Online research in philosophy

Are philosophers of science limited to conducting autopsies on dead scientific theories, or might they also help resolve contemporary methodological disputes in science? This essay (1) gives an overview of thought experiments, especially in mathematics; (2) outlines three major positions on the current dose-response controversy for ionizing radiation; and (3) sketches an original mathematical thought experiment that might help resolve the low-dose radiation conflict. This thought experiment relies on the assumptions that radiation "hits'' are Poisson distributed and (...) that background conditions cause many more radiation-induced cancers than human activities. The essay closes by responding to several key objections to the position defended here. (shrink)

Commenting on Atkinson's paper I argue that leading to a successful real experiment is not the only scale on which a thought experiment's value is judged. Even the path from the original EPR thought experiment to Aspect's verification of the Bell inequalities was long-winded and involved considerable input from the sides of technology and mathematics. Von Neumann's construction of hidden variables was, moreover, a genuinely mathematical thought experiment that was successfully criticized by Bell. Such thought (...) experiments are also possible in string theory, where any (non-trivial) empirical corroboration seems to be out of reach. Yet appraising mathematical thought experiments and their contribution to physical thought experiments requires a dynamical account which in the spirit of Mach and Lakatos attributes due weight to informal mathematical reasoning or empirical intuition. (shrink)

Most disciplines make use of thought experiments, but physics and philosophy lead the pack with heavy dependence upon them. Often this is for conceptual clarification, but occasionally they provide real theoretical advances. In spite of their importance, however, thought experiments have received rather little attention as a topic in their own right until recently. The situation has improved in the past few years, but a mere generation ago the entire published literature on thought experiments could have been (...) mastered in a long weekend. Now the subject is beginning to flourish. Given the relative newness of the field, it might be useful to have several examples at one’s finger tips, so a number of great ones will be described. Attention will also be drawn outside physics and philosophy. In mathematics there is something analogous to thought experiments -- visual reasoning and picture proofs. I will look briefly at this class of thought experiments and try using them to make a case for possibly settling the continuum hypothesis. After this, I will return to thought experiments in the sciences and propose an account of how they work. Finally, I will end with a sketch of a topic I am currently working on, a kind of progress report which, I hope, will be an inducement to others. (shrink)

Abstract This paper explains how mathematical computation can be constructed from weaker recursive patterns typical of natural languages. A thought experiment is used to describe the formalization of computational rules, or arithmetical axioms, using only orally-based natural language capabilities, and motivated by two accomplishments of ancient Indian mathematics and linguistics. One accomplishment is the expression of positional value using versified Sanskrit number words in addition to orthodox inscribed numerals. The second is Panini’s invention, around<br>the fifth century BCE, of a (...) formal grammar for spoken Sanskrit, expressed in oral verse extending ordinary Sanskrit, and using recursive methods rediscovered in the twentieth century. The Sanskrit positional number compounds and Panini’s formal system are construed as linguistic grammaticalizations relying on tacit cognitive models of symbolic form. The thought experiment shows that universal computation can be constructed from natural language structure and skills, and shows why intentional capabilities needed for language use play a role in computation across all<br>media. The evolution of writing and positional number systems in Mesopotamia is used to transfer the thought experiment of “oral arithmetic” to inscribed computation. The thought experiment and historical evidence combine to show how and why mathematical computation is a cognitive technology extending generic symbolic skills associated with language structure, usage, and change. (shrink)

Stimulating, thought-provoking analysis of a number of the most interesting intellectual inconsistencies in mathematics, physics and language. Delightful elucidations of methods for misunderstanding the real world of experiment (Aristotle’s Circle paradox), being led astray by algebra (De Morgan’s paradox) and other mind-benders. Some high school algebra and geometry is assumed; any other math needed is developed in text. Reprint of 1982 ed.

This paper explores the semiotic status of algebraic variables. To do that we build on a structuralist and post-structuralist train of thought going from Mauss and L vi-Strauss to Baudrillard and Derrida. We import these authors' semiotic thinking from the register of indigenous concepts (such as mana), and apply it to the register of algebra via a concrete case study of generating functions. The purpose of this experiment is to provide a philosophical language that can explore the openness of (...) mathematical signs to reinterpretation, and bridge some barriers between philosophy of mathematics and critical approaches to knowledge. (shrink)

P. M. S. Hacker 1. The poverty of philosophy as a science Throughout its history philosophy has been thought to be a member of a community of intellectual disciplines united by their common pursuit of knowledge. It has sometimes been thought to be the queen of the sciences, at other times merely their under-labourer. But irrespective of its social status, it was held to be a participant in the quest for knowledge – a cognitive discipline. Cognitive disciplines may (...) be a priori or empirical. The distinction between what is a priori and what is empirical is epistemological. It turns, as Frege noted, on the ultimate justification for holding something to be true.1 If the truths which a cognitive discipline attains are warranted neither by observation nor by experiment (nor by inference therefrom), then they are a priori. Otherwise they are empirical. The natural and moral sciences (the Geisteswissenschaften) strive for and attain empirical knowledge.2 The mathematical sciences are a priori. Cognitive disciplines have a distinctive subject matter, concerning which they aim to add to human knowledge. Physics deals with matter, motion, and energy, chemistry with the constitution of stuffs out of elements, biology with the nature of living beings, history with ‘the crimes, follies and misfortunes of mankind’ (Gibbon), and so forth. The empirical sciences aim not only to discover truths but also to explain the phenomena they study. The natural sciences produce theories (typically with predictive powers) to explain the facts and laws they discover. The moral sciences too aim to explain the phenomena they study – although not to the same extent by way of theory and general laws; and their predictive powers, if any, are more limited. Mathematics and logic strive to produce theorems by means of proofs, and are.. (shrink)

Examples of classic thought experiments are presented and some morals drawn. The views of my fellow symposiasts, Tamar Gendler, John Norton, and James McAllister, are evaluated. An account of thought experiments along a priori and Platonistic lines is given. I also cite the related example of proving theorems in mathematics with pictures and diagrams. To illustrate the power of these methods, a possible refutation of the continuum hypothesis using a thought experiment is sketched.

Predicativity requirements of explicit presentability of objects and predicatively acceptable proof are distinguished from predicativist theses of a philosophical character. Familiar among these are expressions of skepticism about the objectivity of full power sets of infinite sets. Articulation of strong, limitative theses, however, turns out to be problematic: impredicative commitments creep into the very formulations, e.g. that “predicative definability'' marks a limit of “intelligibility''. A thought experiment is proposed to undermine the predicativist idea that arbitrary parts of an infinite (...) whole of atoms are “mind- or language-dependent''. On the other hand, weaker claims, e.g., that predicative mathematics is “more secure'' than impredicative, are nearly platitudinous. The interesting philosophical force of predicativism seems to be negative, in its challenge to indispensability arguments, à la Gödel-Friedman, for the transfinite in pure mathematics, and à la Quine-Putnam, for abstract mathematics in the sciences. Evidence is mounting in favor of Gödel-Friedman, e.g. impredicativity of free-variable formulations of theorems such as Kruskal's and Graph Minor, and more far-reaching, recent work in Boolean Relation Theory. This may lead to a realization of Gödel's idea of justifying strong axioms of infinity through their unifying, explanatory role, in analogy with theoretical physics. (shrink)

Mainstream philosophy of science has embraced an “empiricist” approach to scientific method. To be slightly more precise, I venture that most philosophers of science today would endorse the view that experience is the source of most scientific knowledge. The aim of this essay will be to challenge the consensus, by showing how we cannot and should not abandon all elements of the “rationalist” tradition, a tradition often identified with philosophers such as Descartes. There are several elements frequently identified with “rationalist” (...) science (Stump, 2005): questioning of sense experience, the attempt to rethink the “metaphysical” foundations of one’s science, using either thought experiment, or appealing to demonstrative arguments purporting to establish ‘necessary’ truths, often using either mathematics or geometry, and appeal to “virtues” not usually considered “strictly empirical,” such as simplicity. This essay explores the effective deployment of such considerations in the history and current practice of science. (shrink)

