The goal of this paper is to highlight the fact that the Philosophy for Children Approach can be used to stimulate pupil’s reflection within the framework of school subjects such as mathematics. First we situate P4C within the field of socio-constructivist epistemology. Then, P4C as adapted to mathematics is introduced. Finally, we describe an experiment linked to five types of exchanges, manifested between the beginning and the end of a school year while the pupils were learning to (...) philosophize about mathematics. In the discussion, emphasis is placed on the educational interventions of the teachers who facilitate the P4CM workshops. (shrink)

Kitcher's philosophical approach has moved from the reflection on the nature of mathematical knowledge to an explicit social concern about science, because he considers seriously the relevance of democratic values to scientific activity. Focal issues in this trajectory - from the internal perspective to the external - have been naturalism and scientific progress, which includes studies of the uses of scientific findings in the social milieu. Within this intellectual context, the chapter pays particular attention to his epistemological (...) and methodological evolution. The analysis of Philip Kitcher's contents on progress begins with mathematics, a conception that follows a naturalist perspective. Thereafter, the growth of science comes to the front line, an advancement that he views according to realism and cognitive naturalism. Later, the social concern about science receives a visible consideration, when his vision of scientific undertaking is characterized following modest realism and social naturalism. After these four steps (philosophical context, progress in mathematics, the growth in science, and the social concern about science), there is an analysis of his philosophical-methodological framework in retrospective. This is continued by the presentation of the origins of this book and the bibliography related to this thinker. (shrink)

In 1974, John Maynard Smith wrote in his little book Models in Ecology, A theory of ecology must make statements about ecosystems as a whole, as well as about particular species at particular times, and it must make statements that are true for many species and not just for one… For the discovery of general ideas in ecology, therefore, different kinds of mathematical description, which may be called models, are called for. Whereas a good simulation should include as (...) much detail as possible, a good model should include as little as possible. (1974. (shrink)

In the paper the views and tendencies in the philosophical reflection on mathematics in Poland between the wars are analyzed. Views of most outstanding representatives of Lvov–Warsaw Philosophical School and of Polish Mathematical School are presented. Their influence on logical and mathematical researches is considered.

This edited volume explores the previously underacknowledged 'pre-history' of mathematical structuralism, showing that structuralism has deep roots in the history of modern mathematics. The contributors explore this history along two distinct but interconnected dimensions. First, they reconsider the methodological contributions of major figures in the history of mathematics. Second, they re-examine a range of philosophical reflections from mathematically-inclinded philosophers like Russell, Carnap, and Quine, whose work led to profound conclusions about logical, epistemological, and metaphysic.

This paper discusses some major similarities and differences between community of philosophical inquiry and community of mathematical inquiry , and offers a few examples of the implementation of CMI in the context of a school mathematics classroom. Three modes of CMI are suggested. The first mode facilitates inquiry into mathematical problems - that is, it provides a medium for “doing and talking mathematics.” In this case, CMI is primarily an avenue for problem solving—defining problems, interpreting them, working (...) with different methods to solve them, reflecting on suggested alternative methods, verifying solutions, and drawing conclusions. The second mode leads us to “talk about mathematics” through collaborative inquiry into mathematical concepts such as axioms, theorems, algorithms, infinity, and the posing of philosophical questions that concern mathematics as a system--particular structures and rules and their relation to human experience. The third mode makes use of CMI for meta-inquiry into our collective experience in “doing and talking mathematics” and “talking about mathematics,” and may be characterized as “talking about doing mathematics.”. (shrink)

This chapter contains sections titled: Introduction Platonism in Mathematics Nominalism in Mathematics Conclusion References.

In the scholarship on Wittgenstein’s later philosophy of mathematics, the dominant interpretation is a theoretical one that ascribes to Wittgenstein some type of ‘ism’ such as radical verificationism or anti-realism. Essentially, he is supposed to provide a positive account of our mathematical practice based on some basic assertions. However, I claim that he should not be read in terms of any ‘ism’ but instead should be read as examining philosophical pictures in the sense of unclear conceptions. The contrast (...) here is that basic assertions that frame philosophical ‘isms’ are propositional such that they are subject to normal argumentative evaluation, while pictures in Wittgenstein’s sense are non-propositional—they lack a clear truth condition. They, therefore, need clarification rather than argumentation. In this paper, I provide a detailed analysis of Wittgenstein’s treatment of philosophical pictures with special focus on his argument on contradiction. I begin by explaining the problem with this trend of theoretical interpretation, taking Steve Gerrard’s otherwise excellent interpretation as a representative example and pointing out why it is problematic. Next, I will argue that those problems do not arise if we take Wittgenstein’s task as the clarification of philosophical pictures. I do this, first, by explaining Wittgenstein’s method using his argument concerning the Augustinian Picture in Philosophical Investigations and then pointing out that the same method can be identified in the crucial arguments in his philosophy of mathematics. Finally, in order to connect my interpretation with the current scholarship, I will explain the relation of my interpretation with those of New Wittgensteinian scholars. (shrink)

A personal account is presented for the present status of mathematical chemistry, with emphasis on non-numerical applications. These use mainly graph-theoretical concepts. Most computational chemical applications involve quantum chemistry and are therefore largely reducible to physics, while discrete mathematical applications often do not. A survey is provided for opinions and definitions of mathematical chemistry, and then for journals, books and book series, as well as symposia of mathematical chemistry.

