Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or theorem (...) and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers. (shrink)

In recent years, philosophers have used expressions of Wittgenstein’s (e.g. “language-games,” “form of life,” and “family resemblance”) in attempts to conceive of the discipline of philosophy in a broad, open, and perhaps global way. These Wittgenstein-inspired approaches indicate an awareness of the importance of cultural and historical diversity for approaching philosophical questions. While some philosophers have taken inspiration from Wittgenstein in embracing contextualism in philosophical hermeneutics, Wittgenstein himself was more instrumental than contextual in his treatment of other philosophers; his focus (...) in his writings was on his own philosophical problems. Does this mean that Wittgensteinian philosophy is a poor resource after all for comparative, cross-cultural, or globally-engaged philosophy (i.e. if it is properly Wittgensteinian)? In this article, I examine the relevance of Wittgenstein to contextually-sensitive philosophy through studies of his conceptions of history and culture, his interest in Spengler’s philosophy of history, and recent scholarship by Hans-Johann Glock and Hans Sluga on the place of contextualism in Wittgenstein’s analysis of philosophical problems. Ultimately, this article advances the view that there are strong resources in Wittgenstein’s philosophy for those seeking a more globally-engaged approach to the field. (shrink)

While Garfinkel’s early work, captured in Studies in Ethnomethodology, has received a lot of attention and discussion, this has not been the case for his later work since the 1970s. In this paper, we critically examine the aims of Garfinkel’s later ethnomethodological studies of work programme and evaluate key ideas such as the ‘missing what’ in the sociology of work, ‘the unique adequacy requirements of methods’, and the notion of ‘hybrid studies’. We do so through a detailed engagement with a (...) study that has frequently been singled out as exemplary by Garfinkel for his studies of work programme, namely Livingston’s The Ethnomethodological Foundations of Mathematics. We show how Livingston uses the proof of Gödel’s Incompleteness Theorem as a way to exhibit the work involved in understanding mathematical proofs. We then discuss how Livingston uses this example to introduce a distinction between the written ‘proof account’ and the associated ‘lived work’ of working through that proof, which we argue allows Livingston to provide a powerful critique of a formalist understanding of the objectivity of mathematics. We then discuss three aspects that we find problematic in Livingston’s and Garfinkel’s claims about Livingston’s study. Firstly, we question whether written proofs are best conceived of as descriptions or accounts. Secondly, we interrogate whether an ethnomethodological study could teach mathematicians how to make discoveries. Thirdly, we throw doubt on Garfinkel’s claim that Livingston’s results are results in mathematics. We conclude this paper with a discussion of how Livingston’s study Gödel’s proof highlights key tensions in Garfinkel’s later work. Firstly, we argue that there exists an ambiguity in Garfinkel’s treatment of texts as ‘incompetent’. Secondly, we show that Garfinkel’s attempt to extend his idea of classical studies from sociology to other professions and disciplines is problematic. Finally, we question Garfinkel’s proposals to reorient ethnomethodological studies away from sociological audiences and ask whether ethnomethodological studies promise the delivery of a ‘large prize’ or, rather, provide something like ‘helpful therapy’. (shrink)

In the early-to-mid 1930s, Wittgenstein investigated solipsism via the philosophy of language. In this paper, I want to reopen Wittgenstein's ‘grammatical’ examination of solipsism.Wittgenstein begins by considering the thesis that only I can feel my pains. Whilst this thesis may tempt us towards solipsism, Wittgenstein points out that this temptation rests on a grammatical confusion concerning the phrase ‘my pains’. In Section 1, I unpack and vindicate his thinking. After discussing ‘my pains’, Wittgenstein makes his now famous suggestion that the (...) word ‘I’ has two distinct uses: a subject-use and an object-use. The purpose of Wittgenstein's suggestion has, however, been widely misunderstood. I unpack it in Section 2, explaining how the subject-use connects with a phenomenological language, and so again tempts us into solipsism. In Sections 3–4, I consider various stages of Wittgenstein's engagement with this kind of solipsism, culminating in a rejection of solipsism via reflections on private languages. (shrink)

