Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has (...) to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work. (shrink)
Natural deduction (for short: nd-) calculi have not been used systematically as a basis for automated theorem proving in classical logic. To remove objective obstacles to their use we describe (1) a method that allows to give semantic proofs of normal form theorems for nd-calculi and (2) a framework that allows to search directly for normal nd-proofs. Thus, one can try to answer the question: How do we bridge the gap between claims and assumptions in heuristically motivated ways? This informal (...) question motivates the formulation of intercalation calculi. Ic-calculi are the technical underpinnings for (1) and (2), and our paper focuses on their detailed presentation and meta-mathematical investigation in the case of classical predicate logic. As a central theme emerges the connection between restricted forms of nd-proofs and (strategies for) proof search: normal forms are not obtained by removing local "detours", but rather by constructing proofs that directly reflect proof-strategic considerations. That theme warrants further investigation. (shrink)
Church's and Turing's theses dogmatically assert that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computability. I present an analysis of calculability that is embedded in a rich historical and philosophical context, leads to precise concepts, but dispenses with theses.To investigate effective calculability is to analyze symbolic processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work of (...) Gandy, I formulate boundedness and locality conditions for two types of calculators, namely, human computing agents and mechanical computing devices (discrete machines). The distinctive feature of the latter is that they can carry out parallel computations. The analysis leads to axioms for discrete dynamical systems (representing human and machine computations) and allows the reduction of models of these axioms to Turing machines. Cellular automata and a variety of artificial neural nets can be shown to satisfy the axioms for machine computations. (shrink)
Two young logicians, whose work had a dramatic impact on the direction of logic, exchanged two letters in early 1931. Jacques Herbrand initiated the correspondence on 7 April and Kurt Gödel responded on 25 July, just two days before Herbrand died in a mountaineering accident at La Bérarde (Isère). Herbrand's letter played a significant role in the development of computability theory. Gödel asserted in his 1934 Princeton Lectures and on later occasions that it suggested to him a crucial part of (...) the definition of a general recursive function. Understanding this role in detail is of great interest as the notion is absolutely central. The full text of the letter had not been available until recently, and its content (as reported by Gödel) was not in accord with Herbrand's contemporaneous published work. Together, the letters reflect broader intellectual currents of the time: they are intimately linked to the discussion of the incompleteness theorems and their potential impact on Hilbert's Program. (shrink)
Any thorough discussion of computing machines requires the examination of rigorous concepts of computation and is facilitated by the distinction between mathematical, symbolic and physical computations. The delicate connection between the three kinds of computations and the underlying questions, "What are machines?" and "When are they computing?", motivate an extensive theoretical and historical discussion. The relevant outcome of this..
Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ (...) -definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of effective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Gödel's recursiveness for his Thesis. In Section 5, I contrast Church's analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekind's investigation of continuity. (shrink)
Machines were introduced as calculating devices to simulate operations carried out by human computors following fixed algorithms: this is true for the early mechanical calculators devised by Pascal and Leibniz, for the analytical engine built by Babbage, and the theoretical machines introduced by Turing. The distinguishing feature of the latter is their universality: They are claimed to be able to capture any algorithm whatsoever and, conversely, any procedure they can carry out is evidently algorithmic. The study of such "paper machines" (...) by mathematical means is the topic of our contribution. This is not only in accord with its usual understanding in computer science, but conceptually and historically right, when we recall the purpose for which Turing machines were introduced. (shrink)
This is a summary of developments analysed in my (1997A). A first version of that paper was presented at the workshop Modern Mathematical Thought in Pittsburgh (September 21-24, 1995).
The identification of an informal concept of ‘effective calculability’ with a rigorous mathematical notion like ‘recursiveness’ or ‘Turing computability’ is still viewed as problematic, and I think rightly so. I analyze three different and conflicting perspectives Gödel articulated in the three decades from 1934 to 1964. The significant shifts in Gödel's position underline the difficulties of the methodological issues surrounding the Church-Turing Thesis.
The Carnegie Mellon Proof Tutor project was motivated by pedagogical concerns: we wanted to use a "mechanical" (i.e. computerized) tutor for teaching students..
ââ¬â via appropriate substitutions ââ¬â syntactically identical. The method can be applied directly to quantifierfree formulae and, in this paper, will b e extended in a natural and strai ghlforward way to quantified formulae.
