. I doubt, however, whether this book will become such a classic as his celebrated The Middle Platonists (Ithaca, 1977).My first worry is anticipated by Dillon himself: "This is perforce a rather speculative book, and no doubt it will be criticized for that" (vii). Dillon defends himself against this critique: I have tried, he says, "to put the most favourable construction on the evidence available to us," his assumption being that the figures he is dealing with "were not philosophical imbeciles"(vii). My reservations are not about Dillon's attempt to construct a coherent doctrine using all relevant information on an author, but about his method of reconstruction, in particular his speculation and his preference for often questionable later material to fill the gaps in our information. "With a good deal of speculation, our exiguous information on Polemo's philosophical position can be fleshed out somewhat" (176). Thus, in his exposition of Speusippus's doctrine of the first principles, Dillon does not start from Aristotle, considering him too biased, but from chapter 4 of Iamblichus's treatise De communi mathematica scientia. In this Neo-Pythagorean treatise, Speusippus is not even mentioned. Even if there are arguments, as Dillon believes, to attribute the doctrine of Chapter 4 to Speusippus, one should... (shrink)

This paper aims to bring together the study of normative judgments in mathematics as studied by the philosophy of mathematics and verbal hygiene as studied by sociolinguistics. Verbal hygiene (Cameron 1995) refers to the set of normative ideas that language users have about which linguistic practices should be preferred, and the ways in which they go about encouraging or forcing others to adopt their preference. We introduce the notion of mathematical hygiene, which we define in a parallel way as the (...) set of normative discourses regulating mathematical practices and the ways in which mathematicians promote those practices. To clarify our proposal, we present two case studies from 17th century France. First, we exemplify a case of mathematical hygiene proper: Descartes' algebraic geometry and Newton's subsequent criticism of it, a case of (im)purity in mathematics. Then, we compare Descartes' and Newton's mathematical hygiene with verbal hygiene from this period, as exemplified by the work of the grammarian Claude Favre de Vaugelas (Ayres-Bennett 1987). We argue that these early modern normative discourses on mathematics and language respectively can be seen as emanating from a common socio-political program: the development of a new bourgeois intellectual class. We conclude that the study of mathematical hygiene has the potential to yield new understandings of the social aspects of mathematical practice, and that similarities between mathematical and verbal hygiene at certain time periods, such as 17th century France, open up a new area of inquiry at the borders of linguistics and the philosophy of mathematics. (shrink)

In the first argument of Metaphysics Μ.2 against the Platonist introduction of separate mathematical objects, Aristotle purports to show that positing separate geometrical objects to explain geometrical facts generates an ‘absurd accumulation’ of geometrical objects. Interpretations of the argument have varied widely. I distinguish between two types of interpretation, corrective and non-corrective interpretations. Here I defend a new, and more systematic, non-corrective interpretation that takes the argument as a serious and very interesting challenge to the Platonist.

We are used to seeing foundations linked to the mainstream mathematics of the late nineteenth century: the arithmetization of analysis, non-Euclidean geometry, and the rise of abstract structures in algebra. And a growing number of case studies bring a more philosophy-of-science viewpoint to the latest mathematics, as in [Carter, 2005; Corfield, 2006; Krieger, 2003; Leng, 2002]. Mac Lane's autobiography is a valuable bridge between these, recounting his experience of how the mid- and late-twentieth-century mainstream grew especially through Hilbert's school.An autobiography (...) at age 94 obviously has a lot of ground to cover. Mac Lane entered Yale in 1926 to study chemistry as a good practical career field but by the end of his first year he had won a $50 prize in mathematics and learned that you could make a living with it—as an actuary. He decided to do that. The next year he was excited to learn from his philosophy professor F.S.C. Northrop that you can also pursue new mathematical discoveries! Northrop had studied with A.N. Whitehead and sold Mac Lane on the excitement of Principia Mathematica. In a pattern that foreshadowed his career Mac Lane bought and annotated a copy of the first volume of Principia but his planned tutorial study of the book turned into a study of [Hausdorff, 1914] applying set theory to topology and analysis. As a graduate student at Chicago he was powerfully influenced by E.H. Moore, who taught the new axiomatic methods as tools for unifying and advancing the most classical analysis . Then he went to Göttingen. He heard Hilbert lecture on philosophy. He studied mathematics and its philosophy with Weyl. He followed Noether's lectures on abstract algebra for number theory. He had gone there to study logic and he wrote a …. (shrink)

An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.

