Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. In this survey article, the view is clarified and distinguished from some related views, and arguments for and against the view are discussed.
Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...) description of Mars. But whereas Mars is a physical object, the number 3 is (according to platonism) an abstract object. And abstract objects, platonists tell us, are wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal. Thus, on this view, the number 3 exists independently of us and our thinking, but it does not exist in space or time, it is not a physical or mental object, and it does not enter into causal relations with other objects. This view has been endorsed by Plato, Frege (1884, 1893-1903, 1919), Gödel (1964), and in some of their writings, Russell (1912) and Quine (1948, 1951), not to mention numerous more recent philosophers of mathematics, e.g., Putnam (1971), Parsons (1971), Steiner (1975), Resnik (1997), Shapiro (1997), Hale (1987), Wright (1983), Katz (1998), Zalta (1988), and Colyvan (2001). (shrink)
One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. (...) Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics. (shrink)
If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...) with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics. (shrink)
Kurt Gödel (1906 - 1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computability theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his (...) discovery of unusual cosmological models for Einstein's equations, in theory permitting time travel into the past. The Collected Works is a landmark resource that draws together a lifetime of creative thought and accomplishment. The first two volumes were devoted to Gödel's publications in full (both in original and translation), and the third volume featured a wide selection of unpublished articles and lecture texts found in Gödel's Nachlass. These long-awaited final two volumes contain Gödel's correspondence of logical, philosophical, and scientific interest. Volume IV covers A to G, with H to Z in volume V; in addition, Volume V contains a full inventory of Gödel's Nachlass. L All volumes include introductory notes that provide extensive explanatory and historical commentary on each body of work, English translations of material originally written in German (some transcribed from the Gabelsberger shorthand), and a complete bibliography of all works cited. Kurt Gödel: Collected Works is designed to be useful and accessible to as wide an audience as possible without sacrificing scientific or historical accuracy. The only comprehensive edition of Gödel's work available, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science and all others who wish to be acquainted with one of the great minds of the twentieth century. (shrink)
The prevailing pedagogical approach in business ethics generally underestimates or even ignores the powerful influences of situational factors on ethical analysis and decision-making. This is due largely to the predominance of philosophy-oriented teaching materials. Social psychology offers relevant concepts and experiments that can broaden pedagogy to help students understand more fully the influence of situational contexts and role expectations in ethical analysis. Zimbardo's Stanford Prison Experiment is used to illustrate the relevance of social psychology experiments for business ethics instruction.
P. Kyle Stanford (2000) attempts to offer a truth-linked explanation of the success of science which, he thinks, can be welcome to antirealists. He proposes an explanation of the success of a theory T1 in terms of its predictive similarity to the true theory T of the relevant domain. After raising some qualms about the supposed antirealist credentials of Stanford's account, I examine his explanatory story in some detail and show that it fails to offer a satisfactory explanation (...) of the success of science. (shrink)
In this entry, the authors outline the goals of a "dynamic reference work", and explain how the Stanford Encyclopedia of Philosophy has been designed to achieve those goals.
In this short letter to Ed Zalta we raise a number of issues with regards to his version of Neo-Logicism. The letter is, in parts, based on a longer manuscript entitled “What Neo-Logicism could not be” which is in preparation. A response by Ed Zalta to our letter can be found on his website: http://mally.stanford.edu/publications.html (entry C3).
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A comparison of the engineering schools at UC Berkeley and Stanford during the 1940s and 1950s shows that having an excellent academic program is necessary but not sufficient to make a university entrepreneurial (an engine of economic development). Key factors that made Stanford more entrepreneurial than Cal during this period were superior leadership and a focused strategy. The broader institutional context mattered as well. Stanford did not have the same access to state funding as (...) public universities (such as Cal in the period under consideration) and some private universities (such as the Massachusetts Institute of Technology and the Johns Hopkins University in their early histories). Therefore, in order to gather resources, Stanford was forced to become entrepreneurial first, developing business skills (engaging with high-tech industry) at the same time Cal was developing political skills (protecting and increasing its state appropriation). Stanford’s early development of entrepreneurial business skills played a crucial role in the development of Silicon Valley. (shrink)
The origin of my article lies in the appearance of Copeland and Proudfoot's feature article in Scientific American, April 1999. This preposterous paper, as described on another page, suggested that Turing was the prophet of 'hypercomputation'. In their references, the authors listed Copeland's entry on 'The Church-Turing thesis' in the Stanford Encyclopedia. In the summer of 1999, I circulated an open letter criticising the Scientific American article. I included criticism of this Encyclopedia entry. This was forwarded (by Prof. Sol (...) Feferman) to Prof. Ed Zalta, editor of the Encyclopedia, and after some discussion he invited me to submit an entry on 'Alan Turing.'. (shrink)
The Stanford Encyclopedia of Philosophy is an open access, dynamic reference work designed to organize professional philosophers so that they can write, edit, and maintain a reference work in philosophy that is responsive to new research. From its inception, the SEP was designed so that each entry is maintained and kept up to date by an expert or group of experts in the field. All entries and substantive updates are refereed by the members of a distinguished Editorial Board before (...) they are made public. (shrink)
Feelings and experiences vary widely. For example, I run my fingers over sandpaper, smell a skunk, feel a sharp pain in my finger, seem to see bright purple, become extremely angry. In each of these cases, I am the subject of a mental state with a very distinctive subjective character. There is something it is like for me to undergo each state, some phenomenology that it has. Philosophers often use the term ‘qualia’ (singular ‘quale’) to refer to the introspectively accessible, (...) phenomenal aspects of our mental lives. In this standard, broad sense of the term, it is difficult to deny that there are qualia. Disagreement typically centers on which mental states have qualia, whether qualia are intrinsic qualities of their bearers, and how qualia relate to the physical world both inside and outside the head. The status of qualia is hotly debated in philosophy largely because it is central to a proper understanding of the nature of consciousness. Qualia are at the very heart of the mindbody problem. (shrink)
Notice: This PDF version was distributed by request to members of the Friends of the SEP Society and by courtesy to SEP content contributors. It is solely for their fair use. Unauthorized distribution is prohibited. To learn how to join the Friends of the..
