This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects (...) of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat. (shrink)

Proof requires a dialogue between agents to clarify obscure inference steps, fill gaps, or reveal implicit assumptions in a purported proof. Hence, argumentation is an integral component of the discovery process for mathematical proofs. This work presents how argumentation theories can be applied to describe specific informal features in the development of proof-events. The concept of proof-event was coined by Goguen who described mathematical proof as a public social event that takes place in space and (...) time. This new meta-methodological concept is designed to cover not only “traditional” formal proofs but all kinds of proofs and inference steps, including incomplete or purported proofs. Our approach attempts to make proof-events more comprehensive to express the complete trajectory of a mathematical proof-event until the ultimate validation of the proving outcome. Thus, we advance an extended version of proof-event calculus which is built on argumentation theories designed to capture the internal and external structure of collaborative mathematical practice and highlight the relationship between proof, human reasoning, and cognitive processes. In addition, another area in which argumentation can make a significant contribution is dealing with the defeasible knowledge of the Web which is a product of its open and ubiquitous nature. This approach seems to be sufficient for the presentation of Web-based proving processes as manifested in the case of the Mini-Polymath 4 project. (shrink)

This book features a unique selection of works presented at the 2019 annual international conference of the International Association for Computing and Philosophy (IACAP). Every contribution has been peer-reviewed, revised, and extended. The included chapters are thematically diverse; topics include epistemology, dynamic epistemic logic, topology, philosophy of science and computation, game theory and abductive inferences, automated reasoning and mathematical proofs, computer simulations, scientific modelling, applied ethics, pedagogy, human-robot interactions, and big data, algorithms, and artificial intelligence. The volume (...) is a testament to the value of interdisciplinary approaches to the computational and informational turn. We live in a time of tremendous development, which requires rigorous reflection on the philosophical nature of these technologies and how they are changing the world. How can we understand these technologies? How do these technologies change our understanding of the world? And how do these technologies affect our place as humans in the world? These questions, and more, are addressed in this volume which is of interest to philosophers, engineers, and computer scientists alike. (shrink)

I see the question what it is that makes an inference valid and thereby gives a proof its epistemic power as the most fundamental problem of general proof theory. It has been surprisingly neglected in logic and philosophy of mathematics with two exceptions: Gentzen’s remarks about what justifies the rules of his system of natural deduction and proposals in the intuitionistic tradition about what a proof is. They are reviewed in the paper and I discuss to (...) what extent they succeed in answering what a proof is. Gentzen’s ideas are shown to give rise to a new notion of valid argument. At the end of the paper I summarize and briefly discuss an approach to the problem that I have proposed earlier. (shrink)

The goals of reduction andreductionism in the natural sciences are mainly explanatoryin character, while those inmathematics are primarily foundational.In contrast to global reductionistprograms which aim to reduce all ofmathematics to one supposedly ``universal'' system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistency-proof rationale and the foundational (...) reduction rationale. However, recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foundational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper,various reduction relations betweensystems are explained and compared, and arguments against proof-theoretic reduction as a ``good'' reducibilityrelation are taken up and rebutted. (shrink)

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

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, (...) such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians (...) already there, ready to meet them, who have explicitly invoked virtues in discussing what is necessary for a mathematician to succeed. In both ethics and epistemology, virtue theory tends to emphasize character virtues, the acquired excellences of people. But people are not the only sort of thing whose excellences may be identified as virtues. Theoretical virtues have attracted attention in the philosophy of science as components of an account of theory choice. Within the philosophy of mathematics, and mathematics itself, attention to virtues has emerged from a variety of disparate sources. Theoretical virtues have been put forward both to analyse the practice of proof and to justify axioms; intellectual virtues have found multiple applications in the epistemology of mathematics; and ethical virtues have been offered as a basis for understanding the social utility of mathematical practice. Indeed, some authors have advocated virtue epistemology as the correct epistemology for mathematics (and perhaps even as the basis for progress in the metaphysics of mathematics). This topical collection brings together several of the researchers who have begun to study mathematical practices from a virtue perspective with the intention of consolidating and encouraging this trend. (shrink)

The ten contributions in this volume range widely over topics in the philosophy of mathematics. The four papers in Part I (entitled "Set Theory, Inconsistency, and Measuring Theories") take up topics ranging from proposed resolutions to the paradoxes of naïve set theory, paraconsistent logics as applied to the early infinitesimal calculus, the notion of "purity of method" in the proof of mathematical results, and a reconstruction of Peano's axiom that no two distinct numbers have the same successor. Papers (...) in the second part ("The Challenge of Nominalism") concern the nominalistic thesis that there are no abstract objects. The two contributions in Part III ("Historical Background") consider the contributions of Mill, Frege, and Descartes to the philosophy of mathematics. (shrink)

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity”. This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. (...) The starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs. (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)

In the present paper I wish to regard constructivelogic as a self-contained system for the treatment ofepistemological issues; the explanations of theconstructivist logical notions are cast in anepistemological mold already from the outset. Thediscussion offered here intends to make explicit thisimplicit epistemic character of constructivism.Particular attention will be given to the intendedinterpretation laid down by Heyting. This interpretation, especially as refined in the type-theoretical work of Per Martin-Löf, puts thesystem on par with the early efforts of Frege andWhitehead-Russell. This quite (...) recent work, however,has proved valuable not only in the philosophy andfoundations of mathematics, but has also foundpractical application in computer science, where thelanguage of constructivism serves as an implementableprogramming language, and within the philosophy oflanguage.\footnote{Nordstr\"{o}m et al. give an overview of the work in computerscience, whereas Ranta provides an impressiveconstructivist alternative to Montague Grammar usingthe richer type structure of Martin-L\"{o}f in placeof the simple classical type theory of Church.} Mypresentation will be carried out through a contrastwith standard metamathematical work.\footnote{Troelstra and van Dalen give an encyclopedictreatment of the metamathematics of constructivism.}In the course of the development I have occasion tooffer some novel considerations on thenature of proof and inference. (shrink)

Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on rule-following skepticism. We argue, with the help of C. S. Peirce's distinction between corollarial and theorematic proofs, that his intuitions are better explained by resistance to what we call conceptual omniscience, treating meaning as fixed content specified in advance. We interpret the distinction in the context of modern epistemic logic (...) and semantic information theory, and show how removing conceptual omniscience helps resolve Wittgenstein's paradoxes and explain the puzzle of deduction, its ability to generate new knowledge and meaning. (shrink)

Richard Foley has suggested that the search for a good theory of epistemic justification and the analysis of knowledge should be conceived of as two distinct projects. However, he has not offered much support for this claim, beyond highlighting certain salutary consequences it might have. In this paper, I offer some further support for Foley’s claim by offering an argument and a way to conceive the claim in a way that makes it as plausible as its denial, and thus levelling (...) the playing field. The burden of proof then lies with those who seek to deny Foley’s radical suggestion. (shrink)

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

The various papers and short "discussions" contained in this latest addition to the "Studies in Logic" series were presented at the 1965 International Colloquium in the Philosophy of Science, in London. Of the nine "problems" considered in this symposium, seven have directly to do with philosophy, one is an historical study of the origins of Euclid's axiomatics, and the last is an interesting—if one-sided—discussion of the "new math" controversy in the pre-college curriculum. Happily, this book demonstrates that the (...) important issues in the philosophy of mathematics somewhat outrun the usual ism-mongering debate between representatives of various schools. Unhappily, it also demonstrates that a good deal of undeserving material seems to find its way between the covers of scholarly books. After reading certain of the articles in this volume one is left with the feeling that the authors have not come to grips with the issues which they raise, that a lot of effort has been expended on peripheral matters, and that, in some instances, a sledge hammer has been brought down upon an egg. Abraham Robinson's article, "The Metaphysics of the Calculus," deals with a series of important linguistic and ontological problems concerning the foundations of analysis. Moreover, these are problems which can no longer be ignored, especially in view of the evolution of non-standard analysis. But, unfortunately, the interested reader would do far better to turn to Robinson's recent book on the subject, and pay special attention to the second, third, and last chapters therein. "On a Fregean Dogma" by Fred Sommers gives a rather curious non-quantificational algorithm for the monadic fragment of syllogistic. This Sommers does in the interest of "a conservative effort to restore categorical statements to predicative status." Despite the heat and the occasional interesting remarks which the discussion generated, one is inclined to ask, "So what?" The sequence of three papers by Mostowski, Bernays, and Körner is best considered as a group, since each of these is concerned with the philosophical significance of recent results in set theory and logic. Students of the philosophy of mathematics would do well to read this part of the book very carefully, although one wishes that the authors would elaborate a bit on their specific philosophical views. Finally, there is Georg Kreisel's difficult and amazing paper, "Informal Rigor and Completeness Proofs," in which is presented a mathematical treatment of the problem of selecting the primitive rules and concepts of mathematical systems. Kreisel's ideas are refreshing, unorthodox, dubious, and a genuine contribution to the philosophy of mathematics. In sum, this book is a mixed bag, uneven in the quality of its contents, but worth reading.—H. P. K. (shrink)

