According to Wittgenstein, mathematics is embedded in, and partly constituting, a form of life. Hence, to imagine different, alternative forms of elementary mathematics, we should have to imagine different practices, different forms of life in which they could play a role. If we tried to imagine a radically different arithmetic we should think either of a strange world or of people acting and responding in very peculiar ways. If such was their practice, a calculus expressing the norms of representation they (...) applied could not be called false. Rather, our criticism could only be to dismiss such a practice as foolish and to dismiss their norms as too different from ours to be called ‘mathematics’. (shrink)

The aim of this paper is to discover whether or not a solitary individual, a human being isolated from birth, could become a rule-follower. The argumentation against this possibility rests on the claim that such an isolate could not become aware of a normative standard, with which her actions could agree or disagree. As a consequence, theorists impressed by this argumentation adopt a view on which the normativity of rules arises from corrective practices in which agents engage in a community. (...) However, it has been suggested that an isolated individual could engage in such a practice by herself. Three prospective examples of such cases are considered, and the possibility of solitary rule-following is vindicated. Furthermore, the nature of the goals at which rule-following practices generally aim is clarified. (shrink)

In a famous passage from the Philosophical Investigations, Wittgenstein describes a pupil who has been learning to write out various sequences of numbers in response to orders such as “+1” and “+2”. He has shown himself competent for numbers up to 1000, but when we have him continue the “+2” sequence beyond 1000, he writes the numerals 1004, 1008, 1012. As Wittgenstein describes the case: We say to him, “Look what you’re doing!” — He doesn’t understand us. We say “You (...) should have added two; look how you began the series!” — He answers: “Yes! Isn’t it right? I thought that’s how I should [sollen] do it. — Or suppose he were to say, pointing to the series, “But I went on in the same way!”1The passage continues:— It would now be no use to us to say “But don’t you see…?” — and repeat for him the old explanations and examples. — In such a case, we might perhaps say: this person naturally understands our order, once given our explanations, as we would understand the order “Add 2 up to 1000, 4 up to 2000, 6 up to 3000, and so on.”. (shrink)

The later Wittgenstein’s perspective on mathematical sentences as norms is motivated for sentences belonging to Hilbertian axiomatic systems where the axioms are treated as implicit definitions. It is shown that in this approach the axioms are employed as norms in that they function as standards of what counts as using the concepts involved. This normative dimension of their mode of use, it is argued, is inherited by the theorems derived from them. Having been motivated along these lines, Wittgenstein’s perspective on (...) mathematical language may appeal also to those who are not friends of or experts on Wittgenstein’s later philosophy of mathematics. (shrink)

There is a basic division in the philosophy of mathematics between realist, ‘platonist’ theories and anti-realist ‘constructivist’ theories. Platonism explains how mathematical truth is strongly objective, but it does this at the cost of invoking mind-independent mathematical objects. In contrast, constructivism avoids mind-independent mathematical objects, but the cost tends to be a weakened conception of mathematical truth. Neither alternative seems ideal. The purpose of this paper is to show that in the philosophical writings of Henri Poincaré there is a coherent (...) argument for an interesting position between the two traditional poles in the philosophy of mathematics. Relying on a semi-Kantian framework, Poincaré combines an epistemological and metaphysical constructivism with a more realist account of the nature of mathematical truth. (shrink)

Many now accept the thesis that norms are somehow constitutively involved in people's contentful intentional states. I distinguish three versions of this normative thesis that disagree about the type of norms constitutively involved. Are they objective norms of correctness, subjective norms of rationality, or intersubjective norms of social practices? I show the advantages of the third version, arguing that it improves upon the other two versions, as well as incorporating their principal insights. I then defend it against two serious challenges: (...) (1) If content is constituted by others' normative judgments, how can content be causally efficacious? (2) This account appears to make having contentful thoughts a matter of people having contentful thoughts about your thoughts. That appears to be viciously circular and so can't be naturalistic. (shrink)

Mark Jay Steiner, a brilliant and influential philosopher of mathematics, whose interests and accomplishments extended beyond that field as well, passed away on.

In his _Being Known_ Peacocke sets himself the task of answering how we come to know about metaphysical necessities. He proposes a semantic principle-based conception consisting of, first, his Principles of Possibility which pro­vide necessary and sufficient conditions for a new concept 'admissibility', and second, characterizations of possibility and of necessity in terms of that new con­cept. I focus on one structural feature; viz. the recursive application involved in the specification of 'admissibility'. After sketching Peacocke’s proposal, I intro­duce a fictional (...) protagonist, Cautious Peacocke, whose coherence I claim shows that Peacocke s proposal cannot be made good. I conclude with the conjecture that similar failure will attend any such semantic-based attempts to ground the epistemology of metaphysical necessity. -/- Dans son livre intitulé Being Known, C. Peacocke se propose de repondre à la question de savoir comment nous en venons a connaitre des nécessités métaphysiques. Il développe une conception sémantique reposant sur des principes comprenant, d’une part, ses Principes de Possibilité — qui fournissent les conditions nécessaires et suffisantes pour un nouveau concept, « l’admissibilité » — et d’autre part des caractérisations de la possibilité et de la nécessité a l’aide de ce nouveau concept. Je me concentre sur une caractéristique structurelle, a savoir l’application récursive a l’œuvre dans la spécification de «l’admissibilité ». Apres avoir esquisse la proposition de Peacocke, j’introduis un personnage fictif, Peacocke-prudent. Je soutiens que la cohérence de ce personnage montre que la proposition de Peacocke ne peut pas être satisfaite. Je conclus en conjecturant qu’un échec similaire se présentera pour toutes les tentatives de fonder l’épistemologie de la nécessité métaphysique sur de telles bases semantiques. (shrink)

Modality and Anti-Metaphysics critically examines the most prominent approaches to modality among analytic philosophers in the twentieth century, including essentialism. Defending both the project of metaphysics and the essentialist position that metaphysical modality is conceptually and ontologically primitive, Stephen McLeod argues that the logical positivists did not succeed in banishing metaphysical modality from their own theoretical apparatus and he offers an original defence of metaphysics against their advocacy of its elimination. -/- Seeking to assuage the sceptical worries which underlie modal (...) anti-realism, McLeod provides an original contribution to essentialist epistemology, engaging with current debates about modality and suggesting that standard essentialist approaches to some issues in the philosophies of logic and language require revision. -/- This book offers valuable insights to professional philosophers, postgraduates and advanced undergraduates interested in metaphysics, philosophy of logic or the history of twentieth-century analytic philosophy. (shrink)

outrageous remarks about contradictions. Perhaps the most striking remark he makes is that they are not false. This claim first appears in his early notebooks (Wittgenstein 1960, p.108). In the Tractatus, Wittgenstein argued that contradictions (like tautologies) are not statements (Sätze) and hence are not false (or true). This is a consequence of his theory that genuine statements are pictures.