A new variational method, the principle of least radix economy, is formulated. The mathematical and physical relevance of the radix economy, also called digit capacity, is established, showing how physical laws can be derived from this concept in a unified way. The principle reinterprets and generalizes the principle of least action yielding two classes of physical solutions: least action paths and quantum wavefunctions. A new physical foundation of the Hilbert space of quantum mechanics is then accomplished and it is (...) used to derive the Schrödinger and Dirac equations and the breaking of the commutativity of spacetime geometry. The formulation provides an explanation of how determinism and random statistical behavior coexist in spacetime and a framework is developed that allows dynamical processes to be formulated in terms of chains of digits. These methods lead to a new foundation for Lorentz transformations and special relativity. The Parker-Rhodes combinatorial hierarchy is encompassed within our approach and this leads to an estimate of the interaction strength of the electromagnetic and gravitational forces that agrees with the experimental values to an error of less than one thousandth. Finally, it is shown how the principle of least-radix economy naturally gives rise to Boltzmann’s principle of classical statistical thermodynamics. A new expression for a general nonequilibrium entropy is proposed satisfying the Second Law of Thermodynamics. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:After Gödel: Platonism and Rationalism in Mathematics and LogicMark C. R. SmithRichard Tieszen. After Gödel: Platonism and Rationalism in Mathematics and Logic. Oxford-New York: Oxford University Press, 2011. Pp. xi + 245. Cloth, $75.00.Tieszen’s new book offers a synthesis and extension of his longstanding project of bringing the method of Husserl’s phenomenology to bear on fundamental questions—both epistemological and ontological—in the philosophy of mathematics. Gödel held Husserl’s (...) philosophy in high regard, and thought that it offered the most promising avenue for an adequate understanding of the foundations of mathematical thought and practice, but Gödel himself did not give a full systematic statement of how phenomenology might carry out that project. Tieszen takes up the task in this book, which is structured around two main guiding themes. The first is to locate the sources of Gödel’s philosophical views—principally [End Page 303] in Plato, Leibniz, and of course Husserl, along with Kant as a sort of backcloth—and the second is to argue that from these materials, a promising rationalism in mathematics and logic can be consistently articulated. Along the way we also encounter a good many of Gödel’s arguments against an array of reductive positions, Hilbert’s formalism and Carnap’s “logical syntax” project among them, as well as arguments against the prospects of a mechanical or purely computational understanding of the human mind. Many of the arguments (both negative and positive) stem from a reflection on the consequences of Gödel’s landmark incompleteness results of 1931.The core of Tieszen’s positive proposals comes in chapters 4, 5, and 6: here is where we meet with “constituted Platonism,” Tieszen’s name for the rationalist, realist synthesis of Husserl’s method and Gödel’s mathematical insights. Methodologically, the starting point is the actual practice of mathematics as a human undertaking, and its conditions of possibility. At the most general level, it is characteristic of thought to be intentional, that is, to be (or try to be) about something, to have a target, to be a bearer of meaning. So it goes for thought about items in sensory awareness, about fictional objects, or about mathematical subject-matter. Concerning the latter, what we encounter in mathematical experience are frequently “phenomena that resist our will, so that we are ‘forced’ in certain directions” (170); what accounts for this resistance to our will are invariants, which have the further characteristics of repeatability—both for a single subject, and among many subjects—and availability for convergence (221), which, I take it, means availability as structures or concepts upon which many independent rational investigators will come to rest. Thinking mathematical thoughts is quite unlike thinking thoughts about fictional items, because fictions are indefinitely plastic in the face of human will. It is also unlike thought about sensory items, in that the intentional objects at issue are not available to outer sense; but from another angle it is more like thought about real material items than imaginary beings, because our mathematical thoughts are exposed to the possibility of error or misrepresentation in various ways. (An elementary example: most people think that 1 is distinct from 0.9999... repeating: this is a misrepresentation, because they are in fact identical.)The phenomenology of mathematical practice is such that we assay various conjectures and proposals about invariants already given to consciousness by previous practice, and discover what further invariant structures or concepts are revealed as necessary constraints upon the various possibilities that we can experimentally float for ourselves. Sometimes our conjectures or intentions are fulfilled, sometimes they are frustrated; but it is the very fact of fulfillment and frustration which points us to a Platonic reality. It is not the noumenal realm of the “naive” Platonist, in Tieszen’s phrase, but nevertheless an objective reality marked off “on the basis of what stabilizes or becomes invariant in our experience, on the basis of what has a history and a sedimentation that permits of further elaboration, development, constraints, surprises, and so on” (101). In something like this way, human reason and consciousness “constitute the meaning of being” of mathematical objects... (shrink)

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could mathematics be knowledge (...) about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)

Glaucon in Plato's _Republic_ fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hypotheses. He offers his version of geometry and astronomy as refuting the charge that he impiously ‘ponders things up in the sky and investigates things under the earth and makes the weaker argument the stronger’. We relate his account (...) briefly to mathematical developments by Plato's associates Theaetetus and Eudoxus, and then to the past 200 years' developments in geometry. (shrink)

The Geometrical Method The Geometrical Method is the style of proof that was used in Euclid’s proofs in geometry, and that was used in philosophy in Spinoza’s proofs in his Ethics. The term appeared first in 16th century Europe when mathematics was on an upswing due to the new science of mechanics. … Continue reading Geometrical Method →.

