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Switch to: References
Citations of:
Wittgenstein on the Foundations of Mathematics
Cambridge, Mass.: Harvard University Press (1980)
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In “Truth by Convention” W.V. Quine gave an influential argument against logical conventionalism. Even today his argument is often taken to decisively refute logical conventionalism. Here I break Quine’s arguments into two— the super-task argument and the regress argument—and argue that while these arguments together refute implausible explicit versions of conventionalism, they cannot be successfully mounted against a more plausible implicit version of conventionalism. Unlike some of his modern followers, Quine himself recognized this, but argued that implicit conventionalism was explanatorily (...) |
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This article offers an explanation of perhaps Wittgenstein’s strangest and least intuitive thesis – the semantical mutation thesis – according to which one can never answer a mathematical conjecture because the new proof alters the very meanings of the terms involved in the original question. Instead of basing our justification on the distinction between mere calculation and proofs of isolated propositions, characteristic of Wittgenstein’s intermediary period, we generalize it to include conjectures involving effective procedures as well. |
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I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can (...) |
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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. |
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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 (...) |
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This paper offers a new interpretation for Wittgenstein`s treatment of mathematical identities. As it is widely known, Wittgenstein`s mature philosophy of mathematics includes a general rejection of abstract objects. On the other hand, the traditional interpretation of mathematical identities involves precisely the idea of a single abstract object – usually a number –named by both sides of an equation. |
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In 1982 Saul A. Kripke proposed a reconstruction of the central insights of Ludwig Wittgenstein's remarks on rule-following. The reconstruction prominently featured a sceptical challenge which soon was recognised as a new and very radical form of scepticism. According to the challenge there is no fact of the matter which constitutes meaning. As there is no such fact, the first-person authority people intuitively seem to have concerning what they mean is also baseless. In response to the sceptic, many solutions have (...) |
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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 (...) |
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Against a broad consensus within contemporary analytic philosophy, Hattiangadi (Mind and Language 21(2):220–240, 2006 , 2007 ) has recently argued that linguistic meaning is not normative, at least not in the sense of being prescriptive. She maintains, more specifically, that standard claims to the effect that meaning is normative are usually ambiguous between two readings: one, which she calls Prescriptivity , and another, which she calls Correctness . According to Hattiangadi, though meaning is normative in the uncontroversial sense specified in (...) |
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It is argued that the finitist interpretation of wittgenstein fails to take seriously his claim that philosophy is a descriptive activity. Wittgenstein's concentration on relatively simple mathematical examples is not to be explained in terms of finitism, But rather in terms of the fact that with them the central philosophical task of a clear 'ubersicht' of its subject matter is more tractable than with more complex mathematics. Other aspects of wittgenstein's philosophy of mathematics are touched on: his view that mathematical (...) |
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The two main challenges of the theory of conceptual content presented by Robert Brandom in Making It Explicit are to account for a referential dimension of conceptual content and to account for the objectivity of conceptual norms. Brandom tries to meet both these challenges in chapter 8 of his book. I argue that the accounts presented there can only be understood if seen against the background of Brandom's theory of communication developed in chapter 7. This theory is motivated by the (...) |
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This paper argues that most of the alleged straight solutions to the sceptical paradox which Kripke ascribed to Wittgenstein can be regarded as the first horn of a dilemma whose second horn is the paradox itself. The dilemma is proved to be a by‐product of a foundationalist assumption on the notion of justification, as applied to linguistic behaviour. It is maintained that the assumption is unnecessary and that the dilemma is therefore spurious. To this end, an alternative conception of the (...) |
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To explain an intentional action one must exhibit the agent’s reasons. Donald Davidson famously argued that the only clear way to understand action explanation is to hold that reasons are causes. Davidson’s discussion conflated two issues: whether reasons are causes and whether reasons causally explain intentional action. Contemporary work on explanation and normativity help disentangle these issues and ground an argument that intentional action explanations cannot be a species of causal explanation. Interestingly, this conclusion is consistent with Davidson’s conclusion that (...) |
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Alberto Coffa used the phrase "the Copernican turn in semantics" to denote a revolutionary transformation of philosophical views about the connection between the meanings of words and the acceptability of sentences and arguments containing those words. According to the new conception resulting from the Copernican turn, here called "the Copernican view", rules of use are constitutive of the meanings of words. This view has been linked with two doctrines: (A) the instances of meaning-constitutive rules are analytically and a priori true (...) |
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Normativism in the theory of meaning and content is the view that linguistic meaning and/or intentional content are essentially normative. As both normativity and its essentiality to meaning/content can be interpreted in a number of different ways, there is now a whole family of views laying claim to the slogan “meaning/content is normative”. In this essay, we discuss a number of central normativist theses, and we begin by identifying different versions of meaning normativism, presenting the arguments that have been put (...) |
