ABSTRACTWittgenstein’s remark that one cannot follow a rule only once has generated two puzzles: how can everyone accept it to be true? and why does Wittgenstein advance it? These two puzzles have tormented commentators for decades. In this paper I put forward a new interpretation and explain away the two puzzles. I shall show that Wittgenstein’s remark is plain truth and that his motivation behind making it is to dissolve the picture theory of meaning propounded in the Tractatus. This interpretation (...) also helps us to see that Wittgenstein’s other remark, that to follow a rule is a custom, is also a grammatical one. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

According to some scholars, such as Rodych and Steiner, Wittgenstein objects to Gödel’s undecidability proof of his formula $$G$$, arguing that given a proof of $$G$$, one could relinquish the meta-mathematical interpretation of $$G$$ instead of relinquishing the assumption that Principia Mathematica is correct. Most scholars agree that such an objection, be it Wittgenstein’s or not, rests on an inadequate understanding of Gödel’s proof. In this paper, I argue that there is a possible reading of such an objection that is, (...) in fact, reasonable and related to Gödel’s proof. (shrink)

Peter Winch often returned to questions about the nature of logic. In the context of his work on Wittgenstein and political philosophy in the 1990s, Winch described a contrast between ‘Aristotelian’ and ‘Socratic’ reasoning. Aristotelian conceptions of reasoning, attributed to Frege and Russell, would see logic as a formal science and rationality as consistency with pre‐existent rules of inference. The Socratic conception, attributed to Wittgenstein, understands rational argument as a form of socially embedded dialogue that involves moral relationships and a (...) dimension of depth. Rational persuasion may also involve use of persuasive images and examples. (shrink)

This paper seeks to widen the dialogue between the “epistemology of peer disagreement” and the epistemology informed by Wittgenstein’s last notebooks, later edited as On Certainty. The paper defends the following theses: not all certainties are groundless; many of them are beliefs; and they do not have a common essence. An epistemic peer need not share all of my certainties. Which response to a disagreement over a certainty is called for, depends on the type of certainty in question. Sometimes a (...) form of relativism is the right response. Reasonable, mutually recognized peer disagreement over a certainty is possible.—The paper thus addresses both interpretative and systematic issues. It uses Wittgenstein as a resource for thinking about peer disagreement over certainties. (shrink)

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, (...) set theory has proceeded in the opposite direction, from a web of intensions to a theory of extensionpar excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical structures and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression ofmathematicalmoves, whatever and sometimes in spite of what has been claimed on its behalf.What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by current set theory. The whole transfinite landscape can be viewed as the result of Cantor's attempt to articulate and solve the Continuum Problem. (shrink)

It is argued that the finitist interpretation of wittgenstein fails to take seriously his claim that philosophy is a descriptive activity. Wittgenstein's concentration on relatively simple mathematical examples is not to be explained in terms of finitism, But rather in terms of the fact that with them the central philosophical task of a clear 'ubersicht' of its subject matter is more tractable than with more complex mathematics. Other aspects of wittgenstein's philosophy of mathematics are touched on: his view that mathematical (...) propositions are 'grammatical propositions' (and so, Strictly speaking, Not genuine propositions at all) and his view that in mathematics 'everything is algorithm, Nothing meaning'. His views on consistency and his anti-Foundationalism are linked with this central thesis. (shrink)

It is argued in this article that the concept of practice is one of the key concepts in Wittgenstein's later philosophy. It partly replaces his earlier talk about the inexpressible. ?The practice has to speak for itself, as Wittgenstein succinctly puts it. The concept of practice not only points to the ways in which the unity of our concepts are underpinned, as Gordon Baker has it, it also comprises the skills involved in handling the conceptualized phenomena, our prereflective familiarity with (...) them, expressed in the sureness in our behaviour towards them, and the judgmental power exercised in applying or withholding a given concept on a particular occasion. These factors are all relevant to the establishment of knowledge, but they cannot themselves be fully and straightforwardly articulated by verbal means. Nevertheless, they represent what we go by when we apply concepts and other types of rules. To follow a rule is what Wittgenstein calls a practice. The sketched analysis of this concept makes us understand better how it is possible to apply a rule without the support of another rule. It also makes us realize in what sense one is justified in talking about tacit knowledge in connection with the application of concepts and rule?following in general. Quite a lot hangs on seeing the world aright at this point. (shrink)

Readers beware! This book is other than it first seems. Ludwig Wittgenstein’s latest philosophical offering is unlike anything that we have had from him to date. Its preface warns that the Tractatus is no textbook. This is an extreme understatement; really it is a deep puzzle—one that must be handled with great care. As the first lines signal there has been a radical change in the author’s characteristic style. Gone are the ingenious, probing explorations of topics undertaken in his highly (...) fragmentary writings that have been Wittgenstein’s trademark to date. In his new book Wittgenstein abandons the interrogative that his signature vision of and approach to philosophy found in his many masterpieces—The Blue and Brown Books, The Philosophical Investigations, and On Certainty. His previous masterworks have prosecuted a continuous philosophical enterprise designed to expose, resist and reject certain deeply compelling pictures that attract, mesmerize and blind us when it comes to thinking .. (shrink)

