This pioneering book demonstrates the crucial importance of Wittgenstein's philosophy 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.

The development of symbolic logic is often presented in terms of a cumulative story of consecutive innovations that led to what is known as modern logic. This narrative hides the difficulties that this new logic faced at first, which shaped its history. Indeed, negative reactions to the emergence of the new logic in the second half of the nineteenth century were numerous and we study here one case, namely logic at Oxford, where one finds Lewis Carroll, a mathematical teacher who (...) promoted symbolic logic, and John Cook Wilson, the Wykeham Professor of Logic who notoriously opposed it. An analysis of their disputes on the topic of logical symbolism shows that their opposition was not as sharp as it might look at first, as Cook Wilson was not so much opposed to the « symbolic » character of logic, but the intrusion of mathematics and what he perceived to be the futility of some of its problems, for logicians and philosophers alike. (shrink)

Frank Plumpton Ramsey (1903–30) made seminal contributions to philosophy, mathematics and economics. Whilst he was acknowledged as a genius by his contemporaries, some of his most important ideas were not appreciated until decades later; now better appreciated, they continue to bear an influence upon contemporary philosophy. His historic significance was to usher in a new phase of analytic philosophy, which initially built upon the logical atomist doctrines of Bertrand Russell and Ludwig Wittgenstein, raising their ideas to a new level of (...) sophistication, but ultimately he became their successor rather than remain a mere acolyte. (shrink)

In this paper, we provide a detailed critical review of current approaches to ecthesis in Aristotle’s Prior Analytics, with a view to motivate a new approach, which builds upon previous work by Marion & Rückert (2016) on the dictum de omni. This approach sets Aristotle’s work within the context of dialectic and uses Lorenzen’s dialogical logic, hereby reframed with use of Martin-Löf's constructive type theory as ‘immanent reasoning’. We then provide rules of syllogistic for the latter, and provide proofs of (...) e-conversion, Darapti and Bocardo and e-subalternation, while showing how close to Aristotle’s text these proofs remain. (shrink)

In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of free (...) variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgenstein's criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgenstein's remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum. (shrink)

John Cook Wilson (1849–1915) was Wykeham Professor of Logic at New College, Oxford and the founder of ‘Oxford Realism’, a philosophical movement that flourished at Oxford during the first decades of the 20th century. Although trained as a classicist and a mathematician, his most important contribution was to the theory of knowledge, where he argued that knowledge is factive and not definable in terms of belief, and he criticized ‘hybrid’ and ‘externalist’ accounts. He also argued for direct realism in perception, (...) criticizing both empiricism and idealism, and argued for a moderate nominalist view of universals as being in rebus and only ‘apprehended’ by their particulars. His influence helped swaying Oxford away from idealism and, through figures such as H. A. Prichard, Gilbert Ryle, or J. L. Austin, his ideas were also to some extent at the origin of ‘moral intuitionism’ and ‘ordinary language philosophy’ which defined much of Oxford philosophy until the second half of the twentieth-century. Nevertheless, his name and legacy were all but forgotten for generations after World War II. Still, his views on knowledge are with us today, being in part at work in the writings of philosophers as diverse as John McDowell, Charles Travis, and Timothy Williamson. (shrink)

In this paper, I examine the transmission of some ideas of the pragmatist tradition to Wittgenstein, in his ‘middle period,’ through the intermediary of F. P. Ramsey, with whom he had numerous fruitful discussions at Cambridge in 1929. I argue more specifically that one must first come to terms with Ramsey’s own views in 1929, and explain how they differ from views expressed in earlier papers from 1925-27, so a large part of this paper is devoted to this task. One (...) is then in a better position to understand the impact of Ramsey’s astute critique of Wittgenstein’s Tractatus Logico-Philosophicus in conjunction with his pragmatism, and explain how it may have set into motion the ‘later’ Wittgenstein. I then argue that Ramsey introduced his notion of ‘variable hypothetical’ as a rule, not a proposition, on pragmatist grounds and that Wittgenstein picked this up in 1929, along with a more ‘dynamic’ view of meaning than the ‘static’ view of the Tractatus, and that this explains in part Wittgenstein’s turn to his ‘later philosophy.’. (shrink)

After presenting the rules of Eleatic antilogic, i.e., dialectic, I argue that Zeno was a practitioner, and, on the basis of key passages from Plato’s Parmenides, that his paradoxes of divisibility and movement were notreductio ad absurdum, but simple derivation of impossibilities meant to ridicule Parmenides’ adversaries. Thus, Zeno did not try to prove that there is no motion, but simply derived this consequence from premises held by his opponents. I argue further that these paradoxes were devised, in accordance with (...) Eleatic antilogic, following a scheme that included hypotheses and their contradictories, within which the subject is to be treated both “in relation to itself,” and “in relation to other things”. (shrink)

This volume portrays the Polish or Lvov-Warsaw School, one of the most influential schools in analytic philosophy, which, as discussed in the thorough introduction, presented an alternative working picture of the unity of science.

