In his Remarks on the Foundations of Mathematics, Wittgenstein claims, puzzlingly, that ‘the proof creates a new concept’ (RFM III-41). This paper aims to contribute to clarifying this idea, and to showing how it marks a major break with the traditional conception of proof. Moreover, since the most natural way to understand his claim is open to criticism, a secondary goal of what follows is to offer an interpretation of it that neutralizes the objection. The discussion proceeds by analysing a (...) well-known geometrical proof. (shrink)

In this paper, I take issue with a core commitment of logical conventionalism: that we impose a logic on ourselves by adopting general linguistic conventions governing our use of logical terms, thereby determining the meanings of the logical constants and which of our inferences are valid. Drawing on Kripke’s ‘adoption problem’, I argue that general logical principles cannot be adopted, either explicitly or implicitly. I go on to argue that the meanings of our logical terms, and the validity of our (...) inferences, cannot depend on our adoption of logico-linguistic conventions. (shrink)

This work explores the later Wittgenstein’s philosophy of mathematics in relation to Lakatos’ philosophy of mathematics and the philosophy of mathematical practice. I argue that, while the philosophy of mathematical practice typically identifies Lakatos as its earliest of predecessors, the later Wittgenstein already developed key ideas for this community a few decades before. However, for a variety of reasons, most of this work on philosophy of mathematics has gone relatively unnoticed. Some of these ideas and their significance as precursors for (...) the philosophy of mathematical practice will be presented here, including a brief reconstruction of Lakatos’ considerations on Euler’s conjecture for polyhedra from the lens of late Wittgensteinian philosophy. Overall, this article aims to challenge the received view of the history of the philosophy of mathematical practice and inspire further work in this community drawing from Wittgenstein’s late philosophy. (shrink)

Conventionalism about mathematics has much in common with two other views: fictionalism and the multiverse view (aka plenitudinous platonism). The three views may differ over the existence of mathematical objects, but they agree in rejecting a certain kind of objectivity claim about mathematics, advocating instead an extreme pluralism. The early parts of the paper will try to elucidate this anti‐objectivist position, and question whether conventionalism really offers a third form of it distinct from fictionalism and the multiverse view. The paper (...) then turns to anti‐objectivism about logic, and suggests that here too conventionalism offers no distinct alternative, and that there are limits on the extent of pluralism/anti‐objectivism. It also argues that these limits can't be used to argue for more extensive objectivity within mathematics, and also that they do allow for more limited pluralism within logic, e.g. as regards resolution of semantic and property‐theoretic paradoxes. (shrink)

Rudolf Carnap’s principle of tolerance states that there is no need to justify the adoption of a logic by philosophical means. Carnap uses the freedom provided by this principle in his philosophy of mathematics: he wants to capture the idea that mathematical truth is a matter of linguistic rules by relying on a strong metalanguage with infinitary inference rules. In this paper, I give a new interpretation of an argument by E. W. Beth, which shows that the principle of tolerance (...) does not suffice to remove all obstacles to the employment of infinitary rules. (shrink)

Establishing Dispositionalism as a viable theory of modality requires the successful fulfilment of two tasks: showing that all modal truths can be derived from truths about actual powers, and offering a suitable metaphysics of powers. These two tasks are intertwined: difficulties in one can affect the chances of success in the other. In this paper, I generalise an objection to Dispositionalism by Jessica Leech and argue that the theory in its present form is ill-suited to account for de re truths (...) about merely possible entities. I argue that such difficulty is rooted in a problem in the metaphysics of powers. In particular, I contend that the well-known tension between two key principle of powers ontology, namely Directedness and Independence has received an unsatisfactory solution so far, and that it is this unsatisfactory solution concerning the status of “unmanifested manifestations” that makes it hard for Dispositionalism to account for mere possibilia. I develop a novel account of the status of unmanifested manifestations and an overall metaphysics of powers which allows to better respond to Leech's objection and handle mere possibilia. The central idea of the proposal is that unmanifested manifestations are akin to mere logical existents, and are best characterised as non-essentially non-located entities. (shrink)

