If ontic dependence is the basis of explanation, there cannot be mathematical explanations. Accounting for the explanatory dependency between mathematical properties and empirical phenomena poses insurmountable metaphysical and epistemic difficulties, and the proposed amendments to the counterfactual theory of explanation invariably violate core commitments of the theory. Instead, mathematical explanations are either abstract mechanistic constitutive explanations or reconceptualizations of the explanandum phenomenon in which mathematics as such does not have an explanatory role. Explanation-like reasoning within mathematics, distinction between explanatory and nonexplanatory proofs, and comparative judgments of mathematical depth can be fully accounted for by a concept of formal understanding.

For computer simulation models to usefully inform climate risk management, uncertainties in model projections must be explored and characterized. Because doing so requires running the model many times over, and because computing resources are finite, uncertainty assessment is more feasible using models that demand less computer processor time. Such models are generally simpler in the sense of being more idealized, or less realistic. So modelers face a trade-off between realism and uncertainty quantification. Seeing this trade-off for the important epistemic issue that it is requires a shift in perspective from the established simplicity literature in philosophy of science.

Early philosophical accounts of explanation mistook the function of boundary conditions for that of contingent facts. I diagnose where this misunderstanding arose and establish that it persists. I disambiguate between two uses of the term “boundary conditions”? and argue that boundary conditions are explanatory via their roles as components of models. Using case studies from fluid mechanics and the physics of waves, I articulate four explanatory functions for boundary conditions in physics: specifying the scope of a model, enabling stable descriptions of phenomena in a model, generating descriptions of novel phenomena, and connecting models from differing theoretical backgrounds.

Researchers often pursue proof of concept research, but criteria for evaluating such research remain poorly specified. This article proposes a general framework for proof of concept research that knits together and augments earlier discussions. The framework includes prototypes, proof of concept demonstrations, and post facto demonstrations. With a case from theoretical evolutionary genetics, the article illustrates the general framework and articulates some of the reasoning strategies used within that field. This article provides both specific tools with which to understand how researchers evaluate models in theoretical evolutionary genetics and general tools that apply to proof of concept research more generally.

The thermal time hypothesis (TTH) is a proposed solution to the problem of time: a coarse-grained state determines a thermal dynamics according to which it is in equilibrium, and this defines the flow of time in generally covariant quantum theories. This article raises a series of objections to the TTH as developed by Alain Connes and Carlo Rovelli. Two technical challenges concern the relationship between thermal time and proper time and the possibility of implementing the TTH in classical theories. Three conceptual problems concern the flow of time in nonequilibrium states and the extent to which the TTH is background independent and gauge invariant.

In many ways, entropy and probability are two concepts that complement each other. But it has been argued that there is no ‘straightforward connection’ between them with a no-go thesis from Kevin Davey. However, this skeptical conclusion rests on counterexamples that fail to do justice to the entropy principle and the equivocality of the notion of probability. Proceeding from the disambiguation of probability, and acknowledging the explanatory goal of the entropy principle, it is argued that the Boltzmannian statistical mechanics account can be vindicated with a justification of the explanandum, the reference class, and the standard uniform probability measure.

The universal optimality theorem for metainduction works for epistemic agents faced with a choice among finitely many prediction methods. Eckhart Arnold and Tom Sterkenburg objected that it breaks down for infinite or unboundedly growing sets of methods. In this article the metainductive approach is defended against this challenge by extending the optimality theorem (i) to unboundedly growing sets of methods whose number grows less than exponentially in time, (ii) to sequences of methods with an application to Goodman's problem, and (iii) to infinite sets of methods whose number of predictive equivalence classes grows less than linearly in time.

A new justification of probabilism is developed that pays close attention to the structure of the underlying space of possibilities. Its central assumption is that rational numerical degrees of belief ought to be partition invariant. By means of a representation theorem, one can prove that if graded belief satisfies the resulting set of postulates, rational degrees of belief may be identified with probabilities.