According to the realist, the meaning of a declarative, non-indexical sentence is the condition under which it is true and the truth-condition of an undecidable sentence can obtain or fail to obtain independently of our capacity, even in principle, to recognize that it obtains or that fails to do so.1 In a series of papers, beginning with “Truth” in 1959, Michael Dummett challenged the position that the classical notion of truth-condition occupied as the central notion of a theory of meaning, (...) and proposed that it should be replaced by the anti-realist (and intuitionistic) notion of assertability-condition. Taken together with normalization results obtained by Dag Prawitz, Dummett’s work truly opened up the anti-realist challenge at the level of proof-theoretical semantics.2 There has been since numerous rejoinders from partisans of classical logic, which were at times met with by attempts at watering down the anti-realist challenge, e.g., by arguing that anti-realism does not necessarily entail the adoption of intuitionistic logic. Only a few anti-realists, such as Crispin Wright and Neil Tennant, tried to look instead in the other direction, towards a more radical version of anti-realism which would entail deeper revisions of classical logic than those recommended by intuitionists.3 In this paper, which is largely programmatic, we shall also argue in favour of a radical anti-realism which would be a genuine alternative to the traditional anti-realism of Dummett and Prawitz. The debate about anti-realism has by now more or less run out of breath and we wish to provide it with a new lease on life, by taking into account the profound changes that took place in proof theory during the intervening years. We have in mind in particular the considerable development within Gentzen-style proof theory of non-classical, substructural logics other than intuitionistic logic, which seriously opens up the possibility that anti-realism, when properly understood, might end up justifying another logic, and the development of closer links between proof theory and computational complexity theory that has renewed interest in a radical form of anti-realism, namely strict finitism. (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.

In this paper, I present a summary of the philosophical relationship betweenWittgenstein and Brouwer, taking as my point of departure Brouwer's lecture onMarch 10, 1928 in Vienna. I argue that Wittgenstein having at that stage not doneserious philosophical work for years, if one is to understand the impact of thatlecture on him, it is better to compare its content with the remarks on logics andmathematics in the Tractactus. I thus show that Wittgenstein's position, in theTractactus, was already quite close to (...) Brouwer's and that the points of divergence are the basis to Wittgenstein's later criticisms of intuitionism. Among the topics of comparison are the role of intuition in mathematics, rule following, choice sequences, the Law of Excluded Middle, and the primacy of arithmetic over logic. (shrink)

We will discuss a mathematical proof found in Wittgenstein’s Nachlass, a constructive version of Euler’s proof of the infinity of prime numbers. Although it does not amount to much, this proof allows us to see that Wittgenstein had at least some mathematical skills. At the very last, the proof shows that Wittgenstein was concerned with mathematical practice and it also gives further evidence in support of the claim that, after all, he held a constructivist stance, at least during the transitional (...) period of his thought (1929-33). (shrink)

These are commented on in this volume, which will also include a publication of new or previously scattered material and an overview of Waismann 's life.

The relation between logic and knowledge has been at the heart of a lively debate since the 1960s. On the one hand, the epistemic approaches based their formal arguments in the mathematics of Brouwer and intuitionistic logic. Following Michael Dummett, they started to call themselves `antirealists'. Others persisted with the formal background of the Frege-Tarski tradition, where Cantorian set theory is linked via model theory to classical logic. Jaakko Hintikka tried to unify both traditions by means of what is now (...) known as `explicit epistemic logic'. Under this view, epistemic contents are introduced into the object language as operators yielding propositions from propositions, rather than as metalogical constraints on the notion of inference. The Realism-Antirealism debate has thus had three players: classical logicians, intuitionists and explicit epistemic logicians. The editors of the present volume believe that in the age of Alternative Logics, where manifold developments in logic happen at a breathtaking pace, this debate should be revisited. Contributors to this volume happily took on this challenge and responded with new approaches to the debate from both the explicit and the implicit epistemic point of view. (shrink)

The thesis according to which the meaning of a mathematical sentence is given by its proof was held by both Wittgenstein and the intuitionists, following Heyting and Dummett. In this paper, we clarify the meaning of this thesis for Wittgenstein, showing how his position differs from that of the intuitionists. We show how the thesis originates in his thoughts, from the middle period, about proofs by induction, and we sketch his answers to a number of objections, including the idea that, (...) given the particular meaning he gives to this thesis, he cannot account for mathematical conjectures. We conclude by showing how his views find a favourable echo today in the paradigm of “proposition-as-type” and extensions of the Curry-Howard isomorphism from which this paradigm originates. (shrink)

