This paper has two main purposes: first to compare Wittgenstein's views to the more traditional views in the philosophy of mathematics; second, to provide a general outline for a Wittgensteinian reply to two objections against Wittgenstein's account of mathematics: the objectivity objection and the consistency objections, respectively. Two fundamental thesmes of Wittgenstein's account of mathematics title the first two sections: mathematical propositions are rules and not descritpions and mathematics is employed within a (...) form of life. Under each heading, I examine Wittgenstein's rejection of alternative views. My aim is to make clear the differences and to suggest some similarities. As will become clear, Wittgenstein often rejects opposing views for the same or similar reasons. This comparison will provide the necessary background for better understanding Wittgenstein'sphilosophy of mathematics, for appreciating its many unappreciated advantages and, finally, for defending a conventionalist account of mathematics. (shrink)
Wittgenstein played a vital role in establishing mathematics as one of this century's principal areas of philosophic inquiry. In this book, Pasquale Frascolla examines the three phases of Wittgenstein's reflections on mathematics, considering them as a progressive whole rather than as separate entities. Frascolla discusses the development of Wittgenstein's views on mathematics from the Tractatus up to 1944. He looks at the presentation of arithmetic in the theory of logical operations, the presence of a (...) strong verificationist orientation and the rule-following considerations in Wittgenstein's writings. Frascolla identifies a unifying key--a "quasi-formalism"--to the development of Wittgenstein's reflections on mathematics. (shrink)
During the course of about ten years, Wittgenstein revised some of his most basic views in philosophy of mathematics, for example that a mathematical theorem can have only one proof. This essay argues that these changes are rooted in his growing belief that mathematical theorems are ‘internally’ connected to their canonical applications, i.e. , that mathematical theorems are ‘hardened’ empirical regularities, upon which the former are supervenient. The central role Wittgenstein increasingly assigns to empirical regularities had profound implications (...) for all of his later philosophy; some of these implications (particularly to rule following) are addressed in the essay. (shrink)
This paper attempts to elucidate Wittgenstein’s remark about the “strange resemblance between a philosophical investigation (especially in mathematics) and an aesthetic one” from 1937 by looking at its textual and philosophical context. The conclusion is that the remark can be seen both as a description of a particular conception of philosophy, a prescription or declaration of intent (to proceed in a particular way), and a reminder (to Wittgenstein himself) about the form of a philosophical investigation. Furthermore, it is (...) concluded that the Darstellungsform he has in mind is the one that finds expression especially in the first part of the PI. (shrink)
An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...) with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question. (shrink)
Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition (...) of number turns out to be valid on its own terms, even though it depends on two epistemological principles logicist philosophers of mathematics may find too ‘constructivist’. (shrink)
Wittgenstein'sphilosophy of mathematics has long been notorious. Part of the problem is that it has not been recognized that Wittgenstein, in fact, had two chief post-Tractatus conceptions of mathematics. I have labelled these the calculus conception and the language-game conception. The calculus conception forms a distinct middle period. The goal of my article is to provide a new framework for examining Wittgenstein's philosophies of mathematics and the evolution of his career as a whole. (...) I posit the Hardyian Picture, modelled on the Augustinian Picture, to provide a structure for Wittgenstein's work on the philosophy of mathematics. Wittgenstein's calculus period has not been properly recognized, so I give a detailed account of the tenets of that stage in Wittgenstein's career. Wittgenstein's notorious remarks on contradiction are the test case for my theory of his transition. I show that the bizarreness of those remarks is largely due to the calculus conception, but that Wittgenstein's later language-game account of mathematics keeps the rejection of the Hardyian Picture while correcting the calculus conception's mistakes. (shrink)