IntroductionIn May 2006, the small group of doctoral students working on ecophilosophy at the Higher Institute of Philosophy at K.U.Leuven invited the Dutch environmental philosopher Martin Drenthen to a workshop to discuss his writings on the concept of wilderness, its metaphysical and moral meaning, and the challenge social constructivism poses for ecophilosophy and environmental protection. Drenthen’s publications on these topics had already been the subject of intense discussions in the months preceding the workshop. His presentation on the workshop and (...) the three critical responses by Wim Bollen, Glenn Deliège, and Richard Kover are published here in a thoroughly revised form on the basis of further and ongoing discussions about the issues involved. Three further contributions to the discussion are added: one by Nathan Edward Kowalski, a young Canadian philosopher who studied in Leuven and wrote his as yet unpublished dissertation on “Evil in Nature,” a second one by Kingsley Goodwin who is currently writing a dissertation on deep ecology at University College in Dublin and, finally, my own unsystematic thoughts inspired by Martin Drenthen’s provocative and stimulating hermeneutics of our postmodern understanding of nature and wilderness, as well as the diverse critical responses to Drenthen’s views by the other contributors. I owe gratitude to all of them for enriching my own ecophilosophical thought and enlarging its horizon although as my fragmentary reflections show at the cost of yet greater uncertainty and more questions about the possibilities of arguing for a radical environmentalist position. Such a position would somehow connect or ground our moral obligation to preserve genuine non-human otherness, wild and free nature, in the non-human meaning of such nature and in our essential need to be in direct and concrete contact with that field of wild non-human meaning because we ourselves belong to nature, because we are not only and purely civilized but are also and most fundamentally wild and free. Three fundamental questions need to be distinguished: the question about our moral obligations regarding nature and its preservation, the metaphysical question about the nature of nature or the meaning of nature, and the question of the metaphysical grounding of our morals in general, and of our moral obligation regarding nature in particular, in the nature or meaning of nature. The complex intertwinement between these three questions is at the heart of Martin Drenthen’s reflections.The authors contributing to the present discussion approach the various issues from markedly different philosophical backgrounds and concerns: critical theory, anthropology, analytic philosophy, phenomenology, hermeneutics, and deep ecology are all to some degree or other brought to bear on the question of what wilderness means for us, why we are fascinated by it and what it actually and in the first place is.Another interesting difference between the five respondents to Martin Drenthen’s wilderness-philosophy relates to the place from which they write: while the Canadians Richard Kover and Nathan Edward Kowalski have wilderness of some sort still close at hand, the Belgians Wim Bollen and Glenn Deliège write from a place which has been thoroughly domesticated, cultivated, and industrialised for a long time. Martin Drenthen himself, of course, writes from the same kind of place as these last two, which explains why he only refers to the wilderness that he has read about in novels, seen in movies, or experienced as recreated in the reserves of the Netherlands. The Irish Kingsley Goodwin writes perhaps from a third, intermediate place that is not wild anymore but still much more natural than Belgium or the Netherlands are.The differences in philosophical approach and concern combined with the difference of place generate a rich tapestry of perspectives and voices wrestling with the fundamental questions of our perceived homelessness and alienation from nature, our cultural discontent, our longing for and need for radical alterity, of immanence and transcendence, of who we are and where we belong, and more concretely of whether there is more to ground our efforts to preserve nature than enlightened self-interest and subjective preference. Martin Drenthen deserves praise for inaugurating this fruitful discussion with his intriguing and highly paradoxical effort to hybridize a postmodern deconstruction of the idea and ideal of wilderness as unspoilt and true nature on the one hand with a new metaphysics of wilderness on the other. Arne Naess, who is generally regarded as the founding father of deep ecology, pointed out the positive and stimulating role that vagueness in the expression of ideas can play in political and philosophical debate. Such general and vague but highly suggestive expressions as ‘all life is one’ are not empty at all, as is often claimed; they are rather full of possibilities of ‘precisation’ and differentiation. Drenthen’s wilderness-philosophy is certainly not vague to the degree of ‘all life is one,’ but his expositions are similarly pregnant with a wealth of further questions, precisations, distinctions, and conceptual clarifications. Good examples of different kinds are to be found in the contributions of Nathan Edward Kowalski and Richard Kover. Kowalski points out that Drenthen’s notions of culture and wilderness need to be made more precise by distinguishing them from the notions of civilization and wildness such that wild cultures become a conceptual possibility. Richard Kover takes the lead from Drenthen’s question about the reason for our postmodern fascination for wilderness to confront it with our primordial fascination in terms of the dependence in the evolution of our subjectivity and our consciousness on the focused perceptual attention on wild forms.Glenn Deliège situates Martin Drenthen’s wilderness-philosophy in its original context of the Dutch debate on the preservation of nature, in particular on the new wilderness paradigm. According to this paradigm, the nature to be preserved should be true and pure, truly natural and wild nature, that is nature outside of and free from the cultivating and domesticating influence of humans. In the Netherlands, however, such nature no longer exists; it can only be preserved after it has first been artificially recreated. Deliège gives an informative summary account of the highly charged three-cornered debate between the adepts of this wilderness-paradigm, the proponents of the more traditional form of conservation of remnants of the typical European landscapes created by pre-industrial farming practices and the modern technocratic functionalists. This debate makes obvious that there are quite different views of the nature of nature and its value. Enter social constructivism: the contested nature of nature is nothing more than a social construct, an idea in a collective mind. The contestation itself is a struggle for authority and power between different such social constructs, any claim to scientific objectivity and objective truth for one’s image of nature is an arrogation of a truth which doesn’t exist and is potentially oppressive by silencing other views. Nature then is up for the grabs of politics, political negotiation, and compromise between various interest groups who all want to get some benefit from nature, some use-value.The problem with Drenthen’s effort to articulate a normative concept of nature which cannot be deconstructed in this way is the extreme formality and emptiness of his metaphysical concept of wilderness as that which is always ‘beyond’, always ‘more’ and which can never be exhausted by any image and construction of nature. Any more concrete determination is mere appearance; the thing in itself is a border concept of that which in fact cannot be grasped conceptually. Glenn Deliège is critical of this formality and emptiness from which it will be impossible to derive any concrete guidance for our conservation practices. He then goes on to defend the Arcadian tradition of our appreciation of nature as giving us a historically well-grounded orientation for those practices. Of course that Arcadian tradition is obviously a cultural construction but that doesn’t make it completely arbitrary and contingent since it is itself ultimately grounded in and open to modification by our concrete sensuous experiences of nature.Deliège here raises the very important issue of the relation between our concrete perceptual and bodily experiences in and with nature and our concepts and metaphysical images of nature. Drenthen himself approaches this issue from the hermeneutical perspective of our need to make sense of our experiences and to articulate their meaning. His confrontation with social constructivism and his effort to salvage something from its relativizing acid lead him, however, into the direction of a highly abstract and empty concept of wilderness.Richard Kover shows that perception is, indeed, the key to a radical critique of social constructivism. He does this with the help of the ecological anthropology of Paul Shepard, the structuralist anthropology of Claude Levi-Strauss and the phenomenology of Maurice Merleau-Ponty. It needs to be said here that Kover takes his leave from an unpublished paper by Martin Drenthen with the title “How to appropriate wilderness appropriately?” – the paper is accessible though on Drenthen’s web-site - in which Drenthen comments and reflects upon three contemporary wilderness-tales, one of them being Werner Herzog’s much-discussed documentary on Timothy Treadwell, an American environmentalist drop-out who lived thirteen summers among wild grizzlies in the Alaskan wilderness before being killed together with his girlfriend by a grizzly bear.Kover tries to show that we are not imprisoned in the mirror-cabinet of our mind or of our human sociality. To make his case, he starts from Paul Shepard’s thoughtful observation that the evolutionary emergence of the mind did not remove us from the natural world but actually connected us deeper with it. Through our mind we evolved more fully into it because our mind is primordially in our senses and in the ability to articulate a perceptual world. The natural world in the form of the system of natural species and their relations is the primary model for perceptual discriminations, for the fundamental discovery of order with its twin-aspects of distinction and discontinuity on the one side and relations and continuity on the other side. Our cognition is essentially perceptually based, our subjectivity is primordially informed by our inherence in the sensible world. Perceptual cognition is inherently intentional and relational, it is not an autonomous mental construction, it is rather in the world and informed by that which is perceived. The other is always intertwined with the subject. We are always already hooked up with the world, with the others, before we start reflecting, thinking, and conceptualizing. This is one of the major themes in Shepard’s work: what this insight means with regard to our relations to non-human otherness, in particular to our perceptual encounter with wild animals. In our encounter with non-human otherness, we are confronted with an essentially ambiguous sense of both affinity and disjuncture. Wild animals demonstrate to us that we are not homeless or alien to this world, that we fundamentally belong to it and are part of it. Their difference, however, shows us that there are different perspectives and ways of being in the world. “Thus the encounter with non-human others inform humans that we are both of the world and simultaneously not the world and there are horizons of significance beyond our own perspective.”Nathan Edward Kowalski comes to a similar conclusion when he situates our morality in the larger moral order of wild nature. According to Kowalski it is indeed a moral order though not our own, an Other to our morality but not simply a negative one. It is, as Drenthen says, a highly paradoxical amoral morality. But what can it mean to speak of morality here? For Kowalski it means that the issue is not merely the theoretical recognition and acknowledgment of axiological transcendence, of non-human values out there, but whether we will appropriately conform to and comport ourselves towards that axiological transcendence so that we can find a human homeland in wild nature instead of turning it into a wasteland. The question then is, how to fit our own human morality into this larger order and greater reality, to subordinate it to that greater reality as a part of it and encompassed but not denied by it so that we can be moral and natural at the same time, tender carnivores hunting the sacred game. Indigenous people follow a moral protocol in their relationship with non-human others which acknowledge and respects their autonomy, their alterity, and the sacred mystery of their different ways of being. They are not befriended but some of them are hunted and eaten – as an exchange in the gift-economy of nature.The problem of the homelessness and alienation from nature is discussed by Wim Bollen from the radically different vantage point of critical theory. In Adorno’s and Horkheimer’s Dialectic of Enlightenment the human story unfolds as a tragic effort to liberate ourselves from nature and to become fully human through rational control when in the end this effort turns against ourselves and instead of liberation and human flourishing we get social domination and repression of our natural needs. The frustrations and mutilations imposed upon us by a repressive civilization give rise to the regressive desire for a return to a natural existence. Referring to the myth of Odysseus and Circe as interpreted by Adorno and Horkheimer, Wim Bollen equates such a natural existence with animal life. In the light of contemporary knowledge about primitive existence, however, this equation is highly problematic. Palaeolithic foragers were already fully human in all respects. A life without civilization need not be an animal-existence.According to Adorno/Horkheimer, our subjectivity and autonomy have to be wrested from animal-nature through an exertive struggle that cannot but proceed by a threefold strategy of domination: the domination of non-human otherness, of human others, and of our own inner nature. The siren song of nature consists in the temptation to give up that struggle for the promise of an unreflective, non-alienated natural existence without toil.When Odysseus’ men give in to Circe’s seduction they are, however, not transformed into wild animals but into swine. According to Bollen this signifies that we cannot re-wild ourselves, that we remain imprisoned in civilization, either as master-subjects or as domesticated animals. Bollen, by the way, like Drenthen, does not distinguish conceptually between civilization and culture. Since re-wilding for Bollen means the impossibility of returning to animal-existence, which is actually the impossibility of becoming another species since biologically we are animals already, to be imprisoned in civilisation means nothing else than to be imprisoned in our humanity. The fact that we cannot become wild animals again – according to Paul Shepard, however, we are still wild animals – is indeed, as Bollen stresses, a trivial form of alienation. A more meaningful concept of alienation which can be put to critical use must refer to a culturally and historically determined frustration and deficit inside the cultural order that is to be overcome not by an impossible regression but by progressive politics leading beyond the three intertwined forms of domination. Not back to nature but forward towards a reconciliation of culture with nature! Such a reconciliation would neither collapse culture into the immanence of nature nor submerge nature into the transcendence of culture. According with the three forms of domination it would be a three-dimensional form of reconciliation.Odysseus’ men did not got what they thought were promised them: the complete liberation from the burden of being autonomous subjects, of being civilized, but they got something worthwhile, namely a certain from of pleasure, a limited and provisional relief and escape. Bollen sees a critical potential in these piggish pleasures after all and consequently even in the siren song of nature since these pleasures were only experienced because Odysseus’ men gave in to it. But one has to give in to this siren song critically if it is to unfold its liberating potential. The problem with pleasure though in bourgeois society is that it is necessarily betrayed, instrumentalized and sold.Kingsley Goodwin’s paper starts by defending deep ecology against the charge of asserting a naïve and dogmatic metaphysics of nature. For convenience sake, but not quite unproblematically, he identifies deep ecology with the environmental thought of Arne Naess. A metaphysics of nature for Naess is always part of a personal worldview. Worldviews which are partly inspired by ecology Naess calls ecosophies. Naess always stresses the personal character of such a worldview and the irreducible plurality of them. It used to be a common misunderstanding of the platform of deep ecology which Naess conceived together with George Sessions that it expressed in a condensed form the metaphysical beliefs of deep ecology. Naess’ intention, however, was to try to articulate in a general fashion the basic viewpoints on which the supporters of the deep ecology movement agreed. It would be possible then to derive these viewpoints from quite different worldviews. In light of this, I think, that Goodwin’s characterisation of the platform as “pragmatic and activist” is not quite appropriate.Goodwin goes on to compare Drenthen’s conception of wilderness and wildness with the similar position developed by Neil Evernden in his book The Social Creation of Nature. He like Naess in his personal ecosophy stresses the crucial role of perceptual experience of wild otherness to find an antidote to the relativity and contingency of our interpretations of nature. Evernden refers to studies on child development to argue for a non-dualistic and relational conception of the self, similar to Naess’ relational ontology. According to Goodwin, Drenthen, in spite of his hermeneutical approach remains too much indebted to the Western subject/object-dualism. The object, in this case wild nature, is pure otherness.With regard to a compost heap, Goodwin shows that we are not separated from nature by an inseparable gulf, that nature is close at hand, and that we can experience and explore it, appreciate the complex play of identity and difference it consists of, the exchanges and transformations and that we can identify with the organisms involved in these exchanges.*The proceeding remarks only highlight some of the prominent themes and considerations in the various contributions to this topical issue of Ethical Perspectives. Hopefully the reader will feel invited and enticed to read the articles themselves, starting, preferably of course, with the essay by Martin Drenthen, who sparked off this lively discussion on the moral meaning of wild nature. I am honoured and pleased to be able to introduce these rich explorations by such talented young environmental philosophers who are eager to develop environmental philosophy further and lead it into new directions. They all have to wrestle with and find answers to the challenge postmodern deconstruction and social constructivism pose for environmental philosophy. In the face of that challenge, it is even more difficult than it already was to provide solid and convincing arguments for the defence of wild and free nature beyond merely pragmatic arguments in terms of its utilitarian value for us. The reason for that defence is not clear-cut and proves to be just as evasive as the moral meaning of nature itself. But neither that reason nor that meaning are mere phantoms and mirages that vanish when we approach them and try to grasp them. They are more substantial, even if they will never have the solidity of a mathematical proof or a scientific measurement. Their locale is the lifeworld which from the point of view of scientific and logical precision and clarity is a messy place where subject and object, reason and heart, fact and value, experience and thought and, yes, nature and culture cannot be clearly separated. This has important consequences for environmental philosophy and its methodology. Conceptual analysis and logical argumentation have to become sensitive to the richness of experience in the lifeworld and to the diverse efforts of its articulation: i.e., they have to open themselves up and maybe even to subordinate themselves to phenomenological and hermeneutical approaches. For me personally this is one of the most important conclusions I draw from the discussion presented here. (shrink)