The starting point of this paper is Sellars’s rejection of foundationalist empiricism as found in his discussion of the Myth of the Given. Sellars attacks the Myth from two main angles, corresponding to the two elements of empiricism: the idea that our beliefs are justified by the world, and the idea that our concepts are derived from experience. In correctly attacking the second, Sellars is also, incorrectly, led to attack the first. Thus, Sellars rejects the commonsensical idea that at least (...) some of our ideas can be justified by appeal to the empirical world. My purpose is to examine why Sellars is led to this point, and how the same assumptions that lead him there also operate in his followers, such as Brandom, Rorty and McDowell. I then show how a rejection of these assumptions gives us a way around this problem that does not fall back into foundationalism. (shrink)
A recent review of his work describes Wilfred Carr as 'one of the most brilliant philosophers now working in the rich British tradition of educational philosophy ... His work is rigorous, refreshing and original ... and examines a number of fundamental issues with clarity and penetration'. In For Education Wilfred Carr provides a comprehensive justification for reconstructing educational theory and research as a form of critical inquiry. In doing this, he confronts a number of important philosophical questions. What (...) is educational theory? What is an educational practice? How are theory and practice related? What is the role of values in educational research? Is a genuinely educational science possible? By appealing to developments in critical theory, the philosophy of science and the philosophy of the social sciences, Wilfred Carr provides answers to these questions which vindicate the idea of an educational science that is not 'on' or 'about' education but 'for education' - a science genuinely committed to promoting educational values and ideals. (shrink)
I have practiced psychotherapy, family therapy, and hypnotherapy for over 25 years without a single board complaint or lawsuit by a client. For over 3 years, however, a group of proponents of the false memory syndrome (FMS) hypothesis, including members, officials, and supporters of the False Memory Syndrome Foundation, Inc., have waged a multimodal campaign of harassment and defamation directed against me, my clinical clients, my staff, my family, and others connected to me. I have neither treated these harassers or (...) their families nor had any professional or personal dealings with any of them; I am not related in any way to the disclosures of memories of sexual abuse in these families. Nonetheless, this group disrupts my professional and personal life and threatens to drive me out of business. In this article, I describe practicing psychotherapy under a state of siege and place the campaign against me in the context of a much broader effort in the FMS movement to denigrate, defame, and harass clinicians, lecturers, writers, and researchers identified with the abuse and trauma treatment communities. (shrink)
How do rational minds make contact with the world? The empiricist tradition sees a gap between mind and world, and takes sensory experience, fallible as it is, to provide our only bridge across that gap. In its crudest form, for example, the traditional idea is that our minds consult an inner realm of sensory experience, which provides us with evidence about the nature of external reality. Notoriously, however, it turns out to be far from clear that there is any viable (...) conception of experience which allows it to do the job. The original problem is to show that thought is rationally constrained by external reality. If sensory experience is to provide the solution--in particular, if it is to provide the answer to sceptical challenges--it must therefore meet two criteria. First, it must itself be `receptive'--i.e., appropriately constrained by external reality. Second, it must be the kind of thing that can enter into a logical or rational relationship with belief--it must already be `conceptual,' in other words. In arguing against the idea that anything could serve both roles, Wilfred Sellars termed this conception of experience "the Myth of the Given.". (shrink)
This article examines the benefits and burdens of the debate between Paul Hirst and Wilfred Carr over a set of issues to do with philosophy and education specifically and theory and practice more generally. Hirst and Carr, in different ways, emphasise the importance of Aristotelian practical philosophy as an antidote to the theory-oriented confined method of ‘conceptual analysis’ that has haunted the philosophy of education. Despite their proper recognition of the irreducible character of practice to theory, they fail to (...) provide a satisfying account of their interpenetrating relation. Hirst falls into error by fencing off ‘forms of theoretical knowledge’ from ‘forms of practice’; Carr's dismissive attitude to theory is saturated with internal tensions in his own discourse. This article contends that what is left unaddressed both in Hirst's and Carr's arguments is the most fundamental sense of ‘social’, which is prior to relative differences in the standards of knowledge among societies and which reminds us that theory is not a socially disembodied enterprise. A lively appreciation of this point encourages us to see the prevailing outlook towards the relation between philosophy and education quite differently. (shrink)