One of the many virtues of Martin Seel’s Aesthetics of Appearing is that it lays its cards on the table at the very outset. The final three chapters consist in a series of complex digressions from the main discussion: one on the aesthetic significance of ‘resonating’(p. 139), one organized around the metaphysics of pictures, and one charged with defending the implausible claim that the artistic representation of violence is uniquely capable of revealing ‘what is violent about violence’ (p. 191). But (...) the thesis of the book and its main arguments are stated in the preface, preceding even the acknowledgements. Seel writes, ‘[t]his book makes the proposal of having aesthetics begin not with concepts of being‐so or semblance but with a concept of appearing’ (p. xi). This might initially seem opaque, as though reducing aesthetics to a subtlety involving the meaning of the Greek word phainomai, but Seel immediately clarifies the stance that he wishes to advance. Seel’s position is that aesthetics is distinguished by attention to the indeterminable particularity of sensory experience; aesthetics so considered comprises the philosophy of art as well as non‐art experience; aesthetic experience is a legitimate mode of world‐encounter by virtue of its immediacy or ‘presence’ (p. xi)(Gegenwart – i.e., a contrary of ‘past’, rather than of ‘absence’); and because the presence of our experience reveals the presence of our lives, aesthetic experience constitutes an important form of self‐knowledge. The subsequent chapters are devoted to explicating this position in extraordinary detail. Seel’s position depends on a somewhat implicit account of subjectivity. In this account, what we fundamentally perceive, conditioned by conceptual activity but transcending any possible determinate content, is a ‘play’ of sensuous qualities (p. 47). Since it is ‘unfettered’ (p. 51) by theoretical interest, this form of perception is far qualitatively richer than our more structured experiences: here one is ‘able to perceive.... (shrink)
This book introduces a new approach to the issue of radical scientific revolutions, or "paradigm-shifts," given prominence in the work of Thomas Kuhn. The book articulates a dynamical and historicized version of the conception of scientific a priori principles first developed by the philosopher Immanuel Kant. This approach defends the Enlightenment ideal of scientific objectivity and universality while simultaneously doing justice to the revolutionary changes within the sciences that have since undermined Kant's original defense of this ideal. Through a modified (...) Kantian approach to epistemology and philosophy of science, this book opposes both Quinean naturalistic holism and the post-Kuhnian conceptual relativism that has dominated recent literature in science studies. Focussing on the development of "scientific philosophy" from Kant to Rudolf Carnap, along with the parallel developments taking place in the sciences during the same period, the author articulates a new dynamical conception of relativized a priori principles. This idea applied within the physical sciences aims to show that rational intersubjective consensus is intricately preserved across radical scientific revolutions or "paradigm-shifts and how this is achieved. (shrink)
Logical AI involves representing knowledge of an agent’s world, its goals and the current situation by sentences in logic. The agent decides what to do by inferring that a certain action or course of action is appropriate to achieve the goals. We characterize briefly a large number of concepts that have arisen in research in logical AI. Reaching human-level AI requires programs that deal with the common sense informatic situation. This in turn requires extensions from the way logic has been (...) used in formalizing branches of mathematics and physical science. It also seems to require extensions to the logics themselves, both in the formalism for expressing knowledge and the reasoning used to reach conclusions. A large number of concepts need to be studied to achieve logical AI of human level. This article presents candidates. The references, though numerous, to articles concerning these concepts are still insuf- ficient, and I’ll be grateful for more, especially for papers available on the web. This article is available in several forms via http://www-formal.stanford.edu/jmc/conceptsai.html. (shrink)
The story goes that Epimenides, a Cretan, used to claim that all Cretans are always liars. Whether he knew it or not, this claim is odd. It is easy to see it is odd by asking if it is true or false. If it is true, then all Cretans, including Epimenides, are always liars, in which case what he said must be false. Thus, if what he says is true, it is false. Conversely, suppose what Epimenides said is false. Then (...) some Cretan at some time speaks truly. This might not tell us anything about Epimenides. But if, to make the story simple, he were the only Cretan ever to speak, and this was the only thing he ever said, then indeed, he would have to speak truly. And we would then have shown that if what he said was false, it must be true. (shrink)
Religious violence is a function of deep philosophical and psychological belief-behavior. This article explores the issue in light of Hinduism, Islam, Christianity, and Psychology of evil.