The nature of this book is fourfold: First, it provides comprehensive education in ontology, epistemology, logic, and ethics. From this perspective, it can be treated as a philosophical textbook. Second, it provides comprehensive education in mathematical analysis and analytic geometry, including significant aspects of set theory, topology, mathematical logic, number systems, abstract algebra, linear algebra, and the theory of differential equations. From this perspective, it can be treated as a mathematical textbook. Third, it makes a student and a researcher (...) in philosophy and/or mathematics capable of developing a holistic approach to reality, of undertaking interdisciplinary endeavors, of understanding (and possibly contributing to) advances and research projects in different academic disciplines, and of having more sources of inspiration and pleasure. From this perspective, it can be treated as a contribution to pedagogy and as an attempt to refresh and, indeed, revitalize modern philosophy. Fourth, it seeks to defend, refresh, and enrich philosophical and scientific structuralism and dynamical philosophy (known also as dynamism). From this perspective, this book can be treated as a research monograph on structuralism and dynamism, tackling the fundamental problems of reality, truth, and consciousness. In this context, Nicolas Laos expounds and proposes: (i) the concepts of dynamized time and dynamized space; (ii) a theory and method that he calls the "dialectic of rational dynamicity"; and (iii) his attempt to consider the fundamental problems of philosophy and science from the perspective of the dialectic of rational dynamicity. Thus, this book pertains to every field that is controlled by the function of consciousness, namely, being, knowing, and acting. The philosophy of rational dynamicity, as the author explains in this book, is a way of contemplating the laws of motion of nature, history, and spirit. (shrink)

The paper focuses on the review of current literature on formal philosophy. Special attention is paid to the review of the book «Introduction to Formal Philosophy» [Hansson, Hendricks, 2018]. The book is a consistent introduction to the problems of formal philosophy, a research tradition that relies on the precise mathematical tools in order to study traditional philosophical problems. The methods of formal philosophy are successfully applied not only to the problems of ontology, epistemology and philosophy (...) of language but also relevant for the problems of ethics, axiology and social philosophy. The book demonstrates that it is not correct to identify formal philosophy with another area of study – philosophical logic, since formal philosophy uses not only logical methods of analysis, but also uses the tools of game theory, decision theory, probability theory, Bayesian statistics, and other theories. Although the book has a propaedeutic character, it also contains some open problems. These problems include the aggregation of the opinions of the group under the condition of a conflicting base of premises in the theory of public choice, there are still open problems in the interpretation of Arrow’s impossibility theorem and others. Certainly, formalization in itself is not a general solution to the particular philosophical problem, but only a tool that allows to formulate a problem in a more rigorous and precise way, which sometimes allows to reveal some unexpected consequences, some implicit contradictions and new solutions. Despite the importance of the concept of coherence in ethics, decision theory, philosophy of law, Bayesian epistemology, philosophy of science, the existing formalizations of the concept of coherence are highly specialized for epistemology, researchers recognize the lack of the relevant explanatory models. Overall, the book is an excellent introduction in to the field of formal philosophy, which provides a general overview of different aspects of formal philosophy and the opportunity to study particular research topics by means of an extensive bibliography accompanying each of the chapters. (shrink)