I aim to present a solution to the apparent paradox of the Tractatus by means of a minimalist reading grounded in the idea that the correct logical symbolism alone “finally solves” in essentials the philosophical problems. I argue that although the sentences of the Tractatus are nonsensical, rules presented in its symbolism are not. The symbolism itself expresses only a priori rules of logic through schematic variables that do not say anything. I argue that this reading correctly expresses the ladder (...) structure of the book, and that it can account for Wittgenstein's critique of the Tractatus in Some Remarks on Logical Form. (shrink)

According to A-theories of time, the metaphysical ground of change and dynamicity is provided by a continuous shifting in which events are past, present and future (A-determinations). It is often claimed that these theories make better sense of our experience of dynamicity than their rival, the B-theories; according to the latter, dynamicity is grounded solely in the irreducible earlier-than relations (B-relations) which obtain between events or states of affairs. In this paper, I argue that the experience of time's dynamicity, on (...) the contrary, cannot be accounted for solely in terms of representations of irreducible A-determinations, because any representation which is adequate to ground these experiences must itself involve representation of irreducible B-relations, while it needs not involve representation of A-determinations. Even if, as a matter of contingent fact, our experiences of dynamicity consisted of representations of successions of A-determinations, what would account for them being experiences of dynamicity would be solely the B-theoretic relations of succession, rather than the irrelevant A-theoretic nature of the relata. (shrink)

Ordinary cases of object seeing involve the visual perception of space and spatial location. But does seeing an object require such spatial perception? An empirical challenge to the idea that it does comes from reflection upon Bálint's syndrome, for some suppose that in Bálint's syndrome subjects can see objects without seeing space or spatial location. In this article, I question whether the empirical evidence available to us adequately supports this understanding of Bálint's syndrome, and explain how the aforementioned empirical challenge (...) can be resisted. (shrink)

Kant, Wittgenstein, and Husserl all held that visual awareness of objects requires visual awareness of the space in which the objects are located. There is a lively debate in the literature on spatial perception whether this view is undermined by the results of experiments on a Balint’s syndrome patient, known as RM. I argue that neither of two recent interpretations of these results is able to explain RM’s apparent ability to experience motion. I outline some ways in which each interpretation (...) may respond to this challenge, and suggest which way of meeting the challenge is preferable. I conclude that RM retains some awareness of the larger space surrounding the objects he sees. (shrink)

My aim in this paper is twofold: to establish that Kierkegaard's so‐called theory of the leap strongly anticipates a line of argument that is central to Wittgenstein's so‐called rule‐following considerations; and to begin to show how Kierkegaard's work has fruitful contributions of its own to make to on‐going discussions about rules and rule‐following. The paper focuses throughout on the question of how, if at all, human rule‐following can be distinguished from behaviour that is merely mechanical or instinctual. I identify a (...) central line of argument in Wittgenstein that problematizes this question by establishing the basis our ability to follow rules in our spontaneous responses. I argue that Kierkegaard not only strongly anticipates this line of argument but also offers a distinctive story about what, at bottom, makes the difference between human rule‐following and behaviour that is merely mechanical or instinctual. In a word, on Kierkegaard's story, what makes the difference is: anxiety. (shrink)

Three kinds of knowledge usually recognised by epistemologists are identified and their relevance for curriculum design is discussed. These are: propositional knowledge, know-how and knowledge by acquaintance. The inferential nature of propositional knowledge is argued for and it is suggested that propositional knowledge in fact presupposes the ability to know how to make appropriate inferences within a body of knowledge, whether systematic or unsystematic. This thesis is developed along lines suggested in the earlier work of Paul Hirst. The different kinds (...) of know-how and their relationships are discussed and it is suggested that they occupy different places and different relationships in any curricular hierarchy. The changing role that knowledge by acquaintance plays within this hierarchy is also discussed. Implications of this account for the current National Curriculum and for curriculum design more generally are discussed, looking at History, Science and Design Technology as examples. (shrink)