The late scholastics, from the fourteenth to the seventeenth centuries, contributed to many fields of knowledge other than philosophy. They developed a method of conceptual analysis that was very productive in those disciplines in which theory is relatively more important than empirical results. That includes mathematics, where the scholastics developed the analysis of continuous motion, which fed into the calculus, and the theory of risk and probability. The method came to the fore especially in the social sciences. In legal theory (...) they developed, for example, the ethical analyses of the conditions of validity of contracts, and natural rights theory. In political theory, they introduced constitutionalism and the thought experiment of a “state of nature”. Their contributions to economics included concepts still regarded as basic, such as demand, capital, labour, and scarcity. Faculty psychology and semiotics are other areas of signifi cance. In such disciplines, later developments rely crucially on scholastic concepts and vocabulary. (shrink)

Machine generated contents note: -- Introduction -- Setting the Scene -- Plato and Aristotle -- From the Roman Empire to the Empire of Islam -- The Western Middle Ages -- The Renaissance -- New Methods of Science -- Bringing Mathematics and Natural Philosophy Together -- Practice and Theory in Renaissance Medicine: William Harvey and the Circulation of the Blood -- The Spirit of System: Rene; Descartes and the Mechanical Philosophy -- The Royal Society and Experimental Philosophy -- Experiment, Mathematics, and (...) Magic: Isaac Newton -- Newton's Legacy: Forces and Fluids (electricity and heat) -- The Chemical Revolution: From Newton to John Dalton, via Priestley and Lavoisier -- Natural Theology and Natural Order: Newtonian Optimism and the History of Science -- The Making of Geology: From James Hutton to Charles Lyell via Catastrophism -- The History of Plants and Animals: Successive Emergence or Evolution? -- Religion and Progress in Victorian Britain: Robert Chambers versus Hugh Miller -- Bringing it All Together?: Charles Darwin's Evolution -- Darwinian Aftermaths: Religion; Social Science; Biology -- Beyond Newton: Energy and Thermodynamics -- Newton deposed: Einstein and Relativity Theory -- Mathematics instead of a World Picture: From Atomism to Quantum Theory -- Afterword. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals (...) are Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

Abstract Thought experimentation is part of accepted scientific practice, and this makes it surprising that philosophers of science did not seriously engage with it for a very long time. The situation changed in the 1990s, resulting in a highly intriguing debate over thought experiments. Initially, the discussion focused mostly on thought experiments in physics, philosophy, and mathematics. Other disciplines have since become the subject of interest. Yet, nothing substantial has been said about the role of thought (...) experiments in nonphilosophical theology. This paper discusses the role of thought experiments in Christian theology in comparison to their role in quantum physics, as mentioned by John Polkinghorne in Quantum Physics and Theology. We first look briefly at the history of the inquiry into thought experiments and then at Polkinghorne's remarks about the role of thought experimentation in quantum physics and Christian eschatology. To determine the actual importance of thought experiments in Christian theology a number of new examples are introduced in a third step. In the light of these examples, in a fourth step, we address the question of what it is that explains the cognitive efficacy of thought experiments in quantum physics and Christian theology. (shrink)

Originally published in 1991, The Laboratory of the Mind: Thought Experiments in the Natural Sciences, is the first monograph to identify and address some of the many interesting questions that pertain to thought experiments. While the putative aim of the book is to explore the nature of thought experimental evidence, it has another important purpose which concerns the crucial role thought experiments play in Brown’s Platonic master argument.In that argument, Brown argues against naturalism and empiricism (Brown (...) 2012), for mathematical Platonism (Brown 2008), and from the Platonist-friendly, abstract universals posited by the Dretske-Tooley-Armstrong (DTA) account of the laws of nature to a more general, physical Platonism. The Laboratory of the Mind is where he takes this final step. (shrink)

Those inquiring into the nature of mind have long been interested in the foundations of mathematics, and conversely this branch of knowledge is distinctive in that our access to it is purely through thought. A better understanding of mathematical thought should clarify the conceptual foundations of mathematics, and a deeper grasp of the latter should in turn illuminate the powers of mind through which mathematics is made available to us. The link between conceptions of mind and of mathematics (...) has been a central theme running through the great competing philosophies of mathematics of the twentieth century, though each has refashioned the connection and its import in distinctive ways. The present collection will be of interest to students of both mathematics and of mind. Contents include: "Introduction" by Alexander George; "What is Mathematics About?" by Michael Dummett; "The Advantages of Honest Toil over Theft" by George Boolos; "The Law of Excluded Middle and the Axiom of Choice" by W.W. Tait; "Mechanical Procedures and Mathematical Experience" by Wilfried Sieg; "Mathematical Intuition and Objectivity" by Daniel Isaacson; "Intuition and Number" by Charles Parsons; and "Hilbert's Axiomatic Method and the Laws of Thought" by Michael Hallett. (shrink)

Platonism is the most pervasive philosophy of mathematics. Indeed, it can be argued that an inarticulate, half-conscious Platonism is nearly universal among mathematicians. The basic idea is that mathematical entities exist outside space and time, outside thought and matter, in an abstract realm. In the more eloquent words of Edward Everett, a distinguished nineteenth-century American scholar, "in pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to (...) exist there when the last of their radiant host shall have fallen from heaven." In What is Mathematics, Really?, renowned mathematician Rueben Hersh takes these eloquent words and this pervasive philosophy to task, in a subversive attack on traditional philosophies of mathematics, most notably, Platonism and formalism. Virtually all philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Mathematical objects are created by humans, not arbitrarily, but from activity with existing mathematical objects, and from the needs of science and daily life. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of the book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Plato, Descartes, Spinoza, and Kant, to Bertrand Russell, David Hilbert, Rudolph Carnap, and Willard V.O. Quine--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, Peirce, Dewey, and Lakatos. In his epilogue, Hersh reveals that this is no mere armchair debate, of little consequence to the outside world. He contends that Platonism and elitism fit well together, that Platonism in fact is used to justify the claim that "some people just can't learn math." The humanist philosophy, on the other hand, links mathematics with geople, with society, and with history. It fits with liberal anti-elitism and its historical striving for universal literacy, universal higher education, and universal access to knowledge and culture. Thus Hersh's argument has educational and political ramifications. Written by the co-author of The Mathematical Experience, which won the American Book Award in 1983, this volume reflects an insider's view of mathematical life, based on twenty years of doing research on advanced mathematical problems, thirty-five years of teaching graduates and undergraduates, and many long hours of listening, talking to, and reading philosophers. A clearly written and highly iconoclastic book, it is sure to be hotly debated by anyone with a passionate interest in mathematics or the philosophy of science. (shrink)

Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Le;vi-Strauss, and Albert Einstein. It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence (...) of unconscious mental processes in mathematical invention and other forms of creativity. Written before the explosion of research in computers and cognitive science, his book, originally titled The Psychology of Invention in the Mathematical Field , remains an important tool for exploring the increasingly complex problem of mental life. The roots of creativity for Hadamard lie not in consciousness, but in the long unconscious work of incubation, and in the unconscious aesthetic selection of ideas that thereby pass into consciousness. His discussion of this process comprises a wide range of topics, including the use of mental images or symbols, visualized or auditory words, "meaningless" words, logic, and intuition. Among the important documents collected is a letter from Albert Einstein analyzing his own mechanism of thought. (shrink)

There has been a significant shift in the discussion of a priori knowledge. The shift is due largely to the influence of Quine. The traditional debate focused on the epistemic status of mathematics and logic. Kant, for example, maintained that arithmetic and geometry provide clear examples of synthetic a priori knowledge and that principles of logic, such as the principle of contradiction, provide the basis for analytic a priori knowledge. Quine’s rejection of the analytic-synthetic distinction and his holistic empiricist account (...) of mathematic and logical knowledge undercut the traditional defenses of the a priori in two ways. First, one could no longer defend the view that mathematical and logical knowledge is a priori solely by rejecting Mill’s inductive empiricism. Moreover, holistic empiricism proved to be a more challenging position to refute than inductive empiricism. Second, the rejection of the analytic-synthetic distinction blocked an alternative defense of the a priori status of mathematics and logic that appealed to their alleged analyticity. (shrink)

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late nineteenth century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis (in the sense of the infinitesimal calculus) received much attention in the nineteenth century. They helped instigate what Hans Hahn called a “crisis of intuition”, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this “crisis” (...) as follows: Mathematicians had for a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof... (p. 67) Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called “logicization”, “when the discipline requires nothing but purely logical fundamental concepts and propositions for its development.” On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals to visual representations. This would be a radical reformation of past practice, necessary, according to its advocates, for avoiding “unclarities and mistakes” like the one exposed by Peano. (shrink)

