In order to address to the relation between philosophy and mathematics it is first necessary to distinguish the grand style and the little style. The little style painstakingly constructs mathematics as the object for philosophical scrutiny. It is called the little style for a precise reason, because it assigns mathematics to the subservient role of that which supports the definition and perpetuation of a philosophical specialisation. This specialisation is called the ‘philosophy of mathematics’, where (...) the ‘of’ is objective. The philosophy of mathematics can in turn be inscribed under the area of specialisation that supports the name ‘epistemology and history of science’, an area to which corresponds a specialised bureaucracy in the academic authorities and committees whose role it is to manage the personnel of researchers and teachers. But in philosophy, specialisation is invariably the means by which the little style insinuates itself. In Lacan terms, this occurs through the collapse of the discourse of the Master, which is rooted in the signifier of the same name, the S1 that gives rise to a signifying chain, onto the discourse of the University, that perpetual commentary which adequately represents the second moment of all speech, that is, the S2 which only exists by making the Master disappear under the commentary which exhausts it. The little style of the philosophy of mathematics, and of its epistemology, strives for such a disappearance of the ontological sovereignty of mathematics, its instituting aristocratism, its unrivalled mastery, by confining its dramatic and almost incomprehensible existence to a generally dusty compartment of academic specialisation. (shrink)

One of the greatest revolutions in mathematics occurred when Georg Cantor promulgated his theory of transfinite sets. This revolution is the subject of Joseph Dauben's important studythe most thorough yet writtenof the philosopher and mathematician who was once called a "corrupter of youth" for an innovation that is now a vital component of elementary school curricula.Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of (...) great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradoxes in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by recurring attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory. (shrink)

Contemporary analytic philosophy recognizes few principled constraints on its subject matter. When other disciplines also lay claim to a particular topic, however, important questions arise concerning the relation between these other disciplines and philosophy. A case in point is mathematics: traditional philosophy of mathematics defines a set of problems and certain general answers to those problems. However, mathematics is a subject matter that can be studied in many other ways: historically, sociologically, or even aesthetically, (...) for example. Given this, we may ask what implications discoveries in these disciplines have for the philosophy of mathematics. (shrink)

Like his Greek and Arab predecessors, Avicenna’s research in mathematics concerned the development of methods of exposition, proof procedures and analytical tools. But Avicenna belonged to a new era of mathematics, and the question that this paper seeks to examine is how Avicenna applied this new mathematical knowledge in his philosophy. A treatment of the subject of Avicenna and mathematics cannot, then, confine itself to generalities, important as these are, but must start from the degree of (...) knowledge he had of the various branches and the use he actually made of them. I shall restrict myself here to a number of examples which seem to me to possess evidential value and which all reveal a new conception of the connections between philosophy and mathematics. It will turn out that what Avicenna in fact did was to develop an analytical philosophy of mathematical concepts. (shrink)

One of the greatest revolutions in mathematics occurred when Georg Cantor (1845-1918) promulgated his theory of transfinite sets.

This paper consists of three main sections. In the first section, we consider how early attempts at understanding the relationship between mathematics and philosophy in Leibniz’s thought were often made within the framework of grand reconstructions guided by intellectual trends such as the search for “the ideal of system”. In the second section, we proceed to recount Leibniz’s first encounter with contemporary mathematics during his four years of study in Paris presenting some of the earliest mathematical successes (...) which he made there. In particular, we argue that recently published letters and papers reveal how his youthful mathematical reflexions were deeply intertwined with important philosophical insights that, in turn, acted as guiding ideas for his mathematical research. Finally, in the third section, we situate the central themes of the essays of the present volume within the new understanding of the interrelations between philosophy and mathematics in Leibniz’s thought briefly indicated in the opening section. (shrink)

Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics. However, many of these disciplines use the term ‘duality’ in vastly different senses, and while some of these senses are intimately connected to each other, others seem to be entirely … Continue reading Duality in Logic and Language →.

The purpose of this note is to examine the relationship between the practice of mathematics and the philosophy of mathematics, ontology in particular. One conclusion is that the enterprises are (or should be) closely related, with neither one dominating the other. One cannot 'read off' the correct way to do mathematics from the true ontology, for example, nor can one ‘read off’ the true ontology from mathematics as practiced.

In this paper a number of oppositions which have haunted mathematics and philosophy are described and analyzed. These include the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable.

The Compactness Theorem The compactness theorem is a fundamental theorem for the model theory of classical propositional and first-order logic. As well as having importance in several areas of mathematics, such as algebra and combinatorics, it also helps to pinpoint the strength of these logics, which are the standard ones used in mathematics and arguably … Continue reading Compactness →.