Machine generated contents note: Part I. General: 1. The Gödel editorial project: a synopsis Solomon Feferman; 2. Future tasks for Gödel scholars John W. Dawson, Jr., and Cheryl A. Dawson; Part II. Proof Theory: 3. Kurt Gödel and the metamathematical tradition Jeremy Avigad; 4. Only two letters: the correspondence between Herbrand and Gödel Wilfried Sieg; 5. Gödel's reformulation of Gentzen's first consistency proof for arithmetic: the no-counter-example interpretation W. W. Tait; 6. Gödel on intuition and on Hilbert's finitism W. (...) W. Tait; 7. The Gödel hierarchy and reverse mathematics Stephen G. Simpson; 8. On the outside looking in: a caution about conservativeness John P. Burgess; Part III. Set Theory: 9. Gödel and set theory Akihiro Kanamori; 10. Generalizations of Gödel's universe of constructible sets Sy-David Friedman; 11. On the question of absolute undecidability Peter Koellner; Part IV. Philosophy of Mathematics: 12. What did Gödel believe and when did he believe it? Martin Davis; 13. On Gödel's way in: the influence of Rudolf Carnap Warren Goldfarb; 14. Gödel and Carnap Steve Awodey and A. W. Carus; 15. On the philosophical development of Kurt Gödel Mark van Atten and Juliette Kennedy; 16. Platonism and mathematical intuition in Kurt Gödel's thought Charles Parsons; 17. Gödel's conceptual realism Donald A. Martin. (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)
Fifty years ago the philosopher Wilfred Sellars identified two images of “man”, which he called respectively the “manifest image” and the “scientific image”; and he considered whether and how these two images could be reconciled. In this paper, I will very briefly look at the distinction drawn by Sellars and at his suggestions for reconciliation of these images. I will suggest that a broad distinction as suggested by Sellars can indeed usefully be drawn, but that the distinction can be (...) more helpfully characterised than it was by Sellars. I will argue that there are more ways of reconciling the two images than those proposed by Sellars. And I will elaborate on what I think are the most promising lines along which the reconciliation could take place. (shrink)
Turing's (1936) analysis of effective symbolic procedures is a model of conceptual clarity that plays an essential role in the philosophy of mathematics. Yet appeal is often made to the effectiveness of human procedures in other areas of philosophy. This paper addresses the question of whether Turing's analysis can be applied to a broader class of effective human procedures. We use Sieg's (1994) presentation of Turing's Thesis to argue against Cleland's (1995) objections to Turing machines and we evaluate her (...) proposal to understand the effectiveness of procedures in terms of their reliability and precision. A number of conditions for effectiveness are identified and these are used to provide a general argument against the possibility of a Leibnizian decision procedure. (shrink)
This paper defends the traditional conception of Church's Thesis (CT), as unprovable but true, against a group of arguments by Gandy, Mendelson, Shapiro and Sieg. The arguments here considered urge that CT is provable or proved. This paper argues, first, that contra-Mendelson, CT does connect a mathematically precise concept (Turing computability) with an intuitive notion (effective calculability). Second, the various ‘proofs’ of (all or half of) CT fail to undermine the traditional conception of CT as unprovable. Either they do (...) not conform to the sense of proof imbedded in the standard conception, or they prove something other than CT. (shrink)
∗A special thanks to those who have assisted my archival research, including Aldo Antonelli, John Burgess, Michael Della Rocca, Herbert Enderton, Bernard Linsky, Heidi Lockwood, Ruth Barcan Marcus, Julien Murzi and Bas van Fraassen. An extra special thanks to Julien Murzi, who as my research assistant in the Fall of 2005 helped me to identify and think more clearly about the famous anonymous referee reports, which are central to the present paper. For discussion and/or assistance I am also grateful to (...) many others, including Scott Berman, Berit Brogaard, Judy Crane, Susan Brower- Toland, David Chalmers, Solomon Feferman, Nick Griffin, Michael Hand, Monte Johnson, Jon Kvanvig, Matthias Lutz-Bachmann, Robert Meyer, Andreas Niederberger, Gualtiero Piccinini, Graham Priest, Krister Segerberg, Wilfried Sieg, Roy Sorensen, Kent Staley, Jim Stone, Neil Tennant, Achille Varzi, Nick Zavediuk, anonymous readers for OUP, and audience members at the Pacific APA in Portland (March 24, 2006), the Goethe University of Frankfurt (May 15, 2006), the Institute for Logic, Language and Computation at the University of Amsterdam (May 23, 2006), and the Namicona Epistemology Workshop, at the University of Copenhagen (August 22, 2006). Thanks also to my department at Saint Louis University for granting time and resources to research and write the paper. (shrink)