In the debate over whether mathematical facts, properties, or entities explain physical events (in what philosophers call “extra-mathematical” explanations), Aidan Lyon’s (2012) affirmative answer stands out for its employment of the program explanation (PE) methodology of Frank Jackson and Philip Pettit (1990). Juha Saatsi (2012; 2016) objects, however, that Lyon’s examples from the indispensabilist literature are (i) unsuitable for PE, (ii) nominalizable into non-mathematical terms, and (iii) mysterious about the explanatory relation alleged to obtain between the PE’s mathematical explanantia (...) and physical explananda. In this paper, I propose a counterexample to Saatsi’s objections. My counterexample is Frank Jackson’s (1998a) program explanation for color experience, which I argue needs recasting as an extra-mathematical PE due to its implicit reliance on reflectance, a property that suffers conceptual regress unless redefined with Fourier harmonics. Pace Saatsi, I argue that this recast example is an authoritative PE, non-nominalizable, and minimally esoteric. Important for the indispensability debate at large, moreover, is that my counterexample reifies Fourier harmonics without the Enhanced Indispensability Argument (an argument to which Lyon applies PE as a premise). Indispensabilists have long overlooked the conditionalization of a limited mathematical realism on property realism, and my counterexample to Saatsi exploits this conditionalization. (shrink)

Common opinion ascribes to Immanuel Kant the view that psychology cannot become a science properly so called, because it cannot be mathematized. It is equally common to claim that this reflects the state of the art of his times; that the quantification of the mind was not achieved during the eighteenth century, while it was so during the nineteenth century; or that Kant's so-called “impossibility claim” was refuted by nineteenth-century developments, which in turn opened one path for psychology to become (...) properly scientific. These opinions are often connected, but they are misguided nevertheless. -/- In Part I, I show how the issue of a quantification of the mind was discussed before Kant, and I analyze the philosophical considerations both of pessimistic and optimistic authors. This debate reveals a certain progress, although it remains ultimately undecided. In Part II, I present actual examples of measuring the mind in the eighteenth century and analyze their presuppositions. Although these examples are limited in certain ways, the common view that there was no such measurement is wrong. In Part III, I show how Kant's notorious “ impossibility claim” has to be viewed against its historical background. He not only accepts actual examples of a quantitative treatment of the mind, but also takes steps toward an explanation of their possibility. Thus, he does not advance the claim that the mind as such cannot be mathematized. His claim is directed against certain philosophical assumptions about the mind, assumptions shared by a then-dominating, strongly introspectionist conception of psychology. This conception did and could not provide an explanation of the possibility of quantifying the mind. In concluding, I reflect on how this case study helps to improve the dispute over when and why psychology became a science. (shrink)

In this paper the reader is asked to engage in some simple problem-solving in classical pure number theory and to then describe, on the basis of a series of questions, what it is like to solve the problems. In the recent philosophy of mind this “what is it like” question is one way of signaling a turn to phenomenological description. The description of what it is like to solve the problems in this paper, it is argued, leads to several morals (...) about the epistemology and ontology of classical pure mathematical practice. Instead of simply making philosophical judgments about the subject matter in advance, the exercise asks the reader to briefly engage in a mathematical practice and to then reflect on the practice. (shrink)

This paper studies Paul Cohen’s philosophy of mathematics and mathematical practice as expressed in his writing on set-theoretic consistency proofs using his method of forcing. Since Cohen did not consider himself a philosopher and was somewhat reluctant about philosophy, the analysis uses semiotic and literary textual methodologies rather than mainstream philosophical ones. Specifically, I follow some ideas of Lévi-Strauss’s structural semiotics and some literary narratological methodologies. I show how Cohen’s reflections and rhetoric attempt to bridge what he (...) experiences as an uncomfortable tension between reality and the formal by means of his notion of intuition. (shrink)

