In this paper, I plan to analyse Wittgenstein and Feferman ideas about the definition of mathematical open problems, and compare them against current mathematical practice. Defining the right mathematical problem, and the mathematical problem in the right way, is the fundamental first step in mathematical practice. Of course, the importance of open problems in mathematics is not a new fashion, but was always recognised (see for example Hilbert’s Problems, that already show quite the mature approach to open problems, or Wittgenstein remarks in his Nachlass that “[...] the correct formulation of the question is already the answer.” (Wittgenstein Nachlass Ms-305,1[4])). Wittgenstein’s positions evolved during his life (see Säätelä (2011) for an overview): at first, the early Wittgenstein position was that mathematical open problems were not genuine propositions, and have no proper sense (see 208, 50r of Wittgenstein (2012)), since their sense would be connected to a logical procedure to check their truth that we still don’t know. I plan to show that this (extreme) position seems similar to Feferman (2014)’s position regarding “undefined problems”, i.e. the problems that are not well defined and don’t satisfy the Law of the Excluded Middle (or have been already settled). in which the interest of finding a solution to CH is challenged on the grounds that it is not a correctly formulated mathematical problem, and it is instead “senseless”. Wittgenstein position later evolved towards a more balanced approach, in which the propositions forming a mathematical open problem are not completely devoid of meaning, and can in some sense still be understood (see Wittgenstein (2009)). The reason why Wittgenstein can claim that there is some sort of understanding stems from the fact that mathematicians usually have beliefs about open problems that tend to guide their practice (whether those beliefs are justified or not it’s another story). Sense, content, and meaning of propositions stem from the use of those proposition (Wittgenstein (2009) and Wittgenstein (1958)), and since we do use open problems as our guide to mathematical practice, they get their meaning, sense, and content from that. I argue that this position is certainly closer to mathematical practice, and to Feferman’s approach to mathematical and logical problems (i.e. problems that have a clear mathematical content without mentioning meta-mathematical and logical notions, and problems that mention them). However, I further argue that Feferman’s approach to “undefined problems” is problematic, and not currently in line with mathematical practice, since even those problems that Feferman considers non well-defined, like CH, are still used as a guide for mathematical (and in particular set-theoretic) research. So it seems that Wittgenstein position is closer to current mathematical practice, in all of its forms, than Feferman’s one (surprisingly!). Nevertheless, I will conclude this paper by moderating this conclusion, by arguing that it is possible to recover Feferman’s position regarding undefined problems by noticing that his remarks are not to be taken in the most general sense (they apply mainly to set theory, and already when talking about arithmetic the Feferman’s approach changes), and by considering Hamkins’ pluralist view on independent statements (that form the bulk of Feferman’s undefined problems) that are to be considered “settled” in the set- theoretic multiverse. In this way, it is possible to argue that Feferman’s position becomes as close as Wittgenstein’s position regarding undefined problems, and even closer in the case of the most recent trends in set-theoretic practice.