The article examines Wittgenstein’s thought on the language of pain in first and third person. Relevant grammatical differences, according to the typical analytical method of this philosopher, are highlighted not only in relation to the two perspectives, but also regarding the use of cognitive verbs such as ‘feeling’ and ‘knowing’. The exam of many texts suggests some issues concerning the relationship between personal experiences, empathic grasping of other’s feelings and their conceptual translation. A brief comparison with some Thomas Aquinas’ texts (...) complements the proposal of the existence of non intentional knowledge. (shrink)

This paper offers a reconstruction of Wittgenstein's discussion on inductive proofs. A "algebraic version" of these indirect proofs is offered and contrasted with the usual ones in which an infinite sequence of modus pones is projected.

This article explains Wittgenstein's distinction between good, bad, and vacuous modern music which he introduced in a diary entry from January 27, 1931. I situate Wittgenstein's discussion in the context of Oswald Spengler's ideas concerning the decline of Western culture, which informed Wittgenstein's philosophical progress during his middle period, and I argue that the music theory of Heinrich Schenker, and Wittgenstein's critique thereof, served as an immediate link between Spengler's cultural pessimism and Wittgenstein's threefold distinction. I conclude that Wittgenstein's distinction (...) between bad and vacuous modern music is analogous to Schenker's distinction between the compositional fallacies of the progressive and the reactionary composers of his time. Concomitantly, Wittgenstein's philosophically problematic notion of good modern music transcended the conceptual framework of both Schenker and Spengler. In this context, I examine Wittgenstein's remarks on Gustav Mahler as well as his remark on the music of the future as monophony, which, I conclude, should be understood ultimately as an ellipsis of his much later view of musical meaning and intelligibility. (shrink)

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 (...) be thought of as a sort of logicism for the logical expressivist---a person who believes that the purpose of logical language is to make explicit commitments and entitlements that are implicit in ordinary practice. The thesis that mathematical statements are expression of norms is a kind of logicism, not because it says that mathematics can be reduced to logic, but because it says that mathematical statements play the same role as logical statements. ;I contrast my position with two sets of views, an empiricist view, which says that mathematical knowledge is acquired and justified through experience, and a cluster of nativist and apriorist views, which say that mathematical knowledge is either hardwired into the human brain, or justified a priori, or both. To develop the empiricist view, I look at the work of Kitcher and Mill, arguing that although their ideas can withstand the criticisms brought against empiricism by Frege and others, they cannot reply to a version of the critique brought by Wittgenstein in the Remarks on the Foundations of Mathematics. To develop the nativist and apriorist views, I look at the work of contemporary developmental psychologists, like Gelman and Gallistel and Karen Wynn, as well as the work of philosophers who advocate the existence of a mathematical intuition, such as Kant, Husserl, and Parsons. After clarifying the definitions of "innate" and "a priori," I argue that the mechanisms proposed by the nativists cannot bring knowledge, and the existence of the mechanisms proposed by the apriorists is not supported by the arguments they give. (shrink)

I argue that it is possible and useful for moral philosophy to provide surveyable representations of moral vocabulary. I proceed in four steps. First, I present two dominant interpretations of the concept “surveyable representation”. Second, I use these interpretations as a background against which I present my own interpretation. Third, I use my interpretation to support the claim that Wittgenstein’s “Lecture on Ethics” counts as an example of a surveyable representation. I conclude that, since the lecture qualifies as a surveyable (...) representation, it is possible to provide surveyable representations of moral vocabulary. Fourth, I argue that it is useful for contemporary moral philosophy to provide surveyable representations, because it may help to dissolve problems in current debates. I provide an example of such a debate, namely, the debate between cognitivists and non-cognivitists. (shrink)

Review of the translation of Ludwig Wittgenstein book: Observações Filosóficas. São Paulo: Edições Loyola, 2005. Tradução (do inglês) por Adail Sobral e Maria Stela Gonçalves.