In one of his last writings, Life: Experience and Science, Michel Foucault argued that twentieth-century French philosophy could be read as dividing itself into two divergent lines: on the one hand, we have a philosophical stream which takes individual experience as its point of departure, conceiving it as irreducible to science. On the other hand, we have an analysis of knowledge which takes into account the concrete productions of the mind, as are found in science and human practices. In order (...) to account for this division, Foucault opposed epistemologists such as Cavaillès and Canguilhem to phenomenologists such as Merleau-Ponty and Sartre but, also, and more particularly, he opposed Poincaré to Bergson. The latter was presented by Foucault as being the key-figure of the ?philosophy of experience? at the beginning of the twentieth century. Fifteen years later, in his Deleuze and in the Logics of Worlds, Alain Badiou again uses this dual structure in his interpretation of the past hundred years of French thought. He employs a series of oppositional couples: himself and Deleuze, Lautmann and Sartre, and, finally, Brunschvicg and Bergson. On the one hand a ?mathematical Platonism? and on the other a ?philosophy of vital interiority.? This Manichean reading of philosophy, and the strategic use of the figure of Bergson has, itself, a long tradition. It was also proposed by Althusser who, following Bachelard, opposed his standpoint to any form of ?empiricism.? Althusser developed his thought from a tradition of Marxist thinkers and ideologists, which included Politzer's and Nizan's critique of bourgeois philosophy and, even before that, neo-Kantians such as the philosophers of the Revue de métaphysique et de morale. The aim of this essay is to deconstruct and to put into its precise context of production this series of genealogies which entails the mobilization of Bergsonism and of the name ?Bergson.? By doing so, I hope to weight the importance of Bergsonism in twentieth-century French philosophy, in both its ?positive? and its ?negative? aspect. The essay will proceed regressively, taking into account figures such as Althusser, Badiou, Deleuze, Foucault, Canguilhem, Cavaillès, Sartre, Merleau-Ponty, but also Polizer, Brunschvicg and Alain. The conclusion of the essay is an attempt at reading the ?Bergson renaissance? in the light of new discoveries in genetics and the cognitive sciences and to tie it to the renewal of studies in the history of French philosophy. (shrink)

A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (“everything (...) is number”), continuing with the geometrization of mathematics after the discovery of irrational numbers and once again, during the nineteenth century returning to an arithmeticistic position. The third option, never explored during the history of mathematics, guides our analysis: instead of reducing space to number or number to space it is argued that both the uniqueness of these two aspects and their mutual coherence ought to direct mathematics. The presence of different schools of thought is highlighted and then the argument proceeds by distinguishing numerical and spatial facts, while accounting for the strict correlation of operations on the law side of the numerical aspect and their correlated numerical subjects (numbers). Discussing the examples of 2 + 2 = 4 and the definition of a straight line as the shortest distance between two points provide the background for a brief sketch of the third alternative proposed (inter alia against the background of an assessment of infinity and continuity and the vicious circles present in contemporary mathematical arithmeticistic claims). (shrink)

Much of the current thought concerning mathematical ontology and epistemology follows Quine and Putnam in looking to the indispensable application of mathematics in science. A standard assumption of the indispensability approach is some version of confirmational holism, i.e., that only "sufficiently large" sets of beliefs "face the tribunal of experience." In this paper I develop and defend a distinction between a pure mathematical theory and a mathematized scientific theory in which it is applied. This distinction allows for the possibility (...) that pure mathematical theories are systematically insulated from such confirmation in virtue of their being distinct from the "sufficiently large" blocks of scientific theory that are empirically confirmed. (shrink)

After more than 60 years, Shannon’s research continues to raise fundamental questions, such as the one formulated by R. Luce, which is still unanswered: “Why is information theory not very applicable to psychological problems, despite apparent similarities of concepts?” On this topic, S. Pinker, one of the foremost defenders of the widespread computational theory of mind, has argued that thought is simply a type of computation, and that the gap between human cognition and computational models may be illusory. In (...) this context, in his latest book, titled Thinking Fast and Slow, D. Kahneman provides further theoretical interpretation by differentiating the two assumed systems of the cognitive functioning of the human mind. He calls them intuition (system 1) determined to be an associative (automatic, fast and perceptual) machine, and reasoning (system 2) required to be voluntary and to operate logical-deductively. In this paper, we propose a mathematical approach inspired by Ausubel’s meaningful learning theory for investigating, from the constructivist perspective, information processing in the working memory of cognizers. Specifically, a thought experiment is performed utilizing the mind of a dual-natured creature known as Maxwell’s demon: a tiny “man–machine” solely equipped with the characteristics of system 1, which prevents it from reasoning. The calculation presented here shows that the Ausubelian learning schema, when inserted into the creature’s memory, leads to a Shannon-Hartley-like model that, in turn, converges exactly to the fundamental thermodynamic principle of computation, known as the Landauer limit. This result indicates that when the system 2 is shut down, both an intelligent being, as well as a binary machine, incur the same minimum energy cost per unit of information (knowledge) processed (acquired), which mathematically shows the computational attribute of the system 1, as Kahneman theorized. This finding links information theory to human psychological features and opens the possibility to experimentally test the computational theory of mind by means of Landauer’s energy cost, which can pave a way toward the conception of a multi-bit reasoning machine. (shrink)

It is a common thought that mathematics can be not only true but also beautiful, and many of the greatest mathematicians have attached central importance to the aesthetic merit of their theorems, proofs and theories. But how, exactly, should we conceive of the character of beauty in mathematics? In this paper I suggest that Kant's philosophy provides the resources for a compelling answer to this question. Focusing on §62 of the ‘Critique of Aesthetic Judgment’, I argue against the common (...) view that Kant's aesthetics leaves no room for beauty in mathematics. More specifically, I show that on the Kantian account beauty in mathematics is a non-conceptual response felt in light of our own creative activities involved in the process of mathematical reasoning. The Kantian proposal I thus develop provides a promising alternative to Platonist accounts of beauty widespread among mathematicians. While on the Platonist conception the experience of mathematical beauty consists in an intellectual insight into the fundamental structures of the universe, according to the Kantian proposal the experience of beauty in mathematics is grounded in our felt awareness of the imaginative processes that lead to mathematical knowledge. The Kantian account I develop thus offers to elucidate the connection between aesthetic reflection, creative imagination and mathematical cognition. (shrink)

Lines of Thought addresses how we are able to think about abstract possibilities: How can we think about math, despite the immateriality of numbers, sets, and other mathematical entities? How are we able to think about what might have happened if history had taken a different turn? Questions like these turn up in nearly every part of cognitive science, and they are central to our human position of having only limited knowledge concerning what is or might be true. Because (...) we cannot experience hypothetical or future events or abstract concepts, we cannot use our ordinary sense of perception or memory to think about these subjects, so what underlies our ability to make these assumptions? Lance Rips explores people's beliefs about possibilities as they arise in the context of basic concepts, including numbers, causality, and reasons. He argues that beliefs about these concepts cannot be meaningfully reduced to perceptual information, remembered instances, or probabilities. He also claims that analogies to cognitive perception models are equally unhelpful in understanding what makes thinking of possibilities possible. Instead, he makes the case that our abilities here depend on the intrinsic hardwiring of the human mind. Lines of Thought provides an overview and a point of view on research in higher-level cognitive science, integrating theories from psychology, philosophy, and linguistics. The book is written in an accessible style that will provide students with essential background for their own thoughts about this domain. (shrink)

Philosophers have devoted a great deal of discussion to the question of whether an inverted spectrum thought experiment refutes functionalism. (For a review of the inverted spectrum and its many philosophical applications, see Byrne, 2004.) If Ho?man is correct the matter can be swiftly and conclusively settled, without appeal to any empirical data about color vision (or anything else). Assuming only that color experiences and functional relations can be mathematically represented, a simple mathematical result.