Within contemporary philosophy of literature, understood as a subfield of analytic aesthetics, there have been several attempts to draw on Wittgenstein’s philosophy in order to address the questions that structure its current agenda, in particular the problem of the cognitive value of literature. I argue that such approaches, in most cases, pay insufficient attention to the methodological and stylistic dimensions of Wittgenstein’s philosophy. In abstracting what they take to be Wittgensteinian insights, scholars press them into the service of theoretical aims that stand in tension with Wittgenstein’s own philosophical practice. Specifically, the problem of the cognitive value of literature, in the form it has taken in recent debates, seems to express a transcendental anxiety regarding literature’s capacity to convey knowledge, truth, understanding, or, more broadly, to be illuminating about our world and ourselves. Here, the appeal to Wittgenstein serves to articulate a non-propositional conception of knowledge, sometimes tied to the notions of “grammatical display” and “showing”, meant to secure for literature a form of understanding alternative (and irreducible) to a set of true propositions. The trouble with these Wittgenstein-inspired neo-cognitivist positions, however, lies in the explanatory role assigned to their claims. Indeed, the theoretical framework of contemporary philosophy of literature calls for systematic explanations in response to its challenges (such as the anti-cognitivist challenge, which demands a satisfactory grounding of the cognitive value of literature), thereby favoring a problem-solving style that results in a range of competing theoretical options. Transplanted into this context, references to Wittgensteinian ideas sit uneasily with Wittgenstein’s own anti-theoretical commitments and with his view of philosophy as a clarificatory and descriptive activity rather than an explanatory and justificatory one. By contrast, attending more closely to Wittgenstein’s method suggests that we should resist the foundational urge to dig ever deeper and the will to explain, learning instead to recognize the rough ground before us (RFM VI §31). A greater sensitivity to the methodological, stylistic, and metaphilosophical aspects of Wittgensteinian practice makes it possible to invoke Wittgenstein (as read by Cavell, Diamond, and within the so-called New Wittgenstein tradition) diagnostically, that is, to reflect critically on the very modes and status of philosophy of literature as a discipline. Much of this project, in fact, risks losing touch with the reality and sense of literary practices. In Cavellian and Diamondian terms, it exhibits a tendency best understood as a form of deflection: a turning away from our sense of the depth of literature by constraining it within the shape of theoretical puzzles, thereby displacing the very sources of significance that initially give rise to our interest in literary works. If the standard treatment of the problem of the cognitive value of literature is one instance of this tendency, I argue that a better way of inheriting Wittgenstein lies in redirecting our gaze toward the place that ordinary engagement with literature occupies in our lives, one that contrasts the laying down of philosophical requirements with “looking at the use, at what we do” (Diamond, The Realistic Spirit, 21).

Wittgenstein’s alleged relativism or anti-relativism has been widely discussed among scholars, without reaching a firm consensus on the nature of his philosophy in this regard. This problem arises from the fact that among his works it is easy to find both remarks that seemingly support a relativist interpretation, and others that point towards anti-relativism. In my talk, I argue that the questions about his alleged relativism can be misguided, and that if we want to understand why some of Wittgenstein’s remarks seem to contradict one another, we could try to look at his metaphilosophy. To see why questions such as “Was Wittgenstein a relativist (or anti-relativist)?” are misguided, I articulate two difficulties – one methodological, one hermeneutical – that the scholarly debate does not properly address. On the methodological side, it is problematic to ask this question, as it shapes our inquiry in a way that presupposes he must have been either a relativist or an anti-relativist. Framing the issue in these terms risks overlooking the possibility that Wittgenstein’s philosophy was, in an important sense, orthogonal to the terms of the relativism debate as it is typically understood. The hermeneutical difficulty, meanwhile, is that we are inclined to conceive “relativism” and “anti-relativism” as conflicting philosophical theories, whereas Wittgenstein held that philosophy cannot advance any theory (see PI §109). While some have suggested that Wittgenstein’s relativism or anti-relativism can be interpreted as something different from a theory, I suggest that a thorough examination of his metaphilosophy can overcome the difficulties above. This move, however, risks introducing a further self-refuting problem: treating metaphilosophy itself as a theory. I address this in the talk by conceiving philosophy as a language game, a human activity to be described rather than theorized. To overcome the difficulties, I provide an account of what Wittgenstein meant when he stated that philosophy is a descriptive endeavor, focusing on the concept of synoptic representation (übersichtliche Darstellung; PI §122). If the goal of Wittgenstein’s philosophy is to display linguistic phenomena so that we may see the connections among them, then philosophy itself, conceived as a language game, could likewise be displayed synoptically. Following this insight, philosophy can be seen as concerned with depicting both positions, relativism and anti-relativism, to gain clarity about the issues, rather than endorsing either side of the debate. The solution I propose can also suggest why some of Wittgenstein’s remarks seem to contradict one another. I understand Wittgenstein’s famous remark that philosophy “simply puts everything before us” (PI §126) as suggesting that what makes a thinker a relativist or an anti-relativist is not (Wittgensteinian) philosophy, but something else (consider, for instance, one’s taste or existential commitments). Therefore, as a synoptic endeavor, philosophy needs to engage with both sides of the debate and refrain from taking either side. To put the point in a slogan: Wittgenstein’s philosophy concerns itself with both sides of the coin, rather than with how the coin happens to land in any particular instance.

