We introduce a framework that can be used to model both mathematics and human reasoning about mathematics. This framework involves stochastic mathematical systems (SMSs), which are stochastic processes that generate pairs of questions and associated answers (with no explicit referents). We use the SMS framework to define normative conditions for mathematical reasoning, by defining a “calibration” relation between a pair of SMSs. The first SMS is the human reasoner, and the second is an “oracle” SMS that can be interpreted as (...) deciding whether the question–answer pairs of the reasoner SMS are valid. To ground thinking, we understand the answers to questions given by this oracle to be the answers that would be given by an SMS representing the entire mathematical community in the infinite long run of the process of asking and answering questions. We then introduce a slight extension of SMSs to allow us to model both the physical universe and human reasoning about the physical universe. We then define a slightly different calibration relation appropriate for the case of scientific reasoning. In this case the first SMS represents a human scientist predicting the outcome of future experiments, while the second SMS represents the physical universe in which the scientist is embedded, with the question–answer pairs of that SMS being specifications of the experiments that will occur and the outcome of those experiments, respectively. Next we derive conditions justifying two important patterns of inference in both mathematical and scientific reasoning: (i) the practice of increasing one’s degree of belief in a claim as one observes increasingly many lines of evidence for that claim, and (ii) abduction, the practice of inferring a claim’s probability of being correct from its explanatory power with respect to some other claim that is already taken to hold for independent reasons. (shrink)

The notions of belief and truth frequently interact in philosophical discourse but, surprisingly, a coherent semantics for such discourse is still wanting. Indeed, a number of puzzles stand in way of a satisfactory semantic account of the notion of truth in doxastic contexts. In this paper we discuss these puzzles and develop a more satisfactory semantic account that combines ideas from contextualist theories of attitude reports and Awareness semantics for non-idealized belief.

Conditionalization governs how to reallocate credence in light of new evidence. One prominent argument in favor of Conditionalization holds that an agent who violates it is vulnerable to a diachronic Dutch book: a series of acceptable bets offered at multiple times that inflict a sure loss. van Fraassen argues that an agent who violates the Principle of Reflection is likewise vulnerable to a diachronic Dutch book. He concludes that agents should conform to both Conditionalization and Reflection. Some authors reply that (...) Reflection is implausible and hence that there must be something wrong with diachronic Dutch book arguments. Other authors try to isolate a principled difference between the Dutch book argument for Conditionalization and the Dutch book argument for Reflection, such that the former argument may succeed even though the latter fails. I pursue a version of this strategy. I contend that, once we properly elucidate the notion of sure loss, non‐reflectors are not vulnerable to a sure loss. An agent who violates Reflection is not thereby subject to a diachronic Dutch book. Appearances to the contrary result from an unmotivated focus upon an overly narrow set of gambling scenarios. (shrink)

So-called “traditional epistemology” and “Bayesian epistemology” share a word, but it may often seem that the enterprises hardly share a subject matter. They differ in their central concepts. They differ in their main concerns. They differ in their main theoretical moves. And they often differ in their methodology. However, in the last decade or so, there have been a number of attempts to build bridges between the two epistemologies. Indeed, many would say that there is just one branch of philosophy (...) here—epistemology. There is a common subject matter after all. In this paper, we begin by playing the role of a “bad cop,” emphasizing many apparent points of disconnection, and even conflict, between the approaches to epistemology. We then switch role, playing a “good cop” who insists that the approaches are engaged in common projects after all. We look at various ways in which the gaps between them have been bridged, and we consider the prospects for bridging them further. We conclude that this is an exciting time for epistemology, as the two traditions can learn, and have started learning, from each other. (shrink)