The heroic era in philosophy of mathematics being taken to extend from Frege to Quine, more recent developments of the past twenty or twenty-five years are widely felt to be disappointing by comparison, suggesting even that the original impulse may well have exhausted itself. A kind of hunkering down is nowhere so evident as in the eminently sober if not somber papers of Charles Parsons to which "philosophy of mathematics" as an ongoing discipline has been heavily (...) indebted during these lean years. Transitional in a personal as well as a historical sense, the Selected Essays consist in the main of pre-1978 material by Parsons, though there are various postscripts and, especially, the introduction to the volume that advance us to the cutting-edge of the present. (shrink)

ARE MATHEMATICAL IDEAS INVENTED OR DISCOVERED? This question has been repeatedly posed by philosophers through the ages, and will probably be with us forever. We shall not be concerned with the answer. What matters is that by asking the question, we acknowledge the fact that mathematics has been leading a double life.

In this paper a number of oppositions which have haunted mathematics and philosophy are described and analyzed. These include the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable.

This volume will be of particular interest to researchers working in the history, and in the philosophy, of logic and mathematics, and more generally, to ...

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

This encyclopedia article provides a procedural account of explication outlining each step that is part of the overall explicative effort (2). It is prefaced by a summary of the historical development of the method (1). The latter part of the article includes a rough structural theory of explication (3) and a detailed presentation of an examplary explication taken from the history of philosophy and the foundations of mathematics (4).

History and Philosophy of Modern Mathematics was first published in 1988. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions. The fourteen essays in this volume build on the pioneering effort of Garrett Birkhoff, professor of mathematics at Harvard University, who in 1974 organized a conference of mathematicians and historians of modern mathematics to examine how the two disciplines approach the (...) history of mathematics. In History and Philosophy of Modern Mathematics, William Aspray and Philip Kitcher bring together distinguished scholars from mathematics, history, and philosophy to assess the current state of the field. Their essays, which grow out of a 1985 conference at the University of Minnesota, develop the basic premise that mathematical thought needs to be studied from an interdisciplinary perspective. The opening essays study issues arising within logic and the foundations of mathematics, a traditional area of interest to historians and philosophers. The second section examines issues in the history of mathematics within the framework of established historical periods and questions. Next come case studies that illustrate the power of an interdisciplinary approach to the study of mathematics. The collection closes with a look at mathematics from a sociohistorical perspective, including the way institutions affect what constitutes mathematical knowledge. (shrink)

The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philosopher Edmund Husserl, namely as a science that deals with entities that can be exemplified in all spheres and domains of reality. It is a dynamic, processual, and (...) non-substantial ontology in which all entities can be treated as transformations, and in which objects are merely the sources and aims of these transformations. Thus, in a rather surprising way, when employed as a formal ontology, category theory can unite seemingly disparate disciplines in contemporary science and the humanities, such as physics, mathematics and philosophy, but also computer and complex systems science. (shrink)

The Polish philosophy of mathematics in the 19th century is not a well-researched topic. For this period, only five philosophers are usually mentioned, namely Jan Śniadecki, Józef Maria Hoene-Wroński, Henryk Struve, Samuel Dickstein, and Edward Stamm. This limited and incomplete perspective does not allow us to develop a well-balanced picture of the Polish philosophy of mathematics and gauge its influence on 19th- and 20th-century Polish philosophy in general. To somewhat complete our picture of the history (...) of the Polish philosophy of mathematics in those times, we here present the profiles of some lesser-known Polish Romantic philosophers of the 19th century, namely Karol Libelt, Bronisław Trentowski, and Józef Kremer. We discuss their contributions to the philosophy of mathematics and their metaphysical perspectives, and we also show how their metaphysical ideas have found some continuity in the studies of some Catholic philosophers. (shrink)

The main claim of this talk is that mathematical physics and philosophy of physics are not different. This claim, so formulated, is obviously false because it is overstated; however, since no non-tautological statement is likely to be completely true, it is a meaningful question whether the overstated claim expresses some truth. I hope it does, or so I’ll argue. The argument consists of two parts: First I’ll recall some characteristic features of von Neumann’s work on mathematical foundations of quantum (...) mechanics and will claim that von Neumann’s motivation and results are essentially philosophical in their nature; hence, to the extent von Neumann’s work exemplifies what is considered to be mathematical physics, mathematical physics appears as formally explicit philosophy of physics. The second argument is based on a rather trivial interpretation of what mathematical physics is. That interpretation implies that mathematical physics shares some key characteristic features with philosophy of physics which make the two almost indistinguishable. (shrink)

This book of sixteen original essays is the first to explore this range of new developments in the philosophy of mathematics, in a language accessible to ...