The paper outlines a project in the philosophy of mathematics based on a proposed view of the nature of mathematical reasoning. It also contains a brief evaluative overview of the discipline and some historical observations; here it points out and illustrates the division between the philosophical dimension, where questions of realism and the status of mathematics are treated, and the more descriptive and looser dimension of epistemic efficiency, which has to do with ways of organizing the mathematical material. The (...) paper’s concern is with the first. The grand tradition in the philosophy of mathematics goes back to the foundational debates at the end of the 19th and the first decades of the 20th century. Logicism went together with a realistic view of actual infinities; rejection of, or skepticism about actual infinities derived from conceptions that were Kantian in spirit. Yet questions about the nature of mathematical reasoning should be distinguished from questions about realism (the extent of objective knowledge– independent mathematical truth). Logicism is now dead. Recent attempts to revive it are based on a redefinition of “logic”, which exploits the flexibility of the concept; they yield no interesting insight into the nature of mathematics. A conception of mathematical reasoning, broadly speaking along Kantian lines, need not imply anti–realism and can be pursued and investigated, leaving questions of realism open. Using some concrete examples of non–formal mathematical proofs, the paper proposes that mathematics is the study of forms of organization—-a concept that should be taken as primitive, rather than interpreted in terms of set–theoretic structures. For set theory itself is a study of a particular form of organization, albeit one that provides a modeling for the other known mathematical systems. In a nutshell: “We come to know mathematical truths through becoming aware of the properties of some of the organizational forms that underlie our world. This is possible, due to a capacity we have: to reflect on some of our own practices and the ways of organizing our world, and to realize what they imply.. (shrink)

Modal logics, originally conceived in philosophy, have recently found many applications in computer science, artificial intelligence, the foundations of mathematics, linguistics and other disciplines. Celebrated for their good computational behaviour, modal logics are used as effective formalisms for talking about time, space, knowledge, beliefs, actions, obligations, provability, etc. However, the nice computational properties can drastically change if we combine some of these formalisms into a many-dimensional system, say, to reason about knowledge bases developing in time or moving objects. To (...) study the computational behaviour of many-dimensional modal logics is the main aim of this book. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied many-dimensional systems (e.g., multi-agent systems, description logics with epistemic, temporal and dynamic operators, spatio-temporal logics, etc.) fit in perfectly with this theoretical framework, and so their computational behaviour can be analyzed using the developed machinery. We start with concrete examples of applied one- and many-dimensional modal logics such as temporal, epistemic, dynamic, description, spatial logics, and various combinations of these. Then we develop a mathematical theory for handling a spectrum of 'abstract' combinations of modal logics - fusions and products of modal logics, fragments of first-order modal and temporal logics - focusing on three major problems: decidability, axiomatizability, and computational complexity. Besides the standard methods of modal logic, the technical toolkit includes the method of quasimodels, mosaics, tilings, reductions to monadic second-order logic, algebraic logic techniques. Finally, we apply the developed machinery and obtained results to three case studies from the field of knowledge representation and reasoning: temporal epistemic logics for reasoning about multi-agent systems, modalized description logics for dynamic ontologies, and spatio-temporal logics. The genre of the book can be defined as a research monograph. It brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). On the other hand, well-known results from modal and first-order logic are formulated without proofs and supplied with references to accessible sources. The intended audience of this book is logicians as well as those researchers who use logic in computer science and artificial intelligence. More specific application areas are, e.g., knowledge representation and reasoning, in particular, terminological, temporal and spatial reasoning, or reasoning about agents. And we also believe that researchers from certain other disciplines, say, temporal and spatial databases or geographical information systems, will benefit from this book as well. Key Features: Integrated approach to modern modal and temporal logics and their applications in artificial intelligence and computer science Written by internationally leading researchers in the field of pure and applied logic Combines mathematical theory of modal logic and applications in artificial intelligence and computer science Numerous open problems for further research Well illustrated with pictures and tables. (shrink)

In this paper, we describe "metaphysical reductions", in which the well-defined terms and predicates of arbitrary mathematical theories are uniquely interpreted within an axiomatic, metaphysical theory of abstract objects. Once certain (constitutive) facts about a mathematical theory T have been added to the metaphysical theory of objects, theorems of the metaphysical theory yield both an analysis of the reference of the terms and predicates of T and an analysis of the truth of the sentences of T. The well-defined terms and (...) predicates of T are analyzed as denoting abstract objects and abstract relations, respectively, in the background metaphysics, and the sentences of T have a reading on which they are true. After the technical details are sketched, the paper concludes with some observations about the approach. One important observation concerns the fact that the proper axioms of the background theory abstract objects can be reformulated in a way that makes them sound more like logical axioms. Some philosophers have argued that we should accept (something like) them as being logical. (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)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

We give an overview of recent results in ordinal analysis. Therefore, we discuss the different frameworks used in mathematical proof-theory, namely "subsystem of analysis" including "reverse mathematics", "Kripke-Platek set theory", "explicit mathematics", "theories of inductive definitions", "constructive set theory", and "Martin-Löf's type theory".

This chapter provides broad coverage of the notion of logical consequence, exploring its modal, semantic, and epistemic aspects. It develops the contrast between proof-theoretic notion of consequence, in terms of deduction, and a model-theoretic approach, in terms of truth-conditions. The main purpose is to relate the formal, technical work in logic to the philosophical concepts that underlie reasoning.

_Dialectic_ is now widely regarded as a classic of contemporary philosophy. This book, first published in 1993, sets itself three main aims: the development of a general theory of dialectic, of which Hegelian dialectic can be seen to be a special case; the dialectical enrichment and deepening of critical realism, viz. into the system of dialectical critical realism; and the outline of the elements of a totalizing critique of Western philosophy. The first chapter clarifies the rational core of (...) Hegelian dialectic. Chapter two then proceeds to develop a general theory of dialectic. Isolating the fallacy of 'ontological monovalence', Roy Bhaskar then shows how absence and other negating concepts such as contradiction have a legitimate and necessary ontological employment. He then goes on to give a synoptic account of key dialectical concepts such as the concrete universal; to sketch the further dialectical development of critical naturalism through an account of what he calls four-planar social being; and following consideration of the dialectical critique of analytical reason, he moves on to the real definition of dialectic as absenting absence and in the human sphere, the axiology of freedom. Chapter three extends and deepens critical realism’s characteristic concerns with ontology, science, social science and emancipation not only into the realms of negativity and totality, but also into the fields of reference and truth, spatio-temporality, tense and process, the logic of dialectical universalizability and on to the plane of ethics, where it articulates a combination of moral realism and ethical naturalism, whereby consideration of elemental desire involves commitment to the eudaimonistic society. This is then followed by a sublime discussion of key moments in the trajectory of Western philosophy, the tradition of which can now be seen to be based on what the author calls the unholy trinity of the epistemic fallacy or the reduction of being to knowledge, primal squeeze or the collapse of structure and alethic truth, and ontological monovalence. (shrink)

A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.

In a career that spans 60 years so far, W.W. Tait has made many highly influential contributions to logic, the philosophy of mathematics, and their history. The present collection of new essays - contributed by former students, colleagues, and friends - is a Festschrift, i.e., a celebration of his life and work. The essays address a variety of themes prominent in his work or related to it. The collection starts with an introduction in which Tait's contributions are sketched and (...) put into context. The eleven essays that follow are arranged in three parts: Part I. Proof Theory and its History; Part II. Logic and Philosophy of Mathematics; and Part III. History of Logic and Philosophy of Mathematics. Each of the essays contributes substantially to one or several of these areas. The authors included are: Steve Awodey, Solomon Feferman, Michael Friedman, Warren Goldfarb, Geoffrey Hellman, William Howard, Stephen Menn, Rebecca Morris, Charles Parsons, Erich Reck, Thomas Ricketts, and Wilfried Sieg. The editor, Erich H. Reck is Professor of Philosophy at the University of California at Riverside. (shrink)

