On Certainty,Change, and “Mathematical Hinges”

Annalisa Coliva (Int J Study Skept 10(3–4):346–366, 2020) asks, “Are there mathematical hinges?” I argue here, against Coliva’s own conclusion, that there are. I further claim that this affirmative answer allows a case to be made for taking the concept of a hinge to be a useful and general-purpose tool for studying mathematical practice in its real complexity. Seeing how Wittgenstein can, and why he would, countenance mathematical hinges additionally gives us a deeper understanding of some of his latest thoughts on mathematics. For example, a view of how mathematical hinges relate to Wittgenstein’s well-known river-bed analogy enables us to see how his way of thinking about mathematics can account nicely for a “dynamics of change” within mathematical research—something his philosophy of mathematics has been accused of missing (e.g., by Robert Ackermann (Wittgenstein’s city, The University of Massachusetts Press, Amherst, 1988) and Mark Wilson (Wandering significance: an essay on conceptual behavior, Oxford University Press, Oxford, 2006). Finally, the perspective on mathematical hinges ultimately arrived at will be seen to provide us with illuminating examples of how our conceptual choices and theories can be ungrounded but nevertheless the right ones (in a sense to be explained).

     

Minds and Dialogues in Analytic Spaces—Negotiating Certainty and Uncertainty Reply to Commentary

This paper is a response to a critique by C. Brooks Brenneis of the written account of a clinical enactment that featured countertransference dissociative effects in treatment and in supervision. The response focuses on the need to analyze power in overlapping clinical dyads, the effect of transpersonal transmission of affect states and disavowed experience, and the relation of power dynamics to transference and to meaning making.     

Children's and Adults’ Judgments of the Certainty of Deductive Inferences,Inductive Inferences,and Guesses

Children and adults rated their own certainty following inductive inferences, deductive inferences, and guesses. Beginning in kindergarten, participants rated deductions as more certain than weak inductions or guesses. Deductions were rated as more certain than strong inductions beginning in Grade 3, and fourth-grade children and adults differentiated strong inductions, weak inductions, and informed guesses from pure guesses. By Grade 3, participants also gave different types of explanations for their deductions and inductions. These results are discussed in relation to children's concepts of cognitive processes, logical reasoning, and epistemological development.     

What Can Artificial Intelligence Learn from Wittgenstein＇s On Certainty？.

Meta-philosophically speaking, the philosophy of artificial intelligence （AI） is intended not only to explore the theoretical possibility of building ＂thinking machines,＂ but also to reveal philosophical implications of specific AI approaches. Wittgenstein＇s comments on the analytic/empirical dichotomy may offer inspirations for AI in the second sense. According to his ＂river metaphor＂ in On Certainty, the analytic/empirical boundary should be delimited in a way sensitive to specific contexts of practical reasoning. His proposal seems to suggest that any cognitive modeling project needs to render the system context-sensitive by avoiding representing large amounts of truisms in its cognitive processes, otherwise neither representational compactness nor computational efficiency can be achieved. In this article, different AI approaches （like the Common Sense Law of Inertia approach, the Bayesian approach and the connectionist approach） will be critically evaluated under the afore-mentioned Wittgensteinian criteria, followed by the author＇s own constructive suggestion on what AI needs to try to do in the near future.     

What is wrong with Allais' Certainty Effect?

Several choice situations are constructed to explore whether the violation of expected utility theory in an Allais paradox choice situation can be attributed to what Tversky and Kahneman (1986) describe as the Allais certainty effect. Problems are developed where the Allais certainty effect would be expected to occur but results show it does not. Other problems demonstrate that the Allais ‘paradox’ is observed in the absence of the Allais certainty effect. The study concludes that, although expected utility theory is known to be wrong through the Allais paradox, the Allais certainty effect does not appear to be able to rescue it.     

The Combined Effect of the Severity and the Certainty of Threatened Penalties: Additive or Interactive?

Research carried out to date has yielded inconsistent results regarding the manner in which the severity and certainty of threatened penalties combine to affect deterrence. While some studies presented evidence showing an interactive combination, others yielded an additive pattern. The present study tested and supported the proposition that the process whereby the two variables combine depends on the type of penalty with which a person is threatened. In the case of penalties the severity of which can be psychologically weighed in interval values, the combined effect is multiplicative or interactive. On the other hand, when the severity of a penalty can be expressed only ordinally, the combined effect is largely additive.     