All normal human beings alive in the last fifty thousand years appear to have possessed, in Mark Turner's phrase, "irrepressibly artful minds." Cognitively modern minds produced a staggering list of behavioral singularities--science, religion, mathematics, language, advanced tool use, decorative dress, dance, culture, art--that seems to indicate a mysterious and unexplained discontinuity between us and all other living things. This brute fact gives rise to some tantalizing questions: How did the artful mind emerge? What are the basic mental operations that make (...) art possible for us now, and how do they operate? These are the questions that occupy the distinguished contributors to this volume, which emerged from a year-long Getty-funded research project hosted by the Center for Advanced Study in the Behavioral Sciences at Stanford. These scholars bring to bear a range of disciplinary and cross-disciplinary perspectives on the relationship between art (broadly conceived), the mind, and the brain. Together they hope to provide directions for a new field of research that can play a significant role in answering the great riddle of human singularity. (shrink)
The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...) the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences — many indispensable, some startling — and has come to figure prominently in discussions on the foundations of mathematics. It (or its equivalents) have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it. (shrink)
Peirce's Sign Theory, or Semiotic, is an account of signification, representation, reference and meaning. Although sign theories have a long history, Peirce's accounts are distinctive and innovative for their breadth and complexity, and for capturing the importance of interpretation to signification. For Peirce, developing a thoroughgoing theory of signs was a central philosophical and intellectual preoccupation. The importance of semiotic for Peirce is wide ranging. As he himself said, “[…] it has never been in my power to study anything,—mathematics, ethics, (...) metaphysics, gravitation, thermodynamics, optics, chemistry, comparative anatomy, astronomy, psychology, phonetics, economics, the history of science, whist, men and women, wine, metrology, except as a study of semiotic”. (SS 1977, 85–6). Peirce also treated sign theory as central to his work on logic, as the medium for inquiry and the process of scientific discovery, and even as one possible means for 'proving' his pragmatism. Its importance in Peirce's philosophy, then, cannot be underestimated. -/- Across the course of his intellectual life, Peirce continually returned to and developed his ideas about signs and semiotic and there are three broadly delineable accounts: a concise Early Account from the 1860s; a complete and relatively neat Interim Account developed through the 1880s and 1890s and presented in 1903; and his speculative, rambling, and incomplete Final Account developed between 1906 and 1910. The following entry examines these three accounts, and traces the changes that led Peirce to develop earlier accounts and generate new, more complex, sign theories. However, despite these changes, Peirce's ideas on the basic structure of signs and signification remain largely uniform throughout his developments. Consequently, it is useful to begin with an account of the basic structure of signs according to Peirce. (shrink)
entry for the Stanford Encyclopedia of Philosophy (SEP) This entry will attempt to provide a broad overview of the central themes of Leibniz’s philosophy of physics, as well as an introduction to some of the principal arguments and argumentative strategies he used to defend his positions. It tentatively includes sections entitled, The Historical Development of Leibniz’s Physics, Leibniz on Matter, Leibniz’s Dynamics, Leibniz on the Laws of Motion, Leibniz on Space and Time. A bibliography arranged by topic is also (...) included. A working draft is available here (comments, as always welcome): draft. (shrink)
Epistemology is the study of knowledge and justified belief. Belief is thus central to epistemology. It comes in a qualitative form, as when Sophia believes that Vienna is the capital of Austria, and a quantitative form, as when Sophia's degree of belief that Vienna is the capital of Austria is at least twice her degree of belief that tomorrow it will be sunny in Vienna. Formal epistemology, as opposed to mainstream epistemology (Hendricks 2006), is epistemology done in a formal way, (...) that is, by employing tools from logic and mathematics. The goal of this entry is to give the reader an overview of the formal tools available to epistemologists for the representation of belief. A particular focus will be the relation between formal representations of qualitative belief and formal representations of quantitative degrees of belief. (shrink)
Auguste Comte (1798–1857) is the founder of positivism, a philosophical and political movement which enjoyed a very wide diffusion in the second half of the nineteenth century. It sank into an almost complete oblivion during the twentieth, when it was eclipsed by neopositivism. However, Comte's decision to develop successively a philosophy of mathematics, a philosophy of physics, a philosophy of chemistry and a philosophy of biology, makes him the first philosopher of science in the modern sense, and his constant attention (...) to the social dimension of science resonates in many respects with current points of view. His political philosophy, on the other hand, is even less known, because it differs substantially from the classical political philosophy we have inherited. Comte's most important works are (1) the Course on Positive Philosophy (1830-1842, six volumes, translated and condensed by Harriet Martineau as The Positive Philosophy of Auguste Comte); (2) the System of Positive Polity, or Treatise on Sociology, Instituting the Religion of Humanity, (1851-1854, four volumes); and (3) the Early Writings (1820-1829), where one can see the influence of Saint-Simon, for whom Comte served as secretary from 1817 to 1824. The Early Writings are still the best introduction to Comte's thought. In the Course, Comte said, science was transformed into philosophy; in the System, philosophy was transformed into religion. The second transformation met with strong opposition; as a result, it has become customary to distinguish, with Mill, between a “good Comte” (the author of the Course) and a “bad Comte” (the author of the System). Today's common conception of positivism corresponds mainly to what can be found in the Course. (shrink)
The “ethics of belief” refers to a cluster of questions at the intersection of epistemology, philosophy of mind, psychology, and ethics. The central question in the debate is whether there are norms of some sort governing our habits of belief formation, belief maintenance, and belief relinquishment. Is it ever or always morally wrong (or epistemically irrational, or imprudent) to hold a belief on insufficient evidence? Is it ever or always morally right (or epistemically rational, or prudent) to believe on the (...) basis of sufficient evidence, or to withhold belief in the perceived absence of it? Is it ever or always obligatory to seek out all available epistemic evidence for a belief? Are there some ways of obtaining evidence that are themselves immoral or imprudent? -/- . (shrink)