Zalamea’s book is as original as it is belated. It is indeed surprising, if we give it a moment’s thought, just how greatly behind schedule philosophical reflection on contemporary mathematics lags, especially considering the momentous changes that took place in the second half of the twentieth century. Zalamea compares this situation with that of the philosophy of physics: he mentions D’Espagnat’s work on quantum mechanics, but we could add several others who, in the last few decades, have (...) elaborated an extremely timely philosophy of contemporary physics (see for example Bitbol 2000; Bitbol et al. 2009). As was the case in biology, philosophy – since Kant’s crucial observations in the Critique of Judgment, at least – has often “run ahead” of life sciences, exploring and opening up a space for reflections that are not derived from or integrated with its contemporary scientific practice. Some of these reflections are still very much auspicious today. And indeed, some philosophers today are saying something truly new about biology... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Method and Mathematics:Peter Ramus's Histories of the SciencesRobert GouldingPeter Ramus (1515–72) was, at first sight, the least likely person to write an influential history of mathematics. For one thing, he was clearly no great mathematician himself. His sympathetic biographer Nicholas Nancel related that Ramus would spend the mornings being coached in mathematics by a team of experts he had assembled, and in the afternoon would lecture (...) on the very same subjects.1 But there can be no doubt that Ramus held mathematics in particular esteem and even played a crucial role in linking philosophical discourse to mathematics and in promoting the use of quadrivial reasoning in the study of the natural world.2The origins for his enthusiasm for mathematics are to be found in his account of the nature of the arts and critique of the curriculum of the universities. From his earliest career, Ramus was a logician and remained one in all his works, whether writing on mathematics or Virgil, and he conceived [End Page 63] of all the arts, especially mathematics, as unchanging structures of necessarily true propositions—not prima facie a position which lends itself to notions of historical change and development.3 His career as a historian of mathematics, I will argue, was directed by problems that arose in his theoretical understanding of mathematics as he began to immerse himself in the sciences of the ancient world.Ramus and the Reform of DialecticRamus set out his fundamental logical positions in his very first printed work, the Dialecticae institutiones (Education in Dialectic) of 1543, a contribution to the ongoing humanist attack on scholastic logic, in which Ramus rehearsed many of the commonplace criticisms of the university dialectic. Humanists complained that logic no longer concerned itself with real human reasoning; instead, it had become a discipline studied for its own sake, using its own incomprehensible jargon, and was of no practical interest. The new humanist dialectics of Lorenzo Valla and Rudolph Agricola attempted to teach the kind of practical reasoning useful for composing a speech or letter, and they borrowed extensively from rhetorical works of Cicero and Quintilian to develop a highly rhetoricized logic. Questions about the formal validity of arguments were of little interest; what mattered was whether the arguments were persuasive.4In his Dialecticae institutiones, Ramus took this humanist reformulation of dialectic a step further. Dialectic—or, in fact, any art or science—consisted of nature, doctrine, and exercise. The natural workings of the mind formed the most significant element. Doctrine was nothing more than a record of natural reasoning; in importance it paled next to nature and practice.5 The logic of the universities bore no resemblance to the true, natural dialectic, as he argued at length in the companion volume to the Institutiones, the innocuously titled but exuberantly offensive Aristotelicae animadversiones [End Page 64] (Observations on Aristotle).6 The question, then, remained: how can we gain access to this "natural reasoning"?Ramus's answer was surprising. He asks us to find a group of men—not scholars, but completely uneducated vineyard workers. Question them about the coming year: the fertility of the soil, the quality and quantity of the crop. "And then," he writes "from their minds as from a mirror an image of nature will be reflected."7 In the reasoned replies of these uneducated men, says Ramus, we discover every part of logic needed for any purpose, whether everyday discourse or composition of poetry: the invention of arguments, the assessment of their truth and their proper and orderly presentation. Other humanists had praised man's natural logical faculties, elevating them over artificial technique, but Ramus was the first to look beyond the walls of the university and the writings of the ancients to find natural dialectic at work.If even the uneducated have some possession of the arts, then the arts taught at the university should do no more than clear away the misleading junk in the mind, and allow its natural clarity to shine through.8 Logic should be easy to learn—not because this is a pedagogic ideal, but from the very nature of the art. And if logic as it is... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Husserl and Mathematics by Mirja HartimoAndrea StaitiMirja Hartimo. Husserl and Mathematics. Cambridge: Cambridge University Press, 2021. Pp. 214. Hardback, $99.99.Mirja Hartimo has written the first book-length study of Husserl's evolving views on mathematics that takes his intellectual context into full consideration. Most importantly, Hartimo's historically informed approach to the topic benefits from her extensive knowledge of Husserl's library. Throughout the book, she provides references to (...) texts and articles that Husserl read, marked, and annotated, although regrettably, very few of these sources are quoted explicitly in his published works. The overall picture that emerges from Hartimo's book shows that Husserl continued to read and engage with mathematics throughout his career, and, even though extensive discussions of mathematical issues are not frequent in his later works, some of the ideas he develops, for instance, in The Crisis of the European Sciences and Transcendental Phenomenology, can be read against the backdrop of the developments of mathematics in his time.Chapter 1 sets the method for the rest of the book. Hartimo emphasizes the importance of a methodological device that takes center stage in Husserl's work from the 1920s onward: Besinnung. Hartimo claims that Besinnung is a method to reactivate the "goals and purposes" (21) that originally motivated scientists engaging in a particular line of research. According to Husserl, as specialized research progresses and becomes standard practice, such original goals and purposes might be blurred by other concerns. When applied to mathematics, Besinnung thus amounts to a reflection "on what mathematicians should do, on what the genuine goal of their activities should be" (21–22). Hartimo argues that this kind of approach to mathematics comes close to Penelope Maddy's naturalism, which takes its departure from what mathematicians are actually doing and then proceeds to inquire into the ontological status of mathematical objects and theories. Hartimo identifies particularly in David Hilbert, a colleague of Husserl's at Göttingen, a vivid embodiment of what Husserl considered to be exemplary work oriented toward the ideal goal of mathematics.In chapters 2 and 3, Hartimo expands upon Husserl's conception of what the true goal of mathematics should be. She does so, first, by addressing the classical topic of Husserl's psychologism in his first (and only) work entirely devoted to mathematics, The Philosophy of Arithmetic, and Frege's well-known critical review of it. In Hartimo's rendition, Husserl criticizes Frege for his attempt to provide formal definitions of basic notions, such as equality. For Husserl, such basic notions are intuitively available and require no definition—rather, they should be put to work to clarify vague notions. Frege, in return, attacks what he takes to be Husserl's wishy-washy psychological account of numbers, which revolves around the concrete act of collecting items and abstracting from their specific contents. Ultimately, Hartimo argues, Husserl takes Frege's criticism to heart and in his later Prolegomena no longer pursues a psychological foundation of number, but rather emphasizes the distinction between the subjective and the objective sides of mathematics, with philosophers studying the former and mathematicians focusing on the latter. While Hartimo's verdict on the philosophical yield of the debate is in line with the standard interpretation of Frege's review as prompting Husserl's antipsychologistic turn, she makes a remark that might be worthy of further discussion: "While for Husserl the debate was about logicism, Frege shifts the debate to be about psychologism and here commentators have followed Frege's example" (50). It would be interesting to imagine what a Fregean reply to Husserl's view about the uselessness of formal definitions for basic notions like equality might look like. In any case, Hartimo's reconstruction makes it clear that, however motivated, Husserl's rejection of psychologism does not amount to an acceptance of logicism.In chapter 3, Hartimo puts forward two of her central claims: (1) Husserl was a structuralist about mathematics; and (2) for Husserl, the goal of modern mathematics is encapsulated in the concept of definiteness. According to (1), mathematics is about the formal structures of systems, which can be isolated and compared. Definiteness, in turn, is the property of systems... (shrink)

It is often said to-day that mathematics is nothing but an extension or development of logic; indeed, the identity of logic and pure mathematics is alleged so confidently by persons whose mathematical attainments entitle them to consideration when they talk about the subject-matter of mathematics, as to be in danger of being ranked with the truths that an educated man should accept on the authority of the specialist. Yet a little reflection might at least make (...) one hesitate. For whatever else may be said about logic, it is generally allowed to study thinking. Some would say merely that it studies inference; but inference is thinking. Now the mathematician thinks and infers, but he does not study the activity of thinking and inferring; and a study of that activity would never make the discoveries credited to the mathematicians. (shrink)

