The status of mathematical facts has long been taken to be unclear in Wittgenstein's philosophy of mathematics, and often, it seems that he wants to eliminate mathematical facts in favour of facts about our beliefs or behaviour. In this paper, I argue that by reading Wittgenstein as a radical conventionalist, we can give a reading of the relevant passages according to which Wittgenstein doesn't deny that there are mathematical facts, but rather denies that one needs a metaphysical account of what (...) mathematical facts are and how they relate to the world that goes beyond the minimal claims of radical conventionalism—that empirical facts about how we would find natural to project our training into new cases and the constitution of concepts by our agreement are enough. At the end of the paper, I discuss the implications of this reading on how to understand a rule's determination of its own application. (shrink)

Wittgenstein and the logical positivists attempted to explain logical necessity in terms of linguistic conventions. It is often thought that their respective accounts have been conclusively refuted by objections from Quine, Dummett and others. We argue that this verdict is premature. Several of the most popular anti‐conventionalist arguments fail, partly because they misconstrue the idea of truth by convention in Wittgenstein and/or logical positivism. Correctly understood, conventionalism claims that, given certain linguistic conventions, some sentences are unconditionally true, that is true (...) irrespective of how the world happens to be. This claim is difficult to deny, and the corresponding conventionalism about logical necessity remains a viable position. (shrink)

The mature Wittgenstein's groundbreaking analyses of sense and the logical must—and the powerful new method that made them possible—were the result of a multi‐year process of writing, re‐arranging, re‐writing and one large‐scale revision that eventually produced the Philosophical Investigations and RFM I. In contrast, his struggles during the same period with questions of arithmetic and higher mathematics remained largely in first‐draft form, and he drops the topic entirely after 1945. In this paper, I argue that Wittgenstein's new method can be (...) applied to the cases of arithmetic and set theory and that the result is innovative, recognizably Wittgensteinian, and independently appealing. I conclude by acknowledging the reasons Wittgenstein himself might have had to resist applying his own proven method to the case of mathematics—particularly to set theory—and by indicating why I think those reasons are ultimately unsound. (shrink)

In what way does a mathematical proof depend on the notation used in its presentation? This paper examines this question by analysing the computational differences, in the sense of Larkin and Simon's ‘Why a diagram is (sometimes) worth 10,000 words’, between diagrammatic and sentential notations as a means for presenting geometric proofs. Wittgenstein takes up the question of mathematical notation and proof in Section III of Remarks on the Foundations of Mathematics. After discussing his observations on a proof's ‘characteristic visual (...) shape’ in Section III with respect to arithmetical proofs, the paper shows how the notion of a characteristic visual shape illuminates the special effectiveness of diagrammatic notation in geometry. (shrink)

The conditions involved in the applicability of mathematics in science are the subject of ongoing debates. One of the best‐received approaches is the inferential account, which involves structural mappings and pragmatic considerations in a three‐step model. According to the inferential account, these pragmatic considerations happen in the immersion and interpretation stages, but not during derivation (symbol‐pushing in a mathematical formalism). In this work, I draw inspiration from the later Wittgenstein and make the case that the applicability of mathematics also rests (...) on pragmatic considerations at the mathematical derivation level. I make the further case that pragmatic considerations at the mathematical derivation level may substitute pragmatic considerations in the other two stages, thus showing how they are holistically interrelated. I illustrate these two points with the solution of a simple geometrical problem. (shrink)

On the one hand, I show that the later Wittgenstein's practice-based approach to meaning, including the idea that the meaningfulness of mathematics ultimately is rooted in the everyday ‘applications’ it emerged from, as well as his insistence on the variability in and contingency of mathematical and mathematics-like practices, foreshadows more recent work in Philosophy of Mathematical Practice (PMP), although Wittgenstein's approach was more radically practice-based than what is prevalent in present-day PMP. On the other hand, I also show that Wittgenstein's (...) work on mathematics is driven by a pervasive critical attitude towards some key characteristics (monism, foundationalism and exceptionalism) of mainstream philosophical discourse about mathematics (then and now), which is mostly absent from present-day PMP, and I argue that this aspect of Wittgenstein's work on mathematics could inspire present-day practitioners of PMP to rethink the identity of their discipline. (shrink)