The German physicist Heinrich Hertz played a decisive role for Wittgenstein's use of a unique philosophical method. Wittgenstein applied this method successfully to critical problems in logic and mathematics throughout his life. Logical paradoxes and foundational problems including those of mathematics were seen as pseudo-problems requiring clarity instead of solution. In effect, Wittgenstein's controversial response to David Hilbert and Kurt Gödel was deeply influenced by Hertz and can only be fully understood when seen in this context. (...) To comprehend the arguments against the metamathematical programme, and to appreciate how profoundly the philosophical method employed actually shaped the content of Wittgenstein'sphilosophy, it is necessary to make an intellectual biographical reconstruction of their philosophical framework, tracing the Hertzian elements in the early as well as in the later writings. In order to write Wittgenstein's biography, we have to take seriously the coherence of his thought throughout his life, and not let convenient philosophical ideologies be our guidance in drawing up a “Wittgensteinian philosophy”. To do so, we have to take a second look upon what he actually wrote, not only in the already published material, but in the entire Nachlass. Clearly, this is not easily done, but it is a necessary task in the historical reconstruction of Wittgenstein's life and work. (shrink)
Mathematics plays an inordinate role in the work of many of famous Western philosophers, from the time of Plato, through Husserl and Wittgenstein, and even to the present. Why? This paper points to the experience of learning or making mathematics, with an emphasis on proof. It distinguishes two sources of the perennial impact of mathematics on philosophy. They are classified as Ancient and Enlightenment. Plato is emblematic of the former, and Kant of the latter. The Ancient (...) fascination arises from the sense that mathematics explores something ‘out there’. This is illustrated by recent discussions by distinguished contemporary mathematicians. The Enlightenment strand often uses Kant's argot: ‘absolute necessity’, ‘apodictic certainty’ and ‘a priori’ judgement or knowledge. The experience of being compelled by proof, the sense that something must be true, that a result is certain, generates the philosophy. It also creates the illusion that mathematics is certain. Kant's leading question, ‘How is pure mathematics possible?’, is easily misunderstood because the modern distinction between pure and applied is an artefact of the 19th century. As Russell put it, the issue is to explain ‘the apparent power of anticipating facts about things of which we have no experience’. More generally the question is, how is it that pure mathematics is so rich in applications? Some six types of application are distinguished, each of which engenders its own philosophical problems which are descendants of the Enlightenment, and which differ from those descended from the Ancient strand. (shrink)
Discussion of Wittgenstein's Tractatus is currently dominated by two opposing interpretations of the work: a metaphysical or realist reading and the 'resolute' reading of Diamond and Conant. Marie McGinn's principal aim in this book is to develop an alternative interpretative line, which rejects the idea, central to the metaphysical reading, that Wittgenstein sets out to ground the logic of our language in features of an independently constituted reality, but which allows that he aims to provide positive philosophical insights into (...) how language functions. McGinn takes as a guiding principle the idea that we should see Wittgenstein's early work as an attempt to eschew philosophical theory and to allow language itself to reveal how it functions. By this account, the aim of the work is to elucidate what language itself makes clear, namely, what is essential to its capacity to express thoughts that are true or false. However, the early Wittgenstein undertakes this descriptive project in the grip of a set of preconceptions concerning the essence of language that determine both how he conceives the problem and the approach he takes to the task of clarification. Nevertheless, the Tractatus contains philosophical insights, achieved despite his early preconceptions, that form the foundation of his later philosophy. -/- The anti-metaphysical interpretation that is presented includes a novel reading of the problematic opening sections of the Tractatus, in which the apparently metaphysical status of Wittgenstein's remarks is shown to be an illusion. The book includes a discussion of the philosophical background to the Tractatus, a comprehensive interpretation of Wittgenstein's early views of logic and language, and an interpretation of the remarks on solipsism. The final chapter is a discussion of the relation between the early and the later philosophy that articulates the fundamental shift in Wittgenstein's approach to the task of understanding how language functions and reveal the still more fundamental continuity in his conception of his philosophical task. (shrink)
This anthology focuses on the extraordinary contributions Wittgenstein made to several areas in the philosophy of psychology - contributions that extend to psychology, psychiatry, sociology and anthropology. To bring them a richly-deserved attention from across the language barrier, Danièle Moyal-Sharrock has translated papers by eminent French Wittgensteinians. They here join ranks with more familiar renowned specialists on Wittgenstein's philosophical psychology. While revealing differences in approach and interests, this coming together of some of the best minds on the subject (...) discloses a surprising degree of consensus, and gives us the clearest picture yet of Wittgenstein as a philosopher of psychology. (shrink)
From his return to Cambridge in 1929 to his death in 1951, Wittgenstein influenced philosophy almost exclusively through teaching and discussion. These lecture notes indicate what he considered to be salient features of his thinking in this period of his life.
