Manuel Barrantes

The bridges of Königsberg case has been widely cited in recent philosophical discussions on scientific explanation as a potential example of a structural explanation of a physical phenomenon. However, when discussing this case, different authors have focused on two different versions, depending on what they take the explanandum to be. In one version, the explanandum is the _failure_ of a given individual in performing an Eulerian walk over the bridge system. In the other version, the explanandum is the _impossibility_ of performing an Eulerian walk over the bridges. The goal of this paper is to show that only the latter version amounts to a real case of a structural explanation. I will also suggest how to fix the first version, and show how my remarks apply to other purported cases of structural explanations.

"A theoretical juice extractor: The role of mathematics in scientific explanation". There have recently been proposed cases where, supposedly, mathematics would play a genuinely explanatory role in science. These have been divided into those situations where the explanatory role would be played by mathematical operations, and those where it would be played by mathematical entities. In this article, I analyze some of these purported cases and argue that claims that mathematics can be genuinely explanatory are unfounded. Throughout my discussion, I emphasize the representational role of mathematics, as opposed to its supposed explanatory role: the role of mathematics, even in the cases that I discuss, is to represent physical facts and help draw inferences about those facts

In this paper I defend an intermediate position between the ‘bare mathematical results’ view and the ‘transmission’ view of mathematical explanations of physical phenomena (MEPPs). I argue that, in MEPPs, it is not enough to deduce the explanandum from the generalizations cited in the explanans. Rather, we must add information regarding why those generalizations obtain. However, I also argue that it is not necessary to provide explanatory proofs of the mathematical theorems that represent those generalizations. I illustrate this with the bridges of Königsberg case.

“Structuralism, Fictionalism, and the Applicability of Mathematics in Science”. This article has two objectives. The first one is to review some of the most important questions in the contemporary philosophy of mathematics: What is the nature of mathematical objects? How do we acquire knowledge about these objects? Should mathematical statements be interpreted differently than ordinary ones? And, finally, how can we explain the applicability of mathematics in science? The debate that guides these reflections is the one between mathematical realism and anti-realism. On the other hand, the second objective is to discuss the arguments that use the applicability of mathematics in science to justify mathematical realism, and show that none of them reaches its aim. To this end, we will distinguish three aspects of the problem of the applicability of mathematics: the utility of mathematics in science, the unexpected utility of some mathematical theories, and the apparent indispensability of mathematics in our best scientific theories - in particular, in our best scientific explanations. Finally, I argue that none of these aspects constitutes a reason to adopt mathematical realism.

In the Principles of Philosophy, Descartes explains several observable phenomena showing that they are caused by special arrangements of unobservable microparticles. Despite these microparticles being unobservable, many passages suggest that he was very confident that these explanations were correct. In other passages, however, Descartes points out that these explanations merely hold the status of “suppositions” or “conjectures” that could be wrong. My main goal in this chapter is to clarify this apparent conflict. I argue first that for Descartes it was indeed possible to have knowledge of unobservable particles and structures, and that the possibility of natural explanations being wrong should be understood as these explanations not being absolutely certain, but only morally certain. I use the debate in contemporary philosophy of science between scientific realism and antirealism as a framework to understand how Cartesian explanations work. Especifically, I argue that Cartesian explanations rely on what Ernan McMullin calls retroduction, which is a mode of inference that justifies beliefs in concrete unobservable entities and processes, based on considerations such as simplicity, coherence, etc. This kind of justification is of common use among scientific realists. Thus, another goal of this chapter is to highlight the relevance of Descartes’ ideas about explanation in the contemporary debate on scientific realism.

The Enhanced Indispensability Argument appeals to the existence of Mathematical Explanations of Physical Phenomena to justify mathematical Platonism, following the principle of Inference to the Best Explanation. In this paper, I examine one example of a MEPP—the explanation of the 13-year and 17-year life cycle of magicicadas—and argue that this case cannot be used defend the EIA. I then generalize my analysis of the cicada case to other MEPPs, and show that these explanations rely on what I will call ‘optimal representations’, which are representations that capture all that is relevant to explain a physical phenomenon at a specified level of description. In the end, because the role of mathematics in MEPPs is ultimately representational, they cannot be used to support mathematical Platonism. I finish the paper by addressing the claim, advanced by many EIA defendants, that quantification over mathematical objects results in explanations that have more theoretical virtues, especially that they are more general and modally stronger than alternative explanations. I will show that the EIA cannot be successfully defended by appealing to these notions.

We argue that there is no tension between Reid's description of science and his claim that science is based on the principles of common sense. For Reid, science is rooted in common sense since it is based on the idea that fixed laws govern nature. This, however, does not contradict his view that the scientific notions of causation and explanation are fundamentally different from their common sense counterparts. After discussing these points, we dispute with Cobb's and Benbaji's interpretations of Reid's views on causation and explanation. Finally, we present Reid's views from the perspective of the contemporary debate on scientific explanation