Mathieu Marion

Waismann’s writings can be divided into three periods. The fi rst corresponds to his early work in Vienna under the aegis of Schlick, thus mainly to his collaboration with Wittgenstein on the fi rst drafts of Logik, Sprache, Philosophie, out of which came not only the book itself many years later but also transcriptions of conversations with Schlick and Wittgenstein and numerous dictations reworked by Waismann, now published under the title The Voice of Wittgenstein. The Vienna Circle. Waismann also did at that stage independent work, albeit largely infl uenced by Wittgenstein, on probability and identity. The second period runs roughly from the moment relations with Wittgenstein were severed – towards the end of 1934 – to his arrival in Oxford, where he started lecturing in Michaelmas Term 1939

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.

É fácil mostrar que, no Tractatus, atribuições de cor não podem ser proposiçõeselementares. Mas já não é tão fácil determinar que tipo de análise poderia ser feita dejuízos de percepção do tipo “a é vermelho”. Wittgenstein nos dá uma indicação vaga noaforismo 6.3751. Ele pede que o leitor tenha em mente o modo pelo qual lidamos com aexclusão das cores no campo da física. Mesmo assim, é difícil determinar o que exatamenteele estava tentando dizer ali. Ofereço uma interpretação do aforismo que associa o problemada exclusão mútua das cores ao tratamento dado aos números e à noção de “sériede formas” (Formenreihe). Algumas passagens das Philosophische Bemerkungen são usadasem apoio a esta linha interpretativa.

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.

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.

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.

Reuben Louis Goodstein (1912-1985) foi aluno de Wittgenstein em Cambridge de1931 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 umafunção definida por uma função recursiva. Esse último aspecto se encontra no Big Typescriptde Wittgenstein. Também mostro que as idéias fundamentais do cálculo equacionalpodem ser encontradas não apenas no período intermediário, mas, in nuce, nas observaçõessobre matemática do Tractatus Logico-philosophicus. A partir disso, procuro desenvolverum argumento contra uma leitura corrente daquele livro, o assim chamado “NovoWittgenstein”. Outra conexão entre Goodstein e Wittgenstein se encontra na rejeição dateoria da quantificação; na parte final do artigo, recorro às observações críticas de Goodsteinsobre a Lei do Terceiro Excluído (que também incluem uma crítica a Brouwer e à suarejeição “pela metade” dessa lei) para lançar luz sobre as observações do próprioWittgenstein a esse respeito.

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’

It is first argued that dialectic was a form of regimented debate, which grew out of public debates in Ancient Greece. A set of rules for dialectical bouts is then given and their meaning explained. The transition from oral to written arguments is briefly examined, leading to the formulation of a delimitation problem in Plato’s dialogues, as he inserted dialectical arguments within ordinary dialogue contexts, turning them into discussions where one of the participants reasons hypothetically to make the other realize that they are not entitled to their view. Doing so, Plato adjusted dialectic to a variety of dialogue purposes and in order to explore this variety, a study of the early tradition of classifying Plato’s dialogues in terms of their ‘character’ is suggested, the results of which are then compared with types of dialogues in contemporary Argumentation Theory.

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

En logique mathématique, on doit distinguer entre une conception « axiomatique »de la logique, qui fut celle de Frege, Russell et Hilbert, et une conception plus « pragmatique »en termes d’actes de preuves, que l’on retrouve dans les systèmes de déduction naturelle de Gentzen. Des parallèles sont esquissés entre la conception de l’inférence et de la logique dans le Tractatus Logico-philosophicus de Wittgenstein et celle de Gentzen. Ce cadre permet en outre de jeter un regard neuf sur l’argument de Wittgenstein sur « suivre une règle ».

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).