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)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.3 (2003) 426-427 [Access article in PDF] Timothy Smiley, editor. Mathematics and Necessity: Essays in the History of Philosophy. New York: Oxford University Press, 2000. Pp. ix + 166. Cloth, $35.00.Mathematics and Necessity contains essays by M. F. Burnyeat, Ian Hacking, and Jonathan Bennett based on lectures given to the British Academy in 1998. All concern the history of (...) the philosophical treatment of mathematics, necessity, or both, although there is no single thread running through all three essays.Burnyeat focuses on Plato's understanding of mathematics in the Republic, making use of the Timeaus in particular to defend his interpretation. Plato claimed that in the ideal society mathematics should be studied for a full ten years, which many have thought excessive. To explain it, Burnyeat argues that the role of mathematics is pervasive: ethics and politics are mathematical, because in Plato's view mathematics is a constitutive part of ethical understanding. On Burnyeat's interpretation, goodness is part of the objective world, and mathematics is a science of values. The link between mathematics and goodness in the objective world is forged by the concord, harmony, and unity of proportions, which express the goodness of the Divine Craftsman's beneficent design. Proportions are both intrinsically good and the proper study of mathematics.Burnyeat explains why Plato thinks abstract kinematics (the pure non-sensible counterpart to astronomy) and abstract harmonics lead to knowledge of the good. Abstract kinematics concerns the "harmony of the spheres" described in the Republic. Drawing heavily on the Timeaus, Burnyeat argues that the circular motions of the spheres are motions in the intelligence of the world soul and that there are similar motions in human souls. The world soul and human souls are spatial and have circular motions (invisibility and intangibility make them incorporeal). Souls are the most important subject matter of abstract harmonics—the study of good proportions. A soul that understands the various mathematical disciplines and their relations takes on the harmony and hence goodness of the world soul. Such a soul serves as a model for organizing the social world, a model which properly trained rulers can use to shape a community. Burnyeat's well-argued essay is compelling and exciting. [End Page 427]Ian Hacking's views provide a great contrast to Burnyeat's Plato. Hacking conducts a phenomenological inquiry into why some philosophers have become obsessed by mathematics as a source of philosophical inspiration. He holds that mathematics concerns a relatively minor part of human culture and that the influence of mathematics on philosophy is pernicious; he even describes it as an infection.Hacking examines the views of six philosophers: Plato, Leibniz, Mill, Descartes, Lakatos, and Wittgenstein. Hacking's assessment of the six philosophers is at times disputable, but it is also interesting and stimulating, and serves his phenomenological inquiry.Hacking traces philosophy's infection by mathematics to two factors. A mathematical proof can be perspicuous; that is, a proof can be "grasped," which requires seeing directly why something must be true and being able to recapitulate it. A mathematical proof can also anticipate facts about the world. He thinks that these two factors affect us in two ways: they give us the idea of a priori knowledge, and they dazzle us into unreasoned conviction in proofs. One might respond that, setting aside the feeling proofs give us, we are still justified in being impressed by them. Hacking argues that much of what impresses is based on a misunderstanding of their nature, which he attempts to correct.According to Hacking, a mathematical demonstration involves shifts of meaning that analytically bind the proof and the theorem it establishes. They thereby mutually support each other and provide what he calls self-authentication. The correctness of the proof is supported by the "analytified" truth of the theorem and the theorem is considered true because proven.This bootstrapping feature of mathematical proofs supports Hacking's deflationary account of necessity. Hacking suggests that the "must" of logical necessity rests upon our unwillingness to accept empirical counter-examples to the... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Symbolic Mathematics and the Intellect Militant: On Modern Philosophy’s Revolutionary SpiritCarl PageWhat makes modern philosophy different? My question presupposes the legitimacy of calling part of philosophy “modern.” That presupposition is in turn open to question as regards its meaning, its warrant, and the conditions of its applicability. 1 Importance notwithstanding, such further inquiries all start out from the phenomenon upon which everyone agrees: philosophy (...) running through Plato and Aristotle looks significantly different from philosophy running from Descartes to Kant.My concern in this essay is with the phenomenon of the difference itself, rather than with the second-order questions associated with how properly to assign it historical meaning. I take the difference between ancient and modern philosophy to be as significant as differences in philosophy’s history can be: modern philosophy rests on a new interpretation of the nature and fulfillment of human reason, and disputes about the nature of human reason are the ultimate battles of philosophy. But the general thesis is not my main point. 2 The focus of this essay falls on what may be called the integrity of the phenomenon, on the specific interpretation of human reason that lends modern philosophy its peculiar face. [End Page 233]Modern philosophy’s strikingly revolutionary spirit is my point of departure. When Descartes writes in the first of his Meditations that “it was necessary, once in the course of my life, to demolish everything completely and start again right from the foundations if I wanted to establish anything at all in the sciences that was stable and likely to last,” 3 he reveals the same enthusiasm for total reform later found in Kant: “This attempt to alter the procedure which has hitherto prevailed in metaphysics, by completely revolutionizing it in accordance with the example set by the geometers and physicists, forms indeed the main purpose of this critique of pure speculative reason.” 4 How exactly did philosophy become so convinced that its central tradition—a sprawling, disorganized, ugly city, as Descartes has it in the Discourse (I:116)—needed razing to the ground in the interest of some rational town-planning? Moreover, the calls for revolution have not abated, despite contemporary disillusionment with both Cartesian rationalism and Enlightenment philosophy in general; they have grown more shrill. The confidence with which rationalism, foundationalism, universalism, logocentrism, Platonism, and so on are currently set at naught for the sake of contingency, particularity, and difference reveals the same revolutionary and totalizing spirit that marks the earlier phase of philosophy’s modernity. Such enthusiasm is reason’s freedom taken to an extreme. 5 What inspires this march of the intellect militant? What, if anything, justifies its hubristic self-assertions in the domain of philosophy? These are the questions I address.Descartes is commonly identified as the father of modern philosophy. While the full story of modern philosophy’s parentage is more complicated than this, it is fair to say that in Descartes self-consciousness of a new mode of doing philosophy emerges with a focus and revolutionary sense of purpose that caught philosophical imagination in his own time and continues to do so in ours. 6 Motifs of modern philosophy may be found in many places—Machiavelli, Hobbes, Francis Bacon, Nicholas of Cusa, Giordano Bruno, Jakob Boehme—nonetheless, Descartes’s intensely single-minded, even jealous advocacy commends itself to all but the most stubborn antiquarian mentality as modernity’s almost perfect philosophical representative. [End Page 234]That Descartes stands on a remarkable philosophical cusp is apparent in the contrast between the title and the subject matter of his most influential philosophical work: Meditationes de Prima Philosophiae (1641). To that point prima philosophia or First Philosophy had been construed as the metaphysics. It was not concerned with the critical question of how metaphysical sciences are possible and was not directly related to any doctrine of the human soul—except perhaps on the one point of the divinity of nous, the human soul’s highest part. In Descartes’s Meditations, on the other hand, the landscape has altered. Doubt, certainty, knowledge, the ego cogitans and its stream of representations are the new subject matter. What is first in philosophy is no longer what... (shrink)