There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. According to this account, to which I refer as the Gandy–Sieg account, Turing and Church aimed to characterize the functions that can be computed by a human computer. In addition, Turing provided a highly convincing (...) argument for CTT by analyzing the processes carried out by a human computer. I then contend that if the Gandy–Sieg account is correct, then the notion of effective computability has changed after 1936. Today computer scientists view effective computability in terms of finite machine computation. My contention is supported by the current formulations of CTT, which always refer to machine computation, and by the current argumentation for CTT, which is different from the main arguments advanced by Turing and Church. I finally turn to discuss Robin Gandy's characterization of machine computation. I suggest that there is an ambiguity regarding the types of machines Gandy was postulating. I offer three interpretations, which differ in their scope and limitations, and conclude that none provides the basis for claiming that Gandy characterized finite machine computation. (shrink)
Wilfred Sellars once famously described philosophy as "the attempt to say how things, in the most general sense of the term, hang together, in the most general sense of the term." (Sellars, 1962). In the spirit of that suggestion, we can think of philosophy of mind as the attempt to say how minds hang together-how things fit to form minds, and how minds fit with other things. It can hardly be disputed that there are these kinds of fit; in (...) that respect at least, the world is a coherent place. The philosophical challenge is to understand and elucidate that nature of the fit, such as it is. (shrink)
Sieg has proposed axioms for computability whose models can be reduced to Turing machines. This lecture will investigate to what extent these axioms hold for reasoning. In particular we focus on the requirement that the configurations that a computing agent (whether human or machine) operates on must be ’immediately recognisable’. If one thinks of reasoning as derivation in a calculus, this requirement is satisfied; but even in contexts which are only slightly less formal, the requirement cannot be met. Our (...) main example will be the Wason selection task, a propositional reasoning task in which in a typical (undergraduate) subject group only around 5% arrive at the answer dictated by classical logic. The instructions for this task (as well as other standard tasks in the psychology of reasoning, such as syllogisms) do not contain any ’immediately recognisable’ configurations. The subject must try to find an interpretation of the task by making the various elements in the instructions cohere, in effect solving a difficult constraint satisfaction problem, which has no unique solution. The subject has given a complete interpretation of the task if she can formulate the problem posed in the task as a theorem to be proved. The complexity of such theorems can be quite high; e.g. for the propositional Wason selection task the theorem can be in Σ1 3 . This sounds implausible, but we’ll present experimental data confirming this point. (shrink)
When Paul Hirst and Wilfred Carr squared up to each other a few years ago on the issue of the role of philosophical theory in educational practice, it became clear that theory itself had become a troubled term. The very fact that Wilfred Carr could argue for the end of educational theory recalls Paul Feyerabend's fiery argument for the end of theory in natural science and simply deepened the attack that had already appeared in Carr and Kemmis's book, (...) Becoming Critical (1986). In response, Hirst insisted that theory, and particularly the philosophical theory of education, should be defined as a discrete area of study in itself, governed and structured by the axioms of logic. In this way, he argued, the philosophy of education would be no different from philosophy in general (at least in its analytic formulation). Carr, on the other hand, preferred to consider educational theory as a flexible event that took its shape from the landscape explored, and hence precisely not the kind of study that Hirst supported, but one based in action research and reflective practitioner experience. This debate is as yet unresolved. In this piece I begin by making several remarks about the current context for raising the question Hirst and Carr address, and I go on to consider other possible understandings of theoria in a Greek sense before developing this idea through a reading of Aristotle. I eventually conclude that each of the protagonists in the debate has taken a step too far. (shrink)
In this article it is argued that there are notable parallels between all of the different strands within ethics on the one hand, and accountancy on the other that, in teaching, can be drawn upon to enhance students’ understanding of the latter. Accountancy, part of economics, draws on utilitarian ethics, but not solely so. Accounting, in addition, draws on deontological and communitarian strands in ethics. The article suggests that the teaching of accounting – especially to non-economists – would benefit substantially (...) from highlighting and developing these parallels. (shrink)
