According to structuralism in philosophy of mathematics, arithmetic is about a single structure. First-order theories are satisfied by models that do not instantiate this structure. Proponents of structuralism have put forward various accounts of how we succeed in fixing one single structure as the intended interpretation of our arithmetical language. We shall look at a proposal that involves Tennenbaum's theorem, which says that any model with addition and multiplication as recursive operations is isomorphic to the standard model of arithmetic. On (...) this account, the intended models of arithmetic are the notation systems with recursive operations on them satisfying the Peano axioms. [A]m Anfang […] ist das Zeichen. (shrink)

Review of G.P. Baker and P.M.S. Hacker's Wittgenstein: Rules, Grammar and Necessity, the second volume of their analytical commentary on Wittgenstein's Philosophical Investigations.

Tennenbaum's Theorem yields an elegant characterisation of the standard model of arithmetic. Several authors have recently claimed that this result has important philosophical consequences: in particular, it offers us a way of responding to model-theoretic worries about how we manage to grasp the standard model. We disagree. If there ever was such a problem about how we come to grasp the standard model, then Tennenbaum's Theorem does not help. We show this by examining a parallel argument, from a simpler model-theoretic (...) result. (shrink)

I discuss Skolem's own ideas on his ‘paradox’, some classical disputes between Skolemites and Antiskolemites, and the underlying notion of ‘informal mathematics’, from a point of view which I hope to be rather unusual. I argue that the Skolemite cannot maintain that from an absolute point of view everything is in fact denumerable; on the other hand, the Antiskolemite is left with the onus of explaining the notion of informal mathematical knowledge of the intended model of set theory. 1 conclude (...) that one must take seriously the embodiment of the uncountable into countable languages, as a symptom of some specific features which characterize mathematical languages with respect to other languages. (shrink)

This fourth and final volume of the monumental commentary on Wittgenstein's Philosophical Investigations covers pp 428-693 of the book. Like the previous volumes, it consists of philosophical essays and exegesis.

This book provides a novel interpretation of the ideas about language in Ludwig Wittgenstein's Philosophical Investigations. Travis places the "private language argument" in the context of wider themes in the Investigations, and thereby develops a picture of what it is for words to bear the meaning they do. He elaborates two versions of a private language argument, and shows the consequences of these for current trends in the philosophical theory of meaning.

This is the third volume of Hilary Putnam's philosophical papers, published in paperback for the first time. The volume contains his major essays from 1975 to 1982, which reveal a large shift in emphasis in the 'realist'_position developed in his earlier work. While not renouncing those views, Professor Putnam has continued to explore their epistemological consequences and conceptual history. He now, crucially, sees theories of truth and of meaning that derive from a firm notion of reference as inadequate.

This volume collects the most substantial correspondence and documents relating to Wittgenstein's long association with Cambridge between the years 1911 and his death in 1951, including the letters he exchanged with his most illustrious Cambridge contemporaries Russell, Keynes, Moore, and Ramsey. Now expanded to include 200 previously unpublished letters and documents, including correspondence between Wittgenstein and the economist Piero Sraffa, and between Wittgenstein and his pupils Includes extensive editorial annotations Provides a fascinating and intimate insight into Wittgenstein's life and thought.

This volume collects the most substantial correspondence and documents relating to Wittgenstein's long association with Cambridge between the years 1911 and his death in 1951, including the letters he exchanged with his most illustrious Cambridge contemporaries Russell, Keynes, Moore, and Ramsey. Now expanded to include 200 previously unpublished letters and documents, including correspondence between Wittgenstein and the economist Piero Sraffa, and between Wittgenstein and his pupils Includes extensive editorial annotations Provides a fascinating and intimate insight into Wittgenstein's life and thought.

Offers a text of mathematical logic on a sophisticated level, presenting the reader with several of the most significant discoveries, including the independence of the continuum hypothesis, the Diophantine nature of enumerable sets and the impossibility of finding an algorithmic solution for certain problems.