Few philosophers that have been studied as much as Ibn Sīnā have been as much misunderstood. His extraordinary ability to reflect upon and write in a variety of styles about seemingly every topic in every domain has steered his thought from philosophy and theology to mysticism and esoterism. Instead of helping us to learn and understand better Ibn Sīnā than he has previously been understood, the recent surge of Avicennan studies only adds more confusion to the already complex social (...) context which he was living in. (shrink)

A reproach has been done many times to post-modernism: its picking up mathematical notions or results, mostly by misrepresenting their real content, in order to strike the readers and obtaining their assent only by impressing them . In this paper I intend to point out that although Alain Badiou’s approach to philosophy starts with taking distance both from analytic philosophy and from French post-modernism, the categories that he uses for labelling logicism, formalism and intuitionism do not reflect the real content (...) of the foundational schools. Hence, a re-thinking from him would be required about them, otherwise he would risk the same reproaches as post-modernists. (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)

Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.

In his published work and even more in conversations, Tarski emphasized what he thought were important philosophical aspects of his work. The English translation of his more philosophical papers [56m] was dedicated to his teacher Tadeusz Kotarbinski, and in informal discussions of philosophy he often referred to the influence of Kotarbinski. Also, the influence of Leiniewski, his dissertation adviser, is evident in his early papers. Moreover, some of his important papers of the 1930s were initially given to (...) class='Hi'>philosophical audiences. For example, the famous monograph on the concepotf truth ([33”], [35b]) was first given as twol ectures to the Logic Section of the Philosophical Society in Warsaw in 1930. Second, his paper [33], which introduced the concepts of co-consistency and co-completeness as well as the rule of infinite induction: was first given at the Second Conference of the Polish Philosophical Society in Warsaw in 1927. Also [35c] was based upon an address given in 1934 to the conference for the Unity of Science in Prague; C361 and [36a] summarize an address given at the International Congress of Scientific Philosophy in Paris in 1935. The article [44a] was published in a philosophical journal and widely reprinted in philosophical texts. This list is of course not exhaustiveb ut only representative of Tarski’s philosophical interactions as reflected in lectures given to philosophical audiences, which were later embodied in substantial papers. After 1945 almost all of Tarski’s publications and presentations are mathematical in character with one or two minor exceptions. This division, occurring about 1945, does not, however, indicate al oss of interest in philosophical questionbsu t is a result of Tarski’s moving to the Department of Mathematics at Berkeley. There he assumed an important role in the development of logic within mathematics in the United States. (shrink)

This unique volume introduces and discusses the methods of validating computer simulations in scientific research. The core concepts, strategies, and techniques of validation are explained by an international team of pre-eminent authorities, drawing on expertise from various fields ranging from engineering and the physical sciences to the social sciences and history. The work also offers new and original philosophical perspectives on the validation of simulations. Topics and features: introduces the fundamental concepts and principles related to the validation of computer (...) simulations, and examines philosophical frameworks for thinking about validation; provides an overview of the various strategies and techniques available for validating simulations, as well as the preparatory steps that have to be taken prior to validation; describes commonly used reference points and mathematical frameworks applicable to simulation validation; reviews the legal prescriptions, and the administrative and procedural activities related to simulation validation; presents examples of best practice that demonstrate how methods of validation are applied in various disciplines and with different types of simulation models; covers important practical challenges faced by simulation scientists when applying validation methods and techniques; offers a selection of general philosophical reflections that explore the significance of validation from a broader perspective. This truly interdisciplinary handbook will appeal to a broad audience, from professional scientists spanning all natural and social sciences, to young scholars new to research with computer simulations. Philosophers of science, and methodologists seeking to increase their understanding of simulation validation, will also find much to benefit from in the text. (shrink)

The idea that mathematics is reducible to logic has a long history, but it was Frege who gave logicism an articulation and defense that transformed it into a distinctive philosophical thesis with a profound influence on the development of philosophy in the twentieth century. This volume of classic, revised and newly written essays by William Demopoulos examines logicism's principal legacy for philosophy: its elaboration of notions of analysis and reconstruction. The essays reflect on the deployment of these ideas (...) by the principal figures in the history of the subject - Frege, Russell, Ramsey and Carnap - and in doing so illuminate current concerns about the nature of mathematical and theoretical knowledge. Issues addressed include the nature of arithmetical knowledge in the light of Frege's theorem; the status of realism about the theoretical entities of physics; and the proper interpretation of empirical theories that postulate abstract structural constraints. (shrink)

To what extent are the subjects of our thoughts and talk real? This is the question of realism. In this book, Justin Clarke-Doane explores arguments for and against moral realism and mathematical realism, how they interact, and what they can tell us about areas of philosophical interest more generally. He argues that, contrary to widespread belief, our mathematical beliefs have no better claim to being self-evident or provable than our moral beliefs. Nor do our mathematical beliefs have better (...) claim to being empirically justified than our moral beliefs. It is also incorrect that reflection on the "genealogy" of our moral beliefs establishes a lack of parity between the cases. In general, if one is a moral antirealist on the basis of epistemological considerations, then one ought to be a mathematical antirealist as well. And, yet, Clarke-Doane shows that moral realism and mathematical realism do not stand or fall together -- and for a surprising reason. Moral questions, insofar as they are practical, are objective in a sense that mathematical questions are not. Moreover, the sense in which they are objective can be explained only by assuming practical anti-realism. One upshot of the discussion is that the concepts of realism and objectivity, which are widely identified, are actually in tension. Another is that the objective questions in the neighborhood of questions of logic, modality, grounding, and nature are practical questions too. Practical philosophy should, therefore, take center stage. (shrink)

