Realists about laws of nature and their Humean opponents disagree on whether laws ‘govern’. An independent commitment to the ‘governing conception’ of laws pushes many towards the realist camp. Despite its significance, however, no satisfactory account of governance has been offered. The goal of this article is to develop such an account. I base my account on two claims. First, we should distinguish two notions of governance, ‘guidance’ and ‘production’, and secondly, explanatory phenomena other than laws are also candidates for (...) governance. My goal is to develop a unified account which captures both guidance and production as well as the governance of phenomena, such as essence and logical consequence. The account of governance I develop belongs to the family of modal accounts, which was popularized by David Armstrong, but it also employs essentialist resources. If successful, this modal-essentialist account not only reveals the costs that proponents of governance incur, but it also puts that important notion on a solid theoretical foundation. (shrink)

Philosophers have developed several detailed accounts of what makes some mathematical proofs explanatory. Significantly less attention has been paid, however, to what makes some proofs more explanatory than other proofs. That is problematic, since the reasons for thinking that some proofs explain are also reasons for thinking that some proofs are more explanatory than others. So in this paper, I develop an account of comparative explanation in mathematics. I propose a theory of the `at least as explanatory as' relation among (...) mathematical proofs. (shrink)

Michael Weisberg and Kenneth Reisman argue that the Volterra Principle can be derived from multiple predator-prey models and that, therefore, the Volterra Principle is a prime example for robustness analysis. In the current article, I give new results regarding the Volterra Principle, extending Weisberg’s and Reisman’s work, and I discuss the consequences of these results for robustness analysis. I argue that we do not end up with multiple, independent models but rather with one general model. I identify the kind of (...) situation in which this generalization approach may occur, and I analyze the generalized Volterra Principle from an explanatory perspective. (shrink)

The present paper provides an analysis of Euler’s solutions to the Königsberg bridges problem. Euler proposes three different solutions to the problem, addressing their strengths and weaknesses along the way. I put the analysis of Euler’s paper to work in the philosophical discussion on mathematical explanations. I propose that the key ingredient to a good explanation is the degree to which it provides relevant information. Providing relevant information is based on knowledge of the structure in question, graphs in the present (...) case. I also propose computational complexity and logical strength as measures of relevant information. (shrink)

The present paper provides an analysis of Euler’s solutions to the Königsberg bridges problem. Euler proposes three different solutions to the problem, addressing their strengths and weaknesses along the way. I put the analysis of Euler’s paper to work in the philosophical discussion on mathematical explanations. I propose that the key ingredient to a good explanation is the degree to which it provides relevant information. Providing relevant information is based on knowledge of the structure in question, graphs in the present (...) case. I also propose computational complexity and logical strength as measures of relevant information. (shrink)

The author of “Parsimony and inference to the best mathematical explanation” argues for platonism by way of an enhanced indispensability argument based on an inference to yet better mathematical optimization explanations in the natural sciences. Since such explanations yield beneficial trade-offs between stronger mathematical existential claims and fewer concrete ontological commitments than those involved in merely good mathematical explanations, one must countenance the mathematical objects that play a theoretical role in them via an application of the relevant mathematical results. The (...) nominalist’s challenge is thus to undermine the platonistic force of such explanations by way of alternative nominalistic ones. The author’s contention is that such nominalistic explanations should provide a paraphrase of the proofs of the mathematical results being applied. There are reasons to doubt that proofs, construed here as formal derivations, actually contribute to the platonistc force to be undermined and, by parity, that nominalized proofs should bear responsability for the corresponding undermining. A discussion of two examples and of associated arguments by Lange, Pincock, Steiner and Tallant, point to a a wealth of worries concerning the construal of this explanatory role. Among those figure the distinction between the weak and strong role of proofs, the distinction between causal or “ordinary” explanations and genuine mathematical ones, and the unifying role of optimization explanations. More generally, the very idea that the explanatory advantages yielded by applied mathematical claims may be construed as gradual or progressive and the associated notion that the feasibility of their nominalistic paraphrases decreases as the generality and force of these claim increases, deserves a closer attention. (shrink)

Mathematicians judge proofs to possess, or lack, a variety of different qualities, including, for example, explanatory power, depth, purity, beauty and fit. Philosophers of mathematical practice have begun to investigate the nature of such qualities. However, mathematicians frequently draw attention to another desirable proof quality: being motivated. Intuitively, motivated proofs contain no "puzzling" steps, but they have received little further analysis. In this paper, I begin a philosophical investigation into motivated proofs. I suggest that a proof is motivated if and (...) only if mathematicians can identify (i) the tasks each step is intended to perform; and (ii) where each step could have reasonably come from. I argue that motivated proofs promote understanding, convey new mathematical resources and stimulate new discoveries. They thus have significant epistemic benefits and directly contribute to the efficient dissemination and advancement of mathematical knowledge. Given their benefits, I also discuss the more practical matter of how we can produce motivated proofs. Finally I consider the relationship between motivated proofs and proofs which are explanatory, beautiful and fitting. (shrink)

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 i...

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss (...) and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

ABSTRACT We use Gödel’s incompleteness theorems as a case study for investigating mathematical depth. We examine the philosophical question of what the depth of Gödel’s incompleteness theorems consists in. We focus on the methodological study of the depth of Gödel’s incompleteness theorems, and propose three criteria to account for the depth of the incompleteness theorems: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of Gödel’s incompleteness theorems.

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)