The year 2012 has witnessed worldwide celebrations of Alan Turing’s 100th birthday. A great number of conferences and workshops were organized by logicians, computer scientists and researchers in AI, showing the continued flourishing of computer science, and the fruitful interfaces between logic and computer science. Logic is no longer just the concept that Frege had about one hundred years ago, let alone that of Aristotle twenty centuries before. One of the prominent features of contemporary logic is its interdisciplinary character, connecting (...) across mathematics, philosophy, modern computer science, and even the cognitive and social sciences. -/- China is no exception. This special issue explores interfaces of logic with AI and other disciplines, and it attempts to give a bird’s eye view of the developing landscape of logic and AI in China. Our contributors come from computer science, AI, philosophy and linguistics, united by their interest in using formal logical tools and notions to deal with substantial problems in their fields. We are aware that AI is a vast area, and that the limitations of space in a volume like this may provide a somewhat incomplete or biased view. Still we hope that the six papers selected here illustrate the concerns and curiosity driving Chinese researchers, on topics shared with our colleagues around the world. -/- One of the active research areas in AI is planning under uncertainty, with incomplete knowledge, where a well-established logical framework is the action language of Son and Baral (Son and Baral 2001). This language can express the notions of sensing actions and making conditional plans. However, its current semantics is not yet satisfactory, since according to the semantics an agent would not have knowledge of what happens after a plan is executed, which can be a serious problem in a safety-critical environment. In their paper Proof Systems for Planning under Cautious Semantics (Shen and Zhao 2013), Yuping Shen and Xishun Zhao address this problem and propose a cautious and weakly cautious semantics for the action language. This new semantics enables the agent to know exactly what happens after executing an action. Hoare style proof systems are then proposed and proved to be sound and complete. What is more, the proposed new system reduces the reasoning complexity of the action language. -/- In Preferential Semantics for Plausible Subsumption in Possibility Theory (Qi and Zhang 2013), Guilin Qi and Zhizheng Zhang investigate another important issue in AI: how to handle exceptions in a knowledge-based system. This has become a hot topic in recent years, especially in ontological design, where description logics are widely used. The question then arises whether we can get a nonmonotonic extension of description logic. To answer this, the authors first propose three preferential semantics for plausible subsumption in the framework of possibility theory (Dubois and Prade 1985) and study their properties and their relationships. Then they introduce preferential subsumptions to a description logic-based knowledge system, and define an entailment relation of plausible subsumption. Moreover, they study reductions of the new semantics to standard semantics for an expressive description logic. -/- Counterfactual reasoning has been explored extensively by philosophers, cf. Lewis (1973), Stalnaker (1968). But in the last two decades, causal models have been proposed for modeling counterfactual reasoning by researchers in AI. In A Lewisian Logic of Causal Counterfactuals (Zhang 2013), Jiji Zhang studies classes of causal models and provides formal characterizations. In particular, a new logic system is proposed and proved complete for the class of ‘Lewisian models’ satisfying all the principles of David Lewis’s logic of counterfactuals. Moreover, Zhang compares his new framework with those of Lewis and Stalnaker, and also establishes formal connections with more recent approaches by Galles and Pearl, and by Halpern. -/- Consistency is a fundamental notion in logic, but it also plays a key role in many techniques for solving constraint satisfaction problems (Mackworth 1977). In Variable-Centered Consistency in Model RB (Li et al. 2013), Liang Li, Tian Liu and Ke Xu start with a discussion of i-consistency in ‘Model RB’ and ‘Model RD’, two models of random constraint satisfaction problems with growing domain size, then compare it with the notion of t-consistency and strong t-consistency in well-known earlier work by Freuder. After that, the authors define a new kind of variable-centered consistency, and show an upper bound on a suitably chosen parameter in Model RB and Model RD, such that up to this bound, all typical instances of Model RB are variable-centered consistent. -/- Language understanding is a cognitive process that human beings carry out every day. Metaphor is a pervasive phenomenon in natural languages, and its cognitive role calls for interpretation. There is a growing body of literature on cognitive mechanisms underlying metaphor. The paper An Ontology-Based Approach to Metaphor Cognitive Computation (Huang et al. 2013) by Xiaoxi Huang, Huaxin Huang, Beishui Liao, and Cihua Xu proposes a theoretical framework for metaphor understanding based on recent results from cognitive science (Shutova 2010). They then introduce ontology as a knowledge representation method, and present a new ontological model to formalize metaphor-based knowledge. An algorithmic description is provided using a quantitative measure of integrated degree, and its effectiveness is demonstrated. -/- Puzzles and their solution mechanisms are a recurrent theme in many fields. Dynamic epistemic logic (van Ditmarsch et al. 2007); (van Benthem 2011) is a recent framework that can handle information dynamics in knowledge puzzles. In Reasoning About Agent Types and the Hardest Logic Puzzle Ever (Liu and Wang 2013), Fenrong Liu and Yanjing Wang propose a new dynamic-epistemic logic that can specify different types of agents and analyse agents’ knowledge, communication and reasoning. To demonstrate its power, Smullyan’s well-known Knights and Knaves puzzles are formalized, as well as the Hardest Logic Puzzle Ever (HLPE) of Boolos (Boolos 1996). Moreover, a spectrum of new puzzles is found involving knowledge-based agent types, and their solution complexity is shown to be harder than that for existing logic puzzles. It is also shown that a version of HLPE in which the agents do not know the others’ types does not have a solution at all. -/- Finally, we would like to thank the editor of Minds and Machines, Gregory Wheeler, for his efficient assistance and kind support. We also thank Ties Nijssen at Springer for his encouragement to start this project at the first place. And of course, our appreciation goes to our authors, as well as all the colleagues who have helped us review the papers in this collection. (shrink)

This book is dedicated to V.A. Yankov’s seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic. The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov’s results and their applications in algebraic logic, the theory (...) of admissible rules and refutation systems is included in the book. In addition, the reader can find the studies on splitting and join-splitting in intermediate propositional logics that are based on Yankov-type formulas which are closely related to canonical formulas, and the study of properties of predicate extensions of non-classical propositional logics. The book also contains an exposition of Yankov’s revolutionary approach to constructive proof theory. The editors also include Yankov’s contributions to history and philosophy of mathematics and foundations of mathematics, as well as an examination of his original interpretation of history of Greek philosophy and mathematics. (shrink)

PREFACEProf. Göran Sundholm of Leiden University inspired the group of Logic at Lille and Valparaíso to start a fundamental review of the dialogical conception of logic by linking it to constructive type logic. One of Sundholm's insights was that inference can be seen as involving an implicit interlocutor. This led to several investigations aimed at exploring the consequences of joining winning strategies to the proof-theoretical conception of meaning. The leading idea is, roughly, that while introduction rules lay down the (...) conditions under which a winning strategy for the Proponent may be built, the elimination rules lay down those elements of the Opponent's assertions that the Proponent has the right to use for building winning strategy. It is the pragmatic and ethical features of obligations and rights that naturally lead to the dialogical interpretation of natural deduction.During the Visiting Professorship of Prof. Sundholm at Lille in (2012) the group of Lille started delving into the ways of implementing Per Martin-Löf's Constructive Type Theory with the dialogical perspective. In particular, Aarne Ranta's (1988) paper, the first publication on the subject, was read and discussed during Sundholm’s seminar. The discussions strongly suggested that the game-theoretical conception of quantifiers as deploying interdependent moves provide a natural link between CTT and dialogical logic. This idea triggered several publications by the group of Lille in collaboration with Nicolas Clerbout and Juan Redmond at the University of Valparaíso, including the publication of the book (2015) by Clerbout / Rahman that provides a systematic development of this way of linking CTT and the dialogical conception of logic. However the Clerbout / Rahman book was written from the CTT perspective on dialogical logic, rather than the other way round. The present book should provide the perspective from the other side of the dialogue between the Dialogical Framework and Constructive Type Theory. The main idea of our present study is that Sundholm's (1997) notion of epistemic assumption is closely linked to the Copy-Cat Rule or Socratic Rule that distinguishes the dialogical framework from other game-theoretical approaches. One way to read the present book is as a further development of Sundholm’s extension of Austin's (1946, p. 171) remark on acts of assertion to inference. Indeed, Sundholm (2013, p. 17) gave the following forceful formulation: When I say therefore, I give others my authority for asserting the conclusion, given theirs for asserting the premisses.Per Martin-Löf, in recent lectures, have utilized the dialogical perspective on epistemic assumptions to get out of a certain circle that threatens the explanation of the notions of inference and demonstration. A demonstration may be explained as a chain of (immediate) inferences starting from no premisses. That an inference J1... Jn—————Jis valid means that the conclusion J can be made evident on the assumption that J1, …, Jn are known. The notion of epistemic assumption thus enters in the explanation of valid inference. We cannot, however, in this explanation understand 'known' in the sense of demonstrated, for then we are explaining the notion of inference in terms of demonstration, whereas demonstration has been explained in terms of inference. Martin-Löf suggests that we here understand 'known' in the sense of asserted, so that epistemic assumptions are judgements others have made, judgements for which others have taken the responsibility; that the inference is valid then means that, given that others have taken responsbility for the premisses, I can take responsibility for the conclusion:The circularity problem is this: if you define a demonstration to be a chain of immediate inferences, then you are defining demonstration in terms of inference. Now we are considering an immediate inference and we are trying to give a proper explanation of that; but, if that begins by saying: Assume that J1, …, Jn have been demonstrated – then you are clearly in trouble, because you are about to explain demonstration in terms of the notion of immediate inference, hence when you are giving an account of the notion of immediate inference, the notion of demonstration is not yet at your disposal. So, to say: Assume that J1, …, Jn have already been demonstrated, makes you accusable of trying to explain things in a circle. The solution to this circularity problem, it seems to me now, comes naturally out of this dialogical analysis. […]The solution is that the premisses here should not be assumed to be known in the qualified sense, that is, to be demonstrated, but we should simply assume that they have been asserted, which is to say that others have taken responsibility for them, and then the question for me is whether I can take responsibility for the conclusion. So, the assumption is merely that they have been asserted, not that they have been demonstrated. That seems to me to be the appropriate definition of epistemic assumption in Sundholm's sense. One of the main tenets of the present study is that the further move of relating the Socratic Rule rule with judgemental equality provides both a simpler and more direct way to implement the constructive type theoretical approach within the dialogical framework. Such a reconsideration of the Socratic Rule, roughly speaking, amounts to the following. 1.A move from P that brings-forward a play-object in order to defend an elementary proposition A can be challenged by O. 2.The answer to such a challenge, involves P bringing forward a definitional equality that expresses the fact that the play-object chosen by P copies the one O has chosen while positing A. For short, the equality expresses at the object-languaje level that the defence of P relies on the authority given to O to be able to produce the play-objects she brings forward. More generally, according to this view, a definitional equality established by P and brought forward while defending the proposition A, expresses the equality between a play-object (introduced by O) and the instruction for building a play-object deployed by O while affirming A. So it can be read as a computation rule that indicates how to compute O's instructions. Let us recall that from the strategic point of view, O's moves correspond to elimination rules of demonstrations. Thus, the dialogical rules that prescribe how to introduce a definitional equality – correspond – at the strategic level – to the definitional equality rules for CTT as applied to the selector-functions involved in the elimination rules. So, what we are doing here is extending the dialogical interpretation of Sundholm's epistemic assumption to the rules that set up the definitional equality of a type.The present work is structured in the following way:•In the Introduction we state the main tenets that guide our book•Chapter II contains a brief overview of Constructive Type Theory penned by Ansten Klev with exercises and their solutions. •Chapter III offers a novel presentation of the dialogical conception of logic. In fact, the first two sections of the chapter provide an overview of standard dialogical logic – including solved exercises, slightly adapted to the content of the present volume. The rest of the sections develop a new dialogical approach to Constructive Type Theory. •Chapters IV and V contain the main body of our study, namely the relation between the Socratic Rule, epistemic assumptions, and equality. The chapter provides the reader with thoroughly worked out examples with comments. •After the Final Remarks the book containsa.An appendix containing a an overview of all the rules for the new formulation of dialogical approach to constructive type theory developed in the book; b.a brief outline of Per Martin-Löf's informal presentation of the demonstration of the axiom of choice.c.two examples of a tree-shaped graph of an extensive strategy. Acknowledgments: Many thanks to Mark van Atten (Paris1), Giuliano Bacigalupo (Zürich), Charles Zacharie Bowao (Univ. Ma Ngouabi de Brazzaville, Congo), Christian Berner (Lille3), Michel Crubellier (Lille3), Pierre Cardascia (Lille3), Oumar Dia (Dakar), Marcel Nguimbi ((Univ. Marien Ngouabi de Brazzaville, Congo), Adjoua Bernadette Dango (Lille3 / Alassane Ouattara de Bouaké, Côte d'Ivoire), Steephen Rossy Eckoubili (Lille3), Johan Georg Granström (Zürich), Gerhard Heinzmann (Nancy2), Muhammad Iqbal (Lille3), Matthieu Fontaine (Lille3), Radmila Jovanovic (Belgrad), Hanna Karpenko (Lille3), Laurent Keiff (Lille3), Sébastien Magnier (Saint Dennis, Réunion), Clément Lion (Lille3), Gildas Nzokou (Libreville), Fachrur Rozie (Lille3), Helge Rückert (Mannheim), Mohammad Shafiei (Paris1), and Hassan Tahiri (Lisbon), for rich interchanges, suggestions on the main claims of the present study and proof-reading of specific sections of the text. Special thanks to Gildas Nzokou (Libreville), Steephen Rossy Eckoubili (Lille3) and Clément Lion (Lille3), with whom a Pre-Graduate Textbook in French on dialogical logic and CTT is in the process of being worked out – during the visiting-professorship in 2016 of the former. Eckoubili is the author of the sections on exercises for dialogical logic, Nzokou cared of those involving the development of CTT-demonstrations out of winning-strategies. Special thanks also to Ansten Klev (Prague) for helping with technical details and conceptual issues on CTT, particularly so as author of the chapter introducing CTT. (shrink)

