This book offers a detailed account and discussion of Ludwig Wittgenstein's philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege's logicist attempt to provide arithmetic with a foundation and Wittgenstein's criticisms of it, followed by sketches of Wittgenstein's early views of mathematics, in the Tractatus and in the early 1930s. Then (in Part II), Wittgenstein's mature philosophy of mathematics (1937-44) is carefully presented and examined. Schroeder explains that it is based on two (...) key ideas: the calculus view and the grammar view. On the one hand, mathematics is seen as a human activity - calculation - rather than a theory. On the other hand, the results of mathematical calculations serve as grammatical norms. The following chapters (on mathematics as grammar; rule-following; conventionalism; the empirical basis of mathematics; the role of proof) explore the tension between those two key ideas and suggest a way in which it can be resolved. Finally, there are chapters analysing and defending Wittgenstein's provocative views on Hilbert's Formalism and the quest for consistency proofs and on Gödel's incompleteness theorems. (shrink)

Wittgenstein shows that understanding is a capacity, and cannot be accounted for by mental representations of what is understood. But if a person’s understanding or thinking cannot be accounted for by occurrences of mental representations, then understanding that person cannot be a matter of knowing what is going on inside him or her: what representations he or she has in his or her mind. That, I argue, is the point of Wittgenstein’s famous and frequently misunderstood saying, “If a lion could (...) talk we wouldn’t be able to understand it.” Our inability to understand a strange creature would not be overcome if that creature could speak and lay open to us on the mental operations inside its mind. Finally, I explain in what way understanding other people is not an entirely intellectual affair, but also has moral and aesthetic dimensions. -/- . (shrink)

This book offers a lucid and highly readable account of Wittgenstein's philosophy, framed against the background of his extraordinary life and character. Woven together with a biographical narrative, the chapters explain the key ideas of Wittgenstein's work, from his first book, the Tractatus Logico-Philosophicus, to his mature masterpiece, the Philosophical Investigations.

This chapter contains sections titled: I II III IV V Bibliography.

This chapter contains sections titled: Voluntary Action Reasons and Causes References.

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)

Wittgenstein offers three objections to the idea of aesthetics as a branch of psychology: (i) Statistical data about people’s preferences have no normative force. (ii) Artistic value is not instrumental value, a capacity to produce independently identifiable – and scientifically measurable – psychological effects. (iii) While psychological investigations may bring to light the causes of aesthetic preferences, they fail to provide reasons for them. According to Wittgenstein, aesthetic explanations (unlike scientific explanations) are poignant synoptic representations of aspects of a work, (...) and the criterion of success of an aesthetic explanation is that it satisfies the addressee. He repeatedly remarked that they resemble philosophical explanations, which also try to dispel puzzlement or confusion. The difference, however, is that whereas in philosophy we deal with general conceptual problems, aesthetic explanations typically concern individual responses to particular works of art. (shrink)

This chapter contains sections titled: Voluntary Action Reasons and Causes References.

This chapter contains sections titled: I II III IV V Notes References.

Contrary to a widespread interpretation, Wittgenstein did not regard credal statements as merely metaphorical expressions of an attitude towards life. He accepted that Christian faith involves belief in God's existence. At the same time he held that although as a hypothesis, God's existence is extremely implausible, Christian faith is not unreasonable. Is that a consistent view? According to Wittgenstein, religious faith should not be seen as a hypothesis, based on evidence, but as grounded in a proto‐religious attitude, a way of (...) experiencing the world or certain aspects of it. A belief in religious metaphysics is not the basis of one's faith, but a mere epiphenomenon. Given further that religious doctrine is both falsification‐transcendent and that religious faith is likely to have beneficial psychological effects, religious doctrine can be exempt from ordinary standards of epistemic support. An unsupported religious belief need not be unreasonable. However, it is hard to see how one could knowingly have such an unsupported belief, as Wittgenstein seems to envisage. How can one believe what, at the same time, one believes is not likely to be true? This, I argue, is the unresolved tension in Wittgenstein's philosophy of religion. (shrink)

It is widely held that whenever someone φs, that person tries to do φ. I examine arguments by B. O’Shaughnessy and J. Hornsby, and considerations by P. Grice in support of that thesis. I argue that none of them are convincing. The remainder of the paper defends an analysis of the concept of trying along the lines opposed by Grice et al. By speaking of someone’s trying to φ the speaker leaves the room for failure or the possibility of failure. (...) This account of the function of the word ‘try’ makes it clear what is wrong with the claim that whenever someone φs, that person tries to do φ. For that claim amounts to saying that every action is either a failure or a possible failure – which clearly one can’t say of any action that one has already acknowledged to be successful. Furthermore, whether an action is possibly a failure depends on how much one knows about it. Hence the question whether in itself an action does or does not involve trying is as misguided as the question whether it’s intrinsically unexpected. It may be correct for some to describe it as ‘an attempt to φ’, but incorrect for others. (shrink)