In his 1939 Cambridge Lectures on the Foundations of Mathematics, Wittgenstein proclaims that he is not out to persuade anyone to change their opinions. I seek to further our understanding of this point by investigating an exchange between Wittgenstein and Turing on contradictions. In defending the claim that contradictory calculi are mathematically defective, Turing suggests that applying such a calculus would lead to disasters such as bridges falling down. In the ensuing discussion, it can seem as if Wittgenstein challenges Turing's (...) claim that such disasters would occur. I argue that this is not what Wittgenstein is doing. Rather, he is scrutinizing the meaning and philosophical import of Turing's claim—showing how Turing is wavering between making an empirical prediction and a logical observation, and that it is only through this wavering that Turing can believe that he has provided a proper explanation of why contradictory calculi are mathematically defective. (shrink)

This paper offers an evaluation of Wittgenstein's critique of Cantorian set theory, illustrating his broader philosophical stance on mathematics. By emphasizing the constructed nature of mathematical theories, Wittgenstein encourages a reflective approach to mathematics that acknowledges human agency in its development. His engagement with Cantorian set theory provides valuable insights into the philosophical and practical dimensions of mathematics, urging a reconsideration of its foundations and the nature of mathematical proofs. This perspective aligns closely with the philosophy of mathematical practice, which (...) emphasizes the human side of mathematics by its focus on mathematical practice. (shrink)

An American teacher of literature and poetry living in Montréal, Curtis Brown, recovers Theodore Dreiser's An American Tragedy (1925) as a ‘virus novel’, venturing an integrated response to two literary questions occasioned by the recent pandemic: What might the contemporary ‘Covid novel’ look like? And how did the Spanish flu of 1918–1919—which infected a third of the world's population and killed 50 million—leave literary traces so few and faint? Brown tracks the genesis of An American Tragedy in Dreiser's personal life, (...) his research in microbiology and his general interest in plagues to show how a form traditionally associated with the ‘individual moral adventure’ was repurposed for an institutional epic of viral contagion and systemic malaise. Charting the vicissitudes of Dreiser's reputation over a century, and presenting his formal innovations as widely assimilated, largely unrecognized, and more relevant than ever, Brown concludes that a writer's influence, like a virus, can disappear into ubiquity. (shrink)

Though Plato, famously, had Socrates ban the poets from his republic while Wittgenstein seems to aspire to a style of philosophical writing that approaches the literary, there is a troubling similarity between them in their elision of the self as a self. Saying what he means in the mode of the philosopher or meaning what he says in the mode of the poet, how can the self both be and say itself? The misprision of Abraham's binding of Isaac in the (...) thinking of Kierkegaard, who with Plato was one of Wittgenstein's favourite philosophers, is a telling opening into the question of the self in its responsibilities and its discontents. (shrink)

Philosophical Investigations, EarlyView.

In humanities departments across the country Allan Bloom is condemned and reviled—but for what misconduct? The present investigation into our distorted view of Bloom begins by reframing his transgressions. Bloom has offended eminent philosophy professors by boosting students' desire for a philosophic education. He has emboldened promising young people by teaching them to bear up under the deforming forces of convention and corruption and edge their way towards the question, ‘How should I live my life?’ A philosophic education as Bloom (...) sees it ‘means precisely helping students to pose this question to themselves, to become aware that the answer is neither obvious nor simply unavailable, and that there is no serious life in which this question is not a continuous concern’. The philosopher's interests are not shared by most people. Hostility to philosophy is the natural condition of man. This surprising fact, plainly stated, invites so much anger that Bloom—standing in for the philosopher—is forced to defend himself. The story of Bloom, like Socrates', gives an account of philosophy's emergence, distinctive features and always imaginable suppression. Plato's artful writing captured Socrates' way of life for posterity. This essay explores the nature and function of Bloom's bestselling book The Closing of the American Mind (1987), as well as the purpose of Saul Bellow's final novel, Ravelstein (2000), a fictive account of Bloom's life and death. Together these books amount to the true Apology of Allan Bloom. (shrink)

The social structures and thought patterns of the modern world are the fruits of the Enlightenment, which begins by eliminating final causal explanations in favour of purely material and efficient causes. The development and great technical success of Enlightenment procedures has, however, produced a cultural blindness about the good. Camus's novel shows us this cultural blindness through characters who themselves suffer from it; for modern man, it is almost a natural evil—we are born into it. Camus' hope must have been (...) that the vision of the characters' incapacity might help break us from the same incapacity. What the cured life looks like Camus does not say; in the world of our plague, it appears as either odd or mystery. (shrink)