This paper responds to Severin Schroeder's recent charge that Wittgenstein'sphilosophy of religion contains an ‘unresolved tension’ between three propositions, namely: (1) ‘As a hypothesis, God's existence (&c) is extremely implausible’; (2) ‘Christian faith is not unreasonable’; and (3) ‘Christian faith does involve belief in God's existence (&c)’. I argue as follows: that the first of these propositions has no place in Wittgenstein's thinking on religion; that the second is ill-phrased and should be re-worded as the proposition (...) that ‘Christian faith is neither unreasonable nor reasonable’; and that the third proposition (contrary to what Schroeder seems to assume) tells us nothing about the nature of the objects of religious belief. It follows from my argument that Schroeder has not exposed a tension in Wittgenstein's thoughts on religion. I end with some positive remarks about Wittgenstein's method. (shrink)
In his later philosophy, Wittgenstein unlike Russell offers no theories, because he believes that philosophical theories are never explanatory. They try to imitate scientific theories, but they lack the empirical basis that gives science its explanatory power. Two examples of his deconstructive work are discussed. One is his critique of the theory that the direct objects of perception are always sense-data, describable in a radically private language. Austin too criticized the theory of sense-data, but Wittgenstein's critique, unlike Austin's, (...) included an attempt to show what had made it so attractive to its supporters: it presented a picture of the human predicament that appealed to their imaginations. The second example is his critique of the theory of the pure ego, which tended to collapse into solipsism. This critique was developed in his first book, Tractatus Logico-Philosophicus , and his later deconstructive work was modelled on it. (shrink)
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'sphilosophy, 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 role of training in Wittgenstein's later philosophy, the paper offers a tour of key passages in the Investigations and other works to develop an understanding of what Wittgenstein meant by 'mastery of techniques'. In opposition to Luntley's liberal-individual, or his subject as rational agent, the author explores Wittgenstein's non-foundationalist, forms of life approach to how we act with agreement. More effort must be given to differentiating Wittgenstein's view from that of the analytic school, which Luntley appears to echo despite his criticism of the analytic divide. (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)
Wittgenstein's treatment of private language is the dissolution of some of the major problems in traditional philosophy. Philosophical problems, for Wittgenstein, are the conceptual confusion arising due to the abuse of language. They can be fully dispensed with by commanding a clear view of language. Language, for Wittgenstein, is on the one hand, the source of philosophical problems while, on the other hand, it is a means to dispense with them. Private language is one such issue which is (...) ultimately rooted I a mistaken conception of language and is the sources of various philosophical problems/ puzzles. (shrink)
On the basis of historical and textual evidence, this paper claims that (i) after his Tractatus, Wittgenstein was actually influenced by Einstein's theory of relativity and, (ii) the similarity of Einstein's relativity theory helps to illuminate some aspects of Wittgenstein's work. These claims find support in remarkable quotations where Wittgenstein speaks approvingly of Einstein's relativity theory and in the way these quotations are embedded in Wittgenstein's texts. The profound connection between Wittgenstein and relativity theory concerns not only (...) class='Hi'>Wittgenstein's “verificationist” phase (more closely connected with Schlick's work), but also Wittgenstein's later philosophy centred on the theme of rule-following. (shrink)
In this volume, Pears examines the internal organization of Wittgenstein's thought and the origins of his philosophy to provide unusually clear insight into the philosopher's ideas. Part I surveys the whole of Wittgenstein's work, while Part II details the central concepts of his early system; both reveal how the details of Wittgenstein's work fit into its general pattern.
In this paper I show and discuss the relevance of Wittgenstein´s arguments as to the spatial nature of sight for recent issues in the philosophy of mind. The first, bearing upon the dimensionality of the manifolds at play in depiction, plays a critical role in Clark´s attempt to provide an independent account of qualia and of their differentiative properties. The second, pertaining to the properly spatial structure formed by the data of sight, is explicitly appealed to in the debate (...) on the realistic character of any genuinely spatial conceptual scheme. I argue that if Wittgenstein rightly assumes that the simultaneous presence of sensible places in vision is a key condition on objectivity, he fails however to warrant the allegedly realistic character of the conceptual scheme employed in his own search for a phenomenological description of the visual field. (shrink)
This pioneering book demonstrates the crucial importance of Wittgenstein'sphilosophy of mathematics to his philosophy as a whole. Marion traces the development of Wittgenstein's thinking in the context of the mathematical and philosophical work of the times, to make coherent sense of ideas that have too often been misunderstood because they have been presented in a disjointed and incomplete way. In particular, he illuminates the work of the neglected 'transitional period' between the Tractatus and the (...) Investigations. (shrink)
This book provides a novel interpretation of the ideas about language in Ludwig Wittgenstein's Philosophical Investigations. Travis places the "private language argument" in the context of wider themes in the Investigations, and thereby develops a picture of what it is for words to bear the meaning they do. He elaborates two versions of a private language argument, and shows the consequences of these for current trends in the philosophical theory of meaning.
