Lately, philosophers of mathematics have been exploring the notion of mathematical explanation within mathematics. This project is supposed to be analogous to the search for the correct analysis of scientific explanation. I argue here that given the way philosophers have been using “ explanation,” the term is not applicable to mathematics as it is in science.

Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) coherent and places the problem of reliability within the province of the philosophy of mathematics. (shrink)

Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) coherent (in an informal sense) and places the problem of reliability within the province of the philosophy of mathematics. (shrink)

Scientific and ‘intuitive’ or ‘folk’ theories are typically characterized as serving three critical functions: prediction, explanation, and control. While prediction and control have clear instrumental value, the value of explanation is less transparent. This paper reviews an emerging body of research from the cognitive sciences suggesting that the process of seeking, generating, and evaluating explanations in fact contributes to future prediction and control, albeit indirectly by facilitating the discovery and confirmation of instrumentally valuable theories. Theoretical and empirical considerations also suggest (...) why explanations may nonetheless feel intrinsically valuable. The paper concludes by considering some implications of the psychology of explanation for a naturalized philosophy of explanation. (shrink)

Philosophers who regard some mathematical proofs as explaining why theorems hold, and others as merely proving that they do hold, disagree sharply about the explanatory value of proofs by mathematical induction. I offer an argument that aims to resolve this conflict of intuitions without making any controversial presuppositions about what mathematical explanations would be.

Unlike explanation in science, explanation in mathematics has received relatively scant attention from philosophers. Whereas there are canonical examples of scientific explanations, there are few examples that have become widely accepted as exhibiting the distinction between mathematical proofs that explain why some mathematical theorem holds and proofs that merely prove that the theorem holds without revealing the reason why it holds. This essay offers some examples of proofs that mathematicians have considered explanatory, and it argues that these examples suggest a (...) particular account of explanation in mathematics. The essay compares its account to Steiner's and Kitcher's. Among the topics that arise are proofs that exploit symmetries, mathematical coincidences, brute-force proofs, simplicity in mathematics, merely clever proofs, and proofs that unify what other proofs treat as separate cases. (shrink)

To explain the phenomena in the world of our experience, to answer the question “why?” rather than only the question “what?”, is one of the foremost objectives of all rational inquiry; and especially, scientific research in its various branches strives to go beyond a mere description of its subject matter by providing an explanation of the phenomena it investigates. While there is rather general agreement about this chief objective of science, there exists considerable difference of opinion as to the function (...) and the essential characteristics of scientific explanation. In the present essay, an attempt will be made to shed some light on these issues by means of an elementary survey of the basic pattern of scientific explanation and a subsequent more rigorous analysis of the concept of law and of the logical structure of explanatory arguments. (shrink)

Although prediction has been largely absent from discussions of explanation for the past 40 years, theories of explanation can gain much from a reintroduction. I review the history that divorced prediction from explanation, examine the proliferation of models of explanation that followed, and argue that accounts of explanation have been impoverished by the neglect of prediction. Instead of a revival of the symmetry thesis, I suggest that explanation should be understood as a cognitive tool that assists us in generating new (...) predictions. This view of explanation and prediction clarifies what makes an explanation scientific and why inference to the best explanation makes sense in science. *Received August 2009; revised September 2009. †To contact the author, please write to: Department of Philosophy, University of Tennessee, 801 McClung Tower, Knoxville, TN 37920‐0480; e‐mail: [email protected]. (shrink)

The three cardinal aims of science are prediction, control, and explanation; but the greatest of these is explanation. Also the most inscrutable: prediction aims at truth, and control at happiness, and insofar as we have some independent grasp of these notions, we can evaluate science’s strategies of prediction and control from the outside. Explanation, by contrast, aims at scientific understanding, a good intrinsic to science and therefore something that it seems we can only look to science itself to explicate.