The axiological tenet of scientific realism, “science seeks true theories,” is generally taken to rest on a corollary epistemological tenet, “we can justifiably believe that our successful theories achieve (or approximate) that aim.” While important debates have centered on, and have led to the refinement of, the epistemological tenet, the axiological tenet has suffered from neglect. I offer what I consider to be needed refinements to the axiological postulate. After showing an intimate relation between the refined postulate and ten theoretical (...) desiderata, I argue that the axiological postulate does not depend on its epistemological counterpart; epistemic humility can accompany us in the quest for truth. Upon contrasting my axiological postulate against the two dominant non-realist alternatives and the standard realist postulate, I contend that its explanatory and justificatory virtues render it, among the axiologies considered, the richest account of the scientific enterprise. (shrink)

This book explores the results of applying empirical methods to the philosophy of logic and mathematics. Much of the work that has earned experimental philosophy a prominent place in twenty-first century philosophy is concerned with ethics or epistemology. But, as this book shows, empirical methods are just as much at home in logic and the philosophy of mathematics. -/- Chapters demonstrate and discuss the applicability of a wide range of empirical methods including experiments, surveys, interviews, (...) and data-mining. Distinct themes emerge that reflect recent developments in the field, such as issues concerning the logic of conditionals and the role played by visual elements in some mathematical proofs. -/- Featuring leading figures from experimental philosophy and the fields of philosophy of logic and mathematics, this collection reveals that empirical work in these disciplines has been quietly thriving for some time and stresses the importance of collaboration between philosophers and researchers in mathematics education and mathematical cognition. (shrink)

Hao Wang was one of the few confidants of the great mathematician and logician Kurt Gödel. _A Logical Journey_ is a continuation of Wang's _Reflections on Gödel_ and also elaborates on discussions contained in _From Mathematics to Philosophy_. A decade in preparation, it contains important and unfamiliar insights into Gödel's views on a wide range of issues, from Platonism and the nature of logic, to minds and machines, the existence of God, and positivism and phenomenology. The impact of Gödel's theorem (...) on twentieth-century thought is on par with that of Einstein's theory of relativity, Heisenberg's uncertainty principle, or Keynesian economics. These previously unpublished intimate and informal conversations, however, bring to light and amplify Gödel's other major contributions to logic and philosophy. They reveal that there is much more in Gödel's philosophy of mathematics than is commonly believed, and more in his philosophy than his philosophy of mathematics. Wang writes that "it is even possible that his quite informal and loosely structured conversations with me, which I am freely using in this book, will turn out to be the fullest existing expression of the diverse components of his inadequately articulated general philosophy." The first two chapters are devoted to Gödel's life and mental development. In the chapters that follow, Wang illustrates the quest for overarching solutions and grand unifications of knowledge and action in Gödel's written speculations on God and an afterlife. He gives the background and a chronological summary of the conversations, considers Gödel's comments on philosophies and philosophers, and his attempt to demonstrate the superiority of the mind's power over brains and machines. Three chapters are tied together by what Wang perceives to be Gödel's governing ideal of philosophy: an exact theory in which mathematics and Newtonian physics serve as a model for philosophy or metaphysics. Finally, in an epilog Wang sketches his own approach to philosophy in contrast to his interpretation of Gödel's outlook. (shrink)

This volume offers a wide range of both reconstructions of Nikolai Vasiliev’s original logical ideas and their implementations in the modern logic and philosophy. A collection of works put together through the international workshop "Nikolai Vasiliev’s Logical Legacy and the Modern Logic," this book also covers foundations of logic in the light of Vasiliev’s contradictory ontology. Chapters range from a look at the Heuristic and Conceptual Background of Vasiliev's Imaginary Logic to Generalized Vasiliev-style Propositions. It includes works which cover (...) Imaginary and Non-Aristotelian Logics, Inconsistent Set Theory and the Expansion of Mathematical Thinking, Plurivalent Logic, and the Impact of Vasiliev's Imaginary Logic on Epistemic Logic. The Russian logician, Vasiliev, was widely recognized as one of the forerunners of modern non-classical logic. His "imaginary logic" developed in some of his work at the beginning of 20th century is often considered to be one of the first systems of paraconsistent and multi-valued logic. The novelty of his logical project has opened up prospects for modern logic as well as for non-classical science in general. This volume contains a selection of papers written by modern specialists in the field and deals with various aspects of Vasiliev's logical ideas. The logical legacy of Nikolai Vasiliev can serve as a promising source for developing an impressive range of philosophical interpretations, as it marries promising technical innovations with challenging philosophical insights. (shrink)