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.

In this paper, I will defend the claim that there are three existence properties: the second-order property of being instantiated, a substantive first-order property (or better a group of such properties) and a formal, hence universal, first-order property. I will first try to show what these properties are and why we need all of them for ontological purposes. Moreover, I will try to show why a Meinong-like option that positively endorses both the former and the latter first-order property is the (...) correct view in ontology. Finally, I will add some methodological remarks as to why this debate has to be articulated from the point of view of reality, i.e., by speaking of properties, rather than from the point of view of language, i.e., by speaking of predicates (for such properties). (shrink)

Between April and November 1912, Bertrand Russell and Ludwig Wittgenstein were engaged in a joint philosophical program. Wittgenstein‘s meeting with Gottlob Frege in December 1912 led, however, to its dissolution – the joint program was abandoned. Section 2 of this paper outlines the key points of that program, identifying what Russell and Wittgenstein each contributed to it. The third section determines precisely those features of their collaborative work that Frege criticized. Finally, building upon the evidence developed in the preceding two (...) sections, section 4 recasts along previously undeveloped lines Wittgenstein‘s logical–philosophical discoveries in the two years following his encounter with Frege in 1912. The paper concludes, in section 5, with an overview of the dramatic consequences the Frege-Wittgenstein critique had for Russell‘s philosophical development. (shrink)

The unpleasantness of pain motivates action. Hence many philosophers have doubted that it can be accounted for purely in terms of pain’s possession of indicative representational content. Instead, they have explained it in terms of subjects’ inclinations to stop their pains, or in terms of pain’s imperative content. I claim that such “noncognitivist” accounts fail to accommodate unpleasant pain’s reason-giving force. What is needed, I argue, is a view on which pains are unpleasant, motivate, and provide reasons in virtue of (...) possessing content that is indeed indicative, but also, crucially, evaluative. (shrink)

Many philosophers have held that it is not possible to experience a spatial object, property, or relation except against the background of an intact awareness of a space that is somehow ‘absolute’. This paper challenges that claim, by analyzing in detail the case of a brain-damaged subject whose visual experiences seem to have violated this condition: spatial objects and properties were present in his visual experience, but space itself was not. I go on to suggest that phenomenological argumentation can give (...) us a kind of evidence about the nature of the mind even if this evidence is not absolutely incorrigible. (shrink)

The purpose of this paper is twofold: (1) to clarify Ludwig Wittgenstein’s thesis that colours possess logical structures, focusing on his ‘puzzle proposition’ that “there can be a bluish green but not a reddish green”, (2) to compare modeltheoretical and gametheoretical approaches to the colour exclusion problem. What is gained, then, is a new gametheoretical framework for the logic of ‘forbidden’ (e.g., reddish green and bluish yellow) colours. My larger aim is to discuss phenomenological principles of the demarcation of the (...) bounds of logic as formal ontology of abstract objects. (shrink)

This paper is more than a plea for Rhees’ reading of the work of Wittgenstein (particularly of On Certainty). My interest in Rhees’ interpretation lies on its resemblance with my own reading, on the one hand, and on its being (surprisingly) unmentioned by other interpreters, on the other. The two core aims of this paper focus on Rhees’ main ideas. First, I argue that although certain facts that are accepted beyond doubt belong to the method, which in turn is included (...) in grammar, this does not mean that these facts are expressions of rules of grammar. Second, I argue that grammar is not conditioned by a certain class of facts (i.e. general facts of nature), but a language-game is possible because we do not call in question certain facts (i.e. grammar is not conditioned by something like ontology). The point is that those facts that are not called in question are beyond truth and falsity, but this does not mean that these facts must be true. The logical role these facts (and the sentences used to express them) play in a language-game is not that of being true or false. Moreover, grammar itself constitutes what is meant by ‘object’, ‘fact’, or ‘general fact of nature’. (shrink)