It is widely held that whenever someone φs, that person tries to do φ. I examine arguments by B. O’Shaughnessy and J. Hornsby, and considerations by P. Grice in support of that thesis. I argue that none of them are convincing. The remainder of the paper defends an analysis of the concept of trying along the lines opposed by Grice et al. By speaking of someone’s trying to φ the speaker leaves the room for failure or the possibility of failure. (...) This account of the function of the word ‘try’ makes it clear what is wrong with the claim that whenever someone φs, that person tries to do φ. For that claim amounts to saying that every action is either a failure or a possible failure – which clearly one can’t say of any action that one has already acknowledged to be successful. Furthermore, whether an action is possibly a failure depends on how much one knows about it. Hence the question whether in itself an action does or does not involve trying is as misguided as the question whether it’s intrinsically unexpected. It may be correct for some to describe it as ‘an attempt to φ’, but incorrect for others. (shrink)

“Grammar” is Ludwig Wittgenstein's preferred term for the workings of a language: the system of rules that determine linguistic meaning. A philosophical study of language is a study of “grammar”, in this sense, and insofar as any philosophical investigation is concerned with conceptual details, which manifest themselves in language, it is a grammatical investigation. In the Tractatus Logico‐Philosophicus Wittgenstein offered a mathematical picture of language: presenting language as a calculus. Like a calculus, language was claimed to be governed by syntactic (...) rules: formation rules about the licit combination of names to form elementary propositions; formation rules about the licit combination of elementary propositions to make complex propositions; and finally, truth‐table rules, which enable us to identify logical truths and entailments. Logical grammar, for the young Wittgenstein, is only logical syntax and logical syntax can be determined without paying any attention to the meaning of the signs. (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)

Wittgenstein and Contemporary Philosophy of Mind aims to reassess the work of Wittgenstein in terms of its importance to contemporary debates surrounding the philosophy of mind.The first part of this study examines Wittgenstein in the context of current views on the human mind in relation to the body and behavior. The arguments confront the views of Quine and Dennett, as well as functionalism, eliminative materialism, and the current debate about consciousness. The essays that make up the second part focus on (...) a particular psychological concept, thinking, imagining, sensation, knowledge, and reason. This study takes a fresh look at this established thinker and demonstrates both the relevance and power of his arguments in the 21st century. (shrink)

It is widely thought that Bayesian confirmation theory has provided a solution to Hempel's Paradox (the Ravens Paradox). I discuss one well‐known example of this approach, by John Mackie, and argue that it is unconvincing. I then suggest an alternative solution, which shows that the Bayesian approach is altogether mistaken. Nicod's Condition should be rejected because a generalisation is not confirmed by any of its instances if it is not law‐like. And even law‐like non‐basic empirical generalisations, which are expressions of (...) assumed underlying causal regularities, are not so confirmed if they are absurd in the light of our causal background knowledge or if their instances are not also possible instances of the relevant causal claim. (shrink)

This paper explores Wittgenstein's attempts to explain the peculiarities of the first-person use of 'believe' that manifest themselves in Moore's paradox, discussed in, Part II, section x. An utterance of the form 'p and I do not believe that p' is a kind of contradiction, for the second conjunct is not, as it might appear, just a description of my mental state, but an expression of my belief that not-p, contradicting the preceding expression of my belief that p. Thus, 'I (...) believe that p' is just a stylistic variant of 'p'; the word 'believe' doesn't seem to have a substantial role to play in such an utterance. Following Wittgenstein, I discuss why there could not be a first-person present-tense use of the word that was more akin to its use in the third person: why it is impossible to describe one's own current beliefs in a detached manner without thereby expressing them. In the final section, I try to develop Wittgenstein's suggestion that the non-epistemic authority we have regarding the contents of our beliefs can be clarified by considering its link with intention and action. (shrink)

Is the concept of a work of art an evaluative concept: does its application imply a positive evaluation? I shall discuss this question by considering two opposing attempts at defining art, namely the Institutional Theory and the view that art is a functional concept. I shall argue that the concept of art does not imply an unconditionally positive evaluation, but that art is a prestige concept. Moreover, it will be shown that functional definitions of art are flawed.

In the so-called private language argument, Wittgenstein argues both against the alleged epistemological privacy of sensations and against their alleged ontological privacy, that is, the common view that somebody else cannot have my pain. A prominent proponent of the claim of sensations' ontological privacy was Gottlob Frege, whose position has recently been defended by Wolfgang Künne. This paper reconsiders Wittgenstein's objections to ontological privacy and attempts to defend Wittgenstein's position against Künne's Frege-inspired arguments.

