This paper has two aims. First, we argue that Wittgenstein’s notion of petrification can be used to explain phenomena in advanced mathematics, sometimes better than more popular views on mathematics, such as formalism, even though petrification usually suffers from a diet of examples of a very basic nature (in particular a focus on addition of small numbers). Second, we analyse current disagreements on the absolute undecidability of CH under the notion of petrification and hinge epistemology. We argue that in contemporary (...) set theory the usage of construction techniques for set-theoretic models in which the Continuum Hypothesis holds and those in which it fails have petrified into the normative demand that CH remain undecidable. That is, the continuous and successful practices involving the construction of various set-theoretic models now act as a normative hinge shared among practitioners, i.e., have normative force in the discipline. However, not all hinges are universal, which is why we find disagreements in set theory. We will show that this is a refinement of, and partially conflicts with, the arguments presented by set theorist Joel David Hamkins. (shrink)

In this paper, I will defend a communitarian perspective on the so-called “hinge propositions” (hinges, for short). Accordingly, I will argue that hinges play a normative role, in the sense that, among other things, they govern the mechanisms of social inclusion/exclusion. In particular, I will examine the so-called “religious hinges”; and I will argue that such hinges, being the product of mere indoctrination, are particularly effective in shaping boundaries among communities. Finally, with the help of Peter Munz's theory of altruism, (...) I will attempt to explain why religious hinges play the role they do. (shrink)

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). (shrink)

Epistemologists have developed a diverse group of theories, known as hinge epistemology, about our epistemic practices that resort to and expand on Wittgenstein's concept of ‘hinges’ in On Certainty. Within hinge epistemology there is a debate over the epistemic status of hinges. Some hold that hinges are non-epistemic (neither known, justified, nor warranted), while others contend that they are epistemic. Philosophers on both sides of the debate have often connected this discussion to Wittgenstein's later views on mathematics. Others have directly (...) questioned whether there are mathematical hinges, and if so, these would be axioms. Here, we give a hinge epistemology account for mathematical practices based on their contextual dynamics. We argue that 1) there are indeed mathematical hinges (and they are not axioms necessarily), and 2) a given mathematical entity can be used contextually as an epistemic hinge, a non-epistemic hinge, or a non-hinge. We sustain our arguments exegetically and empirically. (shrink)

Susan Stebbing is often portrayed as indebted to G. E. Moore for her ideas concerning the relationship between common sense and philosophy and about analysis. By focusing mostly on her article “The...