John Carroll undertakes a careful philosophical examination of laws of nature, causation, and other related topics. He argues that laws of nature are not susceptible to the sort of philosophical treatment preferred by empiricists. Indeed he shows that emperically pure matters of fact need not even determine what the laws are. Similar, even stronger, conclusions are drawn about causation. Replacing the traditional view of laws and causation requiring some kind of foundational legitimacy, the author argues that these phenomena are inextricably (...) intertwined with everything else. This distinctively clear and detailed discussion of what it is to be a law will be valuable to a broad swathe of philosophers in metaphysics, the philosophy of mind, and the philosophy of science. (shrink)
The ultimate success of Hollywood blockbusters is dependent upon repeat viewings. Fans return to theaters to see films multiple times and buy DVDs so they can watch movies yet again. Although it is something of a received dogma in philosophy and psychology that suspense requires uncertainty, many of the biggest box office successes are action movies that fans claim to find suspenseful on repeated viewings. The conflict between the theory of suspense and the accounts of viewers generates a problem known (...) as the paradox of suspense, which we can boil down to a simple question: If suspense requires uncertainty, how can a viewer who knows the outcome still feel suspense? (shrink)
Over the past three decades, philosophy of science has grown increasingly “local.” Concerns have switched from general features of scientific practice to concepts, issues, and puzzles specific to particular disciplines. Philosophy of neuroscience is a natural result. This emerging area was also spurred by remarkable recent growth in the neurosciences. Cognitive and computational neuroscience continues to encroach upon issues traditionally addressed within the humanities, including the nature of consciousness, action, knowledge, and normativity. Empirical discoveries about brain structure and function suggest (...) ways that “naturalistic” programs might develop in detail, beyond the abstract philosophical considerations in their favor. -/- The literature distinguishes “philosophy of neuroscience” and “neurophilosophy.” The former concerns foundational issues within the neurosciences. The latter concerns application of neuroscientific concepts to traditional philosophical questions. Exploring various concepts of representation employed in neuroscientific theories is an example of the former. Examining implications of neurological syndromes for the concept of a unified self is an example of the latter. In this entry, we will assume this distinction and discuss examples of both. (shrink)
This article examines questions connected with the two features of Locke's intellectual landscape that are most salient for understanding his philosophy of science: (1) the profound shift underway in disciplinary boundaries, in methodological approaches to understanding the natural world, and in conceptions of induction and scientific knowledge; and (2) the dominant scientific theory of his day, the corpuscular hypothesis. Following the introduction, section 2 addresses questions connected to changing conceptions of scientific knowledge. What does Locke take science (scientia) and scientific (...) knowledge to be generally, why does he think that scientia in natural philosophy is beyond the reach of human beings, and what characterizes the conception of human knowledge in natural philosophy that he develops? Section 3 addresses the question provoked by Locke's apparently conflicting treatments of the corpuscular hypothesis. Does he accept or defend the corpuscular hypothesis? If not, what is its role in his thought, and what explains its close connection to key theses of the Essay? Since a scholarly debate has arisen about the status of the corpuscular hypothesis for Locke, Section 3 reviews some main positions in that debate. Section 4 considers the relationship between Locke's thought and that of a figure instrumental to the changing conceptions of scientific knowledge, Isaac Newton. (shrink)
The paradox of knowability is a logical result suggesting that, necessarily, if all truths are knowable in principle then all truths are in fact known. The contrapositive of the result says, necessarily, if in fact there is an unknown truth, then there is a truth that couldn't possibly be known. More specifically, if p is a truth that is never known then it is unknowable that p is a truth that is never known. The proof has been used to argue (...) against versions of anti-realism committed to the thesis that all truths are knowable. For clearly there are unknown truths; individually and collectively we are non-omniscient. So, by the main result, it is false that all truths are knowable. The result has also been used to draw more general lessons about the limits of human knowledge. Still others have taken the proof to be fallacious, since it collapses an apparently moderate brand of anti-realism into an obviously implausible and naive idealism. (shrink)
RESUMEN: Se ofrece un análisis de las transformaciones disciplinares que ha experimentado la lógica matemática o simbólica desde su surgimiento a fines del siglo XIX. Examinaremos sus orígenes como un híbrido de filosofía y matemáticas, su madurez e institucionalización bajo la rúbrica de “lógica y fundamentos”, una segunda ola de institucionalización durante la Posguerra, y los desarrollos institucionales desde 1975 en conexión con las ciencias de la computación y con el estudio de lenguaje e informática. Aunque se comenta algo de (...) la “historia interna”, nos centraremos en la emergencia, consolidación y convoluciones de la lógica como disciplina, a través de varias asociaciones profesionales y revistas, en centros como Turín, Gotinga, Varsovia, Berkeley, Princeton, Carnegie Mellon, Stanford y Amsterdam.ABSTRACT: We offer an analysis of the disciplinary transformations underwent by mathematical or symbolic logic since its emergence in the late 19th century. Examined are its origins as a hybrid of philosophy and mathematics, the maturity and institutionalisation attained under the label “logic and foundations”, a second wave of institutionalisation in the Postwar period, and the institutional developments since 1975 in connection with computer science and with the study of language and informatics. Although some “internal history” is discussed, the main focus is on the emergence, consolidation and convolutions of logic as a discipline, through various professional associations and journals, in centers such as Torino, Göttingen, Warsaw, Berkeley, Princeton, Carnegie Mellon, Stanford, and Amsterdam. (shrink)
