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Switch to: References
Citations of:
Remarks on the foundations of mathematics
Oxford [Eng.]: Blackwell. Edited by G. E. M. Anscombe, Rush Rhees & G. H. von Wright (1967)
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Responding to Michael Luntley's article, ‘Learning, Empowerment and Judgement’, the author shows he cannot successfully make the following three moves: (1) dissolve the analytic distinction between learning by training and learning by reasoning, while advocating the latter; (2) diminish the role of training in Wittgenstein's philosophy, nor attribute to him a rationalist model of learning; and (3) turn to empirical research as a way of solving the philosophical problems he addresses through Wittgenstein. Drawing on José Medina's analysis of the fundamental (...) |
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Over a decade after publication of Thinking Again: Education After Postmodernism (1998) contention still emerges among Foucaultians over whether discursively made‐up things really exist, and whether removal of the constituent subject leaves room for agency within techniques of caring for the self. That these questions are kept alive shows that some readers have not rethought Foucault, finding what possibly comes after postmodernism. Using Wittgenstein to ‘reciprocally illuminate’ Foucault (after Tully and Marshall), I open teacher inspection and reforms to problematization, as (...) |
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Offering a cautionary tale about the abuses of paradigm‐shift rhetoric in secondary school reforms, the paper shows potential misuses and ethical effects of the relativistic language‐game in post‐compulsory education. Those initiating the shift often shelter their reform from the criticism of non‐adepts, marginalizing expert teachers that adhere to ‘antiquated’ or ‘folk’ pedagogies. The rhetoric herds educators uncritically into the citadel of new discourses and policies that often lack practical foundations; consequently, teachers often dissimulate compliance to the reform in order to (...) |
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While Wittgenstein wrote unconventionally and denied that he was advancing philosophical theses, most of his interpreters have attributed conventional philosophical theses to him. But the best recent interpretations have taken the form of his writing and his distinctive way of doing philosophy seriously. The 1980s have also seen the emergence of a body of work on Wittgenstein that makes extensive use of the unpublished Wittgenstein papers. This work on Wittgenstein's method and his way of writing are the main themes of (...) |
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A set of hypotheses is formulated for a connectionist approach to cognitive modeling. These hypotheses are shown to be incompatible with the hypotheses underlying traditional cognitive models. The connectionist models considered are massively parallel numerical computational systems that are a kind of continuous dynamical system. The numerical variables in the system correspond semantically to fine-grained features below the level of the concepts consciously used to describe the task domain. The level of analysis is intermediate between those of symbolic cognitive models (...) |
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I support Rips et al.'s critique of psychology through (1) a complementary argument about the normative, modal, constitutive nature of mathematical principles. I add two reservations about their analysis of mathematical induction, arguing (2) for constructivism against their logicism as to its interpretation and formation in childhood (Smith 2002), and (3) for Piaget's account of reasons in rule learning. |
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A logic focusing on the analytic a priori and explicitly rejecting the synthetic a priori developed in the early decades of the 20th century, largely through the efforts of the Logical Empiricists. This group was very influenced by Wittgenstein's early work Tractatus Logico-Philosophicus. But Wittgenstein himself, later on, departed from the Tractatus in significant ways that the Logical Empiricists did not follow. Wittgenstein came later to accept the synthetic a priori, and out of this insight comes a non-analytic logic that (...) |
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Francesco Berto has recently written “The Gödel Paradox and Wittgenstein's Reasons,” about a paradox first formulated by Graham Priest in 1971. The major reason for disagreeing with Berto's conclusions concerns his elucidation of Wittgenstein's understanding of Gödel's theorems. Seemingly, Wittgenstein was some kind of proto-paraconsistentist. Priest himself has also, though in a different way, tried to tar Wittgenstein with the same brush. But the resolution of other paradoxes is intimately linked with the resolution of the Gödel Paradox, and with understanding (...) |
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The traditional and prevailing definition of lying is that lying is some variation or combination of: “an untruth told with intent to deceive.” I establish that this is the case, and that, as a result, contradictions and injustices arise. An alternative definition is proposed which is shown to avoid these difficulties. It is also shown that and how on the new definition the alleged “Liar paradox” is easily dissolved. |
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Recent accounts of the role of diagrams in mathematical reasoning take a Platonic line, according to which the proof depends on the similarity between the perceived shape of the diagram and the shape of the abstract object. This approach is unable to explain proofs which share the same diagram in spite of drawing conclusions about different figures. Saccheri’s use of the bi-rectangular isosceles quadrilateral in Euclides Vindicatus provides three such proofs. By forsaking abstract objects it is possible to give a (...) |
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Rule following is often made an unnecessary mystery in the philosophy of social science. One form of mystification is the issue of 'rule finitism', which raises the puzzle as to how a learner can possibly extend the rule to applications beyond those examples which have been given as instruction in the rule. Despite the claim that this problem originated in the work of Wittgenstein, it is clear that his philosophical method is designed to evaporate, not perpetuate, such problems. The supposed (...) |
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This paper combines views of Wittgenstein and Heidegger into an account of mind/ action. It does this by suggesting that these two philosophers be viewed in part as descendants of Life?philosophy (Lebensphilosophie). Part I describes the conception of life that informs and emerges from these thinkers. Parts Two and Three detail particular aspects of this conception: Wittgenstein on the constitution of states of life and Heidegger on the flow?structure of the stream of life. The Conclusion offers reasons for believing their (...) |
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According to Wittgenstein, mathematics is embedded in, and partly constituting, a form of life. Hence, to imagine different, alternative forms of elementary mathematics, we should have to imagine different practices, different forms of life in which they could play a role. If we tried to imagine a radically different arithmetic we should think either of a strange world or of people acting and responding in very peculiar ways. If such was their practice, a calculus expressing the norms of representation they (...) |