Volume 9 of the Routledge History of Philosophy surveys ten key topics in the Philosophy of Science, Logic and Mathematics in the Twentieth Century. Each article is written by one of the world's leading experts in that field. The papers provide a comprehensive introduction to the subject in question, and are written in a way that is accessible to philosophy undergraduates and to those outside of philosophy who are interested in these subjects. Each chapter contains (...) an extensive bibliography of the major writings in the field. Among the topics covered are the philosophy of logic; Ludwig Wittgenstein's Tractatus; a survey of logical positivism; the philosophy of physics and of science; probability theory and cybernetics. (shrink)
Th e con fusion a nd b arren ness o f psycho logy is no t to be e xplain ed b y calling it a “yo ung science”; its state is not comparable with that of physics, for instance, in its beginnings. (Rather with that of certain branches of mathematics. Set theory.) For in psychology there are experimental methods and conceptual confusion. (As in the oth er case, con cep tual co nfusion and m ethod s of pro (...) of.) The existence of the experimental method makes us think we have the means of solving the problems that trouble us; though problem and method pass one another by. (PI p. 232). (shrink)
Structure and content of the philosophical investigations -- Wittgenstein's metaphilosophy -- The method of description -- Wittgenstein's distinctive arguments : from mistake to paradox -- Two domains : linguistic mastery vs. initiate learning -- The structure of the book -- Playing the game -- The Fregean picture of language -- Wittgenstein's rejection of Frege's idea -- Builders game : language or signaling? -- Dummett's challenge : sense vs. force -- The domestication of reference -- The problem of (...) normative similarity 1 : ostension -- Rejection of Quine's picture of language -- Objects and paradigms -- Ostensive teaching and social practices -- Logical form and the paradox of thought -- The subliming of logic -- Frege's idea and the paradox of thought -- Davidson's challenge : meaning and logical form -- The limits of systematicity -- Meaning and the paradox of interpretation -- The problem of normative similarity 2 : rules -- Two pleas for interpretation -- The community view and reductionism -- The individualist view and mystification normativity and the threat of regularism -- Rules and regularities -- The public basis of normativity -- The social basis of normativity : the negative argument -- The social basis of normativity : the positive argument -- Necessity and the threat of psychologism -- Two forms of holism -- Stage-setting : conventions without decisions -- Background technique : necessity without metaphysics -- Normativity and "psychologized" necessity -- Learning, trust, and certainty -- The paradoxes of consciousness -- The problem of normative similarity 3 : consciousness -- The epistemology of subjectivity : paradox of self-knowledge -- The ontology of subjectivity : paradox of sensation -- Cartesian thought experiments and the expressivist view -- Criteria, deception, and the new problem of other minds. (shrink)
BL Translated from German with additions and amendments -/- The writings preserved in Wittgenstein's manuscripts from 1945 to 1949, after he had completed the first part of Philosophical Investigations, chiefly concern the nature of certain psychological concepts. Joachim Schulte here uses these manuscripts - not just the selections from them published so far - as a basis for reconstructing the central arguments and conceptual elucidations developed by Wittgenstein during that period.
Did Wittgenstein consider himself a Jew? Should we? Wittgenstein repeatedly wrote about Jews and Judaism in the 1930s, and biographical studies make it clear that this writing about Jewishness was a way in which he thought about the kind of person he was and the nature of his philosophical work. Those who have written about Wittgenstein on the Jews have drawn very different conclusions. But much of this debate is confused, because the notion of being a Jew, of Jewishness, is (...) itself ambiguous and problematic. The paper provides a close reading of leading passages in which Wittgenstein discusses Jews and Jewishness, and argues that previous interpreters have been too quick to condemn or defend him. If we consider what it could mean to say that Wittgenstein was, or was not, a Jew, we will see that Wittgenstein's problems with 'Jewishness' arise out of the philosophically problematic nature of the concept, a philosophical problem he was unable to resolve. (shrink)
The paper presents a formal explication of the early Wittgenstein's views on ontology, the syntax and semantics of an ideal logical language, and the propositional attitudes. It will be shown that Wittgenstein gave a language of thought analysis of propositional attitude ascriptions, and that his ontological views imply that such ascriptions are truth-functions of (and supervenient upon) elementary sentences. Finally, an axiomatization of a quantified doxastic modal logic corresponding to Tractarian semantics will be given.