Edited book containing the following essays: 1 Getting over Gettier, Alan Musgrave.- 2 Justified Believing: Avoiding the Paradox Gregory W. Dawes.- Chapter 3! Literature and Truthfulness,Gregory Currie.- 4 Where the Buck-passing Stops, Andrew Moore.- 5 Universal Darwinism: Its Scope and Limits, James Maclaurin, - 6 The Future of Utilitarianism,Tim Mulgan. 7 Kant on Experiment, Alberto Vanzo.- 8 Did Newton ʻFeignʼ the Corpuscular Hypothesis? Kirsten Walsh.- 9 The Progress of Scotland: The Edinburgh Philosophical Societies and the Experimental Method, Juan Gomez.- 10 (...) Propositions: Truth vs. Existence, Heather Dyke.- 11 Against Advanced Modalizing, Josh Parsons.- 12 Spread Worlds, Plenitude and Modal Realism: A Problem for DavidLewis, Charles R. Pigden and Rebecca E. B. Entwisle.- 13 Defending Quine on Ontological Commitment. 14. The Scandal of Platonism, Vladimír Svoboda.- 15 A Neglected Reply to Prior's Dilemma J. C. Beall. 16 Mathematical and Empirical Concepts, Pavel Materna.- 17 Post-Fregean Thoughts on Propositional Unity, Bjørn Jespersen.- 18 Best-path Theorem Proving: Compiling Derivations, Martin Frické.- 19 Is Imperative Inference Impossible?, Hannah Clark-Younger. ​. (shrink)

Nobel Laureate Eugene Wigner once wondered about "the unreasonable effectiveness of mathematics" in the formulation of the laws of nature. Is God a Mathematician? investigates why mathematics is as powerful as it is. From ancient times to the present, scientists and philosophers have marveled at how such a seemingly abstract discipline could so perfectly explain the natural world. More than that -- mathematics has often made predictions, for example, about subatomic particles or cosmic phenomena that were unknown at the time, (...) but later were proven to be true. Is mathematics ultimately invented or discovered? If, as Einstein insisted, mathematics is "a product of human thought that is independent of experience," how can it so accurately describe and even predict the world around us? Mathematicians themselves often insist that their work has no practical effect. The British mathematician G. H. Hardy went so far as to describe his own work this way: "No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." He was wrong. The Hardy-Weinberg law allows population geneticists to predict how genes are transmitted from one generation to the next, and Hardy's work on the theory of numbers found unexpected implications in the development of codes. Physicist and author Mario Livio brilliantly explores mathematical ideas from Pythagoras to the present day as he shows us how intriguing questions and ingenious answers have led to ever deeper insights into our world. This fascinating book will interest anyone curious about the human mind, the scientific world, and the relationship between them. (shrink)

This is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. Reassessing the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as our understanding of mathematics, Michael Potter places arithmetic at the interface between experience, language, thought, and the world.

In books such as The World Within the World and The Anthropic Cosmological Principle, astronomer John Barrow has emerged as a leading writer on our efforts to understand the universe. Timothy Ferris, writing in The Times Literary Supplement of London, described him as "a temperate and accomplished humanist, scientist, and philosopher of science--a man out to make a contribution, not a show." Now Barrow offers the general reader another fascinating look at modern physics, as he explores the quest for a (...) single, unifying theory that will unlock nature's secrets. Theories of Everything is more than a history of science, more than a popular report on recent research and discoveries. Barrow provides a reflective, intelligent commentary on what a true Theory of Everything would be--its ingredients, its limitations, and what it could tell us about the universe. Never before, he writes, have physicists been so confident and so eager in the hunt for this "cosmic Rosetta Stone," as he calls it: "a single all-embracing picture of all the laws of nature from which the inevitability of all things seen must follow with unimpeachable logic." He lays out eight essential ingredients for a Theory of Everything and then explores each in turn, tracing how our knowledge has developed and how scientific discovery relates to our changing philosophy and religious thought in each area. Some of these ingredients are obvious--the laws of nature must be explained, for example, as well as its organizing principles--but others may be surprising, such as broken symmetries and selection biases. A Theory of Everything must account for the fact that the universe is "messy and complicated," he tells us, and for the limitations imposed by the questions we ask and the information we can obtain. The key lies in the remarkable capacity of mathematics to express the fundamental workings of the physical world--a language that the human mind is uniquely equipped to understand and manipulate. Barrow examines what mathematics actually is and describes how it makes the universe intelligible and provides a path to the underlying coherence in nature--which has led, in fact, to arguments that the universe itself is a vast computer. Yet even the most complete theory, even the most comprehensive mathematical explanation, cannot account for the uncomputable varieties of human experience and thought. "No non-poetic account of reality," he writes, "can be complete." In a field where the authorities converse in equations and mathematical notations, John Barrow speaks with the voice of thoughtful and knowledgeable humanist. Written with eloquence and expertise, Theories of Everything establishes a new perspective on humanity's efforts to explain the universe. (shrink)

Is the self narratively constructed? There are many who would answer yes to the question. Dennett (1991) is, perhaps, the most famous proponent of the view that the self is narratively constructed, but there are others, such as Velleman (2006), who have followed his lead and developed the view much further. Indeed, the importance of narrative to understanding the mind and the self is currently being lavished with attention across the cognitive sciences (Dautenhahn, 2001; Hutto, 2007; Nelson, 2003). Emerging from (...) this work, there appear to be a variety of ways in which we can think of the narrative construction of the self and the relationship between the narrative self and the embodied agent. I wish to examine two such ways in this paper. The first I shall call the abstract narrative account, this is because its proponents take the narrative self to be an abstraction (Dennett, 1991; Velleman, 2006). Dennett, for example, refers to the self as a centre of narrative gravity, to be thought of as analogous to a mathematical conception of the centre of gravity of an object. The second I shall call the embodied narrative account and this is the view that the self is constituted both by an embodied consciousness whose experiences are available for narration and narratives themselves, which can play a variety of roles in the agent’s psychological life. (shrink)

The physicist's conception of space-time underwent two major upheavals thanks to the general theory of relativity and quantum mechanics. Both theories play a fundamental role in describing the same natural world, although at different scales. However, the inconsistency between them emerged clearly as the limitation of twentieth-century physics, so a more complete description of nature must encompass general relativity and quantum mechanics as well. The problem is a theorists' problem par excellence. Experiment provide little guide, and the inconsistency mentioned above (...) is an important problem which clearly illustrates the intermingling of philosophical, mathematical, and physical thought. In fact, in order to unify general relativity with quantum field theory, it seems necessary to invent a new mathematical framework which will generalise Riemannian geometry and therefore our present conception of space and space-time. Contemporary developments in theoretical physics suggest that another revolution may be in progress, through which a new kind of geometry may enter physics, and space-time itself can be reinterpreted as an approximate, derived concept. The main purpose of this article is to show the great significance of space-time geometry in predetermining the laws which are supposed to govern the behaviour of matter, and further to support the thesis that matter itself can be built from geometry, in the sense that particles of matter as well as the other forces of nature emerges in the same way that gravity emerges from geometry. Scientific research is not a process of steady accumulation of absolute truths, which has culminated in present theories, but rather a much more dynamic kind of process in which there are no final theoretical concepts valid in unlimited domains. (David Bohm). (shrink)

Most philosophers writing on the imagination have insisted that we cannot gain knowledge by relying on imagining – in contrast, say, to perception or inference – as our source of knowledge. Their doubts have not concerned the widely acknowledged fact that imagining a situation may help or enable us to acquire certain pieces of knowledge – for instance, when we visualise geometrical figures or patterns of numbers to come to know mathematical facts (cf. Giaquinto (1992) and (2007)), or when we (...) engage in thought experiments or other imaginative projects to gain philosophical knowledge (cf. Gendler (2000), and Gendler & Hawthorne (2002)). Instead, what is traditionally rejected is the idea that mental episodes of imagining can ground or constitute knowledge in the same way in which episodes of perceiving, remembering or judging can do so.1.. (shrink)

: This paper discusses the origins of two key notions in Foucault's work up to and including The Archaeology of Knowledge. The first of these notions is the notion of "archaeology" itself, a form of historical investigation of knowledge that is distinguished from the mere history of ideas in part by its unearthing what Foucault calls "historical a prioris". Both notions, I argue, are derived from Husserlian phenomenology. But both are modified by Foucault in the light of Jean Cavaillès's critique (...) of Husserl's theory of science. On Husserl's view, we demand that propositions holding of scientific objects be intersubjective and invariant, but this demand conflicts with our immediate experience, which is essentially bound to a subject's perspective. Thus the mathematical and physical sciences must utilise formal languages to fix these truths independently of the thoughts of a particular subject. This necessary procedure leads to the sedimentation of these formal systems: we forget their source in the concrete experiences of individuals, and use them as purely technical means. The technique of reactivating the intentional acts in which sedimented formal systems originated is thus, in Fink's terminology, an archaeological method. Foucault and Cavaillès retain the general outlines of this archaeology of the sciences, but they reject its appeal to conscious acts of meaning, to what Cavaillès calls "the philosophy of consciousness". I conclude by discussing the implicit difficulties in the "linguistic transcendentalism" proposed as an alternative by these French critics of Husserl. (shrink)