The book offers an interdisciplinary perspective on finance, with a special focus on stock markets. It presents new methodologies for analyzing stock markets’ behavior and discusses theories and methods of finance from different angles, such as the mathematical, physical and philosophical ones. The book, which aims at philosophers and economists alike, represents a rare yet important attempt to unify the externalist with the internalist conceptions of finance.

In this paper, we approach the problem of the relationship between Greek mathematics and Eleatic philosophy from a new perspective, which leads us to a reappraisal of Szabó’s hypothesis about the origin of mathematics out of Eleatic philosophy. We claim that Parmenidean philosophy, particularly its semantic core, has possibly been shaped by reflexion on the Pythagoreans’ mathematical practice, particularly in arithmetic. Furthermore, Pythagorean arithmetic originates not from another domain outside mathematics but from counting, i.e., (...) it has its roots in man’s practical activity. This interpretation restores the historically inverse relationship between mathematics and philosophy, refuting the attribution of mathematics’ origin to a field outside mathematics, for which Szabó’s hypothesis has been criticised. Moreover, Parmenidean theory of truth is viewed not as a defective predecessor of Aristotle’s classical theory of truth that needs to be remedied but as a semantic conception coordinated with the mathematics of Parmenides’ times. (shrink)

The ancient problem of the relationship of the continuous to the discrete, since its discovery by the Greeks, has posed a range of immensely fruitful challenges to both philosophical and mathematical thought, leading to a variety of mathematical and conceptual innovations whose positive development actively continues today. In this brief section introduction, I selectively outline some significant moments at which this problem has provided important historical occasions for concrete mathematical innovation as well as closely linked philosophical insights, before introducing the (...) particular contributions of the section authors to historical and contemporary research and practice into the nature and structure of the continuum, the construction of real numbers, the foundations of analysis, and the logical, set-theoretical, and computational underpinnings of the structure of continuously valued numbers in the most general sense, in both constructive and nonconstructivist contexts. (shrink)

We give an overview of Lakatos’ life, his philosophy of mathematics and science, as well as of this issue. Firstly, we briefly delineate Lakatos’ key contributions to philosophy: his anti-formalist philosophy of mathematics, and his methodology of scientific research programmes in the philosophy of science. Secondly, we outline the themes and structure of the masterclass Lakatos’ Undone Work – The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of (...) Science, which gave rise to this special issue. Lastly, we provide a summary of the contributions to this issue. (shrink)

The book offers an interdisciplinary perspective on finance, with a special focus on stock markets. It presents new methodologies for analyzing stock markets' behavior and discusses theories and methods of finance from different angles, such as the mathematical, physical and philosophical ones. The book, which aims at philosophers and economists alike, represents a rare yet important attempt to unify the externalist with the internalist conceptions of finance.

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)