Reuben Louis Goodstein (1912-1985) foi aluno de Wittgenstein em Cambridge de 1931 a 1934. Neste artigo, faço uma breve descrição de seu trabalho na lógica matemática, no qual se percebe a influência das idéias de Wittgenstein, inclusive a substituição, em seu cálculo equacional, da indução matemática por uma regra de unicidade de uma função definida por uma função recursiva. Esse último aspecto se encontra no Big Typescript de Wittgenstein. Também mostro que as idéias fundamentais do cálculo equacional podem ser encontradas (...) não apenas no período intermediário, mas, in nuce, nas observações sobre matemática do Tractatus Logico-philosophicus. A partir disso, procuro desenvolver um argumento contra uma leitura corrente daquele livro, o assim chamado “Novo Wittgenstein”. Outra conexão entre Goodstein e Wittgenstein se encontra na rejeição da teoria da quantificação; na parte final do artigo, recorro às observações críticas de Goodstein sobre a Lei do Terceiro Excluído (que também incluem uma crítica a Brouwer e à sua rejeição “pela metade” dessa lei) para lançar luz sobre as observações do próprio Wittgenstein a esse respeito. (shrink)

Dans ce texte, je pars de l’analyse intuitionniste de la vérité mathématique, « A est vrai si et seulement s’il existe une preuve de A » comme cas particulier de l’analyse de la vérité en termes de « vérifacteur », et je montre pourquoi Wittgenstein partageait celle-ci avec les intuitionnistes. Cependant, la notion de preuve à l’oeuvre dans cette analyse est, selon l’intuitionnisme, celle de la « preuve-comme-objet », et je montre par la suite, en interprétant son argument sur le (...) caractère « synoptique » des preuves, que Wittgenstein avait plutôt en tête une conception de la « preuve-comme-trace ».In this paper, I start with the intutionist analysis of mathematical truth, « A is true if and only if there exists a proof of A », as a particular case of the analysis of truth in terms of « truth-makers », and I show why Wittgenstein shared it with the intuitionists. However, the notion of proof at work in this analysis is, according to intuitionism, that of « proof-as-object », and I then show, with an interpretation of his argument on the « surveyability » of proofs, that, instead, Wittgenstein had in mind a notion of « proof-as-trace ». (shrink)

L’opinion est souvent exprimée que Bradley fut un des tout premiers critiques du psychologisme. Dans cet article, j’examine cette thèse en me penchant principalement sur ses Principles of Logic . Je définis le psychologisme au sens étroit comme une thèse portant sur les fondements de la logique, et le psychologisme au sens large comme une thèse plus générale en théorie de la connaissance pour montrer que Bradley a rejeté les deux, même s’il n’avait pas grand chose à dire sur la (...) version étroite. Sa critique de l’autre version est basée sur une distinction entre contenu psychologique et contenu logique, et sur sa défense de la thèse de l’idéalité du contenu logique, avant Frege et Husserl. Cependant, il tient encore à l’idée que le contenu logique provient de la perception. Après une brève présentation de ses critiques de la psychologie associationniste, je montre qu’il fait face à de véritables difficultés en essayant d’éviter de retomber dans le psychologisme en faisant appel à la distinction entre universel abstrait et universel concret. Je termine avec quelques remarques sur la place de Bradley dans l’histoire de la psychologie britannique.One often hears the opinion voiced that Bradley was an early critique of psychologism. In this paper, I investigate that claim, focussing on his Principles of Logic . I define psychologism in the narrow sense as a thesis pertaining to the foundations of logic, and psychologism in the wide sense as a more general thesis concerning the theory of knowledge, and show that Bradley rejected both, although he had little to say on the narrow version. His criticism of the wider version is based on his distinguishing between psychological and logical content and on his defence of the ideality of logical content, before Frege and Husserl. Nevertheless, he still hung to the idea that the latter harks back to ordinary perception. I then review briefly his criticisms of associationism in psychology, to show that he faced some difficulties in trying to avoid lapsing back into psychologism, with an appeal to a distinction between abstract and concrete universals. I conclude with some remarks on the palace of Bradley in the history of British psychology. (shrink)