Wittgenstein is among those who understand art in terms of expression. Yet the way works of art are expressive seems to differ markedly from the way we express our affective or attitudinal states in ordinary life. This paper addresses the question of how works of art can be expressive in the absence of a directly accessible expressing subject. I argue that answering this question requires distinguishing between expression on the one hand, understood as a communicative act through which a subject makes their affective state available to others within a determinate context, and expressivity on the other, as the structural and relational property by virtue of which a form is apt to function intelligibly as an expression within a form of life.

Drawing on a Wittgensteinian account of emotional expression, the paper develops this distinction in three stages. First, it shows that expression is not confined to a narrow class of deliberate expressive actions, but pervades our intentional life as a whole, operating through recognizable structures that are neither arbitrary nor naturally fixed, and oriented toward communication even in its spontaneous and non-voluntary forms. Second, it argues that these structures possess a layer of significance, their expressivity, consisting in the range of possible expressive uses for which they are apt within a form of life. This conception is analogous to the practice-embedded conception of word meaning developed in the Philosophical Investigations. Third, it shows how this framework explains aesthetic expressivity: artworks, encountered sub specie aeternitatis and thus detached from any determinate act of expression, are expressive not by virtue of the intentions behind them but by virtue of what their structure can intelligibly be taken to express within the broader network of artistic practices and forms of life.

The paper concludes that aesthetic expressivity does not say anything propositional but shows potentially revelatory ways of seeing and feeling, which may be imaginatively taken up, modified, or resisted: one value of a work of art lies in the richness of the expressive possibilities that it makes available.

Drawing on Wittgensteinian family resemblances, the underresearched philosopher of science Mary Hesse (1974) puts forward a compromise between the absolute theory of universals (corresponding to a realist position) and the resemblance theory of universals (roughly corresponding to a nominalist position) in her network model of universals. Hesse’s network model centers analogical reasoning; her conceptualization of analogies depends heavily on Aristotle’s treatment of metaphors. In contrast, Ludwig Wittgenstein, whose concept of family resemblances – like Aristotelian analogies – provides a method for identifying similarities, repeatedly tries to demarcate his philosophy of language from Aristotle and metaphors in general. Disregarding this demarcation, Hesse (1974) combines her Aristotelian understanding of analogies with Wittgenstein’s family resemblances in her network model of universals. Also secondary authors (e.g., Hommen, 2025) find that Hesse’s and Wittgenstein’s works combine well, even better than assumed by Hesse herself.

Contrary to what one might assume given their divergent relations to Aristotle, Wittgenstein and Hesse can be fruitfully reconciled. Reconciliation is achieved by two complementary strategies: First, I identify which aspects of Aristotle’s theory of metaphor are most important to get going Hesse’s account of the role of analogies in scientific reasoning and how she adapts her view with the help of Max Black’s (1962) interaction view of metaphor. Second, to approach the network model of universals not only from Hesse’s but also from Wittgenstein’s position, I show how Wittgenstein’s rejection of metaphors is based on a misunderstanding about the scope of metaphors. My analysis explains how pairings of Hesse’s and Wittgenstein’s philosophy can be successful and paves the way for further such combinations. I further demonstrate that Hesse’s network model of universals stands on solid ground what concerns her combination of analogies and family resemblances.

I go about as follows. First, I clarify the scope of metaphors and analogies by providing an overview about the distinctions made in the "Poetics" and explaining the continuous shift of these distinctions in Roman and modern receptions of Aristotle. Second, I show how Hesse builds on Aristotle for her analysis of scientific reasoning and models. Third, I turn to family resemblances, focusing on Wittgenstein’s supposed ignorance towards Aristotle and his attempts to demarcate what he dubbed “secondary meaning” from metaphorical connotations. Subsequently, Hesse’s network model of universals is introduced with an emphasis on how she combines insights from Wittgenstein and Aristotle to explain the function of analogical reasoning in science. I dive into David Hommen’s (2025) strategy for reconciling Hesse with Wittgenstein before transitioning to my arguments, which show how Hesse adds new layers to Aristotelian metaphors in a way that brings her closer to Wittgenstein and how Wittgenstein can be interpreted to be Aristotelian in at least those aspects important to Hesse.

Sources:

• Black, M. (1962). Models and Metaphors. Cornell University Press.

• Hesse, M. B. (1974). The Structure of Scientific Inference. University of California Press.