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In this paper, a Wittgensteinian account of the human sciences is constructed around the notions of the surface of human life and of surface phenomena as expressions. I begin by explaining Wittgenstein's idea that the goal of interpretive social science is to make actions and practices seem natural. I then explicate his notions of the surface of life and of surface phenomena as expressions by reviewing his analysis of mental state language. Finally, I critically examine three ideas: (a) that the (...) |
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This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles’ does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV–V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: Russell's logicism (...) |
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This paper introduces the Special Issue: ‘The Roots of Normativity. Developmental, Comparative and Conceptual issues’. The contributions collected in this volume aim to present a picture of contemporary accounts of normativity that integrate philosophy and developmental and comparative psychology and purport to provide the reader with new insights regarding a classical debate about what makes us human: being governed by norms and being able to orient ourselves in the light of them. This introduction presents a broad picture of the issues (...) |
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According to the Adoption Problem certain basic logical principles cannot be adopted. Drawing on the AP, Suki Finn presents an argument against logical pluralism: Modus Ponens and Universal Instantiation both govern a general structure shared by every logical rule. As such, analogues of these two rules must be present in every meta-logic for any logical system L, effectively imposing a restriction to logical pluralism at the meta-level through their presence constituting a “meta-logical monism”. We find a tension in the dual (...) |
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This article suggests that an epistemological and ontological pluralist perspective may enable human beings to cooperate each other and live with less injustice. Intercultural cooperation may help for a reformulation of the Human rights in order to consider aspects of different ways of life like the variations of moral, political and judicial norms. I expound that Liberal pluralism does not respond adequately to present day multiculturalism. Additionally, I explain that Human rights are not inclusive norms for all ways of life. (...) |
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This article examines Kant’s discussion of the division between reason and unreason in his Anthropology from a Pragmatic Point of View . On the one hand, Kant says that there is a normative, clear, and definite division between reason and unreason. On the other hand, Kant offers three arguments showing that we cannot draw such a division. First, we cannot explain the normative grounds for the division. Second, both reason and unreason are present in everyone to varying degrees in different (...) |
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In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (...) |
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Ludwig Wittgenstein imagines a variety of eccentric social practices—like a tribe trained “to give no expression of feeling of any kind”. But he also speaks of “the common behavior of mankind” that is rooted in “natural/primitive reactions”. This emphasis on the uniformities of human behavior raises questions about the plausibility of some of his imagined language games. Indeed, it suggests the claim of evolutionary psychologists that there are biologically based human universals that shape social practices. But in contrast to E.O. (...) |
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The notion of `truth' is one of the most important concepts within critical thinking and critical pedagogy as well as in other traditions or theories, and truth is seen by many as the outcome of inquiry. In this article I will argue for an alternative notion of truth to those that will be discussed in it and that such a view has to be included in a new critical language in education. I discuss a realist notion, a postmodernist social constructivist (...) |
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Institutions worldwide respond to the need to recognise the value of educating children and young people to handle or solve conflicts in communication. But how do they or we know that an event is correctly interpreted as a conflict? How can people analyse the quality of deliberation when handling or solving conflicts in communication in education? I discuss these questions and argue that the notion of conflict cannot be defined only in terms of incompatibility, clash, opposition and/or disagreement; it also (...) |
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The long‐standing issue of Wittgenstein's controversial remarks on Gödel's Theorem has recently heated up in a number of different and interesting directions [, , ]. In their , Juliet Floyd and Hilary Putnam purport to argue that Wittgenstein's‘notorious’ “Contains a philosophical claim of great interest,” namely, “if one assumed. that →P is provable in Russell's system one should… give up the “translation” of P by the English sentence ‘P is not provable’,” because if ωP is provable in PM, PM is (...) |
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Given (1) Wittgensteins externalist analysis of the distinction between following a rule and behaving in accordance with a rule, (2) prima facie connections between rule-following and psychological capacities, and (3) pragmatic issues about training, it follows that most, even all, future artificially intelligent computers and robots will not use language, possess concepts, or reason. This argument suggests that AIs traditional aim of building machines with minds, exemplified in current work on cognitive robotics, is in need of substantial revision. |
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In this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different communities, which endorse (...) |
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This paper offers a new interpretation for Wittgenstein`s treatment of mathematical identities. As it is widely known, Wittgenstein`s mature philosophy of mathematics includes a general rejection of abstract objects. On the other hand, the traditional interpretation of mathematical identities involves precisely the idea of a single abstract object – usually a number –named by both sides of an equation. |
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In this essay, the tension that Benacerraf identifies for theories of mathematical truth is used as the vehicle for arguing against a particular desideratum for semantic theories. More specifically, I place in question the desideratum that a semantic theory, provided for some area of discourse, should run in parallel with the semantic theory holding for the rest of the language. The importance of this desideratum is also made clear by means of tracing out the subtle implications of its rejection. |
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