Individuals are a prominent part of the biological world. Although biologists and philosophers of biology draw freely on the concept of an individual in articulating both widely accepted and more controversial claims, there has been little explicit work devoted to the biological notion of an individual itself. How should we think about biological individuals? What are the roles that biological individuals play in processes such as natural selection (are genes and groups also units of selection?), speciation (are species individuals?), and (...) organismic development (do genomes code for organisms)? Much of our discussion here will focus on organisms as a central kind of biological individual, and that discussion will raise broader questions about the nature of the biological world, for example, about its complexity, its organization, and its relation to human thought. (shrink)
This article critically reviews an outstanding collection of new essays addressing Edmund Husserl’s Crisis of European Sciences. In Science and the Life-World (Stanford, 2010), David Hyder and Hans-Jörg Rheinberger bring together an impressive range of first-rate philosophers and historians. The collection explicates key concepts in Husserl’s often obscure work, compares Husserl’s phenomenology of science to the parallel tradition of historical epistemology, and provocatively challenges Husserl’s views on science. The explications are uniformly clear and helpful, the comparative work intriguing, and (...) the criticisms interesting but uneven. The article also elaborates on Husserl’s phenomenological method as it relates to the historiography of science, and compares his views on mathematical idealisation to more recent work in the analytical tradition. (shrink)
Increasingly, epistemologists are becoming interested in social structures and their effect on epistemic enterprises, but little attention has been paid to the proper distribution of experimental results among scientists. This paper will analyze a model first suggested by two economists, which nicely captures one type of learning situation faced by scientists. The results of a computer simulation study of this model provide two interesting conclusions. First, in some contexts, a community of scientists is, as a whole, more reliable when its (...) members are less aware of their colleagues' experimental results. Second, there is a robust tradeoff between the reliability of a community and the speed with which it reaches a correct conclusion. ‡The author would like to thank Brian Skyrms, Kyle Stanford, Jeffrey Barrett, Bruce Glymour, and the participants in the Social Dynamics Seminar at University of California–Irvine for their helpful comments. Generous financial support was provided by the School of Social Science and Institute for Mathematical Behavioral Sciences at UCI. †To contact the author, please write to: Department of Philosophy, Baker Hall 135, Carnegie Mellon University, Pittsburgh, PA 15213-3890; e-mail: kzollman@andrew.cmu.edu. (shrink)
Most models of generational succession in sexually reproducing populations necessarily move back and forth between genic and genotypic spaces. We show that transitions between and within these spaces are usually hidden by unstated assumptions about processes in these spaces. We also examine a widely endorsed claim regarding the mathematical equivalence of kin-, group-, individual-, and allelic-selection models made by Lee Dugatkin and Kern Reeve. We show that the claimed mathematical equivalence of the models does not hold. *Received January 2007; revised (...) April 2008. †To contact the authors, please write to: Elisabeth Lloyd, Department of History and Philosophy of Science, 130 Goodbody Hall, Indiana University, Bloomington, IN 47405; e-mail: ealloyd@indiana.edu; Richard Lewontin, Department of Organismic and Evolutionary Biology, Harvard University, 26 Oxford Street, Cambridge, MA 02138; Marcus Feldman, Department of Biological Sciences, Stanford University, Stanford, CA 94305; e-mail: marc@charles.stanford.edu. (shrink)
Three clusters of philosophically significant issues arise from Frege's discussions of definitions. First, Frege criticizes the definitions of mathematicians of his day, especially those of Weierstrass and Hilbert. Second, central to Frege's philosophical discussion and technical execution of logicism is the so-called Hume's Principle, considered in The Foundations of Arithmetic . Some varieties of neo-Fregean logicism are based on taking this principle as a contextual definition of the operator 'the number of …', and criticisms of such neo-Fregean programs sometimes appeal (...) to Frege's objections to contextual definitions in later writings. Finally, a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend is whether the logical structures of the definientia reflect our pre-Fregean understanding of arithmetical terms. It seems that unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes the definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre-definitional understanding. One or more of these topics may be studied in a survey course in the philosophy of mathematics or a course on Frege's philosophy. The latter two topics are obviously central in a seminar in the philosophy of mathematics in general or more specialized seminars on logicism, or on mathematical definitions and concept formation. Author Recommends: 1. Kant, Immanuel. Critique of Pure Reason . Trans. P. Guyer and A. Wood. Cambridge: Cambridge University Press, 1999 [1781, 1787], A7-10/B11-14, A151/B190. In the first Critique , Kant appears to give four distinct accounts of analytic judgments. The initial famous account explains analyticity in terms of the predicate-concept belonging to the subject-concept (A6–7/B11). In this passage, we also find an account of establishing analytic judgments on the basis of conceptual containments and the principle of non-contradiction. (The other accounts are in terms of 'identity' (A7/B1l), in terms of the explicative–ampliative contrast (A7/B11), and by reference to the notion of 'cognizability in accordance with the principle of contradiction' (A151/B190).) 2. Frege, Gottlob. The Foundations of Arithmetic . Trans. J. L. Austin. 2nd ed. Evanston, IL: Northwestern University Press, 1980 [1884], especially sections 1–4, 87–91. Frege here criticizes and reformulates Kant's account of analyticity. Central to Frege's account is the provability of an analytic statement on the basis of (Frege's) logic and definitions that express analyses of (mathematical, especially arithmetical) concepts. 3. Frege, Gottlob. Review of E. G. Husserl. 'Philosophie der Arithmetik I [1894],' in Frege, Collected Papers . Ed. B. McGuinness. Trans. M. Black et al. Oxford: Blackwell, 1984. 195–209. In this review, Frege responds to Husserl's charge that Frege's definitions fail to capture our intuitive pre-analytic arithmetical concepts by claiming that the adequacy of mathematical definitions is measured, not by their expressing the same senses, but merely by their having the same references, as pre-definitional vocabulary. It follows not only that Husserl's criticism is unfounded, but also that there can be alternative, equally legitimate, definitions of mathematical terms. 4. Frege, 'Logic in Mathematics,' in Frege, Posthumous Writings . Trans. P. Long and R. White. Oxford: Blackwell, 1979 [1914]. 