Wittgenstein est mort en 1951 et on attend toujours une édition de ses œuvres complètes. Ce n'est qu'en 1994 que sont parus, accompagnés d'un volume d'introduction à l'ensemble du projet d'édition de la main du directeur de publication, Michael Nedo, les deux premiers d'une série de quinze volumes, les Wiener Ausgabe, qui reproduiront l'intégralité des écrits de Wittgenstein, de son retour à Cambridge en janvier 1929 à la première version du Big Typescript en 1933, avec index et concordances. D'après le (...) catalogue établi par Georg Henrik von Wright, il s'agit des articles suivants: MS 105 à 114 et 153 à 155, TS 208 à 218. Ces deux premiers volumes reproduisent les manuscrits écrits entre janvier ou février 1929 et l'été 1930, soit les MS 105, 106, 107 et 108. Ils constituent l'aboutissement d'une véritable saga entourant l'œuvre posthume de Wittgenstein, dont il vaut la peine de rapporter ici quelques-uns des faits marquants. (shrink)

Mathematics is one of the many domains where the adoption of a form of realism has traditionally been challenged. Behaviorism and phenomenalism, opposed to realism about, respectively, mental entities and the existence of material objects, are other well‐known examples. The realism debate, as initiated by Dummett's challenge, appears to have run its course, at least on its original terms, and difficulties have been raised with respect to both antirealist and anti‐antirealist readings of Wittgenstein within that debate. Realism about meaning did (...) not remain unchallenged, however, as successor debates took its place, surrounding Robert Brandom's “inferentialism” or Price's generalized form of quasi‐realism, called “pragmatism”. Brandom also espouses “non‐representationalism” and predictably made use of Wittgenstein, with a reading akin to Kripke's of the rule‐following argument. He was criticized by McDowell largely for failing to account for Wittgenstein's quietism. (shrink)

In 1956, a series of BBC radio talks was published in London under the title The Revolution in Philosophy . This short book included papers by prominent British philosophers of the day, such as Sir Alfred Ayer and Sir Peter Strawson, with an introduction by Gilbert Ryle. Although there is precious little in it concerning the precise nature of the ‘revolution’ alluded to in the title, it is quite clear that these lectures were meant to celebrate in an insular manner (...) the birth of ‘analytic philosophy’ at the hands of Moore, Russell, and Wittgenstein. The use of the word ‘revolution’ to describe the birth of ‘analytical philosophy’ and its rise to near complete dominance in the British universities thus carries with it connotations of a forcible substitution of one philosophical order by a new one that itself owed nothing to the past. To me, however, this talk of ‘revolution’ carries with it a false view of the history of British philosophy: it isn’t ‘historical’ at all, just ideological; it was introduced for a strategic purpose, namely that of encouraging new generations of students to avoid reading the work of earlier philosophers or becoming interested in the problems that concerned them. I shall sketch the broad outlines of a landscape that involves, when limited to the specific topic of the theory of knowledge, no radical break, where no entirely new order replaces an older one: the options, so to speak, and even some of the arguments are the same before and after the soi-disant ‘revolution’. (shrink)

This volume is a collation of original contributions from the key actors of a new trend in the contemporary theory of knowledge and belief, that we call “dynamic epistemology”. It brings the works of these researchers under a single umbrella by highlighting the coherence of their current themes, and by establishing connections between topics that, up until now, have been investigated independently. It also illustrates how the new analytical toolbox unveils questions about the theory of knowledge, belief, preference, action, and (...) rationality, in a number of central axes in dynamic epistemology: temporal, social, probabilistic and even deontic dynamics. (shrink)

By North-American standards, philosophy is not new in Quebec: the first men tion of philosophy lectures given by a Jesuit in the College de Quebec dates from 1665, and the oldest logic manuscript dates from 1679. In English-speaking universities such as McGill, philosophy began to be taught later, during the second half of the 19th century. The major influence on English-speaking philosophers was, at least initially, that of Scottish Empiricism. On the other hand, the strong influence of the Catholic Church (...) on French-Canadian society meant that the staff of the facultes of the French-speaking universities consisted, until recently, almost entirely of Thomist philosophers. There was accordingly little or no work in modem Formal Logic and Philosophy of Science and precious few contacts between the philosophical communities. In the late forties, Hugues Leblanc was a young student wanting to learn Formal Logic. He could not find anyone in Quebec to teach him and he went to study at Harvard University under the supervision of W. V. Quine. His best friend Maurice L' Abbe had left, a year earlier, for Princeton to study with Alonzo Church. After receiving his Ph. D from Harvard in 1948, Leblanc started his profes sional career at Bryn Mawr College, where he stayed until 1967. He then went to Temple University, where he taught until his retirement in 1992, serving as Chair of the Department of Philosophy from 1973 until 1979. (shrink)