A remarkable theorem by Clifton, Bub and Halvorson (2003) (CBH) characterizes quantum theory in terms of information--theoretic principles. According to Bub (2004, 2005) the philosophical significance of the theorem is that quantum theory should be regarded as a ``principle'' theory about (quantum) information rather than a ``constructive'' theory about the dynamics of quantum systems. Here we criticize Bub's principle approach arguing that if the mathematical formalism of quantum mechanics remains intact then there is no escape route from solving the measurement (...) problem by constructive theories. We further propose a (Wigner--type) thought experiment that we argue demonstrates that quantum mechanics on the information--theoretic approach is incomplete. (shrink)

A new view of the functional role of the left anterior cortex in language use is proposed. The experimental record indicates that most human linguistic abilities are not localized in this region. In particular, most of syntax (long thought to be there) is not located in Broca's area and its vicinity (operculum, insula, and subjacent white matter). This cerebral region, implicated in Broca's aphasia, does have a role in syntactic processing, but a highly specific one: It is the neural (...) home to receptive mechanisms involved in the computation of the relation between transformationally moved phrasal constituents and their extraction sites (in line with the Trace-Deletion Hypothesis). It is also involved in the construction of higher parts of the syntactic tree in speech production. By contrast, basic combinatorial capacities necessary for language processing – for example, structure-building operations, lexical insertion – are not supported by the neural tissue of this cerebral region, nor is lexical or combinatorial semantics. The dense body of empirical evidence supporting this restrictive view comes mainly from several angles on lesion studies of syntax in agrammatic Broca's aphasia. Five empirical arguments are presented: experiments in sentence comprehension, cross-linguistic considerations (where aphasia findings from several language types are pooled and scrutinized comparatively), grammaticality and plausibility judgments, real-time processing of complex sentences, and rehabilitation. Also discussed are recent results from functional neuroimaging and from structured observations on speech production of Broca's aphasics. Syntactic abilities are nonetheless distinct from other cognitive skills and are represented entirely and exclusively in the left cerebral hemisphere. Although more widespread in the left hemisphere than previously thought, they are clearly distinct from other human combinatorial and intellectual abilities. The neurological record (based on functional imaging, split-brain and right-hemisphere-damaged patients, as well as patients suffering from a breakdown of mathematical skills) indicates that language is a distinct, modularly organized neurological entity. Combinatorial aspects of the language faculty reside in the human left cerebral hemisphere, but only the transformational component (or algorithms that implement it in use) is located in and around Broca's area. Key Words: agrammatism; aphasia; Broca's area; cerebral localization; dyscalculia; functional neuroanatomy; grammatical transformation; modularity; neuroimaging; syntax; trace deletion. (shrink)

This volume, honoring the renowned historian of science, Allen G Debus, explores ideas of science - `experiences of nature' - from within a historiographical tradition that Debus has done much to define. As his work shows, the sciences do not develop exclusively as a result of a progressive and inexorable logic of discovery. A wide variety of extra-scientific factors, deriving from changing intellectual contexts and differing social millieus, play crucial roles in the overall development of scientific thought. These essays (...) represent case studies in a broad range of scientific settings - from sixteenth-century astronomy and medicine, through nineteenth-century biology and mathematics, to the social sciences in the twentieth-century - that show the impact of both social settings and the cross-fertilization of ideas on the formation of science. Aimed at a general audience interested in the history of science, this book closes with Debus's personal perspective on the development of the field. Audience: This book will appeal especially to historians of science, of chemistry, and of medicine. (shrink)

_The usual approach in Buddhist-Western writings uses Buddhist perspectives to help answer Western philosophical-psychological questions. This paper reverses the process and uses the Western philosophical perspective of Nietzsche to answer questions of Buddhist-conceived nirvana. Nietzsche's philosophy of will, expounded primarily through the Zarathustra essays, provides an active and affirmative alternative for understanding and attaining nirvana. His ideas of free will and will to power have commonalities with Buddhist practice and thought, including nonattachment, nihilism, no-self, and meditation. Nietzschean will revises (...) the Buddhist notion of right effort to answer questions about coping with inner suffering and outer-world corruption. It shows nirvana to be less a state of passive being and more a state of active becoming. Why approach such important matters as transcendence, power, and God from the standpoint of the 'I'? First, I-centered analysis can clarify egological concepts such as the subject-I, object-self, and conceptualizing-ego and what these concepts contribute to an experience-based metaphysics, for even the most objective factual or mathematical expression must be stated and understood by an active subject-I. Second, I-centered analysis can advance the phenomenological study of the role of the I in the subjective realms of mind. Third, it can help resolve issues in both Western and Buddhist philosophy such as activism-passivism, subjectivity-objectivity, will and freedom, I and other, and secular/sacred presence in consciousness_. (shrink)

This essay interprets the meaning of one of the cards in aTarot deck, "The Magician," in the context of process philosophy in the tradition of Alfred North Whitehead. It brings into the conversation the philosophical legacy of American semiotician Charles Sanders Peirce as well as French poststructuralist Gilles Deleuze. Some of their conceptualizations are explored herein for the purpose of explaining the symbolic function of the Magician in the world. From the perspective of the logic of explanation, the sign of (...) the Magician is an index of nonmechanistic, mutualist or circular, causality that enables self-organization embedded in coordination dynamics. Its action is such as to establish an unorthodox connection crossing over the dualistic gap between mind and matter, science and magic, process and structure, the world without and the world within, subject and object, and human experience and the natural world, thereby overcoming what Whitehead called the paradox of the connectedness of things. The Magician represents a certain quality that acts as a catalytic agent capable of eliciting transmutations, that is, the emergence of novelty. I present a model for process∼structure that uses mathematics on the complex plane and the rules of projective geometry. The corollary is such that the presence of the Magician in the world enables a particular organization of thought that makes pre-cognition possible. (shrink)

Classical physics states that physical reality is local--a point in space cannot influence another point beyond a relatively short distance. However, In 1997, experiments were conducted in which light particles (photons) originated under certain conditions and traveled in opposite directions to detectors located about seven miles apart. The amazing results indicated that the photons "interacted" or "communicated" with one another instantly or "in no time." Since a distance of seven miles is quite vast in quantum physics, this led physicists to (...) an extraordinary conclusion--even if experiments could somehow be conducted in which the distance between the detectors was half-way across the known universe, the results would indicate that interaction or communication between the photons would be instantaneous. What was revealed in these little-known experiments in 1997 is that physical reality is non-local--a discovery that Robert Nadeau and Menas Kafatos view as "the most momentous in the history of science." In The Non-Local Universe, Nadeau and Kafatos offer a revolutionary look at the breathtaking implications of non-locality. They argue that since every particle in the universe has been "entangled" with other particles like the two photons in the 1997 experiments, physical reality on the most basic level is an undivided wholeness. In addition to demonstrating that physical processes are vastly interdependent and interactive, they also show that more complex systems in both physics and biology display emergent properties and/or behaviors that cannot be explained in the terms of the sum of parts. One of the most startling implications of non-locality in human terms, claim the authors, is that there is no longer any basis for believing in the stark division between mind and world that has preoccupied much of western thought since the seventeenth century. And they also make a convincing case that human consciousness can now be viewed as emergent from and seamlessly connected with the entire cosmos. In pursuing this groundbreaking argument, the authors not only provide a fascinating history of developments that led to the discovery of non-locality and the sometimes heated debate between the great scientists responsible for these discoveries. They also argue that advances in scientific knowledge have further eroded the boundaries between physics and biology, and that recent studies on the evolution of the human brain suggest that the logical foundations of mathematics and ordinary language are much more similar than we previously imagined. What this new knowledge reveals, the authors conclude, is that the connection between mind and nature is far more intimate than we previously dared to imagine. What they offer is a revolutionary look at the implications of non-locality, implications that reach deep into that most intimate aspect of humanity--consciousness. (shrink)