• Hommen, D. (2025). Wittgenstein and Hesse on models and metaphors. In P. Gori (Ed.), Mary. B. Hesse (1924–2016): Metaphors, models, and the growth of scientific knowledge. Springer.

In On Certainty, Wittgenstein’s examples of ‘hinges’ (propositions which can’t be doubted) range from the general – ‘The Earth exists’ – to the exceedingly individual: ‘for the last month I have taken a bath every day’. Commentators working on the text – hinge epistemologists – systematically overlook the individual examples, in favour of the general. But this is unwarranted.

My paper starts making the case for a theory of hinges which pays due attention to these individual examples: a particularist hinge epistemology. I show individual hinges have a significant presence in the text of On Certainty. I consider – and dismiss – two putative philosophical objections to countenancing individual hinges: that they can’t be used in a response to scepticism, and that they can’t exhibit the property of epistemic necessity. These points together make a provisional case for a particularist hinge epistemology. They show it to be textually warranted and philosophically viable. I close the essay by making the first move of such a theory, and giving a definition of individual hinges. That definition shows how context-sensitive individual hinges – like ‘I had lunch just now’ – can still exhibit the indubitability required for responding to the sceptic.

The paper’s main contribution is a new approach to a notion of central importance in On Certainty. But a particularist hinge epistemology is not only a fresh textual perspective. It radically expands the range and kind of propositions which can function as hinges. This increased diet of examples would enrich discussion about conflict between hinges, as featured in debates about relativism and deep disagreement.

At the end of the Tractatus, Wittgenstein famously states that “whoever understands me eventually recognizes [my propositions] as nonsensical, when he has used them – as steps – to climb beyond them” (TLP 6.54). Wittgenstein got the metaphor of the ladder that must eventually be discarded from Sextus Empiricus (likely by way of Fritz Mauthner), who says the same about his skeptical arguments. Sextus also employs the metaphor of a laxative, which must remove itself from our bodies along with everything else – and this metaphor is also used by the Buddhist philosophers Candrakīrti and Jizang.

The question what portion of the Tractatus Wittgenstein wants to present as nonsensical is, of course, controversial. But in either case, there is a conceptually interesting feature here: Wittgenstein thinks that at least some of his propositions are nonsensical, but should nevertheless be believed for a while, presumably because they help us remove another group of beliefs. Only after this should his propositions also be discarded.

This is different from most self-defeating propositions. While it is possible to erroneously have self-defeating belief, these propositions normally only remove themselves (once we recognize them as self-defeating). It is also odd that Wittgenstein (as well as Sextus and several Buddhist philosophers) openly tell us that their propositions are self-undermining, but nevertheless expect us to take them seriously.

I suggest that to understand this strategy, we must assume two things. Firstly, Wittgenstein expects us to take on his propositions as a kind of “experimental beliefs” – this is something we often do in philosophical speculation, and is encapsulated in the phrase “for the sake of the argument”. Secondly we must come to recognize something self-undermining about these proposition – but realize that what is self-undermining about them in fact undermines not only Wittgenstein’s propositions, but also a host of other propositions. As a result, all propositions are discarded – those provided by Wittgenstein as well as a number of others.

Sextus explains Pyrrhonian skepticism as an ability to provide certain arguments that will prevent us from believing anything with conviction. This fits well with Wittgenstein’s remark that the correct method of philosophy would be to say nothing until someone else says something philosophical, and then to show him the error of his ways (TLP 6.53). If we take this to be a good way of understanding Wittgenstein’s argumentative strategy in the Tractatus, it will also help us formulate clearer criteria for success. In particular, it is not enough that his particular approach of capturing meaning through the picture theory fails to explain the relation between words and objects; but rather, this failure needs to be such that it would re-occur in any conceivable theory of meaning. If this were the case, the picture theory would have the potential to “cure” us from wanting any such theory at all.