This book presents a philosophical interpretation to numerical cognition based on dual process theories and heuristics. It shows how investigations in cognitive science can shed light on issues traditionally raised by philosophers of mathematics. The analysis will also help readers to better understand the relationship between current neuroscientific research and the philosophical reflection on mathematics. The author seeks to explain the acquisition of mathematical concepts. To accomplish this, he needs to answer two questions. How can (...) the concepts of approximate numerosity become an object of thought that is so accessible to our consciousness? How are these concepts refined and specified in such a way as to become numbers? Unfortunately, there is currently no model that can truly demonstrate the role of language in the development of numerical skills starting from approximate pre-verbal skills. However, the author details a solution to this problem: dual process theories. It is an approach widely used by theorists focusing on reasoning, decision making, social cognition, and consciousness. Here, he applies this approach to the studies on mathematical knowledge. He details the results brought about by psychological and neuroscientific studies conducted on numerical cognition by key neuroscientists. In the process, he develops the foundations of a new, potential philosophical explanation on mathematical knowledge. (shrink)

This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea (...) that mathematics is logic (logicism), the view that the essence of mathematics is the rule-governed manipulation of characters (formalism), and a revisionist philosophy that focuses on the mental activity of mathematics (intuitionism). Finally, Part IV brings the reader up-to-date with a look at contemporary developments within the discipline. This sweeping introductory guide to the philosophy of mathematics makes these fascinating concepts accessible to those with little background in either mathematics or philosophy. (shrink)

This article was written jointly by a philosopher and a mathematician. It has two aims: to acquaint mathematicians with some of the philosophical questions at the foundations of their subject and to familiarize philosophers with some of the answers to these questions which have recently been obtained by mathematicians. In particular, we argue that, if these recent findings are borne in mind, four different basic philosophical positions, logicism, formalism, platonism and intuitionism, if stated with some moderation, are in (...) fact reconcilable, although with some reservations in the case of logicism, provided one adopts a nominalistic interpretation of Plato's ideal objects. This eclectic view has been asserted by Lambek and Scott (LS 1986) on fairly technical grounds, but the present argument is meant to be accessible to a wider audience and to provide some new insights. (shrink)

After delimiting the topic by reflecting on the heuristic function of the concept of “theory” in “Delimiting the Topic” section, the paper considers the literary aspects of karman-theory in the Mahābhārata in “Literary Characteristics” section. “Axioms, Theorems, Domains” section then lists the elements or axioms that fall under the umbrella term “karman-theory.” Next, dealing with contexts and collocations, “Contexts, Collocations” section combines the consideration of literary and theoretical aspects of the matter. “Historical Perspective” section then argues for the inclusion of (...) a historical perspective in the study of karman-theory. The “Conclusion”, section, formulates provisional results and poses further questions. (shrink)

It is well known that the history of Calculus in the nineteenth century coincides with the process of substitution of infinitesimals by the notion of limit. But it is adviseable to keep in mind the ontological implications of that process.We can find a background for this ontological approach in Abraham Robinson’s Non-Standard AnaIysis and “The Metaphysics of the Calculus”. Indeed, by the choice of the word “metaphysics” and by the several recalls of the ontological nature of the arguments, Robinson claims (...) for a filiation which is at the same time fruitful in the mathematical register and necessary for a true philosophical reflection about the infinite. (shrink)

The phrase ‘mathematical foundation’ has shifted in meaning since the end of the nineteenth century. It used to mean a consistent general theory in mathematics, based on basic principles and ideas to which the rest of mathematics could be reduced. There was supposed to be only one foundational theory and it was to carry the philosophical weight of giving the ultimate ontology and truth of mathematics. Under this conception of ‘foundation’ pluralism in foundations of mathematics (...) is a contradiction.More recently, the phrase has come to mean a perspective from which we can see, or in which we can interpret, much of mathematics; it has lost the realist-type metaphysical, essentialist importance. The latter has been replaced with an emphasis on epistemology. The more recent use of the phrase shows a lack of concern for absolute ontology, truth, uniqueness and sometimes even consistency. It is only under the more modern conception of ‘foundation’ that pluralism in mathematical foundations is conceptually possible.Several problems beset the pluralist in mathematical foundations. The problems include, at least: paradox, rampant relativism, loss of meaning, insurmountable complexity and a rising suspicion that we can say anything meaningful about truth and objectivity in mathematics. Many of these are related to each other, and many can be overcome, explained, accounted for and dissolved by concentrating on crosschecking, fixtures and rigour of proof. Moreover, apart from being a defensible position, there are a lot of advantages to pluralism in foundations. These include: a sensitivity to the practice of mathematics, a more faithful account of the objectivity of mathematics and a deeper understanding of mathematics.The claim I defend in the paper is that we stand to learn more, not less, by adopting a pluralist attitude. I defend the claim by looking at the examples of set theory and homotopy type theory, as alternative viewpoints from which we can learn about mathematics. As the claim is defended, it will become apparent that ‘pluralism in mathematical foundations’ is neither an oxymoron, nor a contradiction, at least not in any threatening sense. On the contrary, it is the tension between different foundations that spurs new developments in mathematics. The tension might be called ‘a fruitful meta-contradiction’.I take my prompts from Kauffman’s idea of eigenform, Hersh’s idea of thinking of mathematical theories as models and from my own philosophical position: pluralism in mathematics. I also take some hints from the literature on philosophy of chemistry, especially the pluralism of Chang and Schummer. (shrink)

It has been argued, and not without emotional detachment, that the Holocaust is unlike other events in world and Jewish history. Those who offer such arguments also claim that comparisons between events of ethnic cleansing, mass murder and other sorts of criminal behavior are not meant to purvey a kind of moral one-upmanship. The suffering and harm in one instance is as morally repugnant as those in any other instance, whether it is a Jewish child gassed and cremated by the (...) Nazis, a black child lynched in the southern United States, an Armenian child deliberately poisoned in a Turkish hospital, a Ukrainian child starved to death in a Stalinist blockade, a Native American child shot to death by Conquistadors, a Rwandan child hacked to death, or a Bosnian child blown to pieces by a land mine or sniper. (shrink)

Introduction -- The mathematical roots of the concept of analogy -- Aristotle : the uses of analogy -- Aristotle : analogy and language -- Thomas Aquinas -- Immanuel Kant -- Karl Barth -- Final reflections.