In this paper, I offer a close reading of Wittgenstein's remarks on inconsistency, mostly as they appear in the Lectures on the Foundations of Mathematics. I focus especially on an objection to Wittgenstein's view given by Alan Turing, who attended the lectures, the so-called ‘falling bridges’-objection. Wittgenstein's position is that if contradictions arise in some practice of language, they are not necessarily fatal to that practice nor necessitate a revision of that practice. If we then assume that we have adopted (...) a paraconsistent logic, Wittgenstein's answer to Turing is that if we run into trouble building our bridge, it is either because we have made a calculation mistake or our calculus does not actually describe the phenomenon it is intended to model. The possibility of either kind of error is not particular to contradictions nor inconsistency, and thus contradictions do not have any special status as a thing to be avoided. (shrink)

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 (...) a human if he exists, and for Socrates to accidentally have a beard is for it to be contingent that Socrates has a beard if he exists. According to neo-Aristotelian essentialism, objects have definitions in something like the way words do. For Socrates to be essentially human but accidentally bearded is for it to be part of the definition of Socrates that he is human, but not part of that definition that he is bearded. I argue that both are susceptible to antirealist interpretation. This thesis sets about showing that this is the case. In Chapters One and Two I investigate neo-conventionalist theories of modality, in the hope of using such a position to develop an antirealist modal essentialism. In Chapter Three I discuss the debate between modal and neo-Aristotelian essentialism and conclude that it is by no means settled. In Chapter Four I develop an antirealist neo-Aristotelian essentialism based on the mechanism of one of the neo-conventionalist accounts of modality. In Chapter Five I argue that this account is in a better position to give an essentialist theory of necessity than its realist counterparts. I conclude that, regardless of whether one is a modal or neo-Aristotelian essentialist, antirealist essentialism is a viable theory of essence that is worthy of consideration in contemporary debate. (shrink)

This dissertation is a discussion of Wittgenstein's later philosophy. In it, Wittgenstein's answer to the "going on problem" will be presented: I will give his reply to the skeptic who denies that rule-following is possible. Chapter One will describe this problem. Chapter Two will give Wittgenstein's answer to it. Chapter Three will show how Wittgenstein used this answer to give the standards of mathematics. Chapter Four will compare Wittgenstein's answer to the going on problem to Plato's. Chapter Five will describe (...) Kant's influence on Wittgenstein's later philosophy. The leitmotif of these chapters will be that the way human beings apply a rule is the source of justification for its application. 1. (shrink)

Additional theorizing about mathematical practice is needed in order to ground appeals to truly useful notions of the virtues in mathematics. This paper aims to contribute to this theorizing, first, by characterizing mathematical practice as being epistemic and “objectual” in the sense of Knorr Cetina The practice turn in contemporary theory, Routledge, London, 2001). Then, it elaborates a MacIntyrean framework for extracting conceptions of the virtues related to mathematical practice so understood. Finally, it makes the case that Wittgenstein’s methodology for (...) examining mathematics and its practice is the most appropriate one to use for the actual investigation of mathematical practice within this MacIntyrean framework. At each stage of thinking through mathematical practice by these means, places where new virtue-theoretic questions are opened up for investigation are noted and briefly explored. (shrink)

This paper argues that the problematic of rule following in Wittgenstein's Philosophical Investigations and Heidegger's analysis of anxiety in Being and Time have analogous structures. Working through these analogies helps our interpretation of both of these authors. Contrasting sceptical and anti-sceptical readings of Wittgenstein helps us to resolve an interpretive puzzle about what an authentic response to anxiety looks like for Heidegger. And considering the importance of anxiety to Heidegger's conception of authenticity allows us to locate in Wittgenstein's later philosophy (...) a covert appeal to something resembling Heideggerian authenticity. (shrink)