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Ideas on meaning, rules and mathematical proofs abound in Wittgenstein’s writings. The undeniable fact that they are present together, sometimes intertwined in the same passage of Philosophical Investigations or Remarks on the Foundations of Mathematics, does not show, however, that the connection between these ideas is necessary or inextricable. The possibility remains, and ought to be checked, that they can be plausibly and consistently separated. I am going to examine two views detectable in Wittgenstein’s works: one about proofs, the other (...) |
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forthcoming in R. Cameron, B. Hale and A. Hoffmann (ed.s), The Logic, Epistemology and Metaphysics of Modality, Oxford University Press. Presents a concept-grounding account of modal knowledge. |
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The modal antirealist, as presented here, aims to secure at least some of the benefits associated with talking in genuine modal realist terms while avoiding commitment to a plurality of Lewisian (or ersatz) worlds. The antirealist stance of agnosticism about other worlds combines acceptance of Lewis's account of what world-talk means with refusal to assert, or believe in, the existence of other worlds. Agnosticism about other worlds does not entail a comprehensive agnosticism about modality, but where such agnosticism about modality (...) |
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The paper puts forward the thesis of conceptual pragmatism: that there are pragmatic choices to be made between distinct but similar concepts within various contexts. It is argued that this thesis should be acceptable to all who believe in concepts, whether the believers are platonists, realists or anti-realists. It is argued that the truth of the thesis may help to resolve many long-standing debates, and that in any case it will lead to an extension of philosophical method. The paper then (...) |
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This is a short, and therefore necessarily very incomplete discussion of one of the great questions of modern philosophy. I return to a station at which an interpretative train of thought of mine came to a halt in a paper written almost 20 years ago, about Wittgenstein and Chomsky,[1] hoping to advance a little bit further down the track. The rule-following passages in the Investigations and Remarks on the Foundations of Mathematics in fact raise a number of distinct issues about (...) |
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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 (...) |
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Many people claim that semantic content is normative, and that therefore naturalistic theories of content face a potentially insuperable difficulty. The normativity of content allegedly undermines naturalism by introducing a gap between semantic 'ought's and the explanatory resources of naturalism. I argue here that this problem is not ultimately pressing for naturalists. The normativity thesis, I maintain, is ambiguous; it could mean either that the content of a term prescribes a pattern of use, or that it merely determines which pattern (...) |
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This paper presents a new argument to defend the normativity of meaning, specifically the thesis that there are no meanings without norms. The argument starts from the observation inferentialists have emphasized that incompatibility relations between sentences are a necessary part of meaning as it is understood. We motivate this approach by showing that the standard normativist strategy in the literature, which is developed in terms of veridical reference that may swing free from the speaker’s understanding, violates the ought-implies-can principle, but (...) |
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Brandom (Citation1994) made inferentialism an intensely debated idea in the philosophy of language in the last three decades. Inferentialism is a view that associates the meaning of linguistic expression with the role said expression plays in inferences. It seems rather uncontroversial that the correct theory of meaning should distinguish between linguistic correctness and factual correctness. For instance, speaker S can be wrong in saying ‘I have arthritis’ in two distinct ways: (i) S fails to apply a word correctly to make (...) |
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The main question of metasemantics, or foundational semantics, is why an expression token has the meaning (semantic value) that it in fact has. In his reading of Ludwig Wittgenstein’s later work, Saul Kripke presented a skeptical challenge that threatened to make the foundational question unanswerable. My first contention in this paper is that the skeptical challenge indeed poses an insoluble paradox, but only for a certain kind of metasemantic theory, against which the challenge effectively works as a reductio ad absurdum (...) |
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The extensions of Goodman’s ‘grue’ predicate and Kripke’s ‘quus’ are constructed from the extensions of more familiar terms via a reinterpretation that permutes assignments of reference. Since this manoeuvre is at the heart of Putnam’s model-theoretic and permutation arguments against metaphysical realism (‘Putnam’s Paradox’), both Goodman’s New Riddle of Induction and the paradox about meaning that Kripke attributes to Wittgenstein are instances of Putnam’s. Evidence cannot selectively confirm the green-hypothesis and disconfirm the grue-hypothesis, because the theory of which the green-hypothesis (...) |
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This paper addresses a problem about aesthetic normativity raised by Kant. Can aesthetic experiences be appropriate or inappropriate to their objects? And, if so, how is that possible given that, according to Kant, aesthetic experience is not objective? Kant thought the answer to the first question was yes. But his official answer to the second question, in terms of the free play of the faculties, is obscure. The paper offers a clearer answer, inspired by Kant, which invokes Wittgenstein’s notion of (...) |
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Since the publication of the Remarks on the Foundations of Mathematics, Wittgenstein’s interpreters have endeavored to reconcile his general constructivist/anti-realist attitude towards mathematics with his confessed anti-revisionary philosophy. In this article, the author revisits the issue and presents a solution. The basic idea consists in exploring the fact that the so-called “non-constructive results” could be interpreted so that they do not appear non-constructive at all. The author substantiates this solution by showing how the translation of mathematical results, given by the (...) |
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What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) |