When mathematicians think of the philosophy of mathematics, they probably think of endless debates about what numbers are and whether they exist. Since plenty of mathematical progress continues to be made without taking a stance on either of these questions, mathematicians feel confident they can work without much regard for philosophical reflections. In his sharp–toned, sprawling book, David Corfield acknowledges the irrelevance of much contemporary philosophy of mathematics to current mathematical practice, and proposes reforming the subject (...) accordingly. (shrink)

Stephen Toulmin once observed that ”it has never been customary for philosophers to pay much attention to the rhetoric of mathematical debate’ [Toulmin et al., 1979, An Introduction to Reasoning, Macmillan, London, p. 89]. Might the application of Toulmin’s layout of arguments to mathematics remedy this oversight? Toulmin’s critics fault the layout as requiring so much abstraction as to permit incompatible reconstructions. Mathematical proofs may indeed be represented by fundamentally distinct layouts. However, cases of genuine conflict characteristically reflect an (...) underlying disagreement about the nature of the proof in question. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Kant on the Method of MathematicsEmily Carson1. INTRODUCTIONThis paper will touch on three very general but closely related questions about Kant’s philosophy. First, on the role of mathematics as a paradigm of knowledge in the development of Kant’s Critical philosophy; second, on the nature of Kant’s opposition to his Leibnizean predecessors and its role in the development of the Critical philosophy; and finally, on the specific role (...) of intuition in Kant’s philosophy of mathematics. One of the points that I want to make is that recognizing the importance of these first two issues is essential to giving an adequate account of the third. That is, only by appreciating how Kant uses the example of mathematical knowledge, and in particular the objections he had to previous accounts of such knowledge, can we understand the philosophical role that he ascribes to intuition in his own account of mathematics.I don’t propose to explain the details of Kant’s philosophy of mathematics, but rather to compare his views on mathematics in the pre-Critical Prize Essay of 1764 to those of the Critique of Pure Reason, with the aim of showing how Kant’s doctrine of construction in pure intuition arises out of a combination of two factors: first, his attempt to ground the distinction between the respective methods of mathematics and philosophy, and second, his concern to defend the reality of mathematical knowledge against the views of those he called ‘the metaphysicians.’ The first factor demands an account of the certainty of mathematics, the second requires an account of its content. I shall argue that there is a possible conflict between Kant’s accounts of these features in the Prize Essay, a possibility which is finally ruled out only by means of the Critical doctrine of pure intuition. So I hope to show that there is a significant philosophical role for intuition in Kant’s account of mathematical knowledge (over and above the more logical role attributed to it on readings [End Page 629] such as those of Friedman, Beth, and Hintikka1) in reconciling the content of mathematics and its certainty.2. THE MATHEMATICIANS VS. THE METAPHYSICIANSThroughout his early writings, Kant appealed to the evidence and certainty of mathematics; his attitude towards metaphysics, however, underwent a dramatic shift. In the Physical Monadology of 1756, Kant asked how metaphysics and geometry can be united,[f] or the former peremptorily denies that space is infinitely divisible, while the latter, with its usual certainty, asserts that it is infinitely divisible. Geometry contends that empty space is necessary for free motion, while metaphysics hisses the idea off the stage. Geometry holds universal attraction or gravitation to be hardly explicable by mechanical causes but shows that it derives from the forces which are inherent in bodies at rest and which act at a distance, whereas metaphysics dismisses the notion as an empty delusion of the imagination [1:475–6].2He attempted in this work to reconcile the respective positions of metaphysics and geometry regarding infinite divisibility, thereby demonstrating that “it is neither the case that the geometer is mistaken nor that the opinion to be found among metaphysicians deviates from the truth” [1:480]. But in a series of works from 1762–3, Kant reveals a markedly different attitude towards metaphysics. For example, in “The only possible argument in support of a demonstration of the existence of God,” Kant says:the mania for method and the imitation of the mathematician, who advances with a sure step along a well-surfaced road, have occasioned a large number of such mishaps on the slippery ground of metaphysics. These mishaps are constantly before one’s eyes, but there is little hope that people will be warned by them, or that they will learn to be more circumspect as a result [2:71].In contrast, Kant describes the geometer as uncovering “with the greatest certainty” the most secret properties of that which is extended [2:70]. But the less conciliatory attitude to metaphysics comes out most clearly in Kant’s actual procedure in “The only possible argument.” In the Seventh Reflection, he presents an attempt to explain the origin of... (shrink)

The incredible success of quantum theory as a mathematical model makes it especially frustrating that we cannot agree on a plausible philosophical or metaphysical description of it. Some philosophers of science have noticed certain parallels between quantum theory and the philosophy of Thomas Aquinas, and these parallels are deepened and strengthened if the “observer” of modern physics is associated with the “intellect” of scholastic ontology. In this case we are talking about a human observer. But this type of (...) observer has a unique quality that is not considered at all by either physics or scholastic philosophy—the human observer is mimetic and therefore “interdividual.” By taking this fundamental anthropological fact into account, it turns out that the critical gaps still separating Aquinas from modern physicists can be effectively closed, reconciling the realism of Aquinas with the empirical evidence of quantum mechanics. This book explores this new bridge between the physical and the human—a bridge essentially designed by scholastic theory, clarified by mimetic theory, and built by quantum theory—and the path it opens to that metaphysical understanding for which philosophers of modern science have been striving. It is an understanding, not merely of the physical but of physics in the fuller sense of what is real and what is true. Here the reader will find a physics that describes the natural world and our place as mimetic observers within it. (shrink)

Thinking About Judaism: Philosophical Reflections on Jewish Thought examines the light shed by philosophy upon significant areas of Jewish life and academic studies, including Jewish history, Jewish ethics, Jewish law, and Jewish aesthetics. As the author clearly illustrates, the teachings of leading theorists on the subjects of general history, ethics, law, and aesthetics inspire us to think about corresponding subjects related to Judaica.

