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 (...) applied could not be called false. Rather, our criticism could only be to dismiss such a practice as foolish and to dismiss their norms as too different from ours to be called ‘mathematics’. (shrink)

In some logics, anything whatsoever follows from a contradiction; call these logics explosive. Paraconsistent logics are logics that are not explosive. Paraconsistent logics have a long and fruitful history, and no doubt a long and fruitful future. To give some sense of the situation, I’ll spend Section 1 exploring exactly what it takes for a logic to be paraconsistent. It will emerge that there is considerable open texture to the idea. In Section 2, I’ll give some examples of techniques for (...) developing paraconsistent logics. In Section 3, I’ll discuss what seem to me to be some promising applications of certain paraconsistent logics. In fact, however, I don’t think there’s all that much to the concept ‘paraconsistent’ itself; the collection of paraconsistent logics is far too heterogenous to be very productively dealt with under a single label. Perhaps that will emerge as we go. (shrink)

In Contradiction advocates and defends the view that there are true contradictions, a view that flies in the face of orthodoxy in Western philosophy since Aristotle. The book has been at the center of the controversies surrounding dialetheism ever since its first publication in 1987. This second edition of the book substantially expands upon the original in various ways, and also contains the author’s reflections on developments over the last two decades. Further aspects of dialetheism are discussed in the companion (...) volume, Doubt Truth to be a Liar, also published by Oxford University Press in 2006. (shrink)

In Culture and Value Wittgenstein remarks: ‘Thoughts that are at peace. That's what someone who philosophizes yearns for’. The desire for such conceptual tranquillity is a recurrent theme in Wittgenstein's work, and especially in his later ‘grammatical-therapeutic’ philosophy. Some commentators (notably Rush Rhees and C. G. Luckhardt) have cautioned that emphasising this facet of Wittgenstein's work ‘trivialises’ philosophy – something which is at odds with Wittgenstein's own philosophical ‘seriousness’ (in particular his insistence that philosophy demands that one ‘Go the bloody (...) hard way’). Drawing on a number of correlations between Wittgenstein's conception of philosophy and that of the Pyrrhonian Sceptics, in this paper I defend a strong ‘therapeutic’ reading of Wittgenstein, and show how this can be maintained without ‘trivialising’ philosophy. (shrink)

In the scholarship on Wittgenstein’s later philosophy of mathematics, the dominant interpretation is a theoretical one that ascribes to Wittgenstein some type of ‘ism’ such as radical verificationism or anti-realism. Essentially, he is supposed to provide a positive account of our mathematical practice based on some basic assertions. However, I claim that he should not be read in terms of any ‘ism’ but instead should be read as examining philosophical pictures in the sense of unclear conceptions. The contrast here is (...) that basic assertions that frame philosophical ‘isms’ are propositional such that they are subject to normal argumentative evaluation, while pictures in Wittgenstein’s sense are non-propositional—they lack a clear truth condition. They, therefore, need clarification rather than argumentation. In this paper, I provide a detailed analysis of Wittgenstein’s treatment of philosophical pictures with special focus on his argument on contradiction. I begin by explaining the problem with this trend of theoretical interpretation, taking Steve Gerrard’s otherwise excellent interpretation as a representative example and pointing out why it is problematic. Next, I will argue that those problems do not arise if we take Wittgenstein’s task as the clarification of philosophical pictures. I do this, first, by explaining Wittgenstein’s method using his argument concerning the Augustinian Picture in Philosophical Investigations and then pointing out that the same method can be identified in the crucial arguments in his philosophy of mathematics. Finally, in order to connect my interpretation with the current scholarship, I will explain the relation of my interpretation with those of New Wittgensteinian scholars. (shrink)

Although certainty is a fundamental notion in epistemology, it is less studied in contemporary analytic epistemology than other important notions such as knowledge or justification. This paper focuses on Wittgensteinian certainty, according to which the very basic dimension of our epistemic practices, the elements of our world-pictures, are objectively certain, in that we cannot legitimately doubt them. The aim of the paper is to offer the best philosophical way to clarify Wittgensteinian certainty, in a way that is consonant with Wittgenstein's (...) fundamental insights. The paper critiques two alternative proposals for clarifying Wittgensteinian certainty that are philosophically unsatisfying: the rule view and the proposition view. Finally, it instead shows how viewing world-pictures as pictures, in the sense of unclear conceptions, is a more philosophically fruitful approach to understanding world-pictures. (shrink)