The Abhidharma Buddhist revisionary metaphysics aims to provide an intellectually and morally preferred picture of the world that lacks a self. The first part of the paper claims that the Abhidharma ‘no-self’ view can be plausibly interpreted as a no-ownership view, according to which there is no locus or subject of experience and thus no owner of mental or bodily awarenesses. On this interpretation of the no-self view, the Abhidharma Buddhist metaphysicians are committed to denying the ownership of experiences, and (...) thereby apparently obliged to explain our purported experience of ownership. My experiences seemingly come with the sense that I am the one who is undergoing this experience. But is there a really an experience of ownership—namely, an experience of being a subject that underlies our sense of ownership? I argue that there is nothing that it is like to be an owner of experiences, in the sense that there is no experiential phenomenology associated with the ownership of experience. The second part of the paper argues that, since there is no experience of ownership, there is no onus on the Abhidharma philosopher to give an explanation of the sense of ownership. (shrink)

In her Introduction to Wittgenstein’s Tractatus , Elizabeth Anscombe took it to be a fault of the Tractatus that it excluded the statement “‘Someone’ is not the name of someone”, which she took to be obviously true. It is not a bipolar proposition, and its negation, she said, peters out into nothingness. I examine the question whether she is right that the Tractatus excludes such propositions, and I consider her example in relation to other propositions which, arguably at least, have (...) no intelligible negation. In considering the particular case of Frege’s response to Benno Kerry about the concept ‘horse’, I try to develop an account of the place in Wittgenstein’s philosophy for certain sorts of proposition which do not have an intelligible negation. (shrink)

How should we think about the role of visual spatial awareness in perception and perceptual knowledge? A common view, which finds a characteristic expression in Kant but has an intellectual heritage reaching back farther than that, is that an account of spatial awareness is fundamental to a theory of experience because spatiality is the defining characteristic of “outer sense”, of our perceptual awareness of how things are in the parts of the world that surround us. A natural counterpart to this (...) idea is to treat self-consciousness as residing in a kind of sense that is fundamentally “inner”, such as introspection or whatever else gives one privileged access to his own mental states as well as the proprioceptive and kinesthetic awareness of bodily position. This division is compatible, of course, with the idea that inner sense provides an awareness of a distinctive kind of “body space”, but it treats that as importantly different from the awareness of the worldly space around one. -/- In contrast to such a picture, this dissertation proposes an account of visual spatial awareness according to which it is no less a source of self-consciousness than of the awareness of the objects around us, and an account of self-awareness in which visual experience is essentially implicated. I begin by arguing that we should think of visual spatial awareness not as necessary for the individuation of visual sensations but rather as an essential element in the awareness of an experientially objective world. In the subsequent chapters, I argue that in being visually aware of the egocentric positions of the worldly objects around us we are often aware also of our own spatial locations with respect to them, and that the visual experience of the world around one and one’s own situation in it is often an essential component in the knowledge that a human agent will have of his own intentional actions. (shrink)

In On Certainty, §166, Wittgenstein mentions the difficulty of realizing the “groundlessness of our believing.” In the course of reviewing what makes this realization so difficult, I examine a certain way of understanding one of Wittgenstein's techniques for getting us to realize it, his use of the “hinge” metaphor. It implies that hinge-propositions possess that status inherently; for some commentators, this is because of their connection to instinctive and habitual behaviours. I offer an alternative interpretation of the remarks that have (...) been used to support this understanding that better explains their role in Wittgenstein's response to scepticism and other epistemological problems. (shrink)

Some aspects of Wittgenstein’s thought are considered in the light of a remark he makes about the “apocalyptic” view of the world. The influence of Tolstoy on Wittgenstein is discussed and elaborated with reference to the idea of a “form of life” as a locus of order, and also to that of “exceptionality” in an unfolding course of events—the latter setting up a connection with the “apocalyptic” theme. This imaginative backdrop remains discernible in Wittgenstein’s later philosophy, which draws upon it (...) to perhaps unexpected effect in achieving a dialectical balance between the motifs of order and breakdown. (shrink)

