In the last two decades there has been renewed interest in visualization in logic and mathematics. Visualization is usually understood in different ways but for the purposes of this article I will take a rather broad conception of visualization to include both visualization by means of mental images as well as visualizations by means of computer generated images or images drawn on paper, e.g. diagrams etc. These different types of visualization can differ substantially but I am interested in offering a (...) characterization of visualization that is as broad as possible. The article describes and explains (1) the way in which visual thinking fell into disrepute, (2) the renaissance of visual thinking in mathematics over recent decades, (3) the ways in which visual thinking has been rehabilitated in epistemology of mathematics and logic. (shrink)

In this paper we discuss three interrelated questions. First: is explanation in mathematics a topic that philosophers of mathematics can legitimately investigate? Second: are the specific aims that philosophers of mathematical explanation set themselves legitimate? Finally: are the models of explanation developed by philosophers of science useful tools for philosophers of mathematical explanation? We argue that the answer to all these questions is positive. Our views are completely opposite to the views that Mark Zelcer has put forward recently. Throughout this (...) paper, we show why Zelcer’s arguments fail. (shrink)

Scientists and laypeople alike use the sense of understanding that an explanation conveys as a cue to good or correct explanation. Although the occurrence of this sense or feeling of understanding is neither necessary nor sufficient for good explanation, it does drive judgments of the plausibility and, ultimately, the acceptability, of an explanation. This paper presents evidence that the sense of understanding is in part the routine consequence of two well-documented biases in cognitive psychology: overconfidence and hindsight. In light of (...) the prevalence of counterfeit understanding in the history of science, I argue that many forms of cognitive achievement do not involve a sense of understanding, and that only the truth or accuracy of an explanation make the sense of understanding a valid cue to genuine understanding. (shrink)

(2002). Philosophy of Science 72 (January), 198-208.

Achieving understanding of nature is one of the aims of science. In this paper we offer an analysis of the nature of scientific understanding that accords with actual scientific practice and accommodates the historical diversity of conceptions of understanding. Its core idea is a general criterion for the intelligibility of scientific theories that is essentially contextual: which theories conform to this criterion depends on contextual factors, and can change in the course of time. Our analysis provides a general account of (...) how understanding is provided by scientific explanations of diverse types. In this way, it reconciles conflicting views of explanatory understanding, such as the causal-mechanical and the unificationist conceptions. (shrink)

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)

Although in the past three decades interest in mathematical explanation revived, recent literature on the subject seems to neglect the strict connection between explanation and discovery. In this paper I sketch an alternative approach that takes such connection into account. My approach is a revised version of one originally considered by Descartes. The main difference is that my approach is in terms of the analytic method, which is a method of discovery prior to axiomatized mathematics, whereas Descartes’s approach is in (...) terms of the analytic–synthetic method, which is a heuristic pattern in already axiomatized mathematics. (shrink)

Explains how our innate pattern-making abilities allow us to perform mathematical reasoning.

When scientist investigate why things happen, they aim at giving an explanation. But what does a scientific explanation look like? In the first chapter (Theories of Scientific Explanation) of this book, the milestones in the debate on how to characterize scientific explanations are exposed. The second chapter (How to Study Scientific Explanation?) scrutinizes the working-method of three important philosophers of explanation, Carl Hempel, Philip Kitcher and Wesley Salmon and shows what went wrong. Next, it is the responsibility of current philosophers (...) of explanation to go on where Hempel, Kitcher and Salmon failed. However, we should go on in a clever way. We call this clever way the pragmatic approach to scientific explanation and clarify briefly what this approach consists in. The third chapter (A Toolbox for Describing and Evaluating Explanatory Practices) elaborates the pragmatic approach by presenting a toolbox for analysing scientific explanation. In the last chapter (Examples of Descriptions and Evaluations of Explanatory Practices) the approach is illustrated with real-life examples of scientists aiming at explaining. (shrink)

