From A Mathematics Of Necessity To A Mathematics Of Convention

In Wittgenstein\u27s speculation the transition from the philosophical point of view expressed in the Tractatus logico-philosophicus to the later development of his thought, as it is reflected in the Philosophical investigations and On certainty, is relevant also to his conception of mathematics. In particular, while in the Tractatus, mathematics is not given an account of its own, independent of the account which is given to logic, the Remarks on the foundations of mathematics seems to offer sufficient evidence for the belief that, in the late stage of Wittgenstein\u27s speculation, the analysis of the foundations of mathematics received an explicit treatment on its own. This discussion is concerned with the attempt to illustrate the transformation which occurred in Wittgenstein\u27s way of conceiving mathematics; that is to say, with the passage from the original idea of mathematics as a discipline, on a par with logic itself, reflecting the unmodifiable, and therefore necessary, features of the logical structure of the world, to the more intuitionistic conception of mathematics, as a constructive activity, leading to merely conventional truths. The different connotations of the notion of truth, in relation to these two different conceptions of mathematics, are also considered

Hilary Putnam\u27s Consistency Objection against Wittgenstein\u27s Conventionalism in Mathematics

Hilary Putnam first published the consistency objection against Ludwig Wittgenstein’s account of mathematics in 1979. In 1983, Putnam and Benacerraf raised this objection against all conventionalist accounts of mathematics. I discuss the 1979 version and the scenario argument, which supports the key premise of the objection. The wide applicability of this objection is not apparent; I thus raise it against an imaginary axiomatic theory T similar to Peano arithmetic in all relevant aspects. I argue that a conventionalist can explain the consistency of T and suggest that an analogous explanation can be provided for the consistency of Peano arithmetic

On Frege’s Alleged Indispensability Argument

The expression ‘indispensability argument’ denotes a family of arguments for mathematical realism supported among others by Quine and Putnam. More and more often, Gottlob Frege is credited with being the first to state this argument in section 91 of the Grundgesetze der Arithmetik. Frege\u27s alleged indispensability argument is the subject of this essay. On the basis of three significant differences between Mark Colyvan\u27s indispensability arguments and Frege\u27s applicability argument, I deny that Frege presents an indispensability argument in that very often quoted section of the Grundegesetze

Psychological Continuity: A Discussion of Marc Slors’s Account, Traumatic Experience, and the Significance of Our Relations to Others

This paper addresses a question concerning psycho­logical continuity, i.e., which features preserve the same psychological subject over time; this is not the same question as the one concerning the necessary and sufficient conditions for personal identity. Marc Slors defends an account of psychological continuity that adds two features to Derek Parfit’s Relation R, namely narrativity and embodiment. Slors’s account is a significant improvement on Parfit’s, but still lacks an explicit acknowledgment of a third feature that I call relationality. Because they are usually regarded as cases of radical discontinuity, I start my discussion from the experiences of psychological disruption undergone by victims of severe violence and trauma. As it turns out, the challenges we encounter in granting continuity to the experiences of violence and trauma victims are germane to those we encounter in granting continuity to the experiences of subjects in non-traumatic contexts. What is missing in the most popular accounts of psychological continuity is an explicit acknowledgment of the links that tie our psychological lives to other subjects. A more persuasive notion of psychological continuity is not only embodied and narrative, as is Slors’s notion, but also explicitly relationa

Psychological Continuity: A Discussion of Marc Slors\u27 Account, Traumatic Experience, and the Significance of our Relations to Others