Plato is the first scientist whose work we still possess. He is our first writer to interpret the natural world mathematically, and also the first theorist of mathematics in the natural sciences. As no one else before or after, he set out why we should suppose a link between nature and mathematics, a link that has never been stronger than it is today. Mathematical Plato examines how Plato organized and justified the principles, terms, and methods of our mathematical, natural science. (...) "Roger Sworder deserves our gratitude for drawing attention to the significance of mathematics in Plato's thought and writings. He lays the principal discussions out before us with clarity. He also presents Plato as a theorist of nature: of physics and not just metaphysics, to use Aristotle's distinction. Not all readers, we should admit, will be equally convinced of the usefulness of Plato's science for today, but they will all be led more deeply into Plato's vision of reality."--ANDREW DAVISON, Westcott House, Cambridge "Here is Plato for an anti-Platonic age. The author gives careful attention to some of the most important passages in the Platonic dialogues and offers new solutions to some of Plato's most famous mathematical puzzles. He then considers the implications of these penetrating studies for the philosophy of science, and the natural sciences especially. This is a book that revivifies the core themes of Platonism and restores science to worship. It shows Roger Sworder to be one of the foremost students of Plato writing today, and places him in the noble tradition of Thomas Taylor."--RODNEY BLACKHIRST, author of Primordial Alchemy and Modern Religion: Essays on Traditional Cosmology. (shrink)

The dissertation is a detailed analysis of Berkeley's writings on mathematics, concentrating on the link between his attack on the theory of abstract ideas and his philosophy of mathematics. Although the focus is on Berkeley's works, I also trace the important connections between Berkeley's views and those of Isaac Barrow, John Wallis, John Keill, and Isaac Newton . The basic thesis I defend is that Berkeley's philosophy of mathematics is a natural extension of his views on abstraction. The first chapter (...) is devoted to a consideration of Berkeley's treatment of abstraction, including his arguments against the doctrine of abstract ideas and his own account of how the explanatory ideas traditionally assigned to abstract ideas can be filled by a non-abstractionist account of human knowledge. In chapter two I investigate the details of Berkeley's proposed new foundations for geometry, showing how his rejection of abstract ideas led him to a critique of the traditional conception of geometry . Of particular importance in this context is Berkeley's denial of infinite divisibility and his attempts to show that a satisfactory account of geometry does not require that geometric magnitudes be infinitely divisible. Chapter three is concerned with Berkeley's treatment of arithmetic and algebra. Here I argue that Berkeley's denial of the claim that arithmetic is the science of abstract ideas of number ultimately results in his advocacy of a strongly nominalistic conception of arithmetic which has strong similarities to modern fomalism. In chapter four I discuss Berkeley's famous critique of the calculus in The Analyst and other works, concluding that his criticism of the calculus is essentially correct, although his attempted explanation of the success of infinitesimal methods is unconvincing. (shrink)

The uses and interpretation of reductio ad absurdum argumentation in mathematical proof and discovery are examined, illustrated with elementary and progressively sophisticated examples, and explained. Against Arthur Schopenhauer’s objections, reductio reasoning is defended as a method of uncovering new mathematical truths, and not merely of confirming independently grasped mathematical intuitions. The application of reductio argument is contrasted with purely mechanical brute algorithmic inferences as an art requiring skill and intelligent intervention in the choice of hypotheses and attribution of contradictions (...) deduced to a particular assumption in a contradiction’s derivation base within a reductio proof structure. (shrink)

Sobre la base que aportan las notas manuscritas de David Hilbert para cursos sobre geometría, el artículo procura contextualizar y analizar una de las contribuciones más importantes y novedosas de su célebre monografía Fundamentos de la geometría, a saber: el cálculo de segmentos lineales. Se argumenta que, además de ser un resultado matemático importante, Hilbert depositó en su aritmética de segmentos un destacado significado epistemológico y metodológico. En particular, se afirma que para Hilbert este resultado representaba un claro ejemplo de (...) uno de los rasgos más fructíferos y atractivos de su nuevo método axiomático formal, o sea, la capacidad de descubrir y exhibir conexiones estructurales o internas entre diferentes teorías matemáticas. On the basis of a set of unpublished notes for lecture courses on geometry, the paper seeks to contextualize and analyze one of the most important and original contributions of David Hilbert's celebrated monograph Foundations of Geometry, namely its arithmetic of line segments. It is argued that Hilbert attributed to his arithmetic of segments an important epistemological and methodological meaning, in addition to its relevance as an original mathematical result. In particular, it is claimed that for Hilbert his arithmetic of segments represented a clear example of one of the most fruitful and attractive traits of his new formal axiomatic method, i.e., the power to discover and exhibit inner or structural connections among different mathematical theories. (shrink)

The importance of mathematics in the context of the scientific and technological development of humanity is determined by the possibility of creating mathematical models of the objects studied under the different branches of Science and Technology. The arithmetisation process that took place during the nineteenth century consisted of the quest to discover a new mathematical reality in which the validity of logic would stand as something essential and central. Nevertheless, in contrast to this process, the development of mathematical analysis within (...) a framework that largely involves intuition and geometry is a fact that cannot go unnoticed amongst the mathematics community, as we shall show in this paper through the research made by Bernhard Riemann on complex variables. (shrink)

