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 symposium contribution discusses the conception of ontology which is used in Hilary Putnam's book Ethics without Ontology.

This paper portrays the later Wittgenstein’s conception of contradictions and his therapeutic approach to them. I will focus on and give relevance to the Lectures on the Foundations of Mathematics, plus the Remarks on the Foundations of Mathematics. First, I will explain why Wittgenstein’s attitude towards contradictions is rooted in: a rejection of the debate about realism and anti-realism in mathematics; and Wittgenstein’s endorsement of logical pluralism. Then, I will explain Wittgenstein’s therapeutic approach towards contradictions, and why it means that (...) a contradiction is not a problem for logic and mathematics. Rather, contradictions are problematic when we do not know what to infer from them. Once a meaning is established through a new rule of inference, the contradiction becomes a usable expression like many others in our inferential apparatus. Thus, the apparent problem is dissolved. Finally, I will take three examples of dissolved contradictions from Wittgenstein to clarify further his notion. I will conclude considering why his position on contradictions led him to clash with Alan Turing, and whether the latter was convinced by the Wittgensteinian proposal. (shrink)

In the scholarship on Wittgenstein’s later philosophy of mathematics, the dominant interpretation is a theoretical one that ascribes to Wittgenstein some type of ‘ism’ such as radical verificationism or anti-realism. Essentially, he is supposed to provide a positive account of our mathematical practice based on some basic assertions. However, I claim that he should not be read in terms of any ‘ism’ but instead should be read as examining philosophical pictures in the sense of unclear conceptions. The contrast here is (...) that basic assertions that frame philosophical ‘isms’ are propositional such that they are subject to normal argumentative evaluation, while pictures in Wittgenstein’s sense are non-propositional—they lack a clear truth condition. They, therefore, need clarification rather than argumentation. In this paper, I provide a detailed analysis of Wittgenstein’s treatment of philosophical pictures with special focus on his argument on contradiction. I begin by explaining the problem with this trend of theoretical interpretation, taking Steve Gerrard’s otherwise excellent interpretation as a representative example and pointing out why it is problematic. Next, I will argue that those problems do not arise if we take Wittgenstein’s task as the clarification of philosophical pictures. I do this, first, by explaining Wittgenstein’s method using his argument concerning the Augustinian Picture in Philosophical Investigations and then pointing out that the same method can be identified in the crucial arguments in his philosophy of mathematics. Finally, in order to connect my interpretation with the current scholarship, I will explain the relation of my interpretation with those of New Wittgensteinian scholars. (shrink)

Taking its orientation from Peter Winch, this article critiques from a Wittgensteinian point of view some “theoreticist” tendencies within constructivism. At the heart of constructivism is the deeply Wittgensteinian idea that the world as we know and understand it is the product of human intelligence and interests. The usefulness of this idea can be vitiated by a failure to distinguish conceptual from empirical questions. I argue that such a failure characterises two influential constructivist theories, those of Ernst von Glasersfeld and (...) David Bloor. These are considered in turn. Both theories seek to give a general, causal account of knowledge: von Glasersfeld's in term of cognitive subjectivity, Bloor's in terms of social agreement. Ironically, given that both writers cite Wittgenstein as a source of theoretical inspiration, assumptions of both theories run counter to key Wittgensteinian arguments. To show that Wittgenstein's views offer no solace to the realist, the article closes with a brief consideration of John Searle's theory of knowledge. (shrink)

Neo-Fregeanism in the philosophy of mathematics consists of two main parts: the logicist thesis, that mathematics (or at least branches thereof, like arithmetic) all but reduce to logic, and the platonist thesis, that there are abstract, mathematical objects. I will here focus on the ontological thesis, platonism. Neo-Fregeanism has been widely discussed in recent years. Mostly the discussion has focused on issues specific to mathematics. I will here single out for special attention the view on ontology which underlies the neo-Fregeans’ (...) claims about mathematical objects, and discuss this view in a broader setting. (shrink)

In this dissertation on Hilary Putnam's philosophy, I investigate his development regarding meaning and necessity, in particular mathematical necessity. Putnam has been a leading American philosopher since the end of the 1950s, becoming famous in the 1960s within the school of analytic philosophy, associated in particular with the philosophy of science and the philosophy of language. Under the influence of W.V. Quine, Putnam challenged the logical positivism/empiricism that had become strong in America after World War II, with influential exponents such (...) as Rudolf Carnap and Hans Reichenbach. Putnam agreed with Quine that there are no absolute a priori truths. In particular, he was critical of the notion of truth by convention. Instead he developed a notion of relative a priori truth, that is, a notion of necessary truth with respect to a body of knowledge, or a conceptual scheme. Putnam's position on necessity has developed over the years and has always been connected to his important contributions to the philosophy of meaning. I study Hilary Putnam's development through an early phase of scientific realism, a middle phase of internal realism, and his later position of a natural or commonsense realism. I challenge some of Putnam’s ideas on mathematical necessity, although I have largely defended his views against some other contemporary major philosophers; for instance, I defend his conceptual relativism, his conceptual pluralism, as well as his analysis of the realism/anti-realism debate. (shrink)

Discutindo acerca das centenas de detecções de planetas extrassolares, que supostamente aconteceram desde 1989 e que ele considera (incorretamente) como instâncias de observações, Peter Kosso disse, justamente, que segundo os parâmetros de Bas van Fraassen esses objetos celestes seriam observáveis. Ora, tais astros poderiam sem dúvida ser observados diretamente (sem a necessidade de instrumentos), nas condições apropriadas. Mas, acrescenta Kosso, “esse tipo de epistemologia externalista, que permite que a justificação se baseie em informação que não temos a disposição (nós não (...) estamos em condição de ver planetas extrassolares a olho nu [atualmente]), não ajuda a decidir quais particulares afirmações científicas garantem a crença” (Kosso 2006, 225, nota 1). Van Fraassen provavelmente incorre em uma petição de princípio não muito diferente daquelas que ele muitas vezes desmascarou em textos de autores realistas, quando apela para o fato de o objeto detectado ser observável, para atribuir o estatuto de ‘observação’ a uma certa detecção. O ponto é, mais uma vez, entender o que vale como observação, particularmente quando isso envolve o uso de algum instrumento. Endossar aquilo que Otávio Bueno chama de ‘padrão internalista’ parece garantir solidez à qualificação de determinadas ações como observações. Uma das principais consequências dessa adoção é que tal critério, não operando nenhuma distinção relevante entre detecções ‘diretas’ e detecções mediadas por instrumentos, permite que a linha que separa observáveis e inobserváveis seja traçada de maneira diferente de como van Fraassen acredita que deveria ser, abarcando, entre os observáveis, mais (tipos de) entidades do que esse parece estar disposto a admitir. (shrink)