Reading On Certainty through the Lens of Cavell: Scepticism,Dogmatism and the ‘Groundlessness of our Believing’

Abstract

While Cavell is well known for his reinterpretation of the later Wittgenstein, he has never really engaged himself with post-Investigations writings like On Certainty. This collection may, however, seem to undermine the profoundly anti-dogmatic reading of Wittgenstein that Cavell has developed. In addition to apparently arguing against what Cavell calls ‘the truth of skepticism’ – a phrase contested by other Wittgensteinians – On Certainty may seem to justify the rejection of whoever dares to question one’s basic presuppositions. According to On Certainty, or so it seems, the only right response to someone with different certainties is a reproach like ‘Fool!’ or ‘Heretic!’. This article aims to show that On Certainty need not be taken to prove Cavell wrong. It explains that Wittgenstein, in line with the first two parts of The Claim of Reason, does not reject scepticism out of hand but rather questions the sceptic’s self-understanding. Using arguments from Part Three of The Claim, the article moreover argues that a confrontation with divergence calls for self-examination rather than self-righteousness. Precisely because Wittgenstein acknowledges ‘the groundlessness of our believing’ or, in Cavellian terms, ‘the truth of skepticism’, he is not the authoritarian thinker that some have taken him to be.     

A Comfortable Sureness: Knowledge,Animality and Conceptual Investigations in Wittgenstein’s On Certainty

This essay analyses some remarks of Wittgenstein’s On Certainty in which Wittgenstein compares human behaviour to that of animals and says he wants to consider man as an animal. The essay’s main purpose is to show that these remarks are essentially understood as part and parcel of what Wittgenstein calls “conceptual investigations” and that, consequently, they give little support to On Certainty’s naturalistic interpretations. A second purpose of the essay is to show that Wittgenstein does not intend to combat the use of “I know” in contexts such as those evoked by Moore; rather he wants to draw attention to the different ways in which we say or can say “I know.”

     

Parental Influences on Young Women’s Certainty about Their Career Aspirations

Identity control theory (Kerpelman, Pittman, & Lamke, Journal of Adolescent Research, 12: 325–346, 1997) and the literature on possible selves (Markus & Nurius, American Psychologist, 41: 954–969, 1986) and family influences on adolescents’ vocational choices (e.g., Otto, Journal of Career Development, 27: 111–117, 2000; Whiston & Keller, The Counseling Psychologist, 32: 493–568, 2004) were used to guide this study of young women’s career aspirations. How mother–daughter and father–daughter relationships were associated with young women’s certainty about their anticipated future careers was addressed with data from 304 female undergraduates. Findings indicated that parent–daughter connectedness predicted the young women’s anticipated distress, as well as their willingness to change to fit parental views should their parents disagree with their career aspirations. In addition, anticipated distress mediated associations between willingness to change and connectedness with parents, and between willingness to change and father–daughter career discussions. Discussing career goals with mother increased mother’s influence, but decreased father’s influence, on daughter’s career certainty under conditions of father–daughter disagreement about career choice. The ability to separate her own feelings from those of her father was particularly important in reducing the daughter’s willingness to change her career goals to fit maternal or paternal expectations. Collectively, these findings offer additional insights about the role of parent–adolescent relationships in understanding young women’s career aspirations and note the importance of considering the distinct influences of mothers and fathers on their daughters’ career goals and plans.     

LGBTQ Adolescents’ Positive and Negative Emotions and Experiences in U.S. High Schools

In a national survey of more than 19,000 U.S. high school students, we compared how LGBTQ youth and their non-LGBTQ peers felt at school and how they perceived social and academic experiences. We examined differences in emotions and school experiences across gender identities, sexual identities, and their intersections. LGBTQ adolescents reported significantly more frequent negative emotions and bullying, consistent with previous research. LGBTQ students also reported less frequent experiences of positive emotions at school and less frequent positive school experiences (i.e., positive peer and teacher relationships, subjective task value, and persistence support). Students who were both gender and sexual identity minority reported the most frequent negative and least frequent positive experiences at school, compared to students who were neither a gender or sexual identity minority. Analyses of the intersection of gender and sexual identity showed that heterosexual male students experienced more frequent positive emotions and school experiences, and fewer negative emotions and bullying, compared to all other groups. We discuss how these differences might be addressed through school interventions and future research.