203–50. These are a set of lecture notes including, among other things, an account of proper definitions as mere abbreviation of complex signs by simple ones, in contrast to definitions which purport to express the analyses of existing concepts. Frege here claims that if there is any doubt whether a definition purporting to express an analysis succeeds in capturing the senses of the pre-definitional expressions, then the definition fails as an analysis, and should be regarded as the introduction of an entirely new expression abbreviating the definiens . 5. Picardi, Eva. 'Frege on Definition and Logical Proof,' Temi e Prospettive della Logica e della Filosofia della Scienza Contemporanee . i vol. Eds. C. Cellucci and G. Sambin. Bologna: Cooperativa Libraria Universitaria Editrice Bologna, 1988. 227–30. Picardi sets out forcefully the view that unless Frege's definitions capture the meanings of existing arithmetical terms, his logicism cannot have the epistemological significance he takes it to have. 6. Dummett, Michael. 'Frege and the Paradox of Analysis,' in Dummett, Frege and other Philosophers . Oxford: Oxford University Press, 1991. 17–52. Dummett agrees with Picardi's view and analyzes the philosophical pressures that led Frege to the account of definition in 'Logic in Mathematics.' Especially significant is Dummett's claim of the centrality of the transparency of sense – that if one grasps the senses of any two expressions, one must know whether they have the same sense – in Frege's account. 7. Benacerraf, Paul. 'Frege: The Last Logicist,' Midwest Studies in Philosophy . vol. 6. Eds. P. French, T. Uehling, and H. Wettstein. Minneapolis: University of Minnesota Press, 1981. 17–35. Frege's aims, on Benacerraf's reading, are primarily mathematical. Frege was interested in traditional philosophical issues such as the analyticity of arithmetic only to the extent that they can be exploited for the mathematical goal of proving previously unproven arithmetical statements. Hence, Frege never had any serious interest in or need for showing that his definitions of arithmetical terms reflect existing arithmetical conceptions. 8. Weiner, Joan. 'The Philosopher Behind the Last Logicist,' in Frege: Tradition and Influence . Ed. C. Wright. Oxford: Blackwell, 1984. 57–79. Weiner argues that on Frege's view, prior to his definitions of arithmetical terms the references of such expressions are in fact not known by those who use arithmetical vocabulary. Thus, in Foundations , Frege operated with a 'hidden agenda' (263) namely, replacing existing arithmetic with a new science based on stipulative definitions that assign new senses to key arithmetical terms. 9. Tappenden, Jamie. 'Extending Knowledge and 'Fruitful Concepts': Fregean Themes in the Foundations of Mathematics.' Noûs 29 (1995): 427–67. Tappenden argues that Frege takes his crucial innovation over previous practices and accounts of mathematical concept formation to be the role of quantificational structure made possible by his logical discoveries. 10. Horty, John. Frege on Definitions: A Case Study of Semantic Content . Oxford: Oxford University Press, 2007. A useful interpretation of Frege's views of definition, together with suggestive extensions for resolving the issues framing Frege's views. 11. Shieh, Sanford. 'Frege on Definitions,' Philosophy Compass 3/5 (2008): 992–1012. A more detailed account of Frege's views on definition and the philosophical issues they raise, surveying and discussing critically the main substantive and interpretive issues. Online Materials On Frege http://plato.stanford.edu/entries/frege/ On the Paradox of Analysis http://plato.stanford.edu/entries/analysis/ Sample Syllabus The following is a 3-week module that can be incorporated into fairly focused historically oriented graduate-level seminars on logicism or on the paradox of analysis. It is also possible to compress the material into 2 weeks in an undergraduate or graduate class Frege's thought in general. Week I: Background, Kant on Analyticity; Definition in Foundations , Review of Husserl, and 'Logic in Mathematics' Readings Kant, Immanuel. Critique of Pure Reason , A7–10/B11–14. Frege, Gottlob. The Foundations of Arithmetic , sections 1–4, 87–91. Frege, Gottlob. Review of E. G. Husserl, Philosophie der Arithmetik I. Frege, Gottlob. 'Logic in Mathematics.' Optional Proops, Ian. 'Kant's Conception of Analytic Judgment,' Philosophy and Phenomenological Research LXX, 3 (2005): 588–612. Week II: The Supposed Paradox of Analysis, Picardi and Dummett; Bypassing Traditional Epistemological Issues About Mathematics, Benacerraf Readings Picardi, Eva. 'Frege on Definition and Logical Proof.' Dummett, Michael. 'Frege and the Paradox of Analysis.' Benacerraf, Paul. 'Frege: The Last Logicist.' Optional Tappenden, Jamie. 'Extending Knowledge and 'Fruitful Concepts': Fregean Themes in the Foundations of Mathematics.' Week III: Weiner's Hidden Agenda Interpretation Readings Weiner, Joan. 'The Philosopher Behind the Last Logicist.' Optional Weiner, Joan. Frege in Perspective . Ithaca, NY: Cornell University Press, 1990. Focus Questions 1. To what extent is Frege's account of analyticity in Foundations a rejection, and to what extent an updating, of Kant's view of analyticity? 2. According to Picardi it 'would be incomprehensible' how Frege's proofs tells us anything about the arithmetic we already have unless his 'definitions [are] somehow responsible to the meaning of [arithmetical] sentences as these are understood' (228). Why does she hold this? Why does Dummett agree with her? Do you think Frege's logicism needs to address this worry? 3. What are the major differences and continuities in Frege's discussions of definition in mathematics in Foundations , the review of Husserl and 'Logic in Mathematics'? 4. Frege writes that definitions must prove their worth by being fruitful. He also says that nothing can be proven using a proper definition that cannot be proven without it. Are these claims consistent? Why or why not? 5. Weiner held that in Foundations Frege had 'hidden agenda.' What, according to her, is this agenda? How does this fit with Frege's later views of definition? 6. What are Frege's main complaints about Weierstrass's definitions in 'Logic in Mathematics'? Are these criticisms consistent with Frege's account of 'definition proper' in the same text? Seminar/Project Ideas What, if anything, is the relation between Frege's critique of Hilbert's use of definitions and Frege's later views of definitions? (shrink)