At the beginning of the present century, a series of paradoxes were discovered within mathematics which suggested a fundamental unclarity in traditional mathemati­cal methods. These methods rested on the assumption of a realm of mathematical idealities existing independently of our thinking activity, and in order to arrive at a firmly grounded mathematics different attempts were made to formulate a conception of mathematical objects as purely human constructions. It was, however, realised that such formulations necessarily result in a mathematics which lacks (...) the richness and power of the old ‘platonistic’ methods, and the latter are still defended, in various modified forms, as embodying truths about self-existent mathematical entities. Thus there is an idealism-realism dispute in the philosophy of mathematics in some respects parallel to the controversy over the existence of the experiential world to the settle­ment of which lngarden devoted his life. The present paper is an attempt to apply Ingarden’s methods to the sphere of mathematical existence. This exercise will reveal new modes of being applicable to non-real objects, and we shall put forward arguments to suggest that these modes of being have an importance outside mathematics, especially in the areas of value theory and the ontology of art. (shrink)

This chapter contains sections titled: The Academy and Pyrrhonism The New Academy and Epicureanism The New Academy and Stoicism The New Academy and Middle Platonism Bibliography.

How can we invent new certain knowledge in a methodical manner? This question stands at the heart of Salomon Maimon's theory of invention. Chikurel argues that Maimon's contribution to the ars inveniendi tradition lies in the methods of invention which he prescribes for mathematics. Influenced by Proclus' commentary on Elements, these methods are applied on examples taken from Euclid's Elements and Data. Centering around methodical invention and scientific genius, Maimon's philosophy is unique in an era glorifying the artistic genius, known (...) as Geniezeit. Invention, primarily defined as constructing syllogisms, has implications on the notion of being given in intuition as well as in symbolic cognition. Chikurel introduces Maimon's notion of analysis in the broader sense, grounded not only on the principle of contradiction but on intuition as well. In philosophy, ampliative analysis is based on Maimon's logical term of analysis of the object, a term that has yet to be discussed in Maimonian scholarship. Following its introduction, a new version of the question quid juris? arises. In mathematics, Chikurel demonstrates how this conception of analysis originates from practices of Greek geometrical analysis. (shrink)

While not paramount among Kant scholars, issues in the philosophy of mathematics have maintained a position of importance in writings about Kant’s philosophy, and recent years have witnessed a rejuvenation of interest and real progress in interpreting his views on the nature of mathematics. My hope here is to contribute to this recent progress by expanding upon the general tacks taken by Jaakko Hintikka concerning Kant’s writings on geometry.Let me begin by making a vile suggestion: Kant did not have a (...) philosophy of mathematics. When Kant was writing about mathematics, essentially he was reporting the views of others. The texts provide sufficient evidence to make this suggestion plausible. Generally, when Kant writes about mathematics in his mature works, he does so in order to illustrate or argue for a philosophical point. There are important references to mathematical method in the preface to the 1787 edition of Critique of Pure Reason; however, Kant’s purpose is to describe those basic features of a method that he intended to incorporate in his theory of philosophical method: ‘our new method of thought, namely, that we can know a priori of things only what we ourselves put into them.’ Indeed, Kant makes it clear in this preface that he thought there to be no extant problems to be solved in mathematical methodology; such was the state of the science, he thought. It was for this reason that Kant felt some confidence in borrowing from this method to improve the state of metaphysics; it is also for this reason that one should not expect to find Kant engaging in basic research in mathematical methodology. Similarly, the material on syntheticity added to the second edition Introduction to Critique of Pure Reason occurs in the context of a discussion of the syntheticity of metaphysical principles; that the propositions of both disciplines are synthetic a priori lends credence to the extrapolation of some features from the mathematical method for use in developing a metaphysics. Many writers find a philosophy of mathematics in the ‘Transcendental Aesthetic’; it is clear, however, that in this section his concern is to support his theory of the nature of space, time, and sensation. What is said about geometry, for example, is restricted to those of its features relevant to the subjectivity of space. The other major discussion of mathematics and its method is found in the section, ‘Doctrine of Method.’ Here we find Kant’s fullest account of the mathematical method and of constructions. It must be borne in mind, though, that his purpose is to argue against views that the proper methods of mathematics and metaphysics are identical, that the disciplines differ in subject matter alone. The result of the discussion is not a theory of mathematical method, but an account of the method proper to the philosopher.2 Kant simply is mentioning certain features of mathematical method sufficient to support the claim that the philosopher cannot incorporate it lock, stock, and barrel.’ In short, we do not find a systematic theory of mathematics or its method described by Kant in the first Critique, nor do we find discussions of mathematics other than in contexts where philosophical positions are being developed. This holds for Kant’s other works, too. (shrink)