SummaryThe platonist, in affirming the principle of bivalence for sentences for which there is no decision procedure, disconnects their truth‐conditions from conditions that would enable us to prove them ‐ as if Goldbach's conjecture, say, might just happen to be true. According to Dummett, what has gone wrong here is that the meaning of the relevant sentences has been conceived so as to go beyond what could be learned in learning to use them, or displayed in using them competently. Dummett (...) draws the general conclusion that accounts of meaning must traffic only in decidable circumstances. I suggest that Dummett can be right about platonism but wrong in this general conclusion: the centrality of decidable circumstances in competent use of language is a special feature of mathematical language. This means that someone who recoils from the anti‐realism constituted by Dummett's generalized anti‐platonism, in the case of, say, statements about other minds, need not be recoiling into a close analogue of platonism, as Dummett suggests. We can reinstate the intuitive idea that platonism goes wrong by inappropriately modelling the epistemology and metaphysics of mathematics on the epistemology and metaphysics of the natural world. And we make room for the suggestion that anti‐realism makes a converse mistake; in this vein, I propose a picture of Dummettian anti‐realism as a novel expression of familiar and suspect epistemological and metaphysical thoughts. (shrink)

After sketching an argument for radical anti-realism that does not appeal to human limitations but polynomial-time computability in its definition of feasibility, I revisit an argument by Wittgenstein on the surveyability of proofs, and then examine the consequences of its application to the notion of canonical proof in contemporary proof-theoretical-semantics.

The object of this essay is to discuss Ludwig Wittgenstein's remarks inPhilosophical Investigationsand elsewhere in the posthumously published writings concerning the role of therapy in relation to philosophy. Wittgenstein's reflections seem to suggest that there is a kind of philosophy or mode of investigation targeting the philosophical grammar of language uses that gratuitously give rise to philosophical problems, and produce in many thinkers philosophical anxieties for which the proper therapy is intended to offer relief. Two possible objectives of later Wittgensteinian (...) therapy are proposed, for subjectivepsychologicalversus objectivesemanticsymptoms of ailments that a therapy might address for the sake of relieving philosophical anxieties. The psychological in its most plausible form is rejected, leaving only the semantic. Semantic therapy in the sense defined and developed is more general and long-lasting, and more in the spirit of Wittgenstein's project on a variety of levels. A semantic approach treats language rather than the thinking, language-using subject as the patient needing therapy, and directs its attention to the treatment of problems in language and the conceptual framework a language game use expresses in its philosophical grammar, rather than to soothing unhappy or socially ill-adjusted individual psychologies. (shrink)

In the first section, I characterize realism and illustrate the sense in which Wittgenstein's account of mathematics is anti-realist. In the second section, I spell out the above notion of objectivity and show how and anti-realist account of truth, namely, Putnam's idealized rational acceptability, preserves objectivity. In the third section, I discuss the "majority argument" and illustrate how Wittgenstein's anti-realism can also account for the objectivity of mathematics. What Putnam's and Wittgenstein's anti-realisms ultimately show is that this notion of objectivity (...) is distinct from the notion of realism and that an account of objectivity is no reason to be either realist or anti-realist. (shrink)

A survey of Wittgenstein's writings on logic and mathematics; an analytical bibliography of contemporary articles on rule-following, social constructivism, Wittgenstein, Godel, and constructivism is appended. Various historical accounts of the nature of mathematical knowledge glossed over the effects of linguistic expression on our understanding of its status and content. Initially Wittgenstein rejected Frege's and Russell's logicism, aiming to operationalize the notions of logical consequence, necessity and sense. Vienna positivists took this to place analysis of meaning at the heart of philosophy, (...) while Ramey took extensionalism to result. Wittgenstein's interest in rule-following emerged through his reactions to these attempted appropriations. (shrink)

This article is concerned to say something about what the study of logic meant to wittgenstein. It is concerned to bring out why the kind of questions wittgenstein raised about logic and mathematics cannot be pursued in a purely formal and abstract manner-As russell pursued them to a very large extent. It tries to understand the prominence wittgenstein gave to a study of these questions in his philosophical investigations and to appreciate the sense in which he regarded a study of (...) logic to be fundamental in philosophy. Part I is largely about the sense in which russell's study of logic is philosophical in character though it differs very considerably, In both style and conception, From wittgenstein's study of it. Part ii is concerned to indicate wittgenstein's dissatisfaction with russell's view that mathematics are indistinguishable from logic and to say something about why he thought that russell's formal proof, Even if valid, Did not establish the philosophical thesis for which he argued. Part iii is concerned to indicate wittgenstein's dissatisfaction with russell's approach to the contradictions in the foundations of mathematics and to say something about his very different treatment of this question. (shrink)

Some interpreters have ascribed to Wittgenstein the view that mathematical statements must have an application to extra-mathematical reality in order to have use and so any statements lacking extra-mathematical applicability are not meaningful (and hence not bona fide mathematical statements). Pure mathematics is then a mere signgame of questionable objectivity, undeserving of the name mathematics. These readings bring to light that, on Wittgenstein’s offered picture of mathematical statements as rules of description, it can be difficult to see the role of (...) mathematical statements which relate to concepts that are not employed in empirical propositions e.g. set-theoretic concepts. I will argue that Wittgenstein’s picture is more flexible than might at first be thought and that Wittgenstein sees such statements as serving purposes not directly related to empirical description. Whilst this might make such systems fringe cases of mathematics, it does not bring their legitimacy as mathematical systems into question. (shrink)

Hilary Putnam, James Conant; On Wittgenstein's Philosophy of Mathematics, Proceedings of the Aristotelian Society, Volume 97, Issue 1, 1 June 1997, Pages 195–22.