In Necessary Beings, Bob Hale addresses two questions: What is the source of necessity? What is the source of our knowledge of it? He offers novel responses to them in terms of the metaphysical notion of nature or, more familiarly, essence. In this paper, I address Hale’s response to the first question. My assessment is negative. I argue that his essentialist explanation of the source of necessity suffers from three significant shortcomings. First, Hale’s leading example of an essentialist explanation merely (...) asserts that the nature of an entity explains some necessity, but leaves unexplained how it does so. Second, his essentialist explanation of particular necessities introduces new necessities that remain unexplained. Third, Hale’s version of essentialism presupposes a controversial metaphysical theory of properties, for which he offers no defense. (shrink)

Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on rule-following skepticism. We argue, with the help of C. S. Peirce's distinction between corollarial and theorematic proofs, that his intuitions are better explained by resistance to what we call conceptual omniscience, treating meaning as fixed content specified in advance. We interpret the distinction in the context of modern epistemic logic (...) and semantic information theory, and show how removing conceptual omniscience helps resolve Wittgenstein's paradoxes and explain the puzzle of deduction, its ability to generate new knowledge and meaning. (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)

My main aim is to sketch a certain reading (‘genealogical’) of later Wittgenstein’s views on logical necessity. Along the way, I engage with the inferentialism currently debated in the literature on the epistemology of deductive logic.

'Reason, Causation and Compatibility with the Phenomena' strives to give answers to the philosophical problem of the interplay between realism, explanation and experience. This book is a compilation of essays that recollect significant conceptions of rival terms such as determinism and freedom, reason and appearance, power and knowledge. This title discusses the progress made in epistemology and natural philosophy, especially the steps that led from the ancient theory of atomism to the modern quantum theory, and from mathematization to analytic philosophy. (...) Moreover, it provides possible gateways from modern deadlocks of theory either through approaches to consciousness or through historical critique of intellectual authorities. This work will be of interest to those either researching or studying in colleges and universities, especially in the departments of philosophy, history of science, philosophy of science, philosophy of physics and quantum mechanics, history of ideas and culture. Greek and Latin Literature students and instructors may also find this book to be both a fascinating and valuable point of reference. (shrink)

The article argues that modal concepts should be explained in terms of the essences or nature of things: necessarily p if, and because, there is something the nature of which ensures that p; possibly p if, and because, there is nothing whose nature rules out its being true that p. The theory is defended against various objections and difficulties, including ones arising from attributing essences to contingent individuals.

Here I defend dispositionalism about meaning and rule-following from Kripkenstein's infamous anti-dispositionalist arguments. The problems of finitude, error, and normativity are all addressed. The general lesson I draw is that Kripkenstein's arguments trade on an overly simplistic version of dispositionalism.

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 (...) and that he misinterprets the passage which he takes to support a Wittgensteinian case against objectivity. (shrink)

This essay accounts for the notion of _Lebensform_ by assigning it a _logical _role in Wittgenstein’s later philosophy. Wittgenstein’s additions of the notion to his manuscripts of the _PI_ occurred during the initial drafting of the book 1936-7, after he abandoned his effort to revise _The Brown Book_. It is argued that this constituted a substantive step forward in his attitude toward the notion of simplicity as it figures within the notion of logical analysis. Next, a reconstruction of his later (...) remarks on _Lebensformen_ is offered which factors in his reading of Alan Turing’s “On computable numbers, with an application to the _Entscheidungsproblem_“, as well as his discussions with Turing 1937-1939. An interpretation of the five occurrences of _Lebensform_ in the PI is then given in terms of a logical “regression” to _Lebensform_ as a fundamental notion. This regression characterizes Wittgenstein’s mature answer to the question, “What is the nature of the logical?”. (shrink)

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 (...) idle. Against this I show that pace Quine’s claim that implicit conventionalism has no content beyond the claim that logic is firmly accepted, implicit rules of inference can be used to distinguish the firmly accepted from the conventional. As part of my case, I argue that positing syntactic rules of inference as part of our linguistic competence follows from the same methodology that leads contemporary linguists and cognitive scientists to posit rules of phonology, morphology, and grammar. The upshot of my discussion is a diagnosis of the fallacy in Quine’s master critique of logical conventionalism and a re-opening of possibilities for an attractive conventionalist theory of logic. (shrink)