Newton composed several mathematical tracts which remained in manuscript form for decades. He chose to print some of his mathematical tracts in their entirety only after 1704. In this paper I will give information on the dissemination of Newton’s mathematical manuscripts before the eighteenth-century printing stage. I will not consider another important vehicle of dissemination of Newton’s mathematical discoveries, namely his correspondence with other mathematicians or with intermediaries such as Collins and Oldenburg.In a first stage, Newton’s mathematical manuscripts were rendered (...) available to a group of acolytes, who copied and transmitted them, sometimes in mutilated form. The practice of scribal publication helped Newton to establish his mathematical reputation and to form around himself a group of followers.One can divide Newton’s mathematical activity into two periods: an early analytical and modernist period in which he developed a Cartesian and Wallisian ‘new analysis’, followed by a mature synthetic period, beginning in the 1670s, in which he distanced himself from analysis in favour of geometry, from infinitesimals in favour of limits, from the analysis of the moderns in favour of the geometry of the ancients. Newton came to the conclusion that the ‘new analysis’ was not a mathematical language fit for publication.Newton, however, could not restrict himself to scribal publication of his mathematical manuscripts for too long. The priority dispute with Leibniz, the new status of Newton as undisputed master of British mathematicians and the changing canons of mathematical publication introduced by the funding of scientific academies favoured Newton’s option for the printing of his mathematical manuscripts. However, when Newton decided to print his early analytical mathematical work, he tried to modify and present it to bring it in line with the values that characterise the methodological shift of the 1670s, values that he wished to promote among his followers.Author Keywords: Isaac Newton; Manuscripts; Scribal publication; Analysis; Synthesis. (shrink)

The historian of philosophy often encounters arguments that are enthymematic: they have conclusions that follow from their explicit premises only by the addition of "tacit" or "suppressed" premises. It is a standard practice of interpretation to supply these missing premises, even where the enthymeme is "real," that is, where there is no other context in which the philosopher in question asserts the missing premises. To do so is to follow a principle of charity: other things being equal, one interpretation is (...) better than another just to the extent that the one produces a better argument than the other. We show that this principle leads to paradoxical conclusions, including the following: there is no objectively correct interpretation of any real enthymeme found in the text of a major philosopher; an interpreter will not regard a real enthymeme of a major philosopher as adequately interpreted until he has found a way of reading it that makes it into a good argument; every classical philosopher is infallible and omniscient; major philosophers never disagree. These conclusions are preposterous, but there are indications that they are in fact being reached, as we show by means of a case study of recent scholarship on Plato's Third Man Argument. To avoid the overinterpretation and anachronism that result from the unrestrained use of the principle of charity, one must employ a counterbalancing principle of parsimony: to seek the simplest explanation for the text under discussion. We study the role of the principle of parsimony by means of a mathematical case study, involving the suppressed premises in Euclid's Elements. Here the principle of parsimony plays a larger role than it does in the interpretation of philosophical texts, leading to a sharper distinction between Euclid's geometry and Euclidean geometry than we find between Plato and Platonism. We conclude by comparing two models of interpretation, which we call prospective and retrospective. Although the prospective model of interpretation leads to Platonism rather than to Plato, we argue that it still has a place in Platonic scholarship. (shrink)