§1. Introduction. I dedicate this essay to the two-dozen-odd people whose refutations of Cantor's diagonal argument have come to me either as referee or as editor in the last twenty years or so. Sadly these submissions were all quite unpublishable; I sent them back with what I hope were helpful comments. A few years ago it occurred to me to wonder why so many people devote so much energy to refuting this harmless little argument—what had it done to make them (...) angry with it? So I started to keep notes of these papers, in the hope that some pattern would emerge.These pages report the results. They might be useful for editors faced with similar problem papers, or even for the authors of the papers themselves. But the main message to reach me is that there are several points of basic elementary logic that we usually teach and explain very badly, or not at all.In 1995 an engineer named William Dilworth, who had published a refutation of Cantor's argument in the Transactions of the Wisconsin Academy of Sciences, Arts and Letters, sued for libel a mathematician named Underwood Dudley who had called him a crank. (shrink)

The aim of this paper is to discover whether or not a solitary individual, a human being isolated from birth, could become a rule-follower. The argumentation against this possibility rests on the claim that such an isolate could not become aware of a normative standard, with which her actions could agree or disagree. As a consequence, theorists impressed by this argumentation adopt a view on which the normativity of rules arises from corrective practices in which agents engage in a community. (...) However, it has been suggested that an isolated individual could engage in such a practice by herself. Three prospective examples of such cases are considered, and the possibility of solitary rule-following is vindicated. Furthermore, the nature of the goals at which rule-following practices generally aim is clarified. (shrink)

When it comes to Wittgenstein's philosophy of mathematics, even sympathetic admirers are cowed into submission by the many criticisms of influential authors in that field. They say something to the effect that Wittgenstein does not know enough about or have enough respect for mathematics, to take him as a serious philosopher of mathematics. They claim to catch Wittgenstein pooh-poohing the modern set-theoretic extensional conception of a real number. This article, however, will show that Wittgenstein's criticism is well grounded. A real (...) number, as an 'extension', is a homeless fiction; 'homeless' in that it neither is supported by anything nor supports anything. The picture of a real number as an 'extension' is not supported by actual practice in calculus; calculus has nothing to do with 'extensions'. The extensional, set-theoretic conception of a real number does not give a foundation for real analysis, either. The so-called complete theory of real numbers, which is essentially an extensional approach, does not define (in any sense of the word) the set of real numbers so as to justify their completeness, despite the common belief to the contrary. The only correct foundation of real analysis consists in its being 'existential axiomatics'. And in real analysis, as existential axiomatics, a point on the real line need not be an 'extension'. (shrink)

The phrase ‘Lebensform’ had a long and varied history prior to Wittgenstein’s use of it on a mere three occasions in the Philosophical Investigations. It is not a pivotal concept in Wittgenstein’s philosophy. But it is a minor signpost of a major reorientation of philosophy, philosophy of language and logic, and philosophy of mathematics that Wittgenstein instigated. For Wittgenstein sought to replace the conception of a language as a meaning calculus by an anthropological or ethnological conception. A language is not (...) a class of sentences that can be formed from a set of axioms, formation and transformation rules and the meanings of which is given by their truth-conditions, but an open-ended series of interlocking language-games constituting a form of life or way of living. Wittgenstein’s uses of ‘Lebensform’ and its cognates, both in the Investigations and in his Nachlass are severally analysed, and various exegetical misinterpretations are clarified. (shrink)

Wittgenstein expounds his notion of a family resemblance in two important passages. The first is from The Blue Book:This craving for generality is the resultant of a number of tendencies connected with particular philosophical confusions. There is— The tendency to look for something common to entities which we commonly subsume under a general term. We are inclined to think that there must be something common to all games, say, and that this common property is the justification for applying the general (...) term “game” to the various games; whereas the games form a family the members of which have family likenesses …. There is a tendency rooted in our usual forms of expression, to think that a man who has learnt to understand a general term, say, the term “leaf,” has thereby come to possess a kind of general picture of a leaf, as opposed to pictures of particular leaves …. This again is connected with the idea that the meaning of a word is an image, or a thing correlated with the word. (shrink)

In all of its varieties, traditional liberalism is a universalist political theory. Its content is a set of principles which prescribe the best regime, the ideally best institutions, for all mankind. It may be acknowledged — as it is, by a proto-liberal such as Spinoza — that the best regime can be attained only rarely, and cannot be expected to endure for long; and that the forms its central institutions will assume in different historical and cultural milieux may vary significantly. (...) It will then be accepted that the liberal regime's role in political thought is as a regulative ideal, which political practice can hope only to approximate, subject to all the vagaries and exigencies of circumstance. Nonetheless, the content of traditional liberalism is a system of principles which function as universal norms for the critical appraisal of human institutions. In this regard, traditional liberalism — the liberalism of Locke and Kant, for example — represents a continuation of classical political rationalism, as it is found in Aristotle and Aquinas, where it also issues in principles having the attribute of universality, in that they apply ideally to all human beings. (shrink)