Any study of the 'Scientific Revolution' and particularly Descartes' role in the debates surrounding the conception of nature (atoms and the void v. plenum theory, the role of mathematics and experiment in natural knowledge, the status and derivation of the laws of nature, the eternality and necessity of eternal truths, etc.) should be placed in the philosophical, scientific, theological, and sociological context of its time. Seventeenth-century debates concerning the nature of the eternal truths such as '2 + 2 = 4' (...) or the law of inertia turn on the question of whether these truths were created along with nature, or were uncreated and subsisting in God's mind. One's answer to that question has direct consequences for conceptions of the necessity/contingency of mathematical and natural knowledge, how knowledge of such truths is accomplished by humans, and what grounds these truths. In this paper, I review the positions of four successors to Descartes' philosophy on the question of the eternal truths to illustrate how in specific ways that question with its theological, metaphysical, modal, and epistemological dimensions concerned the objectivity and certainty of the discoveries of the new science. Author Recommends: Clarke, Desmond. Descartes' Philosophy of Science . University Park, Penn State Press, 1982. This work provides an account of Descartes as a practicing scientist whose rationalism is mitigated by reliance on experiment and experience. Author re-examines Descartes' philosophical and scientific works in this new light. Dear, Peter. Revolutionizing the Sciences: European Knowledge and its Ambitions, 1500–1700 . Princeton, Princeton University Press, 2001. This work provides a useful overview of the issues and thinkers of the Scientific Revolution. Of particular relevance is chapter 8 on Cartesian and Newtonian science. Funkenstein, Amos. Theology and the Scientific Imagination from the Middle Ages to the Seventeenth Century . Princeton, Princeton University Press, 1986. This work is an advanced study of the theological and metaphysical foundations of early modern science. Discussions include questions of God's nature, God's knowledge in relation to human knowledge, providence, the laws of nature, and the truths of mathematics. In particular, chapter 3 discusses Descartes' account of the eternal truths and divine omnipotence. Garber, Daniel. Descartes' Metaphysical Physics . Chicago, University of Chicago Press, 1992. This work examines how Descartes' metaphysical doctrines of God, soul, and body set the groundwork for his physics. It includes a study of God and the grounds for the laws of physics (chapter 9). Henry, John. The Scientific Revolution and the Origins of Modern Science . 3rd ed. New York, Palgrave, Macmillan Press, 2008. This work provides a brief, general, and informative overview of the Scientific Revolution, including the themes of method, magic, religion, and culture. Osler, Margaret J. Divine Will and the Mechanical Philosophy: Gassendi and Descartes on Contingency and Necessity in the Created World . Cambridge, Cambridge University Press, 1994. This work is an examination and comparison of the mechanical philosophies of Gassendi and Descartes. It offers in-depth discussion of the issue of voluntarism and intellectualism in the period and how that related to conceptions of laws of nature and the eternal truths. Shapin, Steven. The Scientific Revolution . Chicago, University of Chicago Press, 1996. This work provides a critical synthesis of as well as a guide to recent scholarship in the history of science for a general readership. Online Materials Dr. Robert A. Hatch's Scientific Revolution Website: http://web.clas.ufl.edu/users/rhatch/pages/03-Sci-Rev/SCI-REV-Home/ A compendium of resources for the study of Scientific Revolution. Early English Books Online: http://eebo.chadwyck.com/home Early English Books Online (EEBO) contains digital facsimile page images of virtually every work printed in England, Ireland, Scotland, Wales and British North America and works in English printed elsewhere from 1473 to 1700. Early Modern Resources: http://www.earlymodernweb.org.uk/emr/ Early Modern Resources is a gateway for all those interested in finding electronic resources relating to the early modern period in history. Gallica, the Digital Library of the Bibliothèque Nationale de France: http://gallica.bnf.fr/ An ever-growing digital library which includes numerous primary and secondary texts of relevance to Descartes and his role in Scientific Revolution. Hatfield, Gary, 'René Descartes', The Stanford Encyclopedia of Philosophy. Spring 2009 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/spr2009/entries/descartes/ Slowik, Edward, 'Descartes' Physics', The Stanford Encyclopedia of Philosophy. Winter 2008 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/win2008/entries/descartes-physics/ Syllabus Sample Syllabus: Cartesian Science The following is five weeks covering Cartesian Science in a course on Descartes or the Scientific Revolution, or 17th-century theories of matter, or related themes on early modern truth and method, especially on the continent. This material is best suited to a graduate level audience, but it could be modified to suit an upper-division undergraduate course, as the readings are basically primary texts whose context and background can be explained in lectures. Week 1: Cartesian Revolution in France • Scientific method • Role of mathematics and experiment • Certainty of scientific knowledge Readings: Hatfield, Gary, 'René Descartes', The Stanford Encyclopedia of Philosophy. Spring 2009 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/spr2009/entries/descartes/ Descartes, Discourse on Method , Parts 1–3 Descartes, Meditations on First Philosophy , First Meditation. Week 2: Descartes' Scientific Treatises • Mechanization and mathematization of nature • Primary–secondary quality distinction Readings: Discourse on Method, Parts 4–6 Selections from Descartes' Scientific Essays: The World or Treatise on Light (ATXI 3–48); Treatise on Man (ATXI 119–202); Optics (ATVI 82–147). Slowik, Edward, 'Descartes' Physics', The Stanford Encyclopedia of Philosophy. Winter 2008 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/win2008/entries/descartes-physics/ Henry, John, 'The Mechanical Philosophy,' chapter 5. The Scientific Revolution and the Origins of Modern Science . 3rd ed. Macmillan, 2008. Week 3: Descartes' Theory of Nature • Descartes' derivation of the law of conservation and the three laws of motion • God's role in the metaphysics and physics of nature Readings: Selections from Principles of Philosophy, Preface (all); Letter to Elizabeth; Part I: 1–8; Part II: 1–45, 55, 64; Part III: 1–4, 15–19, 45–47; Part IV: 187–207. John Henry, 'Religion and Science,' chapter 6. The Scientific Revolution and the Origins of Modern Science . 