The aim of the dissertation is to propose a new understanding of the philosophy of Charles S. Peirce. Peirce sought to construct a philosophical system applicable to all of human experience, but he never presented this system in a unified work. In the dissertation I attempt to present the strongest possible reconstruction of Peirce’s mature philosophy. My thesis is that Peirce’s philosophy is best understood as an extended exploration and application of his concept of mathematical continuity, which he called "the (...) master-key of philosophy." Many scholars have recognized that Peirce’s concept of continuity is important to his metaphysical theories. The bulk of the dissertation is devoted to examining this concept and explicating its importance throughout his philosophy. I argue that Peirce’s theory of semeiotic provides a general model of experience that elaborates the direct experience of continuity described in phenomenology. This model in turn serves as the basis for his metaphysics and evolutionary cosmology. Part I of the dissertation sketches Peirce’s response to Kant’s philosophy and presents an outline of his classification of the sciences. Part II presents Peirce’s technical conception of continuity, showing its origins in formal logic and in his revision of Cantor’s theory of transfinite sets. Part III examines the role of the continuity principle in phenomenology, esthetics, ethics, and semeiotic, which bridge the rather wide gap between mathematics and metaphysics in Peirce’s system. Part IV presents an overview of Peirce’s cosmology and metaphysics, with particular attention to their methodological dependence upon semeiotic. Part IV includes consideration of two issues that emerge as crucial to the assessment of Peirce’s thought. The first concerns the ontological status of extra-semeiotic entities, and is known as the problem of "semiotic idealism." I argue that Peirce is not a semiotic idealist. The second issue concerns the testability of Peirce’s metaphysical hypotheses.. (shrink)

This paper concerns the role of intuitions in mathematics, where intuitions are meant in the Kantian sense, i.e. the “seeing” of mathematical ideas by means of pictures, diagrams, thought experiments, etc.. The main problem discussed here is whether Platonistic argumentation, according to which some pictures can be considered as proofs (or parts of proofs) of some mathematical facts, is convincing and consistent. As a starting point, I discuss James Robert Brown’s recent book Philosophy of Mathematics, in particular, his primarily (...) examples and analogies. I then consider some alternatives and counterarguments, namely John Norton’s opposite view, that intuitions are just pictorially represented logical arguments and are superfluous; and the Kantian transcendental theory of construction in imagination, as it is developed in the works of Marcus Giaquinto and Michael Friedman. Although I support the claim that some intuitions are essential in mathematical justification, I argue that a Platonistic approach to intuitions is partial and one should go further than a Platonist in explaining how some intuitions can deliver new mathematical knowledge. (shrink)

In the the passage just quoted from the Dialogues concerning Natural Religion, David Hume developed a thought-experiment that contravened his better-known views about "chance" expressed in his Treatise and first Enquiry. For among other consequences of the 'eternal-recurrence' hypothesis Philo proposes in this passage, it may turn out that what the vulgar call cause is nothing but a secret and concealed chance. (In this sentence, I have simply reversed "cause" and "chance" in a well-known passage from Hume's Treatise, p. (...) 130). In the first eight sections of this essay, I develop one topological and model-theoretic analogue of Hume's thought-experiment, in which 'most' ('A-generic') models M of a 'scientific' theory U are both 'eternally recurrent' and topologically random (in a sense which will be made precise), even though they are 'inductively' defined, via a step-by-step ('empirical'?) procedure that Hume might have been inclined to endorse. The last aspect of this model-theoretic thought-experiment also serves to distinguish it from simpler measure-theoretic prototypes that are known to follow from Kolmogorov's Zero-One Law (cf. the Introduction, 5.2, 6.1 and 6.7 below). In the last three sections, I will argue more informally (1) that the metamathematical thought-experiments just mentioned do have a genuine metaphysical relevance, and that this relevance is predominantly skeptical in its implications; (2) that such 'nonstandard' instances of semantic underdetermination and 'pathology' seem to be the metatheoretic rule rather than the exception; and therefore, (3) that metamathematical and metatheoretic 'malign-genius' arguments are quite coherent, contrary (e.g.) to assertions such as that of Putnam (1980), pp. 7-8. In the essay's conclusion, finally, I assimilate (2) and (3) to the familiar datum that 'simplicity', rather than 'pathology', has more often than not turned out to be an anomalous 'special case' in the historical development of scientific and mathematical ontology. (shrink)

Beckett and Badiou offers a provocative new reading of Samuel Beckett's work on the basis of a full, critical account of the thought of Alain Badiou. Badiou is the most eminent of contemporary French philosophers. His devotion to Beckett's work has been lifelong. Yet for Badiou philosophy must be integrally affirmative, whilst Beckett apparently commits his art to a work of negation. Beckett and Badiou explores the coherences, contradictions, and extreme complexities of the intellectual relationship between the two oeuvres. (...) It examines Badiou's philosophy of being, the event, truth, and the subject and the importance of mathematics within his system. It considers the major features of his politics, ethics, and aesthetics and provides an explanation, interpretation, critique, and radical revision of his work on Beckett. It argues that, once revised, Badiou's version of Beckett offers an extraordinarily powerful tool for understanding his work. -/- Badiou and Beckett are instances of a vestigial or melancholic modernism; that is, in the teeth of a contemporary culture that dreams ever more ambitiously of plenitude, they commit themselves to a rigorous concept of limit and intermittency. Truth and value are occasional and rare. It is seldom that the chance event arrives to disturb the inertia of the world. For Badiou, however, it is the event and its consequences alone that matter. Beckett rather insists on the common experience of intermittency as destitution. His art is a series of limit-figures, exquisitely subtle and nuanced forms for a world whose state of seemingly rigid paralysis is also always volatile, delicately balanced. (shrink)

This essay discusses Socrates’ use of hypothetical choices as an early version of what was to become in the twentieth century the discipline of decision theory as expressed by one of its prominent proponents, F. P. Ramsey. Socrates’ use of hypothetical choices and thought experiments in the dialogues is a way of reassuring himself of an interlocutor’s philosophical potential. For example, to assess just how far Alcibiades is willing to go to attain his goal of being a great Athenian (...) leader, we employ Ramsey’s concept of Mathematical Expectation. Mathematical Expectation operates on the assumption that it is not enough to measure probability; we must also measure our belief to apportion our belief to the probability. In other words, it illustrates how strongly or to what degree a person holds a particular belief. If a person’s belief in X lacks enough doubts to cancel the belief out, the probability of his acting on this belief is higher than if his belief in X was plagued by a greater number of doubts. (shrink)

Starting with the Descartes' cogito, "I think, therefore I am"--and taking an uncompromisingly rational, rigorously phenomenological approach--I attempt to derive the basic principles of recursion theory (the backbone of all mathematics and logic), and from that the principles of feedback control theory (the backbone of all biology), leading to the basic ideas of quantum mechanics (the backbone of all physics). What is derived is not the full quantum theory, but a basic framework--derived from a priori principles along with common everyday (...) experience--of how the universe of everyday experience should work if it operates according to rational principles. We find, to our surprise, that the resulting system has all the most puzzling features of quantum physics that make physicists scratch their heads. Far from being "bizarre" and "weird", as is usually thought, the strangest paradoxes of quantum theory turn out to be just what one ought to expect of a rational universe. It is the classical, pre-quantum universe of the nineteenth century that has irrational, mystical components. The quantum-mechanics-like theory that is developed is, furthermore, most compatible with the strictest, most uncompromisingly rationalist of the standard interpretations of quantum mechanics, those which add no ad hoc elements to the theory, and which generally trace their history to the relative state formulation of Everett (also called the "many worlds" interpretation). These interpretations take the universe to be quite literally describable as a quantum wavefunction. As with any project this far-reaching in scope, I confess I have had to make some working assumptions along the way. I have attempted to isolate these, and clearly label them as points of possible future revision--they are marked in the text with an asterisk (*). (shrink)