Many texts on logic are written with a mathematical emphasis, and focus primarily on the development of a formal apparatus and associated techniques. In other, more philosophical texts, the topic is often presented as an indulgent collection of musings on issues for which technical solutions have long since been devised. What has been missing until now is an attempt to unite the motives underlying both approaches. Paul Hoyningen-Huene’s Formal Logic seeks to find a balance between the necessity of formal (...) considerations and the importance of full reflection and explanation about the seemingly arbitrary steps that occasionally confound even the most serious student of logic. Alex Levine’s artful translation conveys both the content and style of the German edition. Filled with examples, exercises, and a straightforward look at some of the most common problems in teaching the subject, this work is eminently suitable for the classroom. (shrink)

This book provides an interdisciplinary approach to one of the most fascinating and important open questions in science: What is quantum mechanics really talking about? In the last decades quantum mechanics has given rise to a new quantum technological era, a revolution taking place today especially within the field of quantum information processing; which goes from quantum teleportation and cryptography to quantum computation. Quantum theory is probably our best confirmed physical theory. However, in spite of its great empirical effectiveness (...) it stands today still without a universally accepted physical representation that allows us to understand its relation to the world and reality. The novelty of the book comes from the multiple perspectives put forward by top researchers in quantum mechanics, from Europe as well as North and South America, discussing the meaning and structure of the theory of quanta. The book comprises in a balanced manner physical, philosophical, logical and mathematical approaches to quantum mechanics and quantum information. Going from quantum superpositions and entanglement to dynamics and the problem of identity; from quantum logic, computation and quasi-set theory to the category approach and teleportation; from realism and empiricism to operationalism and instrumentalism; the book considers from different angles some of the most intriguing questions in the field. From Buenos Aires to Brussels and Cagliari, from Florence to Florianópolis, the interaction between different groups is reflected in the many different articles. This book is interesting not only to the specialists but also to the general public attempting to get a grasp on some of the most fundamental questions of present quantum physics. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:_Russell_ journal (home office): E:CPBRRUSSJOURTYPE2502\REVIEWS.252 : 2006-02-27 11:52 Reviews  HISTORY OF PHILOSOPHICAL ANALYSIS C P Philosophy / Purdue U. West Lafayette,  ,  @. Scott Soames. Philosophical Analysis in the Twentieth Century, Vol. : The Dawn of Analysis; Vol. : The Age of Meaning. Princeton: Princeton U. P., . Pp. xix, ; xxii, . . (hb), . (pb) for each volume. he last twenty years (...) have seen an explosion in books and papers on RusTsell ’s philosophy and its contemporary significance. There is good reason to think that this will continue as the contents of the Collected Papers are digested by Russell scholars and as more specialists contribute to the history of analytic philosophy more generally. Given all this good news, it is disconcerting to find a -page discussion of Russell, in a well-reviewed book by a first-rate philosopher, repeating many of the errors and misconceptions about Russell that scholars have worked so hard against. Soames’ discussion of Russell in the volumes under review is in fact so distressing that it alone compromises the book as a suitable introduction to the history of analytic philosophy. After briefly reviewing the outline of the two volumes, I discuss the errors concerning Russell, and conclude by drawing some lessons for Russell scholarship. Soames’ focus is on what he takes to be the most important and influential work of analytic philosophers, beginning with Moore’s Principia Ethica and ending in  with Kripke’s Naming and Necessity lectures. Kripke, in fact, marks the culmination of one of the two great achievements of analytic philos- _Russell_ journal (home office): E:CPBRRUSSJOURTYPE2502\REVIEWS.252 : 2006-02-27 11:52  Reviews ophy that Soames sees in this period: … the two most important achievements that have emerged from the analytic tradition in this period are (i) the recognition that philosophical speculation must be grounded in pre-philosophical thought, and (ii) the success achieved in understanding, and separating one from another, the fundamental methodological notions of logical consequence, logical truth, necessary truth, and apriori truth. (: xi) Moore is credited with the methodological innovation required by (i), as once we accept that what we think we know prior to philosophical reflection is a constraint on our epistemology, Moore’s response to scepticism, which Soames endorses, inevitably follows. But even Moore, and nearly every other figure that Soames discusses, is guilty of confusing necessity, analyticity and apriority. Soames endorses Kripke’s basic point that necessity is a metaphysical concept that can come apart from the epistemic notion of apriority and the semantic category of analytic propositions. Volume  repeats this charge several times, using it to undermine Moore’s views on ethics in Part One, Russell’s conception of analysis in Part Two, and logical positivism in Part Three. It is noteworthy that Soames takes Ayer’s Language, Truth and Logic as representative of logical positivism, ignoring contemporary scholarship on the Vienna Circle just as much as he ignores Russell scholarship. Part Four reconstructs Wittgenstein’s views in the Tractatus and Volume  ends with a discussion of Quine’s “Two Dogmas of Empiricism”. The second volume begins with a part on Wittgenstein’s Philosophical Investigations. This paves the way for Soames’ discussion of ordinary language philosophy in Parts Two and Three, which Soames sees as closely tied to the later Wittgenstein. Ryle’s Concept of Mind, Strawson’s early views on truth, Hare’s theory of goodness as well as Malcolm’s paradigm-case argument and Austin’s Sense and Sensibilia are investigated in these chapters. Part Four presents Grice’s theory of conversational implicature as the final nail in the coffin of ordinary language philosophy. In Part Five, Soames returns to Quine, this time discussing the ambitious arguments of Word and Object and the more general project of naturalized epistemology. Part Six articulates Davidson’s program for constructing a theory of meaning for natural languages along the lines of a Tarskistyle theory of truth, and Volume  ends with an extended discussion of the promise and limitations of Kripke’s Naming and Necessity. The material on Russell is entirely confined to four chapters in Volume : Chapter : “Logical Form, Grammatical Form, and the Theory of Descriptions ”, Chapter : “Logic and Mathematics: The Logicist Reduction”, Chapter : “Logical Constructions and... (shrink)