In this article, a comparative analysis of Karl Popper’s falsifiability theory and Ludwig Wittgenstein’s theory of meaning in the context of the historical-philosophical approach to the problem of new knowledge formation and justification is undertaken. An assumption is made that the constellation of fact is connected with the possibility of the emergence of an epistemological field. Researchers have repeatedly addressed this issue; however, one important detail received no due attention: Popper’s counter-arguments regarding Wittgenstein’s view on semantic paradoxes show the fundamental (...) difference of these philosophers’ views on the sign and the signified, which contributes to the analysis of new knowledge formation. Arguments that concern both early and late periods of Wittgenstein’s philosophy are used in this study. The uncertainty of the demarcation criterion allowed analyzing Wittgenstein’s position as to how and why different rules of inference form different epistemological fields. (shrink)

According to Chomsky and his followers, language as a biological phenomenon is a property of individual minds and brains; its status as a social phenomenon is merely epiphenomenal and not a proper object of scientific study. On a rival view, the individual's biological capacity for language cannot be properly understood in isolation from the linguistic environment, which it both depends on for its operation and—in collaboration with other speakers—builds and shapes for future generations. I argue here for the rival view (...) by demonstrating firstly its greater consonance with several themes in current biological theory, and secondly its ability to answer two well‐known philosophical challenges to generative linguistics due to Quine and Kripke. (shrink)

Suppose we were to ask some students of philosophy to imagine a typical book of classical German philosophy and describe its general style and character, how might they reply? I suspect that they would answer somewhat as follows. The book would be long and heavy, it would be written in a complicated style which employed only very abstract terms, and it would be extremely difficult to understand. At all events a description of this kind does indeed fit many famous works (...) of German philosophy. Let us take for example Hegel's Phenomenology of Spirit of 1807. The 1977 English translation published by Oxford has 591 pages, and as for style here is a typical passage: Self-consciousness found the Thing to be like itself, and itself to be like a Thing; i.e., it is aware that it is in itself the objectively real world. It is no longer the immediate certainty of being all reality, but a certainty for which the immediate in general has the form of something superseded, so that the objectivity of the immediate still has only the value of something superficial, its inner being and essence being self-consciousness itself. The object, to which it is positively related, is therefore a self-consciousness. It is in the form of thinghood, i.e. it is independent ; but it is certain that this independent object is for it not something alien, and thus it knows that it is in principle recognized by the object. It is Spirit which, in the duplication of its self-consciousness and in the independence of both, has the certainty of its unity with itself. This certainty has now to be raised to the level of truth; what holds good for it in principle , and in its inner certainty, has to enter into its consciousness and become explicit fork. (shrink)

Cultural pluralism is both a fact and a norm. It is a fact that our world, and indeed our society, are marked by a large diversity of cultures delineated in terms of race, class, gender, ethnicity, religion, ideology, and other partly interpenetrating variables. This fact raises the normative question of whether, or to what extent, such diversities should be recognized or even encouraged in policies concerning government, law, education, employment, the family, immigration, and other important areas of social concern.

Hilary Putnam first published the consistency objection against Ludwig Wittgenstein’s account of mathematics in 1979. In 1983, Putnam and Benacerraf raised this objection against all conventionalist accounts of mathematics. I discuss the 1979 version and the scenario argument, which supports the key premise of the objection. The wide applicability of this objection is not apparent; I thus raise it against an imaginary axiomatic theory T similar to Peano arithmetic in all relevant aspects. I argue that a conventionalist can explain the (...) consistency of T and suggest that an analogous explanation can be provided for the consistency of Peano arithmetic. (shrink)

The present article aims at investigating the political aspects of Wittgenstein’s later philosophy, focusing mainly on the Philosophical Investigations. This theme remains rather marginal within Wittgensteinian scholarship, facing the key challenge of the sparsity of explicit discussions of political issues in Wittgenstein’s writings. Based on the broader anthropological and synecdochic character of Wittgenstein’s later philosophy, the main objective of the article is to make explicit the implicit political import of some of the main themes of the Philosophical Investigations. This is (...) done by correlating them with certain political concepts and issues, such as reification, mystification, emancipation, self-institution and the common. In such a way, a new light is shed on Wittgenstein’s influence on and his relevance to contemporary political theory, while at the same time the first steps are taken for the enrichment of the discussion regarding personal and social autonomy and the critique of late capit... (shrink)

In the first paragraph, the focus is on the early Wittgenstein's conception of truth: the Disquotational Schema is shown to be derivable from the semantic and ontological principles of the picture theory. Then, the article scrutinises the way the Disquotational Schema provides the basis for what the later Wittgenstein takes as a philosophically appropriate description of the practice of making assertions. The general abstract notion of truth makes room for a situated notion of warranted assertibility as the key-notion. Last, the (...) issue of how Wright's Argument from Informational Neutrality could be neutralised from the later Wittgenstein's viewpoint is dealt with. (shrink)