This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...) today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

Despite the otherwise-dominant trends towards physicalism and naturalism in philosophy, it has become increasingly common for metaphysicians to accept the existence either of modal facts and properties, or of Lewisian possible worlds. This paper raises the historical question: why did these heavyweight realist views come into prominence? The answer is that they have arisen in response to the demand to find truthmakers for our modal statements. But this demand presupposes that modal statements are descriptive claims in need of truthmakers. This (...) presupposition was, however, rejected by many earlier analytic philosophers, including the logical positivists, Wittgenstein, Ryle and Sellars, all of whom denied that modal statement were descriptive at all. Yet the non-descriptivist approach has largely fallen out of discussion and out of philosophical consciousness. In this paper I examine why non-descriptivist views first came into and then fell out of favor, and consider what the prospects are for reviving this more deflationary approach to modality. (shrink)

Among the many fundamental problems Wittgenstein discussed, two are especially relevant for evolutionary theory. The first one is the problem of negation and its relation to the intentionality of thought. Its resolution answers the question of how thought can anticipate reality though what is thought may not exist, and explains how empirical propositions are distinguishable from mathematical, logical, and conceptual (or what are traditionally called metaphysical) propositions. The second is the problem of the grounds of sensory experience. Wittgenstein’s resolution of (...) this problem is revolutionary since he showed that first-person, present-tense expressions (these are sentences by which we express what goes on in our mind) have no grounds at all, clarifying why we should abandon (all variants of) representative idealism. In combination with the resolution of the first problem, it enables us to develop new explanations about how during the course of development and evolution the mind evolves and evolved. (shrink)

There have recently appeared claims that the influence Heinrich Hertz exerted over Wittgenstein's later work was far more abiding than previously recognised. I critically evaluate such claims by Gordon Baker and Allan Janik. I first show that Hertz was indeed concerned with the same feature, clarity, which often exercised Wittgenstein. But I then argue that Wittgenstein should not be seen as having adopted the conception of philosophical method, which Hertz deployed in The Principles of Mechanics. I show that Hertz ‘clarifies’ (...) the concept of force only in the sense that he alters that concept, and that he is not using the sort of ‘contrastive’ methods characteristic of Wittgenstein's later works. (shrink)

Psychologism in logic is the doctrine that the semantic content of logical terms is in some way a feature of human psychology. We consider the historically influential version of the doctrine, Psychological Individualism, and the many counter-arguments to it. We then propose and assess various modifications to the doctrine that might allow it to avoid the classical objections. We call these Psychological Descriptivism, Teleological Cognitive Architecture, and Ideal Cognizers. These characterizations give some order to the wide range of modern views (...) that are seen as psychologistic because of one or another feature. Although these can avoid some of the classic objections to psychologism, some still hold. (shrink)

Cora Diamond has recently criticised as mere legend the interpretation of a quip of Ramsey's, contained in the epigraph below, which takes him to be objecting to or rejecting Wittgenstein's Tractarian distinction between saying and showing. Whilst I agree with Diamond's discussion of the legend, I argue that her interpretation of the quip has little evidential support, and runs foul of a criticism sometimes made against intuitionism. Rather than seeing Ramsey as making a claim about the nature of propositions, as (...) Diamond does, we should understand him as making a claim about the grammar of the logical connectives. Such a view coheres with the extant evidence of the nature of Wittgenstein's and Ramsey's 1929 philosophical encounters. It is also compatible with attributing to Ramsey a recognition of Wittgenstein's distinction and with denying that criticising it is the lesson of the quip. (shrink)

Wittgenstein's Tractatus carefully distinguished the concept all from\nthe notion of a truth-function, and thereby from the quantifiers.\nI argue that Wittgenstein's rationale for this distinction is lost\nunless propositional functions are understood within the context\nof his picture theory of the proposition. Using a model Tractatus\nlanguage, I show how there are two distinct forms of generality implicit\nin quantified Tractatus propositions. Although the explanation given\nin the Tractatus for this distinction is ultimately flawed, the distinction\nitself is a genuine one, and the forms of generality that (...) Wittgenstein\nindicated can be seen in the quantified sentences of contemporary\nlogic. (shrink)

In recent years, even some of its own practitioners have accused analytic philosophy of lacking historical awareness. My aim is to show that analytic philosophy and history are not such a mismatch after all. Against the objection that analytic philosophers have unduly ignored the past I argue that for the most part they only resist strong versions of historicism, and for good reasons. The history of philosophy is not the whole of philosophy, as extreme historicists maintain, nor is it indispensable (...) to substantive philosophizing, as mainline historicists have it, it is merely advantageous (pragmatic historicism). Against the objection that analytic histories of philosophy are inevitably anachronistic I argue that it is possible to approach past texts with a view to substantive issues and in a critical spirit (contrary to historicist relativism and to misguided interpretations of the principle of charity). Indeed, such an analytic approach makes not just for better philosophy but also for better history. (shrink)

This paper discusses Gordon Baker’s interpretation of the later Wittgenstein, in particular his interpretation of the notion of Wittgensteinian philosophical conceptions and the notions of non-exclusivity, local incompatibility, non-additivity and global pluralism which Baker uses to characterize Wittgensteinian conceptions. On the basis of this discussion, and a critique of certain features of Baker’s interpretation of Wittgensteinian conceptions, I introduce the notion of a multidimensional logical description of language use, explaining how this notion, which Baker’s interpretation excludes, constitutes and important element (...) of the later Wittgenstein’s philosophical method of clarification and perspicuous representation. I conclude by explaining how Baker’s problematic notions of local incompatibility and non-additivity, if they are seen in the light of Wittgenstein’s criticisms of certain views of the completeness of philosophical or logical accounts, nevertheless point in the right direction. (shrink)

The "standard account" of Wittgenstein’s relations with the Vienna Circle is that the early Wittgenstein was a principal source and inspiration for the Circle’s positivistic and scientific philosophy, while the later Wittgenstein was deeply opposed to the logical empiricist project of articulating a "scientific conception of the world." However, this telegraphic summary is at best only half-true and at worst deeply misleading. For it prevents us appreciating the fluidity and protean character of their philosophical dialogue. In retrospectively attributing clear-cut positions (...) to Wittgenstein and his interlocutors, it is very easy to read back our current understanding of familiar distinctions into a time when those terms were used in a much more open-ended way. The paper aims to to provide a broader perspective on this debate, starting from the protagonists’ understanding of their respective positions. Too often, the programmatic statements about the nature of their work that are repeated in manifestoes, introductions, and elementary textbooks have occupied center stage in the subsequent secondary literature. Consequently, I focus on a detailed examination of a turning point in their relationship. That turning point is Wittgenstein's charge, in the summer of 1932, that a recently published paper of Carnap's, "Physicalistic Language as the Universal Language of Science", made such extensive and unacknowledged use of Wittgenstein's own ideas that Wittgenstein would, as he put it in a letter to Schlick, "soon be in a situation where my own work shall be considered merely as a reheated version or plagiarism of Carnap’s." While the leading parties in this dispute shared a basic commitment to the primacy of physicalistic language, and the view that all significant languages are translatable, there was a remarkable lack of mutual understanding between them, and deep disagreement about the nature of the doctrines they disputed. Three quarters of a century later, we are so much more conscious of the differences that separated them than the points on which they agreed that it takes an effort of historical reconstruction to appreciate why Wittgenstein once feared that his own work would be regarded as a pale shadow of Carnap’s. (shrink)