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This project is an investigation into the prospects for an antirealist theory of essence. Essentialism is the claim that at least some things have some of their properties essentially. Essentialist discourse includes claims such as “Socrates is essentially human”, and “Socrates is accidentally bearded”. Historically, there are two ways of interpreting essentialist discourse. I call these positions ‘modal essentialism’ and ‘neo-Aristotelian essentialism’. According to modal essentialism, for Socrates to be essentially human is for it to be necessary that he be (...) |
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Famously, Kant is a transcendental idealist. Yet he also endorses empirical realism, and even boasts that only the transcendental idealist can be an empirical realist. The difficulty of making sense of those commitments together leads many interpreters to begin by attributing to Kant some variant of conventional, subjective idealism. That in turn requires that Kant's empirical realism be at best a merely ersatz or quasi-realism. But that drains Kant's boast of its significance. For any idealist can be a realist if (...) |
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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 (...) |
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The Four-Colour Theorem (4CT) proof, presented to the mathematical community in a pair of papers by Appel and Haken in the late 1970's, provoked a series of philosophical debates. Many conceptual points of these disputes still require some elucidation. After a brief presentation of the main ideas of Appel and Haken’s procedure for the proof and a reconstruction of Thomas Tymoczko’s argument for the novelty of 4CT’s proof, we shall formulate some questions regarding the connections between the points raised by (...) |
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The paper discusses an objection, put forward by - among others - John McDowell, to Kripke’s Wittgenstein’s non-factualist and relativist view of semantic discourse. The objection goes roughly as follows: while it is usually possible to be a relativist about a given domain of discourse without being a relativist about anything else, relativism about semantic discourse entails global relativism, which in turn entails subjective idealism, which we can reasonably assume to be false. The paper’s first section sketches Kripke’s Wittgenstein’s ideas (...) |
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The dissertation attempts to answer the question of how to theorize resistance from within the philosophy of social science. To answer this question we must consider more than just the philosophy of social science; we also must look to political and moral philosophy. Resistance to the social norms of one’s community is possible to theorize from within the philosophy of social science once we develop a sufficiently nuanced account of social and moral communities, according to which membership in a community (...) |
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Is there a plausible middle position in the debate between realists and constructivists about categories or kinds? Such a position may seem to be contained in the account of triangulation that Donald Davidson develops in recent writings. On this account, the kinds we pick out are determined by an interaction between our shared similarity responses and causal relations between us and things in our environment. So kinds and categories are neither imposed on us by the nature of the world, nor (...) |
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In the scholarship on Wittgenstein’s later philosophy of mathematics, the dominant interpretation is a theoretical one that ascribes to Wittgenstein some type of ‘ism’ such as radical verificationism or anti-realism. Essentially, he is supposed to provide a positive account of our mathematical practice based on some basic assertions. However, I claim that he should not be read in terms of any ‘ism’ but instead should be read as examining philosophical pictures in the sense of unclear conceptions. The contrast here is (...) |
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In “Correspondence to Reality in Ethics”, Mario Brandhorst examines the view of ethics that Wittgenstein took in his later years. According to Brandhorst, Wittgenstein leaves room for truth and falsity, facts, correspondence and reality in ethics. Wittgenstein's target, argues Brandhorst, is objectivity. I argue that Brandhorst's arguments in favour of truth, facts, reality and correspondence in ethics invite similar arguments in favour of objectivity, that Brandhorst does not recognise this because his conception of objectivity is distorted by a Platonist picture (...) |
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In its most general form, the issue of intentionality takes the following shape: How can something be about something else? In basic cases, this relation is one between a subjective occurrence and a state of affairs, allowing the problem of intentionality to be articulated in this manner: What makes it the case that a subjective occurrence has the capacity to be about something external to it? The views of John McDowell on intentionality form the focus of this article. They are (...) |
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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 (...) |
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Explores the stable core of Wittgenstein's philosophy as developed from the Tractatus to the Philosophical Investigations. |
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Kalhat has forcefully criticised Wittgenstein's linguistic or conventionalist account of logical necessity, drawing partly on Waismann and Quine. I defend conventionalism against the charge that it cannot do justice to the truth of necessary propositions, renders them unacceptably arbitrary or reduces them to metalingustic statements. At the same time, I try to reconcile Wittgenstein's claim that necessary propositions are constitutive of meaning with the logical positivists' claim that they are true by virtue of meaning. Explaining necessary propositions by reference to (...) |
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In this paper I argue that there can be genuine (as opposed to merely verbal) disputes about whether a sentence form is logically true or an argument form is valid. I call such disputes ?cases of deviance?, of which I distinguish a weak and a strong form. Weak deviance holds if one disputant is right and the other wrong, but the available evidence is insufficient to determine which is which. Strong deviance holds if there is no fact of the matter. (...) |
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The German physicist Heinrich Hertz played a decisive role for Wittgenstein's use of a unique philosophical method. Wittgenstein applied this method successfully to critical problems in logic and mathematics throughout his life. Logical paradoxes and foundational problems including those of mathematics were seen as pseudo-problems requiring clarity instead of solution. In effect, Wittgenstein's controversial response to David Hilbert and Kurt Gödel was deeply influenced by Hertz and can only be fully understood when seen in this context. To comprehend the arguments (...) |
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