The paper provides an interpretation of Wittgenstein’s claim that a mathematical proof must be surveyable. It will be argued that this claim specifies a precondition for the applicability of the word ‘proof’. Accordingly, the latter is applicable to a proof-pattern only if we can come to agree by mere observation whether or not the pattern possesses the relevant structural features. The claim is problematic. It does not imply any questionable finitist doctrine. But it cannot be said to articulate a feature (...) of our actual usage of the word ‘proof’. The claim can be dissociated, however, from two tenable doctrines of Wittgenstein, namely that proofs can be used as paradigms for corresponding proof concepts and that a proof is not an experiment. (shrink)

In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept (...) of ‘denumerability’ as it is presented in set theory as well as his philosophic refutation of Cantor’s Diagonal Argument and the implications of such a refutation onto the problems of the Continuum Hypothesis and Cantor’s Theorem. Throughout, the discussion will be placed within the historical and philosophical framework of the Grundlagenkrise der Mathematik and Hilbert’s problems. (shrink)

Many interpreters of the Tractatus accept that the book endorses an argument for simples based on the reflection that, since complexes exist only contingently, if names referred to complexes the propositions in which they figure would lack sense if their referents went out of existence. More specifically, most interpreters read 2.0211-2.0212 as putting forward this argument. My main goal in this paper is to attack this reading and to put forward an alternative. I argue that there is no good reason (...) for thinking that the Tractatus advances this argument. I argue that 2.0211-2, in particular, should not be read in this way, and put forward an alternative reading of the passage. (shrink)

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 (...) propositions are 'grammatical propositions' (and so, Strictly speaking, Not genuine propositions at all) and his view that in mathematics 'everything is algorithm, Nothing meaning'. His views on consistency and his anti-Foundationalism are linked with this central thesis. (shrink)

When it comes to Wittgenstein's philosophy of mathematics, even sympathetic admirers are cowed into submission by the many criticisms of influential authors in that field. They say something to the effect that Wittgenstein does not know enough about or have enough respect for mathematics, to take him as a serious philosopher of mathematics. They claim to catch Wittgenstein pooh-poohing the modern set-theoretic extensional conception of a real number. This article, however, will show that Wittgenstein's criticism is well grounded. A real (...) number, as an 'extension', is a homeless fiction; 'homeless' in that it neither is supported by anything nor supports anything. The picture of a real number as an 'extension' is not supported by actual practice in calculus; calculus has nothing to do with 'extensions'. The extensional, set-theoretic conception of a real number does not give a foundation for real analysis, either. The so-called complete theory of real numbers, which is essentially an extensional approach, does not define (in any sense of the word) the set of real numbers so as to justify their completeness, despite the common belief to the contrary. The only correct foundation of real analysis consists in its being 'existential axiomatics'. And in real analysis, as existential axiomatics, a point on the real line need not be an 'extension'. (shrink)

What we observe, through our usually limited lens, is that differential growing of space determines forms -characterized by their shape, size and coloration. As non-Euclidean geometrical mathematics have proclaimed: forms are manifestations of the curvature of space. Physics and other natural laws impose mathematical structural restrictions to biological forms. The molecules comprising any living form become arranged in specific ways in response to physical forces as well as chemical and biochemical conditions. Over time, such forms inherit additional historical restrictions that (...) make up their meaning, their semiotics. Color does not escape from this orthogenic reality. In this study, we present the colored pattern of the connexivum of Triatoma maculata, an insect vector of the etiologic agent of Chagas disease. This model is used as an example to appreciate mimicry under a heterodox and provocative approach. We unveil that beyond 1) mathematical and 2) historical semiotic restrictions of T. maculata mimesis of a yellowish-black alternating pattern, there are also 3) artistic interpretations; where metaphysics of life, beyond human perspective, is included. (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)

