Christopher Pincock

This paper concerns the debate on the nature of Rudolf Carnap'sproject in his 1928 book The Logical Structure of the Worldor Aufbau. Michael Friedman and Alan Richardson haveinitiated much of this debate. They claim that the Aufbauis best understood as a work that is firmly grounded inneo-Kantian philosophy. They have made these claims in oppositionto Quine and Goodman's ``received view'' of the Aufbau. Thereceived view sees the Aufbau as an attempt to carry out indetail Russell's external world program. I argue that both sidesof this debate have made errors in their interpretation ofRussell. These errors have led these interpreters to misunderstandthe connection between Russell's project and Carnap's project.Russell in fact exerted a crucial influence on Carnap in the1920s. This influence is complicated, however, due to the factthat Russell and Carnap disagreed on many philosophical issues. Iconclude that interpretations of the Aufbau that ignoreRussell's influence are incomplete.

This paper begins by distinguishing intrinsic and extrinsic contributions of mathematics to scientific representation. This leads to two investigations into how these different sorts of contributions relate to confirmation. I present a way of accommodating both contributions that complicates the traditional assumptions of confirmation theory. In particular, I argue that subjective Bayesianism does best accounting for extrinsic contributions, while objective Bayesianism is more promising for intrinsic contributions.

Conflicting accounts of the role of mathematics in our physical theories can be traced to two principles. Mathematics appears to be both (1) theoretically indispensable, as we have no acceptable non-mathematical versions of our theories, and (2) metaphysically dispensable, as mathematical entities, if they existed, would lack a relevant causal role in the physical world. I offer a new account of a role for mathematics in the physical sciences that emphasizes the epistemic benefits of having mathematics around when we do science. This account successfully reconciles theoretical indispensability and metaphysical dispensability and has important consequences for both advocates and critics of indispensability arguments for platonism about mathematics.

Explanations of three different aspects of the rainbow are considered. The highly mathematical character of these explanations poses some interpretative questions concerning what the success of these explanations tells us about rainbows. I develop a proposal according to which mathematical explanations can highlight what is relevant about a given phenomenon while also indicating what is irrelevant to that phenomenon. This proposal is related to the extensive work by Batterman on asymptotic explanation with special reference to Batterman’s own discussion of the rainbow

Forthcoming, Studies in the History and Philosophy of Science Abstract: The epistemic problem of assessing the support that some evidence confers on a hypothesis is considered using an extended example from the history of meteorology. In this case, and presumably in others, the problem is to develop techniques of data analysis that will link the sort of evidence that can be collected to hypotheses of interest. This problem is solved by applying mathematical tools to structure the data and connect it to the competing hypotheses. I conclude that mathematical innovations provide crucial epistemic links between evidence and theories precisely because the evidence and theories are mathematically described.

This paper identifies one way that a mathematical proof can be more explanatory than another proof. This is by invoking a more abstract kind of entity than the topic of the theorem. These abstract mathematical explanations are identified via an investigation of a canonical instance of modern mathematics: the Galois theory proof that there is no general solution in radicals for fifth-degree polynomial equations. I claim that abstract explanations are best seen as describing a special sort of dependence relation between distinct mathematical domains. This case study highlights the importance of the conceptual, as opposed to computational, turn of much of modern mathematics, as recently emphasized by Tappenden and Avigad. The approach adopted here is contrasted with alternative proposals by Steiner and Kitcher