Much of the philosophical discussion of explanations has centered around two broad conceptions of what sorts of ‘things’ explanations are – namely, the descriptive and ontic conceptions. Defenders of each argue that scientific psychology has at best little to contribute to the study of explanations. These anti-psychologistic arguments come in two main varieties, the metaphysical and the epistemic. Both varieties trace back to Hempel and recur in the more recent writings of prominent mechanists. The metaphysical arguments attempt to combat psychologism (...) about explanation by doubly dissociating explanations themselves from the commonly associated phenomenology of explanation. The epistemic arguments attempt to combat psychologism by doubly dissociating the good-making features of explanations from the CAPE. I focus on the metaphysical arguments here, for only they directly concern the nature of, as opposed to the evaluation conditions for, explanations. The metaphysical arguments are all shown to equivocate over ‘explanation’, and thus they are no threat to the proposal that sometimes explanations just are psychological events. Such arguments do, nonetheless, gesture towards something important – namely, that the CAPE is neither necessary nor sufficient for explanation in any sense. Similar considerations lead another latter-day mechanist to conclude that no phenomenology of any sort is essential for explanation. What all of these thinkers overlook, however, is that there is another important phenomenological correlate of explanation – namely, finding a happening intelligible or understanding how or why-possibly – and it may well be of the very essence of explanation, at least in the psychological sense. I propose a psychomechanistic theory of intelligibility that accounts for how we can, through finite means, make boundless, endlessly qualified inferences about the implications of our explanations. In closing I note that, although finding a happening intelligible does sometimes engender an unjustified confidence in correctness, we should not overlook how attaining this state does at least put us in a kind of epistemic base camp. I also hint at how intelligibility could turn out to be the ultimate source of the descriptive and ontic conceptions of explanation. (shrink)

According to one large family of views, scientific explanations explain a phenomenon (such as an event or a regularity) by subsuming it under a general representation, model, prototype, or schema (see Bechtel, W., & Abrahamsen, A. (2005). Explanation: A mechanist alternative. Studies in History and Philosophy of Biological and Biomedical Sciences, 36(2), 421–441; Churchland, P. M. (1989). A neurocomputational perspective: The nature of mind and the structure of science. Cambridge: MIT Press; Darden (2006); Hempel, C. G. (1965). Aspects of scientific (...) explanation. In C. G. Hempel (Ed.), Aspects of scientific explanation (pp. 331–496). New York: Free Press; Kitcher (1989); Machamer, P., Darden, L., & Craver, C. F. (2000). Thinking about mechanisms. Philosophy of Science, 67(1), 1–25). My concern is with the minimal suggestion that an adequate philosophical theory of scientific explanation can limit its attention to the format or structure with which theories are represented. The representational subsumption view is a plausible hypothesis about the psychology of understanding. It is also a plausible claim about how scientists present their knowledge to the world. However, one cannot address the central questions for a philosophical theory of scientific explanation without turning one’s attention from the structure of representations to the basic commitments about the worldly structures that plausibly count as explanatory. A philosophical theory of scientific explanation should achieve two goals. The first is explanatory demarcation. It should show how explanation relates with other scientific achievements, such as control, description, measurement, prediction, and taxonomy. The second is explanatory normativity. It should say when putative explanations succeed and fail. One cannot achieve these goals without undertaking commitments about the kinds of ontic structures that plausibly count as explanatory. Representations convey explanatory information about a phenomenon when and only when they describe the ontic explanations for those phenomena. (shrink)

“Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a spring-carriage ashore. But the awful lonesomeness is intolerable. The intense concentration of self in the middle of such a heartless immensity, my God! who can tell it? Mark, how when sailors in a dead calm bathe in the open sea—mark how closely they hug their ship and only coast along her sides.” (Herman Melville, Moby Dick, Chapter 94).

By contrasting three general conceptions of scientific explanation, this paper seeks to clarify the explanandum and to exhibit the fundamental philosophical issues involved in the project of explicating scientific explanation. The three conceptions--epistemic, modal, and ontic--have both historical and contemporary importance. In the context of Laplacian determinism, they do not seem importantly distinct, but in the context of irreducibly statistical explanations, the three are seen to diverge sharply. The paper argues for a causal/mechanical version of the ontic conception, and concludes (...) by exhibiting some striking consequences of this approach. (shrink)

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. (shrink)