Coliva (2020a) 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 riverbed 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 Ackermann (1988) and Wilson (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).

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The paper begins by arguing that additional theorizing about mathematical practice is needed in order to ground appeals to truly useful notions of the virtues in mathematics. It then aims to contribute to this theorizing, first, by characterizing mathematical practice as being epistemic and "objectual" in the sense of Knorr Cetina ( 2001). Then, it elaborates a MacIntyrean framework for extracting conceptions of the virtues related to mathematical practice so understood. Finally, it makes the case that Wittgenstein's methodology for approaching mathematics and its practice provides the appropriate perspective from which to undertake the actual investigation of mathematical practice within this MacIntyrean framework for the virtues.During each stage of thinking through mathematical practice by these means, places where new virtue-theoretic questions are opened up for investigation are noted and briefly explored.

Far from being unwelcome or impossible in a mathematical setting, indeterminacy in various forms can be seen as playing an important role in driving mathematical research forward by providing "sources of newness" in the sense of [Hutter and Farías(2017)]. I argue here that mathematical coincidences, phenomena recently under discussion in the philosophy of mathematics, are usefully seen as inducers of indeterminacy and as put to work in guiding research directions. I suggest that to call a pair of mathematical facts (merely) a coincidence is roughly to suggest that the investigation of connections between these facts isn't worthwhile. To say of this pair, "That's no coincidence!" is to suggest just the opposite. I further argue that this perspective on mathematical coincidence, which pays special attention to what mathematical coincidences do, may provide us with a better view of what mathematical coincidences are than extant accounts. I close by reflecting on how understanding mathematical coincidences as generating indeterminacy accords with a conception of mathematical research as ultimately aiming to reduce indeterminacy and complexity to triviality as proposed in [Rota(1997)]. Keywords indeterminacy • mathematical coincidence • mathematical practice3 On mathematical coincidence, see, e.g., [Baker(2009)], [Lange(2010)], and [Lange(2017), Ch. 8]. See also [Davis(1981)].4 [Homer, Odyssey 11.487-503] 5 The idea of "sourcing newness" is drawn from [Hutter and Farías(2017)]. 6 [Dewey(1938), 104-105, emphasis in the original] 7 See [Dewey(1938), 105-106]. Dewey's use of 'indeterminate situation' in this semitechnical sense helps block Russell's "counterexample" that, according to Dewey, a bricklayer's dealings with a pile of bricks is a form of inquiry [Russell(1946[Russell( /1961. See [Gale(1959)] for more on Russell on Dewey on inquiry.

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