In this book, I will discuss three main topics: the roots and aims of scientific knowledge, scientific knowledge in society, and science and values I understand scientific knowledge as being a planned and continuous production of the general and common knowledge of scientific communities. I begin my discussion with a brief analysis of the main differences between sciences, on the one hand, and everyday experience, philosophies, religions, and ideologies, on the other. I define the concept of science as a set (...) of family resemblances (in Wittgenstein's sense). The sciences can be distinguished along three lines, in terms of the difference between empirical and non-empirical sciences, differences in degrees of theoretical and methodological structure among individual sciences and scientific disciplines. Scientific knowledge is based upon some grounds, paradigmatic knowledge bases. Scientific knowledge is founded on a unified paradigmatic knowledge base. The latter includes fundamental theories, basic facts, generally accepted empirical regularities (laws), scientific research methods, scientific explanations, and established relations to other sciences. Scientific knowledge in general is defined well in classical terms by the definition that knowledge is a true and rationally-justified belief. In this connection, scientific truth is relative, depending on the given paradigmatic knowledge base and on the precision and accessibility of the measurements and observations involved. In the book's first part, I consider the production and testing of scientific hypotheses, scientific explanations, the structure of scientific theories, and the problem of scientific realism. I consider explanation an integral part of a wider nexus of heuristic and theoretical activities in science like giving reasons, understanding, clarification, and founding. Continuous production of scientific explanations is a necessary condition for the systematic reproduction of scientific knowledge. Accordingly, it does not suffice to describe them in terms of the logical structure of the arguments on which they are based, but rather, scientific explanations ought to be given complete with an epistemic and pragmatic context of the explanation. I consider especially deductive-nomological, inductive-statistical, dispositional, teleological-functional, and genetic explanations. I emphasize the ever-increasing importance of explanations by analogy and by model. Orthogonal to these distinctions lies the distinction between causal and epistemic explanation. Epistemic explanations relate scientific discoveries to epistemic progress, and causal explanations are these epistemic explanations that relate scientific discoveries to reality too. Causal explanations inform us that certain changes occur in the world as agents of other changes, or rather, that certain changes are real consequences of other changes. Thus, by having explanations of causes, we can intervene in the world to induce by design certain changes in some course of phenomena. I consider scientific theories as logically and epistemically ordered wholeness of scientific knowledge in a realm of research that enables us to construct useful models of reality. After the discussion of different 'statement' views on theories and different interpretations of the theory-experience relation, I introduce the 'model' theoretical (non-statement) views on theories, especially the structural theory of science. The model views on theories transcend the realism-antirealism dualism by introducing models between theory and reality. Therefore, the question of scientific realism is not merely whether and, if so, which element(s), of scientific theories represent reality, but rather which conditions suffice to infer from a theory that one of its models truly expresses reality. In other words, under which conditions do models of a theory represent models of reality? Generally, models are just as much images of reality in a theory as they are images of a theory in reality. The actual problem of scientific realism doesn't originate in the observable-unobservable distinction (even if this distinction is otherwise theoretically loaded), but rather in ontological quandaries, of the sort especially encountered in modern physical theories such as quantum mechanics, relativity theory, and cosmological theories, although they are known to arise in other sciences as well. Scientific theories may be real, even if they describe nothing about the world. However, such theories can be given this status only if their formal, mathematical postulates exhibit some formal homology between reality and mathematical forms. The very fact that it is possible to describe reality with the help of mathematics even in cases, where every (representational, conceptual) language and thought fall short, would indicate that even mathematics itself is, in some sense, a part of reality, although 'outside' of the empirical world. It follows we have to distinguish the non-empirical reality and the empirical world. Formal objectivity and non-arbitrariness of theories do not emerge from either our language or thoughts, or from the world of facts, causes, and effects, but rather from non-empirical reality, which frames the world as some kind of objective transcendental condition for the possibility of the existence of the world, thoughts, and language. In the second part of the book, I consider the incorporation of scientific work and scientific knowledge into modern society and its relations to values and ethics. Production of scientific knowledge goes on as a parallel distributed process of a special kind of social (shared) cognition. The central part of this cognition is the presentation of parts of the world by scientific (theoretical) models which are warranted by the shared and public practice of justification, and in reflective acting in the world as it would. I consider various formal concepts of collective propositional knowledge (distributed, mutual, common knowledge). Scientific knowledge is an achievement to be ascribed to a scientific community or, indirectly, to all those who apply or expand it, rather than to individuals per se. It presents the continuous and partially planned production of non-trivial common knowledge of the research communities in modern societies. However, besides common knowledge, shared dispositional knowledge (knowledge how of scientific teams) and shared knowledge by acquiring (knowledge of fundamental similarities and differences in scientific teams) are parts of the real body of scientific knowledge. Scientific common knowledge implies the projection into idealized universal knowledge of all real or potentially rational and well-informed individuals. The respect of scientific knowledge thus presents the norm of rationality. Science as the intensively shared and socially distributed kind of cognition enters into modern society as a special kind of work (general work), as a productive force (social invention drive), and as a form of capital (human, social, cultural, informational, etc.). However any of these categories can adequately represent the socio-economical role of science in modern societies, and we are still waiting for a more adequate general concept that would theoretically cover the 'scientification' of modern societies. My thesis is that the most important general effect of scientification is (and will be) the growth of social processes that structurally resemble parallel distributed shared social cognition of science, and not so much the rise of the use of scientific knowledge and scientifically based information in all kinds of production. Scientification of modern societies inevitably includes a much stronger need for moral and social responsibility of scientists for their work and its effects in their local society and the global world. This need comes in contradiction with the quite large accepted idea (and even the 'norm') on value-free research. I discuss to some degree different arguments for the absence of non-epistemic values and ethical reasoning from science and find them non-convincing. The sound idea of value-free research lies in the need for the absence of categorical value statements from scientific explanations and theoretical deductions. Values and ethical judgment have, and sometimes they even have to play an important role in another part of research and its application in science or outside it. Not the value-loudness of research is a danger for science but its blind, non-reflective use. It leads to the ideologization and politicization of science. A similar danger lies in the purely value-neutrality of research. It leads to the instrumentalization of science to the technical or bureaucratic means for 'any' aims. (shrink)

Joel Feinberg : In Memoriam. Preface. Part I: INTRODUCTION TO THE NATURE AND VALUE OF PHILOSOPHY. 1. Joel Feinberg: A Logic Lesson. 2. Plato: "Apology." 3. Bertrand Russell: The Value of Philosophy. PART II: REASON AND RELIGIOUS BELIEF. 1. The Existence and Nature of God. 1.1 Anselm of Canterbury: The Ontological Argument, from Proslogion. 1.2 Gaunilo of Marmoutiers: On Behalf of the Fool. 1.3 L. Rowe: The Ontological Argument. 1.4 Saint Thomas Aquinas: The Five Ways, from Summa Theologica. (...) 1.5 Samuel Clarke: A Modern Formulation of the Cosmological Argument. 1.6 William L. Rowe: The Cosmological Argument. 1.7 William Paley: The Argument from Design. 1.8 Michael Ruse: The Design Argument. 1.9 David Hume: Dialogues Concerning Natural Religion. 2. The Problem of Evil. 2.1 Fyodor Dostoyevsky: Rebellion. 2.2 J. L. Mackie: Evil and Omnipotence. 2.3 Peter van Inwagen: The Argument from Evil. 2.4 Michael Murray and Michael Rea: The Argument from Evil. 2.5 B. C. Johnson: God and the Problem of Evil. 3. Reason and Faith. 3.1 W. K. Clifford: The Ethics of Belief. 3.2 William James: The Will to Believe. 3.3 Kelly James Clark: Without Evidence or Argument. 3.4 Blaise Pascal: The Wager. 3.5 Lawrence Shapiro: Miracles and Justification. 3.6 Simon Blackburn: Infini-Rien. Part III. HUMAN KNOWLEDGE: ITS GROUNDS AND LIMITS. 1. Skepticism. 1.1 John Pollock: A Brain in a Vat. 1.2 Michael Huemer: Three Skeptical Arguments. 1.3 Robert Audi: Skepticism. 2. The Nature and Value of Knowledge. 2.1 Plato: Knowledge as Justified True Belief. 2.2 Edmund Gettier: Is Justified True Belief Knowledge? 2.3 James Cornman, Keith Lehrer, and George Pappas: An Analysis of Knowledge. 2.4 Gilbert Ryle: Knowing How and Knowing That. 2.5 Plato: "Meno". 2.6 Linda Zagzebski, Epistemic Good and The Good Life. 3. Our Knowledge of the External World. 3.1 Bertrand Russell: Appearance and Reality and the Existence of Matter. 3.2 René Descartes: Meditations on First Philosophy. 3.3 John Locke: The Causal Theory of Perception. 3.4 George Berkeley: Of the Principles of Human Knowledge. 3.5 G. E. Moore: Proof of an External World. 4. The Methods of Science. 4.1 David Hume: An Inquiry Concerning Human Understanding. 4.2 Wesley C. Salmon: An Encounter with David Hume. 4.3 Karl Popper: Science: Conjectures and Refutations. 4.4 Philip Kitcher: Believing Where We Cannot Prove. Part IV: MIND AND ITS PLACE IN NATURE. 1. The Mind-Body Problem. 1.1 Brie Gertler: In Defense of Mind--Body Dualism. 1.2 Frank Jackson: The Qualia Problem. 1.3 David Papineau: The Case for Materialism. 1.4 Paul Churchland: Functionalism and Eliminative Materialism. 2. Can Non-Humans Think? 2.1 Alan Turing: Computing Machinery and Intelligence. 2.2 John R. Searle: Minds, Brains, and Programs. 2.3 William G. Lycan: Robots and Minds. 3. Personal Identity and the Survival of Death. 3.1 John Locke: The Prince and the Cobbler. 3.2 Thomas Reid: Of Mr. Locke’s Account of Our Personal Identity. 3.3 David Hume: The Self. 3.4 Derek Parfit: Divided Minds and the Nature of Persons. 3.5 Shelly Kagan: What Matters. 3.6 John Perry: A Dialogue on Personal Identity and Immortality. Part V: DETERMINISM, FREE WILL, AND RESPONSIBILITY. 1. Libertarianism: The Case for Free Will and Its Incompatibility with Determinism. 1.1 Roderick M. Chisholm: Human Freedom and the Self. 1.2 Robert Kane: Free Will: Ancient Dispute, New Themes. 2. Hard Determinism: The Case for Determinism and its Incompatibility with Its Incompatibility with Any Important Sense of Free Will. 2.1 James Rachels: The case against Free Will. 2.2 Derk Pereboom: Why We Have No Free Will and Can Live Without It. 3. Compatibilism: The Case for Determinism and Its Compatibility with the Most Important Sense of Free Will. 3.1 David Hume: Of Liberty and Necessity. 3.2 Helen Beebee: Compatibilism and the Ability to do Otherwise. 4. Freedom and Moral Responsibility. 4.1 Galen Strawson: Luck Swallows Everything. 4.2 Harry Frankfurt: Alternate Possibilities and Moral Responsibility. 4.3 Thomas Nagel: Moral Luck. 4.4 Susan Wolf: Sanity and the Metaphysics of Responsibility. Part VI: MORALITY AND ITS CRITICS. 1. Changes to Morality. 1.1 Joel Feinberg: Psychological Egoism. 1.2 Plato: The Immoralist’s Challenge. 1.3 Friedrich Nietzche: Master and Slave Morality. 1.4 Richard Joyce: The Evolutionary Debunking of Morality. 2. Proposed Standards and Right of Conduct. 2.1 Russ Shafer-Landau: Ethical Subjectivism. 2.2 Mary Midgley: Trying Out One’s New Sword. 2.3 Aristotle: Virtue and the Good Life. 2.4 Thomas Hobbes: Leviathan. 2.5 Plato: Euthyphro. 2.6 Immanuel Kant: The Good Will and the Categorical Imperative. 2.7 J.S. Mill: Utilitarianism, Chapters 2 and 4. 2.8 W. D. Ross: What Makes Right Acts Right? 2.9 Hilde Lindemann: What Is Feminist Ethics? 3. Ethical Problems. 3.1 Kwame Anthony Appiah: What Will Future Generations Condemn Us For? 3.2 Peter Singer: Famine, Affluence and Morality. 3.3 John Harris: The Survival Lottery. 3.4 James Rachels: Active and Passive Euthanasia. 3.5 Mary Anne Warren: The Moral and Legal Status of Abortion. 3.6 Don Marquis: Why Abortion Is Immoral. 4. The Meaning of Life. 4.1 Epicurus: Letter to Menoeceus. 4.2 Richard Taylor: The Meaning of Life. 4.3 Richard Kraut: Desire and the Human Good. 4.4 Leo Tolstoy: My Confession. 4.5 Susan Wolf: Happiness and Meaning. 4.6 Thomas Nagel: The Absurd. (shrink)