3rd ed. Macmillan, 2008. Week 4: Post-1650 Cartesian Science: Necessity and Contingency in Nature • Debates on God, Creation, and Causes Readings: Easton, Patricia, 'What is at Stake in the Cartesian Debates on the Eternal Truths?' Philosophy Compass 4.2 (2009): 348–62. Malebranche, Nicolas, 'Elucidation 10', from The Search after Truth (1674). Note: All selections available in Nicolas Malebranche (1992). Philosophical Selections , edited by S. Nadler, Hackett. Gottfried Leibniz (1714) Monadology . Week 5: Causes in Nature and Morals • Theodicy as an explanation of defect and evil in a lawful universe: Malebranche v. Leibniz Readings: Nicolas Malebranche, Elucidation XVI (on occasionalism), and Treatise on Nature and Grace, Discourse One, Part 1. Gottfried Leibniz (1706), Theodicy. Focus Questions Weekly questions can be used to focus the readings. This can be done in a web or e-mail discussion thread, as a weekly assignment, or for in class discussion. I require students to post a short paragraph in response to the question or some posting by a classmate on the question. Students are required to post by 10 a.m. the day before we meet for class on a course website. Week 1: According to Descartes, what role does skepticism play in scientific reasoning? Week 2: Comment on the following: 'But I am supposing this machine to be made by the hands of God, and so I think you may reasonably think it capable of a greater variety of movements than I could possibly imagine in it, and of exhibiting more artistry than I could possibly ascribe to it' [ Treatise on Man ; ATXI 120]. Week 3: What is Descartes' conception of the relation between the metaphysics and physics of nature? Week 4: Critically discuss the positions of Descartes, Malebranche, and Leibniz on what provides the foundation for the certitude of natural knowledge? Week 5: Explain why both Malebranche and Leibniz consider moral sin to be analogous to natural defect? Seminar/Project Idea Hold a debate on the question of the status of the eternal truths. The proposition will be Descartes' position: 'Eternal truths must be both created and necessary if certainty in science is to be possible'. Format: 1. At the beginning of the 5-week module, students will be assigned to one of three roles: Team A, Team B, and judge's panel. Students will be given the debate proposition, but will not be told which team will take the affirmative and which team the negative until the time of the debate. 2. Recommend a variation on the Classic Debate Format to encourage the development of argument: sequence begins with affirmative construction (8 minutes), negative construction (8 minutes), second affirmative construction (8 minutes), second negative construction (8 minutes), first negative rebuttal (4 minutes), first affirmative rebuttal (4 minutes), final negative rebuttal (4 minutes) and final affirmative rebuttal (4 minutes). 3. Judges Panel: will consist of 3–4 judges who will assess the performance of Teams A and B. Judgment should be based on the persuasiveness of the team position. 4. Debate will be held at the end of the fifth week, or semester, whichever makes most sense given the course length and structure. Acknowledgements The author gratefully acknowledges the immensely helpful comments and suggestions by the participants in her graduate seminar on the Scientific Revolution: Benjamin Chicka, Sarah Jacques-Ross, Richard Ross, Marcella Stockstill, and Zohra Wolters. (shrink)
Ve svém článku ‘Je elementární logika totéž co predikátová logika prvního řádu?’ (Pokroky matematiky, fyziky a astronomie 42, 1997, 127-133) klade Jiří Fiala nesmírně zajímavou otázku, zda je opodstatněné ztotožňovat elementární logiku s predikátovou logikou prvního řádu; s pomocí argumentů propagovaných již delší dobu finským logikem a filosofem Jaako Hintikkou (viz již jeho Logic, Language-Games and Information, Clarendon Press, Oxford, 1973; nejnověji jeho The Principles of Mathematics Revisited, Cambridge University Press, Cambridge, 1996) naznačuje, že by tomu tak být nemuselo. Myslím, (...) že uváděná argumentace stojí za bližší rozbor. Hintikka v podstatě říká: Kvantifikované formule predikátové logiky jsou svou podstatou o vybírání prvků z univerza; například ∀x∃yR(x,y) neříká nic jiného než to, že ke každému x můžeme vybrat y, které je k němu ve vztahu R. Obecněji říká Hintikka to, že každá formule je vlastně zápisem určité hry (ve smyslu matematické teorie her), jejíž některé tahy spočívají ve vybírání individuí. Na základě tohoto se pak ptá: je nějaký rozumný důvod, proč se omezovat jenom na hry toho typu, které jsou vyjádřitelné formulemi standardního predikátového počtu? Proč připouštět jen hry s úplnou informací (tj. ty, při kterých jsou při každém tahu k dispozici všechny tahy předchozí), proč vylučovat hry jiné; tudíž proč připouštět jen lineárně uspořádané kvantifikátory, a nepřipustit i kvantifikátory uspořádané třeba jen částečně? Hintikkova argumentace je příkladem argumentace typu formule logiky prvního řádu jsou ve skutečnosti o tom a o tom, tudíž ... . Uveďme pro ilustraci jiný nedávný příklad stejného argumentačního schématu, který pochází od Johana van Benthema (Exploring Logical Dynamics, CSLI, Stanford, 1997). Ten říká: Kvantifikátory jsou v podstatě modality, kvantifikované formule jsou tedy formule modální a jsou tudíž o existenci nějakých alternativ: formule ∀xP(x) říká, že P(x) je nutné, neboli že P(x) platí v každém „dosažitelném možném stavu věcí“, zatímco formule ∃xP(x) říká, že P(x) je možné, neboli že platí alespoň v jednom takovém stavu věcí.. (shrink)
To what extent do nanotechnology researchers discern specific work-related ethical responsibilities that are incumbent upon them? A questionnaire was designed and administered to answer this question. Analysis of responses to 11 ethical responsibility statements (ERSs) by 213 researchers at the Stanford Nanofabrication Facility revealed widespread agreement about a number of work-related ethical responsibilities and substantial divergence in the views about several others. Explanations of this divergence are proposed. A new variable is defined that gauges the respondent’s overall level of (...) discernment of the ethical responsibilities referenced in the ERSs. The mean discernment level score for respondents who had taken a course that included discussion of ethical issues closely related to science, technology, or engineering was significantly higher than for those who had not. Further, among respondents who had taken such a course, the mean discernment level score for those who had taken an ethics course devoted to exploration of ethical issues closely related to science, technology, or engineering was significantly higher than for those who had taken a technical science or engineering course that typically pays only fleeting attention to such issues. Implications of these findings are discussed. (shrink)
The study of logic goes back more than two thousand years and in that time many symbols and diagrams have been devised. Around 300 BC Aristotle introduced letters as term-variables, a "new and epoch-making device in logical technique." (W. & M. Kneale The Development of Logic (1962, p. 61). The modern era of mathematical notation in logic began with George Boole (1815- 1864), although none of his notation survives. Set theory came into being in the late 19th and early 20th (...) centuries, largely a creation of Georg Cantor (1845-1918). See MacTutor's A history of set theory or, for more detail, Set theory from the Stanford Encyclopedia of Philosophy. (shrink)
Definitional and axiomatic theories of truth -- Objects of truth -- Tarski -- Truth and set theory -- Technical preliminaries -- Comparing axiomatic theories of truth -- Disquotation -- Classical compositional truth -- Hierarchies -- Typed and type-free theories of truth -- Reasons against typing -- Axioms and rules -- Axioms for type-free truth -- Classical symmetric truth -- Kripke-Feferman -- Axiomatizing Kripke's theory in partial logic -- Grounded truth -- Alternative evaluation schemata -- Disquotation -- Classical logic -- Deflationism (...) -- Reflection -- Ontological reduction -- Applying theories of truth. (shrink)