The history of mathematics can be read in two ways. On the one hand, unlike the history of physics, it does not proceed by conjectures and refutations. New theories rarely refute old theories, but give them new foundations, generalize them, and reinterpret them through new concepts. This reading is unifying, highlighting the unity of the history of mathematics from its origins, through the permanence of its truths. On the other hand, many contemporary historians of mathematics have insisted on the diversity (...) of methods, objects, and practices throughout history. They have shown that the objects of the past mathematics are not the same as those of today. That second reading shatters the illusion of unity. It disjoins what the first reading unifies.Rather than deciding between these two readings, this chapter examines what makes them both possible and legitimate, based on an analysis of the acts of reading that we have to perform when confronted with the pages of a mathematical text, its signs, diagrams and images, and the different language modalities it employs. Starting with a discussion of Husserl's theses in The Origin of Geometry on the role of writing in the history of mathematics, and using a number of historical examples, the chapter shows that any reading of a mathematical text involves at least three divisions, three delimitations made by the reader’s gaze: divisions between the true and the false, between language and image, and between what is inside mathematics and what is outside. For the same text, these divisions are traced by the gaze in different ways throughout history.The chapter develops an example in greater detail, concerning the status of the equation in Descartes’ Geometry. It discusses the contrasting recent readings of the equation by two historians of science, Henk Bos and Enrico Giusti. It shows the concepts of The Geometry and the ideas the philosopher develops on reading and the nature of the sign. It shows how these ideas can be used to reconstruct how Descartes might have read a mathematical text, and it explains how successive readings of The Geometry can at the same time insert it into the mathematics of later centuries. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Scepticism in the Sixth Century? Damascius’ Doubts and Solutions Concerning First PrinciplesSara RappeThe Doubts and Solutions Concerning First Principles, an aporetic work of the sixth century Neoplatonist Damascius, is distinguished above all by its dialectical subtlety. Although the Doubts and Solutions belongs to the commentary tradition on Plato’s Parmenides, its structure and method make it in many ways unique among such exegetical works. The treatise positions itself, at (...) least in part, as a response to Proclus’ metaphysical system. Thus the first principles alluded to in its title refer to a metaphysical structure consisting of five central elements, the Ineffable, the One, and the Noetic Triad, which Damascius both adumbrates in opposition to Proclus1 as well as subjects to his own, internal critique. In this article, I will be asking whether or not Damascius’ critique of Neoplatonic metaphysics is informed or inspired by ancient Scepticism. No doubt this question catches the reader off guard: if the last exponent of ancient Scepticism is Sextus Empiricus, how could this sixth-century Neoplatonist Scholarch, the last officially appointed Platonic Successor, revert to a tradition that seemingly disappears for well over three centuries?2 In what [End Page 337] follows, I suggest that a plausible historical context for such a reversion can be found in contemporaneous Neoplatonic allusions to Sceptical readings of Plato. I then analyze the Sceptical techniques deployed by Damascius in his critique of Neoplatonist metaphysics, focusing especially on his attacks on Neoplatonic theories of intellect and causation. I conclude by speculating on some ideological factors that shaped Damascius’ Sceptical affiliations, suggesting that here it is possible to see an initial volley in what was to become a centuries-long skirmish between reason and revelation.1. the neoplatonists read the sceptics?R. T. Wallis has suggested that Plotinus borrows from the Sceptics’ arsenal in voicing his own objections against Stoic or Stoicizing epistemology and theology. The seemingly radical stance that Plotinus takes against Stoic notions of divine rationality, virtue, and providence is anticipated by the Sceptics’ refusal to grant that an omniscient mind could stand in need of rational deliberation or exercise virtues that imply the presence of corresponding vices, etc. Not only is it the case that Plotinus makes a detailed use of specific Sceptical arguments in his own polemics against the Stoics, he seems to have been the first philosopher to have grasped the implications of Sceptical challenges to Stoic theology, and to have circumvented Sceptical dilemmas by means of his highly original formulations of nondiscursive thinking and negative theology.That this Platonist might side with Sceptics against dogmatic philosophy would hardly be surprising to the modern student of Plato, but it is also true that historically, this has been an uneasy alliance. In fact, it is just this question, the alliance between Platonists and the Sceptics, that gets rehearsed in the sixth century. Olympiodorus devotes Chapter 10 of his Prolegomena to the Study of Platonic Philosophy to the refutation of an ephectic, or nondogmatic Plato3: “Plato also superseded the philosophy of the New Academy since that school gave precedence to akatalepsia (nonapprehension), while Plato demonstrated that there do exist cognitions grounded in genuine knowledge. Nevertheless, some assert, assimilating Plato to the ephectics and to the Academicians, that he too maintained the doctrine of akatalepsia.” These remarks are curious, for we have no indication that a Sceptical reading of Plato was current or even conceivable at this time. In fact, as the context makes clear, the Sceptics are represented in this passage as a prior philosophical school; chapters 7 through 10 present a concise history of philosophy that is remarkably free from any notions of philosophical currency attributable to any of the views that fall [End Page 338] under its purvue: “there has been no shortage of philosophical haireseis [schools] both before and after Plato, yet he surpassed all of them by his teaching, his thought, and in every possible way”4 (Prolegomena, 7.1).Olympiodorus performs the exegete’s role throughout his refutation of a non-dogmatic Plato, turning first to the grammatical item, Plato’s use of what Olympiodorus terms the “hestitating” adverbs, such as “probably,” “perhaps,” and “as I imagine.” Nevertheless Olympiodorus manages... (shrink)

Reale points out that the good and the demiurgic intelligence are radically distinct, a conclusion denied by J. Seifert in the last paper of the book. Fourteen characteristics of the idea of the good are listed by T. A. Szlezák. It is obvious, he argues, that the theory of principles of Plato’s unwritten doctrines is not identical with what Republic 6 and 7 say about the good, but there is no real opposition. In the next paper, however, H. W. Ausland, (...) after a sober evaluation of the esoteric teachings, suspects that Reale and Szlezák succumb to a certain romanticism. According to L. Brisson, the good belongs to the sphere of being, even if it is not being ; it is not in every way beyond being. Plotinus was the first to place it above being. M. Erler shows what must be the attitude of the philosopher with regard to the information orally received by him. Although Plato did not want to define the good, he pointed out several of its characteristics. The good, M. Migliori notes, combines in itself beauty, proportion, and unity. The Philebus pursues the doctrine of the Republic and goes into the direction of the unwritten doctrines. The ultimate reality, however, is beyond our understanding. E. Cattanei attempts to explain how for Plato mathematics is a way of access to the knowledge of the good. R. Ferber, on the other hand, stresses that Plato nowhere directly says that the essence of the good is the one. Beauty, symmetry, and truth are not the good itself. According to C. Gill, Gadamer demystified the idea of the good, a view which shows some common ground with the position of analytical philosophy: the good is not a transcendent entity. M. Vegetti, however, upholds the transcendent character of the good as a cause of being. G. A. Press argues that over the past hundred years Plato was considered a dogmatic philosopher, and it was believed that the leading speaker in the dialogues states Plato’s own views, but the situation is far more complex. Plato wants us to acquire a vision rather than propositional knowledge. The idea of the good should not be accorded a special position. C. Rowe tries to demystify some current interpretations, whereas F. Ferrari insists on the causality of the good, and sees Plato’s teaching as the origin of the entire tradition of negative theology. Yet some texts give the impression that the good belongs to the ontological order. C. Porebski reminds us that problems of ethics are the starting point of what Plato says about the good in the Republic. According to F. Trabattoni, Aristotle’s description of Plato’s theory of principles is wrong and his philosophy differs fundamentally from Plato’s. The School of Tübingen is a prisoner of Aristotle’s interpretation. Plato’s reluctance to describe the good in detail may mark the beginning of a skeptic trend in the Academy. The Parmenides tells us, E. Berti says, that one has first to remove all hypotheses before a definition of the Good can be given. E. Moutsopoulos draws attention to some simplistic and wrong qualifications of Plato’s thought as related to idealism and mentions Kant and Hegel’s attitudes with regard to Platonism. J. F. Crosby points to a clear leaving behind of the Socratic “no one knowingly does wrong” in the Alcibiades 216b: as Plato explains in the Protagoras, one may live according to one’s lower appetite or follow the insight of the rational soul. G. Santos makes a strong stand in favor of the identity of goodness and reality. L. P. Gerson examines theories of Platos’s development: there is no evidence of a theory of the form of the good earlier than the Republic. Aristotle, J. Dudley writes, upheld an ethical ideal resembling that of Plato, substituting the contemplation of the unmoved mover for that of the idea of the good. In the last paper J. Seifert argues that Plato identified the demiurge and the good. (shrink)

