In a wide range of fields, the word “modular” is used to describe complex systems that can be decomposed into smaller systems with limited interactions between them. This essay argues that mathematical knowledge can fruitfully be understood as having a modular structure and explores the ways in which modularity in mathematics is epistemically advantageous.

Although understanding is the object of a growing literature in epistemology and the philosophy of science, only few studies have concerned understanding in mathematics. This essay offers an account of a fundamental form of mathematical understanding: proof understanding. The account builds on a simple idea, namely that understanding a proof amounts to rationally reconstructing its underlying plan. This characterization is fleshed out by specifying the relevant notion of plan and the associated process of rational reconstruction, building in part on Bratman's (...) theory of planning agency. It is argued that the proposed account can explain a significant range of distinctive phenomena commonly associated with proof understanding by mathematicians and philosophers. It is further argued, on the basis of a case study, that the account can yield precise diagnostics of understanding failures and can suggest ways to overcome them. Reflecting on the approach developed here, the essay concludes with some remarks on how to shape a general methodology common to the study of mathematical and scientific understanding and focused on human agency. (shrink)

In this paper, I argue that explanations just ARE those sorts of things that, under the right circumstances and in the right sort of way, bring about understanding. This raises the question of why such a seemingly simple account of explanation, if correct, would not have been identified and agreed upon decades ago. The answer is that only recently has it been made possible to analyze explanation in terms of understanding without the risk of collapsing both to merely phenomenological states. (...) For the most part, theories of explanation were for 50 years held hostage to the historical accident that they far outstripped in sophistication corresponding accounts of understanding. (shrink)

There is a recent interest within both philosophy of science as well as within epistemology to provide a defensible account of understanding. In the present article I build on insights from previous work in attempt to provide an account of two related forms of understanding in terms of the ability to form rational intentions when using specific types of mental representations. I propose first that “understanding that X” requires that one form a representation of X and, further, that one must (...) be capable of forming rational intentions using this representation across a range of conceivable conditions. I then propose that “understanding why X” requires that one possess a representation of a successful explanation for why X, and that one must be similarly capable of forming rational intentions using this representation across a range of conceivable conditions. I conclude the manuscript by reviewing objections and considering the way this account relates to other literature on explanation and understanding. (shrink)