In the so‐called private language argument, Wittgenstein argues both against the alleged epistemological privacy of sensations and against their alleged ontological privacy, that is, the common view that somebody else cannot have my pain. A prominent proponent of the claim of sensations' ontological privacy was Gottlob Frege, whose position has recently been defended by Wolfgang Künne. This paper reconsiders Wittgenstein's objections to ontological privacy and attempts to defend Wittgenstein's position against Künne's Frege‐inspired arguments.

Peter Kivy’s contour theory provides a promising explanation of the way we describe instrumental music as expressive of emotions. I argue that if, unlike Kivy, we emphasise the metaphorical character of such descriptions, the contour theory, as a strategy for unpacking such metaphors, can be defended convincingly against common objections. This approach is more satisfactory than those of Scruton and Peacocke, who make much of metaphorical experiences, but leave the underlying metaphors unexplained. Moreover, it gives the contour theory a wider (...) scope than Kivy intended, for even very specific narrative descriptions of music in non-musical terms are perfectly legitimate as long as they are presented, and justified, as metaphors, that is, as mere comparisons, rather than as interpretative claims about the music’s actual contents. (shrink)

In the first chapter of his book Logical Foundations of Probability, Rudolf Carnap introduced and endorsed a philosophical methodology which he called the method of ‘explication’. P.F. Strawson took issue with this methodology, but it is currently undergoing a revival. In a series of articles, Patrick Maher has recently argued that explication is an appropriate method for ‘formal epistemology’, has defended it against Strawson’s objection, and has himself put it to work in the philosophy of science in further clarification of (...) the very concepts on which Carnap originally used it (degree of confirmation, and probability), as well as some concepts to which Carnap did not apply it (such as justified degree of belief). We shall outline Carnap’s original idea, plus Maher’s recent application of such a methodology, and then seek to show that the problem Strawson raised for it has not been dealt with. The method is indeed, we argue, problematic and therefore not obviously superior to the ‘descriptive’ method associated with Strawson. Our targets will not only be Carnapians, though, for what we shall say also bears negatively on a project that Paul Horwich has pursued under the name ‘therapeutic’, or ‘Wittgensteinian’ Bayesianism. Finally, explication, as we shall suggest and as Carnap recognised, is not the only route to philosophical enlightenment. (shrink)

This book offers a detailed account and discussion of Ludwig Wittgenstein's philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege's logicist attempt to provide arithmetic with a foundation and Wittgenstein's criticisms of it, followed by sketches of Wittgenstein's early views of mathematics, in the Tractatus and in the early 1930s. Then, Wittgenstein's mature philosophy of mathematics is carefully presented and examined. Schroeder explains that it is based on two key ideas: the calculus (...) view and the grammar view. On the one hand, mathematics is seen as a human activity -- calculation -- rather than a theory. On the other hand, the results of mathematical calculations serve as grammatical norms. The following chapters explore the tension between those two key ideas and suggest a way in which it can be resolved. Finally, there are chapters analysing and defending Wittgenstein's provocative views on Hilbert's Formalism and the quest for consistency proofs and on Gödel's incompleteness theorems. (shrink)

One of the key tenets in Wittgenstein’s philosophy of mathematics is that a mathematical proposition gets its meaning from its proof. This seems to have the paradoxical consequence that a mathematical conjecture has no meaning, or at least not the same meaning that it will have once a proof has been found. Hence, it would appear that a conjecture can never be proven true: for what is proven true must ipso facto be a different proposition from what was only conjectured. (...) Moreover, it would appear impossible that the same mathematical proposition be proven in different ways. — I will consider some of Wittgenstein’s remarks on these issues, and attempt to reconstruct his position in a way that makes it appear less paradoxical. (shrink)

Rudolf Carnap introduced and endorsed a philosophical methodology which he called the method of ‘explication’. P.F. Strawson took issue with this methodology, but it is currently undergoing a revival. In a series of articles, Patrick Maher has recently argued that explication is an appropriate method for ‘formal epistemology’, has defended it against Strawson’s objection, and has himself put it to work in the philosophy of science in further clarification of the very concepts on which Carnap originally used it (degree of (...) confirmation, and probability), as well as some concepts to which Carnap did not apply it (such as justified degree of belief. We outline Carnap’s original idea, plus Maher’s recent application of such a methodology, and then seek to show that the problem Strawson raised for it has not been dealt with. The method is indeed, we argue, problematic and therefore not obviously superior to the ‘descriptive’ method associated with Strawson. (shrink)