This paper addresses a question concerning psychological continuity, i.e., which features preserve the same psychological subject over time; this is not the same question as the one concerning the necessary and sufficient conditions for personal identity. Marc Slors (1998, 2001, 2001a) defends an account of psychological continuity that adds two features to Derek Parfit’s Relation R, namely narrativity and embodiment. Slors’ account is a significant improvement on Parfit’s, but still lacks an explicit acknowledgment of a third feature that I call relationality. Because they are usually regarded as cases of radical discontinuity, I start my discussion from the experiences of psychological disruption undergone by victims of severe violence and trauma. As it turns out, the challenges we encounter in granting continuity to the experiences of violence and trauma victims are germane to those we encounter in granting continuity to the experiences of subjects in non-traumatic contexts. What is missing in the most popular accounts of psychological continuity is an explicit acknowledgment of the links that tie our psychological lives to other subjects. A more persuasive notion of psychological continuity is not only embodied and narrative, as is Slors’ notion, but also explicitly relational

From A Mathematics Of Necessity To A Mathematics Of Convention

In Wittgenstein\u27s speculation the transition from the philosophical point of view expressed in the Tractatus logico-philosophicus to the later development of his thought, as it is reflected in the Philosophical investigations and On certainty, is relevant also to his conception of mathematics. In particular, while in the Tractatus, mathematics is not given an account of its own, independent of the account which is given to logic, the Remarks on the foundations of mathematics seems to offer sufficient evidence for the belief that, in the late stage of Wittgenstein\u27s speculation, the analysis of the foundations of mathematics received an explicit treatment on its own. This discussion is concerned with the attempt to illustrate the transformation which occurred in Wittgenstein\u27s way of conceiving mathematics; that is to say, with the passage from the original idea of mathematics as a discipline, on a par with logic itself, reflecting the unmodifiable, and therefore necessary, features of the logical structure of the world, to the more intuitionistic conception of mathematics, as a constructive activity, leading to merely conventional truths. The different connotations of the notion of truth, in relation to these two different conceptions of mathematics, are also considered

Pieranna Garavaso Interview, 2019

Pieranna Garavaso, professor emeritus of philosophy, talks about her educational and career path that eventually led her to University of Minnesota, Morris. She talks in detail about sexism, feminism and homosexuality on campus. Pieranna also discusses the creation of the Gender, Women and Sexuality Studies major and her time as division chair of the Humanities.https://digitalcommons.morris.umn.edu/stories/1072/thumbnail.jp

Il destino di Karen. Università e donne nel Nord America (Karen’s Fate. Academe and Women in North America)

Una docente italiana all’Università del Minnesota porta la sua testimonianza sull’evoluzione della presenza femminile nel mondo accademico americano

OBJECTIVITY AND CONSISTENCY IN MATHEMATICS: A CRITICAL ANALYSIS OF TWO OBJECTIONS TO WITTGENSTEIN\u27S PRAGMATIC CONVENTIONALISM

Wittgenstein\u27s views on mathematics are radically original. He criticizes most of the traditional philosophies of mathematics. His views have been subject to harsh criticisms. In this dissertation, I attempt to defend Wittgenstein\u27s philosophy of mathematics from two objections: the objectivity objection and the consistency objection. The first claims that Wittgenstein\u27s account of mathematics is not sufficient for the objectivity of mathematics; the second claims that it is only a partial account of mathematics because it cannot explain the semantic properties of mathematical systems. The first chapter outlines Wittgenstein\u27s Philosophy of Mathematics by stressing the differences and similarities with more traditional accounts. The second chapter discusses the objectivity objection. I distinguish epistemic from non-epistemic objectivity and discuss their relation with the issue of realism and anti-realism. I conclude first that either notion of objectivity is insufficient for disparaging anti-realist accounts of mathematics and, second, that Wittgenstein\u27s account is sufficient for both. The third chapter is a detailed discussion of the consistency objection. Some unsuccessful replies are discussed. In the fourth chapter, I reformulate the objection by means of a rather simple axiomatic system that I call theory T. This allows us to see more clearly why the replies considered before are not successful and, furthermore, which view Wittgenstein is actually required to hold in order to reject the objection. In the fifth chapter, I offer a brief outline of Wittgenstein\u27s account of logic and, finally, show how it provides the required answer to the consistency objection