The final two volumes, numbers IV and V, of the Oxford University Press edition of the Collected Works of Kurt Gödel [3]-[7] appeared in 2003, thus completing a project that started over twenty years earlier. What I mainly want to do here is trace, from the vantage point of my personal involvement, the at some times halting and at other times intense development of the Gödel editorial project from the first initiatives following Gödel’s death in 1978 to its completion last (...) year. It may be useful to scholars mounting similar editorial projects for other significant figures in our field to learn how and why various decisions were made and how the work was carried out, though of course much is particular to who and what we were dealing with. My hope here is also to give the reader who is not already familiar with the Gödel Works a sense of what has been gained in the process, and to encourage dipping in according to interest. Given the absolute importance of Gödel for mathematical logic, students should also be pointed to these important source materials to experience first hand the exercise of his genius and the varied ways of his thought and to see how scholarly and critical studies help to expand their significance. Though indeed much has been gained in our work there is still much that can and should be done; besides some indications below, for that the reader is referred to [2]. (shrink)

Academia’s mathematical metaphysics are briefly explored en route to an elaboration of the qualitatively rigorous requirements underpinning the calibration and unambiguous interpretation of quantitative instrumentation in any science. Of particular interest are Gadamer’s emphases on number as the paradigm of the noetic, on the role of play in interpretation, and on Hegel’s sense of method as the activity of the thing itself that thought experiences. These point toward and overlap with (1) Latour’s study of the metrological social networks through (...) which technological phenomena are brought into language as modes of being that can be understood, and (2) the way that Rasch’s models for measurement comprise a potential beginning for metaphysically astute, qualitatively and quantitatively integrated, mathematical methods in the social sciences. The paper closes with observations on the general problem that is philosophy, the need to remain open to multiplicities of meaning even as clear understandings are sought and obtained. (shrink)

The paper deals with an intellectual and historical approach to the changing meanings of the term “model” in life sciences. The author 1st tries to understand how modeling has gradually spread over life sciences then he particularly focus on the birth of mathematical modeling in this field. This quite new practice offers new insights on the old debate concerning the mathematization of life sciences. Nowadays, through computers, mathematics not only analyze or quantify but model things: what does it mean? The (...) question turns out to be dramatic as far as digital simulation is concerned. That is the reason why he choosed to study a particular case: the history of the individual plant mathematical modeling. On this case, one may discern various epistemological standpoints that caused various reactions to the emergence of computer simulation, from the 50s to the 90s. The author shows that philosophical views often play a role in the history of sciences, especially in the choice of supposed proper mathematical formalisms. This will indicate that contemporary discourses tend to echo each other. That is the reason why he feels authorized to address the Foucault’s concept—épistémè—to denote these convergences between the scientific and the philosophical discourses. Finally, it is suggested that this épistémè gradually is changing because one can currently observe the emergence of a “graphical” thought through these simulation experiments, which tends to replace a more functionalist thought. (shrink)

When John von Neumann turned his interest to computers, he was one of the leading mathematicians of his time. In the 1940s, he helped design two of the first stored-program digital electronic computers. He authored reports explaining the functional organization of modern computers for the first time, thereby influencing their construction worldwide (von Neumann, 1945; Burks et al., 1946). In the first of these reports, von Neumann described the computer as analogous to a brain, with an input “organ” (analogous to (...) sensory neurons), a memory, an arithmetical and a logical “organ” (analogous to associative neurons), and an output “organ” (analogous to motor neurons). His experience with computers convinced him that brains and computers, both having to do with the processing of information, should be studied by a new discipline–automata theory. In fact, according to von Neumann, automata theory would cover not only computers and brains, but also any biological or artificial systems that dealt with information and control, including robots and genes. Von Neumann never formulated a full-blown mathematical theory of automata, but he wrote several important exploratory papers (von Neumann, 1951, 1956, 1966). Meanwhile, besides designing hardware, he developed some of the first programs, programming languages, programming techniques, and numerical methods for solving mathematical problems using computers. (Much of his work on computing is reprinted in Aspray and Burks, 1987.) Shortly before his death in 1956, he wrote an informal synthesis of his views about brains. Though von Neumann left his manuscript sketchy and unfinished, Yale University Press published it as The Com- puter and the Brain in 1958. The 2000 reprint of this small but informative book is an opportunity to learn, or be reminded of, von Neumann’s thoughts on the computational organization of the mind-brain. Von Neumann began by explaining computers, which for him were essentially number crunchers: to compute was “to operate on .. (shrink)

Jay Zeman one must keep a bright lookout for unintended and unexpected changes thereby brought about in the relations of different significant parts of the diagram to one another. Such operations upon diagrams, whether external or imaginary, take the place of the experiments upon real things that one performs in chemical and physical research. Chemists have ere now, I need not say, described experimentation as the putting of questions to Nature. Just so, experiments upon diagrams are questions put to the (...) Nature of the relations concerned (4.530). 1 The diagrammatic nature of mathematical reasoning suggests that as my power to create diagrams increases, so too will my capacity for fruitful mathematical reasoning. Peirce's own work involved an unending series of experiments with different diagrammatic notations, all interesting, some difficult, some extremely fruitful. And the diagrammatic notations available are not only a function of some kind of “internal mental activity.” As Dewey has noted, “Breathing is an affair of the air as truly as of the lungs; digesting an affair of food as truly as of tissues of stomach” (Dewey, 15); so analogously is mathematical reasoning an affair of the diagrams available as truly as of the “mind” (which is then not limited to something inside the head, but includes the relevant diagrams, “external” as well as “internal”); so does mathematical reasoning have its “alembics and cucurbits” just as surely as does chemistry. In doing mathematical reasoning, we make of the diagrams “instruments of thought,” and advances in the technology of diagrams can directly affect our patterns of reasoning. I can imagine Peirce spending hours (and dollars) in a modern artists' supply store. (shrink)

The fundamental problem on which Ilya Prigogine and the Brussels-Austin Group have focused can be stated briefly as follows. Our observations indicate that there is an arrow of time in our experience of the world (e.g., decay of unstable radioactive atoms like Uranium, or the mixing of cream in coffee). Most of the fundamental equations of physics are time reversible, however, presenting an apparent conflict between our theoretical descriptions and experimental observations. Many have thought that the observed arrow of (...) time was either an artifact of our observations or due to very special initial conditions. An alternative approach, followed by the Brussels-Austin Group, is to consider the observed direction of time to be a basic physical phenomenon and to develop a mathematical formalism that can describe this direction as being due to the dynamics of physical systems. In part I of this essay, I review and assess an attempt to carry out an approach that received much of their attention from the early 1970s to the mid 1980s. In part II, I will discuss their more recent approach using rigged Hilbert spaces. (shrink)

The Socratic method has a long history in teaching philosophy and mathematics, marked by such names as Karl Weierstra, Leonard Nelson and Gustav Heckmann. Its basic idea is to encourage the participants of a learning group (of pupils, students, or practitioners) to work on a conceptual, ethical or psychological problem by their own collective intellectual effort, without a textual basis and without substantial help from the teacher whose part it is mainly to enforce the rigid procedural rules designed to ensure (...) a fruitful, diversified, open and consensus-oriented thought process. Several features of the Socratic procedure, especially in the canonical form given to it by Heckmann, are highly attractive for the teaching of medical ethics in small groups: the strategy of starting from relevant singular individual experiences, interpreting and cautiously generalizing them in a process of inter-subjective confrontation and confirmation, the duty of non-directivity on the part of the teacher in regard to the contents of the discussion, the necessity, on the part of the participants, to make explicit both their own thinking and the way they understand the thought of others, the strict separation of content level and meta level discussion and, not least, the wise use made of the emotional and motivational resources developing in the group process. Experience shows, however, that the canonical form of the Socratic group suffers from a number of drawbacks which may be overcome by loosening the rigidity of some of the rules. These concern mainly the injunction against substantial interventions on the part of the teacher and the insistence on consensus formation rooted in Leonard Nelson's Neo-Kantian Apriorism. (shrink)

This paper argues that Nicholas of Cusa’s investigation of infinity and incommensurability in De docta ignorantia was shaped by the mathematical innovations and thought experiments of fourteenth-century natural philosophy. Cusanus scholarship has overlooked this influence, in part because Raymond Klibansky’s influential edition of De docta ignorantia situated Cusa within the medieval Platonic tradition. However, Cusa departs from this tradition in a number of ways. His willingness to engage incommensurability and to compare different magnitudes of infinity distinguishes him from his (...) Platonic predecessors, who had appropriated the Pythagorean model of universal harmonies. Cusa’s penchant for representing quantity geometrically suggests not only that he has adopted the fourteenth-century method of latitude measurement, but that he accepts incommensurability as normative. Finally, Cusa’s persistent attention to mathematical inaccuracy and to his own learned ignorance suggests his kinship with the meta-critical, conjectural quality of fourteenth-century thought. (shrink)