When the ancients speak, they do not merely tell us about themselves. They tell us about us. (Williams 1993: 20)

In Bernard Williams’s view, we have errors in our modern ethical thought that the early Greeks could help us correct. Drawing on Homer and the tragedians, Williams argues that the early Greeks developed a complex and, in certain respects, appealing account of ethical life. “If we can come to understand the ethical concepts of the Greeks,” he states, “then that same process may help free us of misunderstandings of ourselves”. (1993: 10) I will briefly outline Williams’s project but refrain from critically engaging with it. Instead, I want to explore a more fundamental point: it is a question to what extent we might meaningfully “make the ancients speak”. What it takes to understand others is one of Wittgenstein’s key contributions to philosophical thought. This paper pits Wittgenstein with Williams. I bring to bear Wittgenstein’s ideas in a piecemeal way, exploring his thoughts on speaking lions, rule following, nonsense, forms of life, and the problem of other minds. I regret the fact that the little I will be able to say here does them scant justice. The particular challenges Wittgenstein poses, I shall argue, affords Williams’s project a new and more profound rationale.

Religious extremism as a phenomenon shows that religious belief systems encounter moral epistemic problems at their very foundation. They face what I call the “moral problem of demarcation” which threatens their rational and moral status. I will illustrate the problem with the example of religious extremism and its epistemology. More specifically, I will rely on the religious epistemology that John Henry Newman (1903) developed in his undervalued Essay in Aid of a Grammar of Assent as a starting point. The Essay offers a framework aligned with an influential current in political epistemology (Ranalli 2022) and it can be used to illustrate how faith can epistemologically underpin extremist beliefs. This reading establishes the religious moral problem of demarcation. This case of Newman and Catholic extremism serves as a particularly vivid illustration of the more general moral problem of demarcation that is in no way limited to religion. There are also non-religious cases of extremism, and, as I will show, the moral problem of demarcation arises for all our belief systems independently of extremism. 

Newman showed that there are also many non-religious certainties that are held in excess of the available evidence. “Everybody has parents”, for instance, is also certain without sufficient evidence. As a consequence, our religious certainties are just as rational as these non-religious certainties. Newman’s framework has been brought into 21st Century epistemology through quasi-fideism, a Wittgenstein-inspired epistemological interpretation of Newman by Duncan Pritchard

 The epistemic structure of Newman’s and Wittgenstein’s certainties blocks anyone from critically examining their basic moral assumptions. As a result, all beliefs that build on these foundations inherit these moral difficulties even though they themselves may have perfect rational and epistemic standing. As a result, the problem has implications for phenomena like moral ignorance, prejudice, and doxastic wronging.

In this talk, I would like to work out the question: What is Wittgenstein’s disagreement with Russell on the issue of solipsism? The disagreement is, in principle, clear-cut. On Russell’s view, the general sentence of solipsism [S] has sense. Yet Wittgenstein relegates S to nonsense. But why is this so? I show that this disagreement is crucially grounded in a private language argument which, together with Diamond (2000), I find in the early Wittgenstein’s Tractatus (1921). It is this private language argument which justifies Wittgenstein’s claim that unique ownership is nonsensical and ultimately his difference with Russell on the status of S.

My plan is as follows. First, I present the issue of solipsism as a self-refuting argument: in its elementary [s] and general [S] form. Then I present an account on which Wittgenstein’s disagreement with Russel on the issue of solipsism is explained by their differing views of generality: Russell’s view of generality allows him to quantify indirectly over private objects in other minds, such that S has sense; whereas this is not possible on Wittgenstein’s view of generality, such that S is relegated to nonsense. I question the explanatory adequacy of this account by noting that it is irrelevant to the status of s, and hence of S, whether the object which it names is private or not. In response, with Diamond (2000), I uncover an affinity between the Tractatus and the PLA invented by the later Wittgenstein in his Philosophical Investigations (1953). By developing the Tractarian PLA, I can demonstrate that Wittgenstein’s claim to the nonsensicality of unique ownership alone entails the nonsensicality of S. This claim is then what crucially grounds his criticism of Russell’s response to solipsism.

As such, I show that Wittgenstein’s contention that S is nonsensical follows no more from his view of generality [the inability to quantify over private objects in other minds] than from his claim to the nonsensicality of unique ownership [that it is incoherent to conceive of experience as belonging to a unique ‘I’]. It might then be said that the mystical is not how, but that, Russell quantifies over private objects in other minds (cf. TLP 6.44).