A combination of epistemic logic and dynamic logic of programs is presented. Although rich enough to formalize some simple game-theoretic scenarios, its axiomatization is problematic as it leads to the paradoxical conclusion that agents are omniscient. A cut-free labelled Gentzen-style proof system is then introduced where knowledge and action, as well as their combinations, are formulated as rules of inference, rather than axioms. This provides a logical framework for reasoning about games in a modular and systematic way, and to (...) give a step-by-step reconstruction of agents omniscience. In particular, its semantic assumptions are made explicit and a possible solution can be found in weakening the properties of the knowledge operator. (shrink)

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture (...) of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

Charles S. Peirce attempted to develop his semiotic theory of cognitive signs interpretation, which are originated in our basic perceptual operations that quasi-prove the truth of perceptual judgment representing reality. The essential problem was to explain how, by a cognitive interpretation of the sequence of perceptual signs, we can represent external physical reality and reflectively represent our cognitive mind’s operations of signs. With his phaneroscopy introspection, Peirce shows how, without going outside our cognitions, we can represent external reality. Hence Peirce (...) can avoid the Berkeleyian, Humean, and Kantian phenomenologies, as well as the modern analytic philosophy and hermeneutic phenomenology. Peirce showed that with the trio of semiotic interpretation – abductive logic of discovery of hypotheses, deductive logic of necessary inference, and inductive logic of evaluation – we can reach a complete proof of the true representation of reality. This semiotic logic of reasoning is the epistemic logic representing human confrontation in reality, with which we can achieve knowledge and conduct our behavior. However, Peirce did not complete his realistic revolution to eliminate previously accepted nominalistic and idealistic epistemologies of formal logic and pure mathematics. Here, I inquire why Peirce did not complete his historical realist epistemological revolution and following that inquiry I attempt to reconstruct it. (shrink)

In this paper, we explore certain exemplifications of transgression in the history and philosophy of mathematics. We recognize transgressive acts in the transition from a “real” to an “imaginary” world. Further, we suggest the concept of proof-events that transgress traditional concepts of mathematical proof. The theory of proof-events provides us with means to identify transgressive acts in the development of a discovery proof-event. These concern the creative understanding of a purported mathematical proof by reconstructing (...) the meaning conveyed by it, eventually supplementing it, surpassing stylistic hindrances in the narrative formulated by a prover, exceeding the boundaries of the (local) logic of a prover, etc. Finally, we identify transgressive acts with discontinuous points of change in the development of proof-events and suggest formal measures for their recognition. (shrink)

Recent years have seen an explosion of interest in evolutionary debunking arguments directed against certain types of belief, particularly moral and religious beliefs. According to those arguments, the evolutionary origins of the cognitive mechanisms that produce the targeted beliefs render these beliefs epistemically unjustified. The reason is that natural selection cares for reproduction and survival rather than truth, and false beliefs can in principle be as evolutionarily advantageous as true beliefs. The present volume brings together fourteen essays that examine evolutionary (...) debunking arguments not only in ethics and philosophy of religion, but also in philosophy of mathematics, metaphysics, and epistemology. The essays move forward research on those arguments by shedding fresh light on old problems and proposing new lines of inquiry. The book will appeal to scholars and graduate students interested in the possible skeptical implications of evolutionary theory in any of the above domains. (shrink)

Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions of (...) such systems from logic to mathematics, and on the connection between the two main forms of structural proof theory - natural deduction and sequent calculus. The authors emphasize the computational content of logical results. A special feature of the volume is a computerized system for developing proofs interactively, downloadable from the web and regularly updated. (shrink)

The concept of truth of the mathematical theory plays an important methodological role in philosophy of mathematics in the context of a link between ontological and gnoseological aspects of a problem of mathematics substantiation. In the article, the author analyzes how the features of mathematical knowledge are reflected in the understanding of concept of truth in the contemporary mathematics. The author claims that the concept of truth of mathematical offers and theorems is metamathematical concept; therefore, underestimation of distinction between (...) mathematics and metamathematics can lead to philosophical misunderstanding. They consist primarily in the fact that theoretically possible level of rigor of contemporary mathematical theories has been virtually represented in the science. In order to resolve the fundamental philosophical and methodological difficulties of the problem of mathematics substantiation, the formal provability in logically consistent theory is necessary. However, contrary to the widespread belief, a rigid formalization of the proof is still not the synonymous with reliability and rigor of mathematical reasoning from the standpoint of the philosophy of contemporary mathematics substantiation. (shrink)

What is mathematics about? Does the subject-matter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics but with some experience of formal logic (...) it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics - Platonism, realism, logicism, formalism, constructivism and structuralism - as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy present-day philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upper-level undergraduate courses and the non-specialist. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Cambridge History of Seventeenth-Century Philosophy by Daniel Garber, Michael AyersDonald RutherfordDaniel Garber, Michael Ayers, editors. The Cambridge History of Seventeenth-Century Philosophy. 2 vols. Cambridge: Cambridge University Press, 1998. Pp. xii + 1616. Cloth, $175.Over a decade in preparation, this latest addition to the Cambridge History of Philosophy is an enormous achievement—both in its size and the contribution it makes to redefining [End Page 165] (...) the landscape of seventeenth-century philosophy. The editors make no bones about their intention to rewrite the history of early modern philosophy, reversing the trend dominant through much of this century of reading seventeenth-century philosophers in the light of twentienth-century “interests and preconceptions” (4). As they note, the movement to historicize our understanding of seventeenth-century philosophy has been underway for some time in the English-speaking world, furthered in many cases by contributors to these volumes. From this perspective, the present work represents, if not the culmination, at least a progress report on these efforts and the success they have achieved.As in previous volumes of the Cambridge History, the subject matter is treated thematically. Consistent with their larger goal, the editors have further attempted to organize the volumes in a way that remains faithful to the expectations of the period. Part I sets the stage with chapters that establish the larger social and historical context: “The institutional setting” (Richard Tuck); “The intellectual setting” (Stephen Menn); and “European responses to non-European culture: China” (D.E. Mungello). For the rest, the editors write, “the structure of the collection corresponds to one way, at any rate, in which an educated European of the seventeenth century might have organized the domain of philosophy” (2). First comes logic (including method and general ontology); then, the primary categories of being (God, body, soul); finally, the basic operations of the soul (understanding, will, action), whose treatment comprehends the topics of epistemology and ethics.Part II, “Logic, Language, and Abstract Objects,” opens with three short chapters by Gabriel Nuchelmans that survey the terrain of seventeenth-century logic according to the traditional divisions of term, proposition/judgment, and argument. These are followed by contributions on “Method and the study of nature” (Peter Dear); “Universal, essences, and abstract entities” (Martha Bolton); and “Individuation” (Udo Thiel).The discussion of metaphysics proper begins in Part III with the most fundamental being: God. Included here are chapters that examine specific theological issues—“The idea of God” (Jean-Luc Marion); “Proofs of the existence of God” (Jean-Robert Armogathe); “The Cartesian dialectic of creation” (Thomas M. Lennon)—as well as ones that deal more broadly with the crucial connection between philosophy and religion: “The relation between theology and philosophy” (Nicholas Jolley) and “The religious background of seventeenth-century philosophy” (Richard Popkin).Part IV, “Body and the Physical World,” forms the single largest part of the book and its intellectual core. It opens with two chapters—“The scholastic background” (Roger Ariew and Alan Gabbey) and “The occultist tradition and its critics” (Brian Copenhaver)—that effectively mark out the boundaries of the “new philosophy” that emerges in conjunction with the development of seventeenth-century science. These are followed by a series of chapters that examine particular features of that philosophy: “Doctrines of explanation in late scholasticism and in the mechanical philosophy” (Steven Nadler); “New doctrines of body and its powers, place, and space” (Daniel Garber, John Henry, Lynn Joy, and Alan Gabbey); “Knowledge of the existence of body” (Charles McCracken); “New doctrines of motion” (Alan Gabbey); “Laws of nature” (J.R. Milton); and “The mathematical realm of nature” (Michael Mahoney). [End Page 166]The final section of Volume I takes up the third main category of being: “Spirit.” It begins with Daniel Garber’s overview of the positions of the major figures, “Soul and mind: Life and thought in the seventeenth century,” followed by chapters on “Knowledge of the soul” (Charles McCracken); “Mind-body problems” (Daniel Garber and Margaret Wilson); “Personal identity” (Udo Thiel); and “The passions in metaphysics and the theory of action” (Susan James).Volume II of the History is devoted to topics that, in one way or another, elaborate operations of the soul. Part V, “The... (shrink)

In the Tractatus, it is stated that questions about logical formatting cannot be meaningfully formulated, since it is precisely the application of logical rules which enables the formulation of a question whatsoever; analogously, Wittgenstein’s celebrated infinite regress argument on rule-following seems to undermine any explanation of deduction, as relying on a logical argument. On the other hand, some recent mathematical developments of the Curry-Howard bridge between proof theory and type theory address the issue of describing the “subjective” side of (...) logic, that is, the concrete manipulation of rules and proofs in space and time. It is advocated that such developments can shed some light on the question of logical formatting and its apparently unintelligible paradoxes, thus reconsidering Wittgenstein’s verdict. (shrink)