Socrates is a man of faith whose love and pursuit of the truth is grounded in religious conviction. Faith, whatever else it may be, involves guiding one's life in terms of a transcendent dimension, recognizing a reality lying behind any particular experience. In Plato's Phaedo, a literary and philosophical masterpiece, we enter the narrative of Socrates' trial and execution on the day of his death, examining arguments for the immortality of the psyche. The dialogue combines logical argument and mythological speculation, (...) and presents Socrates simultaneously as an abstract theorist and an unforgettably singular person. Distinct persons, such as Socrates, have an interest in continuing to exist as the individuals that they are. When we cease to exist, we lose the opportunity to engage in any activity that is worth doing, such as—emphatically, for some of us—doing philosophy. What Socrates is presented as having discovered in the Phaedo—with the ostensibly absent Plato lurking behind every line in this literary gem of a text—is a way of transforming oneself with an eye on immortality. That Socrates dies utterly without fear is the supreme vindication of the philosophical life. (shrink)

What does a famed literary theorist have to say about the interaction between ‘literature’ and ‘philosophy’? Well, if he's Garrett Stewart, the celebrated agent of pyrotechnic style in the service of durable insights across disparate disciplines and media, then we have much reason to lean in and listen. Stewart is the author of 20 books that range with uncanny competency across Victorian narrative, contemporary American fiction, written auralities, poetics and prose stylistics, cinematic evolution from silver oxide to screen pixel, book (...) art, scenes of reading in painting, and most recently, the vocal drama of Barbra Streisand. The distances between his disciplinary terrains are as vast as his close readings of them are coordinated and trenchant. Here, sitting down before a shared conversational canvas, Stewart fields questions from David LaRocca, who asks how he, if he, thinks about the program of literature and philosophy. LaRocca, editor of the recent Attention Spans: Garrett Stewart, a Reader (Bloomsbury, 2024) and the forthcoming Bandwidths: Reading Across Media with Garrett Stewart (Bloomsbury), tracks the urgency of our moment—in which literature and philosophy are independently besieged and yet may concede little rapprochement. Where does philosophy happen in literature (a philosopher's question) is turned around by Stewart such that we see literature conjuring philosophy at the level of the sentence. We are not peering ‘in’ so much as looking ‘at’ the art of literature teaching philosophy through refined reading and the analytical inquest of the syllable and the sayable. Anyone who works at the fraught fissure of these realms will benefit from the ways Stewart makes literature and philosophy sensible to one another—even as he productively unsettles the terms and conditions of their affiliation. (shrink)

F. R. Leavis, the leading literary critic at Cambridge from 1930 to 1960, recounts the time when his friend Ludwig Wittgenstein told him to ‘give up literary criticism!’ The remark came as a surprise to Leavis, and it remains somewhat puzzling to anyone who reads of the encounter, for there are few contextual clues as to why Wittgenstein would say such a thing. But there are clues in Wittgenstein's many remarks on literature, music and other art forms scattered throughout his (...) writings. In this paper, I will present what I think Wittgenstein's concern was with the sort of literary criticism Leavis practised, contrasting it with the sort of reading, and the sort of reader, that Wittgenstein (and others) would consider a literary way of reading; the sort of reader he thought was ‘a better sort of reader’. (shrink)

Philosophical Investigations, EarlyView.

Philosophical Investigations, EarlyView.

In a notorious passage from his Philosophical Investigations, Wittgenstein writes that one can state of the standard metre neither that it is one metre long, nor that it is not one metre long. While many commentators have rejected this claim, it has been commonly assumed that Wittgenstein himself endorsed it. In a recently published article, Thomas Müller not only provides a novel argument against Wittgenstein's claim about the standard metre but also claims that Wittgenstein did not actually endorse that claim. (...) In this paper, I argue that Müller's exegetical thesis is false and that his novel argument against Wittgenstein's claim about the standard metre is unsound. (shrink)

In this essay, I investigate the longstanding philosophical problem of whether we have control over our actions in a deterministic world. In working through a range of everyday situations in which this problem could arise, I come to the realisation that determinism has no bearing on whether we have control over our actions, because having control over our actions and determinism only make sense under different aspects.