Peacocke argues for a ‘generalized rationalism’, holding that ‘all entitlement has a fundamentally a priori component.’ (2) But his rationalism ‘differs from those of Frege and Gödel, just as theirs differ from that of Leibniz.’ He requires both substantive theories of intentional content and of understanding, and systematic formal theories of referential semantics and truth. We need an externalist theory of content: ‘Only mental states with externally individuated contents can make judgements about the external, mind-independent world rational.’ (123) Purely evidential (...) conceptions of meaning and content are inadequate. (34-49) They cannot account for the following: a thinker often has to work out what would be evidence for a content; contents cannot depend, for their identity, on all of the infinitely ramifying evidential connections among them; and thinkers conceive, however tacitly, of (at least some) observed properties as categorical. By contrast with an evidential theory, a truthconditional theory of content can account for all these problematic facts. Peacocke states, develops and defends three principles of rationalism which collectively ‘relate entitlement to truth, to the identity of states and their intentional contents, and to the a priori.’ (3-4) He does not thoroughly explain his central notion of entitlement, but this much is clear: any thinker is entitled to various transitions in, or into, thought. An example of a transition into thought would be that from one’s perceptual experience to an observational judgment. An example of a transition in thought would be a logical inference from certain premises to a conclusion. A transition is rational just in case the thinker is entitled to it. (Note that this aims to explain rationality in terms of entitlement, not the other way round.) It is clear from 28 that Peacocke needs an abstract ontology of entitlements (such as proofs, in the case of mathematics). Yet he does not endorse ‘Gödel’s obscure quasi-perceptual and quasi-causal epistemology of mathematics and the abstract sciences.’ (54.. (shrink)

The search for God is dictated not from without but from a profound sense of one's own moral being and worthiness to be happy. The core of Immanuel Kant's argument remains relevant to the experience of ordinary men and women. He wished to strengthen, not undermine, belief in God and in the spiritual nature of humankind. This 1763 essay is imporrtant in understanding the development of Kant's thought. It exposed the flaw in the Cartesian argument that the existence of (...) a perfect being could be deduced from an idea or concept of such. Similarly, Kant saw the problem inherent in the Leibnizian view of a philosophical system modeled on mathematics: a philosopher who, like a mathematician, began with an arbitrary definition remained trapped in a circle of words. In The One Possible Basis for a Demonstration of the Existence of God , Kant diverged from the familiar forms of ontological argument. The result was a brilliant approach to divine being that anticipated his mature Critique of Pure Reason. With this Bison Book edition, The One Possible Basis appears in paperback for the first time. Gordon Treash's English translation, the only modern one, faces pages containing the original German. Treash, who is a professor of philosophy at Mount Allison University, Sackville, New Brunswick, edited, with Paul A. Bogaard, Metaphysics as Foundation: Essays in Honor of Ivor Leclerc . Also available as a Bison Book is Kant's last major essay, The Conflict of the Faculties (1992). (shrink)

Two fundamental paradigms are in conflict. Expert systems are the creation of the artificial intelligence paradigm which presumes that an objective reality can be understood and controlled by an individual expert intelligence that can be replaced by machinery. The alternative paradigm assumes that reality is the subjective product of human beings striving to collaborate through shared norms and experiences, a process that can be assisted by but never replaced by computers. The first paradigm is appropriate in the domains of natural (...) science and mathematics but dangerous in social sciencet business and, especially, the law. Expert systems are constructed on the basis of a number of metaphysical assumptions that are invalid in the legal domain. These assumptions are assimilated through a number ofcommonplace metaphors that guide the thoughts of the majority of people entering the computing field who are usually trained in first paradigm subjects such as mathematics and the natural science. This inappropriate paradigm hinders our progress in the field of computers and law. We need to adopt a socially orientated view of tbe nature of reality, of language, of meaning, of intelligence, and of reasoning. It will be easier then to build computer systems to facilitate social interactions in the legal domain and easier to understand why boxes that try to imitate legal expertise are intrinsically fraudulent. (shrink)

This article aims to evaluate the purported empirical character of computer-assisted proof, as suggested by Thomas Tymoczko and others. Tymoczko famously argued that the proof of the Four-Color Theorem introduced a new, empirical method of proof, forcing us to modify the traditional conception of mathematical argument as a priori reasoning. Detlefsen and Luker contended that Tymoczko’s suggestion entailed that typically mathematical proofs were empirical. My chief interest is to raise some objections to a line of thought common to both (...) of these arguments, with a view to outlining an account of the a priori which allows thepossibility of a priori knowledge obtained by appeal to computers or through testimony. Drawing on some recent discussions by Tyler Burge, this account gives a broad construal of the non-justificatory, ‘enabling’ role that experience is held to play in knowledge and cognition, allowing us to argue that the purported empirical character of the appeal to computers pertains only to the role experience plays in enabling our access to the a priori warrant provided by computer proof. (shrink)

Many philosophers have claimed that there is a tension between the impenetrability of matter and the possibility of contact between continuous bodies. This tension has led some to claim that impenetrable continuous bodies could not ever be in contact, and it has led others to posit certain structural features to continuous bodies that they believe would resolve the tension. Unfortunately, such philosophical discussions rarely borrow much from the investigation of actual matter. This is probably largely because actual matter is not (...) continuous, and so it might seem as if discussion of the structure of continuous bodies is merely within the realm of philosophical thought experiments rather than actual scientific investigation. However, classical continuum mechanics models actual matter as if it were continuous, and it has implications about the structure of continuous bodies and about what contact and impenetrability are. This paper describes the relevant notions from classical continuum mechanics so as to resolve the alleged tension between contact and impenetrability. (shrink)

In this set of previously unpublished essays, noted scholars from North America and Europe describe how the Irish philosopher George Berkeley (1684-1753) continues to inspire debates about his views on knowledge, reality, God, freedom, mathematics, and religion. Here discussions about Berkeley's account of physical objects, minds, and God's role in human experience are resolved within explicitly ethical and theological contexts. This collection uses debates about Berkeley's immaterialism and theory of ideas to open up a discussion of how divine activity and (...) human experience are reconciled in a recurring appeal to the laws of nature. In that context, objects in the world are linked to one another by means of the perceptions and affections whereby minds come into being. The laws of nature thus become crucial for Berkeley in revealing how objects are unintelligible apart from being apprehended by minds that are themselves connected to one another in virtue of their ideas. -/- Overall, the essays indicate that, for Berkeley, our apprehension of the world as real depends on recognizing how the world expressed by our ideas is not a mere aggregate of disconnected bodies but is rather an integrated unity of the things we experience. This provides an antidote against the loss of unity created by Descartes' isolation of the self from nature and Locke's account of objects in terms of simple, discrete ideas. -/- In juxtaposing discussions of Berkeley's later writings with his earlier works, this volume shows not only how, for Berkeley, mind is intrinsically linked to things in nature as the principle of their determination in law-governed ways, but also how minds are practically related to the objects of the physical world, one another, and ultimately God. (shrink)

Quantum theory can be regarded as a rationally coherent theory of the interaction of mind and matter and it allows our conscious thoughts to play a causally e cacious and necessary role in brain dynamics It therefore provides a natural basis created by scientists for the science of consciousness As an illustration it is explained how the interaction of brain and consciousness can speed up brain processing and thereby enhance the survival prospects of conscious organisms as compared to similar organisms (...) that lack consciousness As a second illustration it is explained how within the quantum framework the consciously experi enced I directs the actions of a human being It is concluded that contemporary science already has an adequate framework for incorporat ing causally e cacious experiential events into the physical universe in a manner that puts the neural correlates of consciousness into the theory in a well de ned way explains in principle how the e ects of consciousness per se can enhance the survival prospects of organisms that possess it allows this survival e ect to feed into phylogenetic de velopment and explains how the consciously experienced I can direct human behaviour.. (shrink)

A demanding introduction to logic and critical thinking, this book offers more traditional means of teaching the art of reasoning at a time when the field has become almost mathematical. Francis Dauer has rethought the framework for teaching reasoning in general and formal logic in particular, the desired epistemological context, and the role of the fallacies. The result is a coherent and very readable work, informed by Dauer's extensive experience teaching and writing on the subject.