It is often held that Wittgenstein had to introduce numbers in elementary propositions due to problems related to the so-called colour-exclusion problem. I argue in this paper that he had other reasons for introducing them, reasons that arise from an investigation of the continuity of visual space and what Wittgenstein refers to as ‘intensional infinity’. In addition, I argue that the introduction of numbers by this route was prior to introducing them _via_ the colour-exclusion problem. To conclude, I discuss two (...) problems that Wittgenstein faced in the writings before _Some Remarks on Logical Form _, problems that are independent of the colour-exclusion problem but dependent on the introduction of numbers in elementary propositions. (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)

Wittgenstein's conception of infinity can be seen as continuing the tradition of the potential infinite that begins with Aristotle. Transfinite cardinals in set theory might seem to render the potential infinite defunct with the actual infinite now given mathematical legitimacy. But Wittgenstein's remarks on set theory argue that the philosophical notion of the actual infinite remains philosophical and is not given a mathematical status as a result of set theory. The philosophical notion of the actual infinite is not to be (...) found in the mathematics of set theory, only in a certain associated philosophy – what Wittgenstein calls a certain kind of “prose”. (shrink)

One of the reasons colours, or better the conceptual organisation of the colour system, could be relevant to the philosophy of logic is that they necessitate some mutilation of truth tables by restricting truth functionality. This paper argues that the so-called ‘Colour Exclusion Problem’, the first great challenge for Wittgenstein’s Tractatus, is a legitimate philosophical motivation for a systematic mutilation of truth tables. It shows how one can express, through these mutilations, some intensional logical relations usually expressed by the Aristotelian (...) Square of Oppositions, as contrariety and subcontrariety. (shrink)

Perception is often analysed as a process in which causal events from the environment act on a subject to produce states in the mind or brain. The role of the subject is an increasing feature of neuroscientific and cognitive literature. This feature is linked to the need for an account of the normative aspects of perceptual competence. A holographic model is offered in which objects are presented to the subject classified according to rules governing concepts and encoded in brain function (...) in that form. This implies that the analysis of perception must consider not only the fact that there is an interaction between the perceiving subject and the perceived object but also that the interaction is shaped by a system of concepts which the subject uses in thought and action. (shrink)

The concept of übersichtliche Darstellung is of fundamental significance for Wittgenstein . Hacker translates übersichtliche Darstellung as ‘surveyable representation’ and equates it with the tabulation of grammar. He asks what surveyability means, whether examples can be found in Wittgenstein’s work, and why this method characterizes the form of account he gives. Ultimately, however, Hacker is unable to answer these questions and he attributes this failure to Wittgenstein. This paper argues that it is Hacker’s interpretation that fails, and presents an alternate (...) understanding of übersichtliche Darstellung ; one that enables us to answer Hacker’s questions in a manner consistent with Wittgenstein’s philosophical practices. Normal 0 false false false EN-US X-NONE X-NONE. (shrink)

Cette publication, la trois cent vingt-troisième de la série, a été rendue possible grâce à la contribution financière du FQRSC (Fonds québécois de recherche sur la société et la culture).

In this paper I contrast conceptions of self from two perspectives: an individualistic orientation and a communitarian approach. In doing so, the philosophical justification is Wittgenstein’s idea that individualism is produced and reinforced as a way of being, thinking and interacting in community. With this contextual frame, I argue that we are shaped by the language practices of our community to ascribe meaning and interpret our own relationships with others through our language lexicon and grammar. To illustrate the communitarian perspective (...) I refer to Māori perspectives in which connectedness is axiomatic: in particular the concepts of whanaungatanga and whakapapa, as described by Ka’ai and Metge. On the basis that such concepts are predicated on language use, I suggest that we can learn the grammar of engagement in a discursive context that emphasises connectedness just as easily as in one which is founded on separateness. I argue that interacting with an other is to engage with the ideas, beliefs, and values that shape their language, even as we expose our own. Engaging between individualistic and communitarian understandings is presented as learning to bridge language gaps between different social constructs and understandings of reality. (shrink)