How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence on the (...) development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as "infinitesimal algebra," and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved. Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field's power and progress. (shrink)

The volume starts with the paper of Lynn Maurice Ferguson Arnold, former Premier of South Australia and former Minister of Education of Australia, concerning the Exposition Internationale des Arts et Techniques dans la Vie Moderne (International Exposition of Art and Technology in Modern Life) that was held from 25 May to 25 November 1937 in Paris, France. The organization of the world exhibition had placed the Nazi German and the Soviet pavilions directly across from each other. Many papers are devoted (...) to the interpretation of this opposition. Arnold’s paper considers the differences in the two totalitarian states’ architectural and design discourse styles. Although each of them communicated a totalitarian language of purposes, permissions, and boundaries, they essentially differed in the styles of discourse represented by the architecture and design of their respective pavilions. They were opposites of each other and the liberal ideals they contested. The internationalist viewpoint reflecting the multi-ethnic mix of the USSR is contrasted to the ein Volk homogeneity represented in the German pavilion. The Soviet pavilion opted for a utopian future to be arrived at by “benign” leadership, whereas the German pavilion anchored itself in the myth of Teutonic history with the nostalgic pride protected by the swastika-bearing eagle. Jean-Yves Béziau is best known as a logician and founder of Universal Logic. However, in this paper, he poses a new philosophical question: why philosophical discourse does not use images? The paper begins with a general analysis of different types of philosophical discourses. Then, he focuses on why images have been and are still rejected in philosophical discourse. He explains various ways to use images in a fruitful way to develop philosophical thinking and discourse and illustrates his view by providing several examples. He concludes with a promising programmatic declaration about founding a new journal entitled World Journal of Pictorial Philosophy to stimulate the usage of images for developing philosophical thinking. Although complaints about the obscurity of many philosophers’ discourse are widespread, Tatiana Denisova, ex-professor of the Surgut University and a research associate of the University of the Aegean undertakes a positive attitude to this problem. She explains that the reasons for the obscurity of philosophical texts, the subsequent complexity in communicating philosophers’ meanings and their eventual incomplete understanding is not a sign of their inferiority. On the contrary, it is a sign of the fruitfulness of philosophical discourse, which can generate new meanings. Thus, the darkness of philosophical discourse is like the life-giving chaos, and the obscurity that it inevitably contains can be the keeper of implicit meanings and even their generator. Katarzyna Gan-Krzywoszyńska and Piotr Leśniewski, authors of a monograph on Polish philosopher Kazimierz Ajdukiewicz (1890 – 1963), make a comparative analysis of his educational style with that of Paulo Reglus Neves Freire (1921 – 1997). The authors highlight that these scholars have distinctly different backgrounds. The former was an educator, philosopher and a leading advocate of critical pedagogy connected with the dialogical Latin-American tradition. The latter was a philosopher and logician, a notable representative of the Lwów–Warsaw school of logic with significant contributions to semantics, model theory and the philosophy of science. Despite their apparent difference, the authors identify some striking similarities regarding their attitudes towards education; notably, their approaches are essentially dialogical. Gilah Yelin Hirsch is a multidisciplinary artist who works as a painter, writer, curator, educator, and filmmaker. In her experiential paper, he explains how and why she uses four channels of communication in her creative expression: writing, painting, filmmaking, and teaching, as dialogic inquiry. She considers that these channels compose a continuum of discourse styles, each reaching a different facet of kaleidoscopic consciousness. She claims that her choice of medium is prompted by a need to communicate her insights profoundly. Thus, for instance, she created painted allegories to express narratives based on dreams and visions emerging from her subconscious. Art and healing is another Tibetan-rooted type of discourse practised by the author. Creating and observing a healing image produces a positive psychophysiological change in both the artist and the viewer. The discourse here is tripartite between creator, image, and viewer. Filmmaking is another type of discourse she practices between the filmmaker/artist as shaman or healer and the viewer as respondent or participant. These films are meant to be experienced frame by frame, physically and emotionally in both the body and mind. Jocelyn Ireson-Paine is a cartoonist and programmer. His paper offers an original attempt to use category theory in cartooning. Category theory is a branch of mathematics that provides concepts and methods to describe general abstract structures in terms of a labelled directed graph called category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms or transformations or mappings). The author explores different ways to define “style” in art using category theory and a relational view of art. Moreover, he examines how “translation” (transformation) between styles could be defined and reveals the difficulties of how it could be implemented in a computer using special software. Jens Lemanski is a philosopher who has revived research in Arthur Schopenhauer’s legacy by exploring his work on mathematical evidence, logic diagrams, and problems of semantics. In this paper, he advances a new approach to eristic as an art of protecting oneself from the one who deliberately violates norms of discourse ethics to win an argument. The author attributes the origins of this view to Schopenhauer, suggesting a new reading of his work. According to the author, eristic is a prohibitive technique that takes effect when the norms of discourse ethics are transgressed and violated. Thus, eristic is viewed as a discipline of Enlightenment philosophy and a correlate of discourse ethics. Roshdi Rashed is an authority on the history of Arabic mathematical sciences. Proceeding from Gilles-Gaston Granger’s definition of mathematical style, he studies the question of whether a mathematical work can be characterized by a single style or by a multitude of styles. This question is explored within the style of a single work, namely Menelaus’s Sphaerica, and through the study of the development of a single problem over time, namely the isoperimetric problem. He indicates Menelaus’s divergence from the Euclidean style of (plane and stereometric) geometry caused by the drop of the Parallel Postulate, which associated it with hyperbolic geometry. Hence, the author concludes that Menelaus’s Sphaerica incorporates the well-known Euclidean style with a variation of the non-Euclidean one. On the other hand, examining the isoperimetric problem reveals another interesting historical picture. A succession of different styles (cosmological, geometric, infinitesimalistic, style of the calculus of variations, of synthetic geometry) can be observed due to the transformation of the research object over time. Boris Shalyutin, a Russian social philosopher, and Ombudsman for Human Rights in the Kurgan Region, explores the origins of discourse in combination with the birth of society and law. He claims that legal discourse marks the historical emergence of discourse in general. Homo Juridicus generated Homo Sapiens, which have created new spheres of discourse, notably moral discourse and further philosophical, political, and scientific discourse. Petros Stefaneas, a logician, computer scientist and novelist, suggests an alternative to the traditional narratology approach for studying (interactive) social media discourse. He claims that style describes how the parts of a narrative are blended into a whole. The author relies upon Goguen’s and Harrel’s concept of style as a choice of blending principles and transfers to the study of the social media narratives elements from the methodology of studying Web-based collaborative search for mathematical proofs like those implemented within the Polymath project. The last paper belongs to Ghil‘ad Zuckermann, a linguist, language revivalist, and proponent of a model of the emergence of Israeli Hebrew, according to which Hebrew and Yiddish were the primary sources of Modern Hebrew. This paper explores the fascinating and multifaceted Yiddish language and its survival in Israeli. Yiddish is characterized by a unique style that embeds psycho-ostensive expressions throughout its discourse. The author highlights the cross-fertilization between Hebrew and Yiddish as it manifests itself in any aspect of the Israeli language. He claims that Yiddish survives beneath Israeli phonetics, phonology, discourse, syntax, semantics, lexis, and even morphology, although traditional and institutional linguists have been most reluctant to admit it. (shrink)