Michael Dummett presents a modus tollens argument against a Wittgensteinian conception of meaning. In a series of papers, Dummett claims that Wittgensteinian considerations entail strict finitism. However, by a “sorites argument”, Dummett argues that strict finitism is incoherent and therefore questions these Wittgensteinian considerations.In this paper, I will argue that Dummett’s sorites argument fails to undermine strict finitism. I will claim that the argument is based on two questionable assumptions regarding some strict finitist sets of natural numbers. It will be (...) shown that strict finitism entails none of these assumptions. Hence, the argument does not prove that the view is internally incoherent, and consequently, Dummett fails to undermine the Wittgensteinian conception of meaning. (shrink)

In this paper I examine Crispin Wright's modal anti-realism as based on the availability of a certain attitude of Caution towards judgements of necessity. I think that Wright's account should be attractive in several ways for modal theorists with an anti-realist bend. However, the attitude of Caution to which it appeals has attracted some controversy. Wright himself has later come to doubt whether Caution is ultimately coherent. Here I first address Wright's worries concerning the coherence of Caution and show that (...) they are unfounded. But then I argue that although the attitude of Caution is coherent, it cannot provide a suitable basis for a non-eliminativist account of necessity. I offer two different objections against Caution. I argue that Wright's appeal to Caution, if successful, would show not only that modal judgement is non-objective but also that it is dispensable. Thus, I claim that appeal to Caution would seem to serve more as a threat against a non-eliminativist account of necessity, rather than as a potential adequate basis for it. However, I argue that Wright's appeal to Caution is unsuccessful, for there is no genuine Caution: Caution is a mere verbal attitude. (shrink)

La tesis de la privacidad linguitica nace con el gesto fundador de la filosofía moderna que apoya toda legitimidad en la subjetividad y la conciencia. Ello da origen a dos problemas filos�ficos fundamentales, concernientes al mundo y al solipsismo. El siglo XX creyó encontrar en el lenguaje una salida a estos problemas. Wittgenstein es allí una pieza clave. Sin embargo las interpretaciones más influyentes de Wittgenstein enfocaron la crítica del lenguaje privado de tal modo que la salida debía permanecer en (...) el terreno epistemológico de las garantías o criterio de corrección (Malcolm, Fogelin, Kenny, Canfield, Tugendhat). Sostenemos que la interpretación propuesta por Kripke se apoya un un agudo diagnóstico de la fuente de estas dificultades, dándose cuenta que la refutación del lenguaje privado requería abandonar toda una concepción, la del realismo, que afectaban por igual al lenguaje privado y al público. Su argumentación, centrada en la semántica, irradia consecuencias más allá de la misma. Su "solución escéptica" al establecer que los juicios de corrección, entendidos como básicos y primitivos, sólo poseen sentido en el marco de una comunidad uniforme en sus juicios, coloca al discurso filosófico en un nuevo terreno. (shrink)

Aesthetic non-cognitivists deny that aesthetic statements express genuinely aesthetic beliefs and instead hold that they work primarily to express something non-cognitive, such as attitudes of approval or disapproval, or desire. Non-cognitivists deny that aesthetic statements express aesthetic beliefs because they deny that there are aesthetic features in the world for aesthetic beliefs to represent. Their assumption, shared by scientists and theorists of mind alike, was that language-users possess cognitive mechanisms with which to objectively grasp abstract rules fixed independently of human (...) responses, and that cognizers are thereby capable of grasping rules for the correct application of aesthetic concepts without relying on evaluation or enculturation. However, in this article I use Wittgenstein’s rule-following considerations to argue that psychological theories grounded upon this so-called objective model of rule-following fail to adequately account for concept acquisition and mastery. I argue that this is because linguistic enculturation, and the perceptual learning that’s often involved, influences and enables the mastery of aesthetic concepts. I argue that part of what’s involved in speaking aesthetically is to belong to a cultural practice of making sense of things aesthetically, and that it’s within a socio-linguistic community, and that community’s practices, that such aesthetic sense can be made intelligible. (shrink)

