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)

This paper revisits Wittgenstein’s heavily criticized claims about the admissibility of inconsistencies in mathematics. It argues from the perspective of mathematics as a tool and combines material from the history and practice of engineering that makes Wittgenstein’s claims about contradiction and inconsistency look much more plausible. Against this background, the paper interprets passages from Wittgenstein, including his exchange with Alan Turing where he highlights that basic laws of thought are at issue and that reflecting on them would be “the most (...) important thing” he has talked about. (shrink)

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)