Wittgenstein’s On Certainty inspired a contemporary epistemology trend called Hinge Epistemology. Regarding the nature of hinges, there are various accounts; this paper challenges one of these, namely Giovanni Mion’s communitarian account. In contrast to other hinge epistemologists like Moyal-Sharrock, Pritchard, and Coliva, who focus on the epistemic dimension of hinges, Mion shifts the focus to the social dimension, claiming that all hinges function as social norms governing the mechanisms of social inclusion and exclusion within linguistic communities, as well as inhibiting our altruistic behaviors. To challenge Mion’s account, I provide a counterexample based on what I term Agape hinges, such as the Christian hinge that “we should love our enemies.” This type of hinge encourages kindness and empathy, advocating for universal love even toward our enemies. Agape hinges can be interpreted in two ways to challenge Mion’s framework. Under the first interpretation, if Christians who embrace the Agape hinge include their enemies as members of their community, this serves as a direct counterexample to Mion’s claim that hinges serve to delineate community boundaries. In this view, the Agape hinge actively works to erase boundaries, embodying an unconditional inclusion that operates outside the mechanisms of inclusion and exclusion as conceived by Mion. Under the second interpretation, Christians who accept this hinge do not consider their enemies as community members; rather, they recognize that they should extend kindness to outsiders. This challenges Mion’s other claim that hinges inherently inhibit our altruism by delineating community boundaries. Furthermore, as population growth and the process of globalization have made our communities increasingly international, incorporating diverse cultures within a single society, this paper highlights that hinges vary in the social roles they play. Because of this diversity, no single account can adequately cover every case. Rather, to understand how hinges function across different social contexts, we need multiple, context-sensitive accounts to distinguish between hinges that regulate social inclusion/exclusion and those that serve other purposes.

The recent debate on moral certainty has renewed interest in the ethical import of Wittgenstein’s later philosophy. Drawing from On Certainty, proponents of the moral certainty framework argue that some moral commitments function as “basic moral certainties” or “hinges” that lie beyond doubt, explanation, or justification (Goodman 1982; Pleasants 2008; Eriksen et al. 2022). Moral certainties ground our conception of morality: they seem impossible to justify and doubt, within the context of a moral discussion. The Wittgensteinian notion of certainty, then, provides a framework to make sense of these unexplainable and unjustifiable beliefs which ground our sense of morality beyond any doubt. However, this framework faces significant hurdles: it struggles to evade the charge of moral relativism (Brice 2013; Rummens 2013); its central examples are still under discussion (Burley 2010; Fairhurst 2019; Glock 2022; Kusch 2022); and it raises difficult metaethical questions regarding the choice between foundationalism and contextualism in moral epistemology (Hermann 2015). Here, I want to explore a different path. I think that Wittgenstein’s philosophy is of crucial relevance for the debate on moral certainty well beyond the scope and limits of On Certainty. The moral certainty framework touches on something very important – namely, that some of our (moral) beliefs are beyond doubt – but this central insight can be better captured, in a way that does not raise the same number of issues, through Wittgenstein’s philosophy of mathematics. Wittgenstein’s philosophy of mathematics revolves around the idea that mathematical propositions are not descriptions of facts or states of affairs, but “apparatus propositions” with a “preparatory use”: that is, they are a sort of grammatical proposition, clarifying and preparing the use of certain terms. Following Diamond (2019), we can consider these propositions as “guides for thought”. Similarly, fundamental moral propositions do not describe sui generis moral facts, but rather function as guides that orient our thinking, judging, and acting. I argue that the grammatical model better captures the moral certainty framework’s central insight, but it also faces a residual difficulty. One peculiarity of grammatical propositions is that they seem “absolutely true”: their negation results in a collapse of meaning (Anscombe 1963). David Wiggins has claimed that the same applies to many ethical propositions, arguing, for example, that «there is nothing else to think but that slavery is unjust and insupportable» (Wiggins 1991): that is, we wouldn’t understand what the concepts “unjust” and “insupportable” mean, if we denied that slavery is unjust and insupportable. In this view, then, moral propositions function as guides for our moral thinking, acting as grammatical remarks that orient our thought in the application of certain crucial moral terms. Consequently, this “grammatical” framework makes sense of the central insight of the moral certainty framework but avoids its most troubling shortcomings. But it is also vulnerable to an important objection: in fact, even our most deeply held moral beliefs are open to real doubt in a way that mathematical or hinge propositions are not, suggesting that the grammar of moral certainty is not simply continuous with its mathematical counterpart. I will explore this objection in the last section of the paper.