It may be a myth that Plato wrote over the entrance to the Academy “Let no-one ignorant of geometry enter here.” But it is a well-chosen motto for his view in the Republic that mathematical training is especially productive of understanding in abstract realms, notably ethics. That view is sound and we should return to it. Ethical theory has been bedevilled by the idea that ethics is fundamentally about actions (right and wrong, rights, duties, virtues, dilemmas and so on). That (...) is an error like the one Plato mentions of thinking mathematics is about actions (of adding, constructing, extracting roots and so on). Mathematics is about eternal relations between universals, such as the ratio of the diagonal of a square to the side. Ethics too is about eternal verities, such as the equal worth of persons and just distributions. Mathematical and ethical verities do both constrain actions, such as the possibility of walking over the seven bridges of Königsberg once and once only or of justly discriminating between races. But they are not themselves about action. In principle, neither mathematical nor ethical verities are subject to historical forces or disagreement among tribes (though they can be better understood as time goes on). Plato is right: immersion in mathematics induces an understanding of the necessities underpinning reality, an understanding that is essential for distinguishing objective ethics from tribal custom. Equality, for example, is an abstract concept which is foundational for both mathematics and ethics. (shrink)

In An Aristotelian Realist Philosophy of Mathematics Franklin develops a tantalizing alternative to Platonist and nominalist approaches by arguing that at least some mathematical universals exist in the physical realm and are knowable through ordinary methods of access to physical reality. By offering a third option that lies between these extreme all-or-nothing approaches and by rejecting the ‘dichotomy of objects into abstract and concrete’, Franklin provides potential solutions to many of these traditional problems and opens up a whole new terrain (...) for debate in the philosophy of mathematics. (shrink)

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...) the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)