The concept of social capital helps to explain relations within and between companies but has not crystallized yet. As such, the nature, development, and effects of such relations remain elusive. How is social capital created, how is it put to use, and how is it maintained? Can it decline, and if so, how? We argue that the concept of social capital remains a black box as the mechanisms that constitute it remain underdeveloped and that it is a black hole as (...) many empirical phenomena are attributed to its presence. We use and develop the literature on gift exchange to provide a firmer theoretical basis for the concept of social capital. (shrink)
Weiming, as a leading spokesman for contemporary New Confucianism, has been reinterpreting the Confucian tradition in the face of the challenges of modernity. Tu takes selfhood as his starting point, emphasizing the importance of cultivating the human mind-and-heart as a deepening and broadening process to realize the anthropocosmic dao. He highlights the concept of a fiduciary community and advocates that, because of it, Confucianism remains a dynamic inclusive humanism. Tuâs mode of thinking tallies well with Wilfred C. Smithâs vision (...) of religion, specifically the latterâs exposition of faith as a universal human quality and proposal of corporate critical self-consciousness. This article details the theories of both scholars, highlights their similarities, and contrasts their differences. It argues that Smithâs world theology provides a heuristic framework through which one understands how Tu has advanced his Confucian humanism from a Chinese philosophical or cultural tradition to the midst of world religions. (shrink)
We give a new elementary proof of the comparison theorem relating $\sum^1_{n + 1}-\mathrm{AC}\uparrow$ and $\Pi^1_n -\mathrm{CA}\uparrow$ ; the proof does not use Skolem theories. By the same method we prove: a) $\sum^1_{n + 1}-\mathrm{DC} \uparrow \equiv (\Pi^1_n -CA)_{ , for suitable classes of sentences; b) $\sum^1_{n+1}-DC \uparrow$ proves the consistency of (Π 1 n -CA) ω k, for finite k, and hence is stronger than $\sum^1_{n+1}-AC \uparrow$ . a) and b) answer a question of Feferman and Sieg.
By examining the theories of justice developed by Joseph Butler and David Hume, the author discloses the conceptual limits of their moral naturalism. Butler was unable to accommodate the possibility that justice is, at least to some extent, a social convention. Hume, who more presciently tried to spell out the conventional character of justice, was unable to carry through that project within the framework of his moral naturalism. These limits have gone unnoticed, largely because Butler and Hume have been misinterpreted, (...) their relation misconstrued. Exegetes have persistently misunderstood the differences that divide them, have misconceived the notion of "public utility" in Hume's account of justice, have wrongly interpreted Butler as a forerunner of Immanuel Kant, and have altogether missed the degree to which Hume stands in the tradition of Thomas Aquinas. (shrink)
It is suggested that the mathematically abstract coordinate frames of reference commonly visualized to be centered at the celestial bodies have real counterparts in the shape of well-defined rigid spatial resonant singularities of infinite extension, which accommodate the matter waves from the superimposition of which the body residing at the coordinate origin results. A universally valid inertial reference frame system is proposed. Qualitative explanations are offered for the inertial and gravitational forces, their observed proportionality, and for the occurrence of second-order (...) gravitational effects in the vicinity of massive bodies. The universal redshift is assumed to result from a closure condition of the eigenspaces introduced. (shrink)
This paper reconvenes Samuel Beckett’s psychotherapy with Wilfred Bion during 1934–1936 during which time Beckett’s conceived and began writing this second novel, Murphy . Based on Beckett’s visits to the Bethlem & Maudsley Hospital and his observation of the male nurses, the climax of Murphy is a chess match between Mr Endon (a male schizophrenic patient) and Murphy (a male psychiatric nurse). The precise notation of the Endon v Murphy chess match tells us that the Beckett intended it to (...) be an exemplar of an anti-match, perhaps a metaphor for the tragedy of being locked into madness. It is also argued that the match offers us insight into Beckett’s experience of the process of psychotherapy with Bion. Based on new information from Beckett’s nephew and Bion’s widow, hypotheses about the long term impact of the Bion-Beckett analysis are advanced as a mutual experience which shaped the lives and later literary output of both men, producing conjoined career writings which continue to offer us stark and sublime condensations of depression, psychosis, and the challenges of therapy and recovery. (shrink)