Wittgenstein is generally supposed to have abandoned in the 1930's a realistic conception of the meaning of mathematical propositions, founded on the idea of tmth-conditions which could in certain cases transcend any possibility of verification, for a realistic one, where the idea of truth-conditions is replaced by that of conditions of justification of assertability. It is argued that for Wittgenstein mathematical propositions, which are, as he says, "grammatical" propositions, have a meaning and a role which differ to a much greater (...) degree from those of ordinary propositions than either platonistic realism or intuitionistic anti-realism would admit, and that is the tendency to assimilate the mathematical proposition to an ordinary descriptive proposition which confers on it an appearance of meaning independent of the possibility of proving it, and not, as Dummett would say, that it is a decision concerning the kind of meaning it has which gives it the status of a proposition describing a determinate objective reality. (shrink)

Wittgenstein is generally supposed to have abandoned in the 1930's a realistic conception of the meaning of mathematical propositions, founded on the idea of tmth-conditions which could in certain cases transcend any possibility of verification, for a realistic one, where the idea of truth-conditions is replaced by that of conditions of justification of assertability. It is argued that for Wittgenstein mathematical propositions, which are, as he says, "grammatical" propositions, have a meaning and a role which differ to a much greater (...) degree from those of ordinary propositions than either platonistic realism or intuitionistic anti-realism would admit, and that is the tendency to assimilate the mathematical proposition to an ordinary descriptive proposition which confers on it an appearance of meaning independent of the possibility of proving it, and not, as Dummett would say, that it is a decision concerning the kind of meaning it has which gives it the status of a proposition describing a determinate objective reality. (shrink)

Graham Priest presents an original exploration of questions concerning the one and the many. He covers a wide range of issues in metaphysics--unity, identity, grounding, mereology, universals, being, intentionality and nothingness--and draws on Western and Asian philosophy as well as paraconsistent logic to offer a radically new treatment of unity.

Paul Horwich presents a bold new interpretation of Wittgenstein's later work. He argues that it is Wittgenstein's radically anti-theoretical metaphilosophy - and not his identification of the meaning of a word with its use - that underpins his discussions of specific issues concerning language, the mind, mathematics, knowledge, art, and religion.

Logical pluralism is the claim that different accounts of validity can be equally correct. Beall and Restall have recently defended this position. Validity is a matter of truth-preservation over cases, they say: the conclusion should be true in every case in which the premises are true. Each logic specifies a class of cases, but differs over which cases should be considered. I show that this account of logic is incoherent. Validity indeed is truth-preservation, provided this is properly understood. Once understood, (...) there is one true logic, relevance logic. The source of Beall and Restall’s error is a recent habit of using a classical metalanguage to analyse non-classical logics generally, including relevance logic. (shrink)

The document before you is by a member of a fanatical sect of heretical Ludwig scholars. Through a twist of fate it has fallen into my hands. I hesitate to make it public, since its circulation may do more harm than good. What speaks against publication is that it has the power to corrupt young minds. I do not take a light view of the dangers it poses in this regard. What speaks in favor of publication is the fact that (...) these people must be stopped. Through their pamphlets and brochures they continue to attract more converts everyday. The importance of this document lies in the fact that it brings to light some of the more esoteric doctrines of the sect, revealing the vulnerable theological underbelly of their creed. It also speaks of quar- relling within the infidel camp. There are even suggestions that the author fears that he himself may be excommunicated by an up-and-coming generation of zealots. He pleads here for a mild interpretation of their creed. (Oblivious to the stench of his own blasphemies, he even imagines entering into dialogue with mainstream Ludwig scholars!) My main aim in making this text generally available is that more learned men than I may make a study of it. A sound theological thrashing of the author’s own (according to him, mild!) version of the creed is devoutly to be wished. But my fondest hope is that, in the hands of one of our finer Ludwig scholars, it might become a weapon that can be turned against the infidel camp. I have a Trojan horse maneuver in mind here. A cunningly crafted pseudonymous publication, addressing some of the niceties of their more peculiar doctrines, under the pretence of attempting to heal the looming schism in their sect, ought to be able to bust it wide open. One particularly confusing feature of the document is that the author occasionally adopts a heavily ironical tone, actually going so far as to refer to himself as an infidel, etc., though without apparently the least appreciation of the fact that the heavier he allows his irony to become, the closer he comes finally to speaking the truth.. (shrink)

According to Wittgenstein, mathematical propositions are rules of grammar, that is, conventions, or implications of conventions. So his position can be regarded as a form of conventionalism. However, mathematical conventionalism is widely thought to be untenable due to objections presented by Quine, Dummett and Crispin Wright. It has also been argued that only an implausibly radical form of conventionalism could withstand the critical implications of Wittgenstein’s rule-following considerations. In this article I discuss those objections to conventionalism and argue that none (...) of them is convincing. (shrink)