Michael Della Rocca has recently argued that Kripkean essentialism is subtly self-defeating: to defend it, certain modal intuitions must be reconstrued in terms of similarity, but reconstruing them in this way threatens the principled rejection of similarity comparisons on which Kripke's essentialism depends. Della Rocca holds that Kripke's strategy must assume the necessity of identity, and that the necessity of identity already presupposes essentialism, which renders the defence circular. Against this, I argue that the necessity of identity may be accepted (...) independently; therefore no circularity need arise. I also argue that Della Rocca fails to rebut an objection raised by Stephen Yablo. (shrink)

The relation between Wittgenstein's philosophy of mathematics and mathematical Intuitionism has raised a considerable debate. My attempt is to analyse if there is a commitment in Wittgenstein to themes characteristic of the intuitionist movement in Mathematics and if that commitment is one important strain that runs through his Remarks on the foundations of mathematics. The intuitionistic themes to analyse in his philosophy of mathematics are: firstly, his attacks on the unrestricted use of the Law of Excluded Middle; secondly, his distrust (...) of non-constructive proofs; and thirdly, his impatience with the idea that mathematics stands in need of a foundation. These elements are Fogelin's starting point for the systematic reconstruction of Wittgenstein's conception of mathematics. (shrink)

An attempt is made to show that Wittgenstein's later philosophy of logic is not the kind of conventionalism which is often ascribed to him. On the contrary, Wittgenstein gives expression to a “mixed” theory which is not only interesting but tends to resolve the perplexities usually associated with the question of the a priori character of logical truth. I try to show that Wittgenstein is better understood not as denying that there are such things as “logical rules” nor as denying (...) that the results of applying such rules are “logically necessary,” but as trying to understand what it is to appeal to a logical rule and what it means to say that the results of applying such a rule are “necessary.” He is not so much overthrowing standard accounts of logical necessity as discovering the limits of the concept. (shrink)

Many contemporary philosophers rate error theories poorly. We identify the arguments these philosophers invoke, and expose their deficiencies. We thereby show that the prospects for error theory have been systematically underestimated. By undermining general arguments against all error theories, we leave it open whether any more particular arguments against particular error theories are more successful. The merits of error theories need to be settled on a case-by-case basis: there is no good general argument against error theories.

This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...) program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals. (shrink)

In his last papers about deontic logic, von Wright sustained that there is no genuine logic of norms. We argue in this paper that this striking statement by the father of deontic logic should not be understood as a death sentence to the subject. Rather, it indicates a profound change in von Wright's understanding about the epistemic and ontological role of logic in the field of norms. Instead of a logical constructivism of deontic systems revealing a necessary structure of prescriptive (...) discourse, which marked his earlier efforts, he adopted the view that such systems should be seem as mere objects of comparison, i.e. as providing practical standards of rationality for normgiving activity. Within such view he proposed an interpretation of standard deontic logic in such a way to free deontic logicians from the philosophical difficulties related to the so-called Jørgensen's dilemma and deontic paradoxes. This effort, as we claim in the present paper, is an application of Wittgenstein's therapeutic method to dissolve philosophical difficulties caused by the use of logical tools to model relations between norms. (shrink)

In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of free (...) variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgenstein's criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgenstein's remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum. (shrink)

The problem of the source of necessity is the problem of explaining what makes necessary truths necessarily true. Simon Blackburn has presented a dilemma intended to show that any reductive, realist account of the source of necessity is bound to fail. Although Blackburn's dilemma faces serious problems, reflection on the form of explanations of necessities reveals that a revised dilemma succeeds in defeating any reductive account of the source of necessity. The lesson is that necessity is metaphysically primitive and irreducible.