Wittgenstein’s interpreters are undivided that the method plays a central role in his philosophy. This would be no surprise if we have in mind the Tractarian dictum: “philosophy is not a body of doctrine but an activity” (4.112). After 1929, Wittgenstein’s method evolved further. In its final form, articulated in Philosophical Investigations, it was formulated as different kinds of therapies of specific philosophical problems that torment our life (§§ 133, 255, 593). In this paper we follow the changes (...) in Wittgenstein’s thinking in four subsequent phases and in three dimensions: (i) in logic and ontology; (ii) in method proper; (iii) in style. (shrink)
I consider the support variously offered for the remark at Philosophical Investigations 246: ‘It can’t be said of me at all (except perhaps as a joke) that I know I am in pain.’ Against the first sort of argument to be found in Wittgenstein and the literature I offer cases in which I learn of pain. Against the second sort of argument I develop the case in which I am persuaded by compelling evidence that I am, contrary to what I (...) imagined, still in an emotional pain about N. I then consider the counter-argument that the mix of sensation and emotion in my second case makes it irrelevant as a criticism of Wittgenstein, but argue that the reverse holds. That ‘sensation’ is quite separate from ‘emotion’ is a Cartesian Mistake which is, I argue, implicit in Wittgenstein’s discussion of ‘I know I am in pain’. (shrink)
Three theses are gleaned from Wittgenstein’s writing. First, extra-mathematical uses of mathematical expressions are not referential uses. Second, the senses of the expressions of pure mathematics are to be found in their uses outside of mathematics. Third, mathematical truth is fixed by mathematical proof. These theses are defended. The philosophy of mathematics defined by the three theses is compared with realism, nominalism, and formalism.
Philosophy of Mathematics is clear and engaging, and student friendly The book discusses the great philosophers and the importance of mathematics to their thought. Among topics discussed in the book are the mathematical image, platonism, picture-proofs, applied mathematics, Hilbert and Godel, knots and notation definitions, picture-proofs and Wittgenstein, computation, proof and conjecture.
It is argued that the Tractatus Project of Logical Atomism, in which the world is conceived of as the totality of independent atomic facts, can usefully be understood by conceiving of each fact as a bit in logical space. Wittgenstein himself thinks in terms of logical space. His elementary propositions, which express atomic facts, are interpreted as tuples of co-ordinates which specify the location of a bit in logical space. He says that signs for elementary propositions are arrangements of names. (...) Here, the names are understood as numerical symbols specifying coordinates. It is argued that, using this approach, the so-called colour-exclusion problem, which was Wittgensteins reason for abandoning the Tractatus, is in fact soluble. However, if logical space is a continuum then some coordinates will need to be expressed by numerical symbols that are infinite in size. How is this to be understood in Tractatus terms? It is shown that, in the Tractatus, Wittgenstein did recognise the possibility of infinite propositions and sentences expressing them. At first sight his approach to infinite sentences, and the approach of the present paper, seem to differ, but it is argued that the difference is superficial. Finally, we address the question of whether Logical Atomism is viable and this raises issues concerning its relationship to natural science. (shrink)
Friedrich Waismann, a little-known mathematician and onetime student of Wittgenstein's, provides answers to problems that vexed Wittgenstein in his attempt to explicate the foundations of mathematics through an analysis of its practice. Waismann argues in favor of mathematical intuition and the reality of infinity with a Wittgensteinian twist. Waismann's arguments lead toward an approach to the foundation of mathematics that takes into consideration the language and practice of experts.
In classical logic, a contradiction allows one to derive every other sentence of the underlying language; paraconsistent logics came relatively recently to subvert this explosive principle, by allowing for the subsistence of contradictory yet non-trivial theories. Therefore our surprise to find Wittgenstein, already at the 1930s, in comments and lectures delivered on the foundations of mathematics, as well as in other writings, counseling a certain tolerance on what concerns the presence of contradictions in a mathematical system. ‘Contradiction. Why just (...) this spectre? This is really very suspicious.’ ( Philosophical Remarks III–56) In the last decades, several authors (e.g. Arrington, Hintikka, Van Heijenoort, Wright, Wrigley) have been digging into Wittgenstein’s rather non-standard standpoint on what concerns the interpretation and import of contradiction in logic and mathematics, and many other authors (e.g. da Costa, Goldstein, Granger, Marconi) have been investigating the possibility of taking Wittgenstein seriously as one of the early forerunners of paraconsistency. While many advances have been made on the first front, the second set of investigations has led almost exclusively to negative results: no, no operational proposal about the construction of a logic in which (some) contradictions are made inoffensive can be read from Wittgenstein’s philosophical work; in fact, it appears that the most one can find there is the exhortation for mathematicians to alter their attitude with respect to contradictions and to consistency proofs. The play is done, and one looks for a resume of the opera. This paper fills that blank, as a thorough investigation of the possible relations between Wittgenstein and paraconsistency. DOI:10.5007/1808-1711.2010v14n1p135. (shrink)