ABSTRACT In a recent essay, Sören Stenlund tries to align Wittgenstein’s approach to the foundations and nature of mathematics with the tradition of symbolic mathematics. The characterization of symbolic mathematics made by Stenlund, according to which mathematics is logically separated from its external applications, brings it closer to the formalist position. This raises naturally the question whether Wittgenstein holds a formalist position in philosophy of mathematics. The aim of this paper is to give a negative answer to this question, defending (...) the view that Wittgenstein always thought that there is no logical separation between mathematics and its applications. I will focus on Wittgenstein’s remarks about arithmetic during his middle period, because it is in this period that a formalist reading of his writings is most tempting. I will show how his idea of autonomy of arithmetic is not to be compared with the formalist idea of autonomy, according to which a calculus is “cut off” from its applications. The autonomy of arithmetic, according to Wittgenstein, guarantees its own applicability, thus providing its own raison d’être. RESUMO Em um recente artigo, Sören Stenlund procura alinhar a abordagem de Wittgenstein em relação aos fundamentos e à natureza da matemática com a tradição da matemática simbólica. A caracterização da matemática simbólica feita por Stenlund, de acordo com a qual a matemática é logicamente separada de suas aplicações externas, a aproxima da posição formalista. Isto naturalmente levanta a questão de se Wittgenstein defende uma posição formalista em filosofia da matemática. O objetivo deste artigo é dar uma resposta negativa a esta questão, ao defender que Wittgenstein sempre pensou não haver separação lógica entre a matemática e suas aplicações. Atenção especial será dada às observações de Wittgenstein sobre a aritmética pertencentes ao seu período intermediário, pois é neste período que uma leitura formalista de seus escritos é mais sedutora. Mostrarei como sua ideia de autonomia da aritmética não deve ser comparada à ideia formalista de autonomia, segundo a qual um cálculo é “cortado” de suas aplicações. A autonomia da aritmética, para Wittgenstein, garante ela própria sua aplicabilidade, provendo, assim, sua própria raison d’être. (shrink)

It is often held that Wittgenstein had to introduce numbers in elementary propositions due to problems related to the so-called colour-exclusion problem. I argue in this paper that he had other reasons for introducing them, reasons that arise from an investigation of the continuity of visual space and what Wittgenstein refers to as ‘intensional infinity’. In addition, I argue that the introduction of numbers by this route was prior to introducing them _via_ the colour-exclusion problem. To conclude, I discuss two (...) problems that Wittgenstein faced in the writings before _Some Remarks on Logical Form _, problems that are independent of the colour-exclusion problem but dependent on the introduction of numbers in elementary propositions. (shrink)

In her book Deviant Logic, Susan Haack argued for a “pragmatist” conception of logic. This conception holds that, logic is a theory on a par with other scientific theories, differing only from such theories by its degree of generality and the choice of a particular logic is to be made based on pragmatist principles, namely, economy, coherence, and simplicity. This view was contrasted, in this book, with an “absolutist” view of logic, according to which logical laws are necessary and immune (...) to revision. Two decades later, however, Haack acknowledged, in the Introduction to an enlarged version of the same book, that she would not approach the question of the revisability of logic in the same way she did earlier. What was missing in her first book was a distinction between the question of the necessity of the laws of logic and the question of our fallibility in recognizing which are the true laws of logic and what structures are essential to representation. She also acknowledged that this change was mainly influenced by Peirce, with whose work she had “only the most superficial acquaintance twenty years ago”. In this context, this paper has two aims: to show that, in Philosophy of Logics, we can find elements that reveal a tension between her early “pragmatic” views and her changing views on the nature of logic; to present some hypotheses about the role Peirce may have had in this change. (shrink)