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme (...) that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

Presenting the history of space-time physics, from Newton to Einstein, as a philosophical development DiSalle reflects our increasing understanding of the connections between ideas of space and time and our physical knowledge. He suggests that philosophy's greatest impact on physics has come about, less by the influence of philosophical hypotheses, than by the philosophical analysis of concepts of space, time and motion, and the roles they play in our assumptions about physical objects and physical measurements. (...) This way of thinking leads to interpretations of the work of Newton and Einstein and the connections between them. It also offers ways of looking at old questions about a priori knowledge, the physical interpretation of mathematics, and the nature of conceptual change. Understanding Space-Time will interest readers in philosophy, history and philosophy of science, and physics, as well as readers interested in the relations between physics and philosophy. (shrink)

Mental model construction is supposed to be a useful cognitive devise for learning. Beyond human capacity of constructing mental models, scientists construct complex explanations about phenomena, named scientific or theoretical models. In this work we revisit three vissions: the first one concern about the polisemic term “model”. Our proposal is to discriminate between “mental models” and “explicit models”, being the former those “imaginistic” ideas constructed in scientists’—o teachers—minds, and the latter those teaching devices expressed in different languages that (...) tend to communicate any “scientific model”. From this point of view, the class is considered a place where teachers’ mental models should be learned by novice students by decoding their teaching devices which are expressed in different languages. Other proposal of this work claims to distinguish the term “representation” with respect to its artistical or instrumental origin, highlighting that they are types of teaching devices and that artistical representations are always analogies. Finally, data about the construction of freshmen’s wrong mental models related to the use of the analogy between the chemical combustion and the global process of cellular respiration from glucose is presented to reinforce previous epistemological reflections. (shrink)

In this essay Nicholas C. Burbules reviews his experiences and the lessons he learned as editor of Educational Theory for more than twenty years, and he explores some of the normative choices that are inevitably made by any editor in carrying out his or her role. Burbules examines the relationship of a journal to its intellectual field; the review process; communications and interactions with authors; the process of editing and revising manuscripts; questions of representativeness in a theoretically pluralistic field; the (...) business of journal publishing; and the dilemmas that confront an editor in terms of his or her own position and identity within a field. In all of these reflections, he examines the ethical and political background of the choices made, and how these in turn reveal deeper assumptions about the nature and purpose of academic publishing. The result is to “lift up the curtain” and reveal how one philosophically reflective and self-questioning editor handled his responsibilities. (shrink)

A case study of quantum mechanics is investigated in the framework of the philosophical opposition “mathematical model – reality”. All classical science obeys the postulate about the fundamental difference of model and reality, and thus distinguishing epistemology from ontology fundamentally. The theorems about the absence of hidden variables in quantum mechanics imply for it to be “complete” (versus Einstein’s opinion). That consistent completeness (unlike arithmetic to set theory in the foundations of mathematics in Gödel’s opinion) can (...) be interpreted furthermore as the coincidence of model and reality. The paper discusses the option and fact of that coincidence it its base: the fundamental postulate formulated by Niels Bohr about what quantum mechanics studies (unlike all classical science). Quantum mechanics involves and develops further both identification and disjunctive distinction of the global space of the apparatus and the local space of the investigated quantum entity as complementary to each other. This results into the analogical complementarity of model and reality in quantum mechanics. The apparatus turns out to be both absolutely “transparent” and identically coinciding simultaneously with the reflected quantum reality. Thus, the coincidence of model and reality is postulated as necessary condition for cognition in quantum mechanics by Bohr’s postulate and further, embodied in its formalism of the separable complex Hilbert space, in turn, implying the theorems of the absence of hidden variables (or the equivalent to them “conservation of energy conservation” in quantum mechanics). What the apparatus and measured entity exchange cannot be energy (for the different exponents of energy), but quantum information (as a certain, unambiguously determined wave function) therefore a generalized law of conservation, from which the conservation of energy conservation is a corollary. Particularly, the local and global space (rigorously justified in the Standard model) share the complementarity isomorphic to that of model and reality in the foundation of quantum mechanics. On that background, one can think of the troubles of “quantum gravity” as fundamental, direct corollaries from the postulates of quantum mechanics. Gravity can be defined only as a relation or by a pair of non-orthogonal separable complex Hilbert space attachable whether to two “parts” or to a whole and its parts. On the contrary, all the three fundamental interactions in the Standard model are “flat” and only “properties”: they need only a single separable complex Hilbert space to be defined. (shrink)

This volume offers a meta-philosophical reflection on feminist philosophies of science. It emphasizes and discusses both the connections and differences between "traditional" philosophies of science and feminist philosophies of science. The collection systematically analyses feminist contributions to the various philosophies of specific sciences. Each chapter is devoted to a specific area of philosophy of science: general philosophy of science, philosophy of biology, philosophy of climate sciences, philosophy of cognitive sciences and neurosciences, philosophy of economics, philosophy of history and (...) archaeology, philosophy of logic and mathematics, philosophy of medicine, philosophy of psychology, philosophy of physics, and philosophy of social sciences. Since some of these areas have so far rarely been addressed by feminist philosophers, this new collection provides new angels and stimulates the debate on pivotal issues that are part and parcel of both "traditional" philosophies of science and feminist philosophies of science. Using a range of different methodologies and styles, the essays all show great clarity in both arguments and contents. (shrink)

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice (ZFC). This framework is, however, laden with philosophical difficulties. One important alternative foundational (...) programme that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)