Blum (Philosophical Investigations 46, 2023, 264) argues that Tarski's T‐schema and the thesis of the necessity of identity are mutually inconsistent. It is argued that his argument fails.

Pierre Hadot is best known as a historian of ancient philosophy and for advocating the relevance of ancient thinking for contemporary lives. What is less well known is that he was one of the first French philosophers to take a serious interest in the work of Wittgenstein, publishing between 1959 and 1962 two essays on the Tractatus and two on the Philosophical Investigations, since republished as Wittgenstein et les limites de langage (Paris: J. Vrin, 2010). Only two of these essays (...) are available (and not widely) in English translation. Part of my goal is to argue that they deserve the attention of anglophone readers. My focus here is on Hadot's remarks about the Philosophical Investigations. Hadot argues that this work produces a self‐defeating paradox because it claims that we can speak intelligibly only within a language‐game, but Wittgenstein, like the philosophers he criticises, tries to transcend language‐games in the presentation of his views. Despite this criticism, Hadot is inspired by Wittgenstein's discussion of the multiplicity of language‐games to argue that the texts of ancient philosophy are not part of the same language‐game as those of modern philosophy and must be approached as ‘spiritual exercises’, rather than as bodies of doctrine or theory. Wittgenstein is thus a key inspiration for Hadot's historiographical method. I argue that Hadot is too impressed by a faulty analogy between the Tractatus and the Investigations and that he gives a problematically reductive interpretation of Wittgenstein's talk of language‐games and implausibly attributes to Wittgenstein a reverence for ‘the ordinary’ that supposedly takes the place of his earlier wonder at the existence of the world. Many commentators since Hadot have made similar errors, so his case remains instructive. I conclude by suggesting, nonetheless, that something like Hadot's proposals about historiographical method may be justified by a better reading of the Investigations. (shrink)

The influence of Heinrich Hertz's The Principles of Mechanics on Ludwig Wittgenstein's Tractatus Logico‐Philosophicus has been studied for decades, but it has never become a mainstream topic in the Wittgensteinian literature. This paper focusses on Tractarian notions of objects, elementary facts and elementary sentences and discusses their similarities with Hertz's concepts of mass, its constituents and their mechanistic images. As the paper demonstrates, the Hertzian context provides some fruitful interpretational leads concerning several controversial ideas endorsed by early Wittgenstein, namely propositional (...) analysis, logical independence of elementary facts, logical independence of elementary sentences, and modalities. (shrink)

In this discussion I argue that Peter Sullivan is wrong to suggest that Wittgenstein's position in the Philosophical Investigations involves a commitment to transcendental idealism. I show that Sullivan's interpretation involves holding that transcendental idealism was employed by Wittgenstein in the attempt to combat a Platonist mythology. I show, through a detailed appraisal of Wittgenstein's discussion of samples, that Wittgenstein's approach to Platonism does not involve any such employment of transcendental idealism. I conclude that there is no such motivation as (...) Sullivan finds in Wittgenstein for endorsing transcendental idealism, and that we ought not, therefore, ascribe to him such a view. (shrink)

This is a reply to ‘Defending Wittgenstein’, Piotr Dehnel's critique of my article, ‘Defending Wittgenstein's Remarks on Cantor from Putnam’. I first show that my position is much more in agreement with Felix Mühlhölzer than Dehnel takes it to be, and that his criticism of me is nothing more than a failure to recognize this. I then show how Dehnel incorrectly reads Wittgenstein as rejecting set theory as false. It is an overemphasis on and a much too narrow picture of (...) ‘applicability’ which leads him to this view. Finally, I conclude by rejecting Dehnel's view that Wittgenstein was a finitist about mathematics. (shrink)

Is ethics about anything? Cora Diamond has famously argued that ethics lacks a subject matter by providing a variety of examples of ethical discourse, which, she claims, are ethically significant without being about anything ethical. They do not have a moral subject matter, but are nonetheless instances of moral thinking. This raises the question what it means for a piece of discourse to be moral. What does Diamond mean by the concept of ethics? Diamond never gives this question a direct (...) answer, which is a cause for concern, among other things because there is reason to worry that any workable account of the concept of ethics will lead back to the idea that ethics has a subject matter. I explain how this worry can be averted by construing the concept of ethics as a family resemblance concept. (shrink)