The concept of a rule that primarily interests Wittgenstein is one that is central to our understanding of 'what is possible before all new discoveries and inventions'. Wittgenstein's investigations of the concept of a rule run 'criss‐cross in every direction'. As Wittgenstein points out, 'any interpretation still hangs in the air along with what it interprets, and cannot give it any support'. Like our inner mental picture of a cube, what we think of as an interpretation of a rule doesn't (...) by itself predetermine the steps that accord with the rule it supposedly interprets. In his great work, Philosophical Investigations, Wittgenstein aims both to uncover the sources of these difficulties and to show his readers how to overcome them. In particular, he aims to expose the emptiness of tempting philosophical explanations of the notion of grounds for correct continuation of a rule. (shrink)

Ludwig Wittgenstein in his later works often implies commitment to a doctrine of the autonomy or arbitrariness of grammar. This chapter discusses the conception of grammar that is presupposed in this doctrine and then explains the doctrine itself. The chapter also explains a sense in which grammar is not autonomous or arbitrary for Wittgenstein and discusses some possible criticisms of the doctrine. It should be noted at the outset that this whole area of exegetical concern is one in which the (...) fairly widespread conception in the secondary literature that the later Wittgenstein is a complete “quietist” in philosophy ‐ i.e., someone who advances no positive doctrines of his own, although encouraged by some of his own statements, seems impossible to sustain. Wittgenstein's doctrine of the autonomy or arbitrariness of grammar presupposes a highly distinctive conception of grammar, one which at least prima facie has little to do with our everyday conception of grammar. (shrink)

Wittgenstein's conception of logical and conceptual truth is often thought to rival that of the logical positivists. This paper argues that there are important respects in which these conceptions complement each other. Analyticity, in the positivists' sense, coincides, not with Wittgenstein's notion of a grammatical proposition, but rather with his notion of a tautology. Grammatical propositions can usually be construed as analyticity postulates in Carnap's sense of the term. This account of grammatical and analytic propositions will be illustrated by appeal (...) to logical, conceptual and arithmetic truths. Its consequences for an analysis of corresponding modal notions will be indicated. (shrink)

This paper argues that there are three reasons why we should regard Wittgenstein's Tractatus as a forerunner of formal semantics: Wittgenstein is convinced that we can apply formal notions to natur...

This paper portrays the later Wittgenstein’s conception of contradictions and his therapeutic approach to them. I will focus on and give relevance to the Lectures on the Foundations of Mathematics, plus the Remarks on the Foundations of Mathematics. First, I will explain why Wittgenstein’s attitude towards contradictions is rooted in: a rejection of the debate about realism and anti-realism in mathematics; and Wittgenstein’s endorsement of logical pluralism. Then, I will explain Wittgenstein’s therapeutic approach towards contradictions, and why it means that (...) a contradiction is not a problem for logic and mathematics. Rather, contradictions are problematic when we do not know what to infer from them. Once a meaning is established through a new rule of inference, the contradiction becomes a usable expression like many others in our inferential apparatus. Thus, the apparent problem is dissolved. Finally, I will take three examples of dissolved contradictions from Wittgenstein to clarify further his notion. I will conclude considering why his position on contradictions led him to clash with Alan Turing, and whether the latter was convinced by the Wittgensteinian proposal. (shrink)

According to some scholars, such as Rodych and Steiner, Wittgenstein objects to Gödel’s undecidability proof of his formula $$G$$, arguing that given a proof of $$G$$, one could relinquish the meta-mathematical interpretation of $$G$$ instead of relinquishing the assumption that Principia Mathematica is correct. Most scholars agree that such an objection, be it Wittgenstein’s or not, rests on an inadequate understanding of Gödel’s proof. In this paper, I argue that there is a possible reading of such an objection that is, (...) in fact, reasonable and related to Gödel’s proof. (shrink)

This book presents an in-depth interpretation of three important parts of Leibniz's metaphysics: the metaphysical part of Leibniz's philosophy of logic, his essentially theological treatment of the central issues of ontology, and his theory of substance.

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

Tractatus, 5.62 famously says: ‘… what the solipsist means is quite correct; only it cannot be said but makes itself manifest. The world is my world: this is manifest in the fact that the limits of language mean the limits of my world.’ The later part of this repeats what was said in summary at 5.6: ‘the limits of my language mean the limits of my world’. And the key to the problem ‘how much truth there is in solipsism’ has (...) been provided by the reflections of TLP, 5.61. (shrink)

Sebastian Grève interviews Felix Mühlhölzer on his work on the philosophy of mathematics.