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chapter title verso 7 Understanding in Physics and Biology From the Abstract to the Concrete m a R g a R e t m o R R i S o n L It is commonly thought that the greater the degree of abstraction used in describing phenomena the less understanding we have with respect to their concrete features. I want to challenge that myth by showing how mathematical abstraction—the characterization of phenomena using mathematical descriptions that seem to bear little or no relation to concrete physical entities /systems—can aid our understanding in ways that more empirically based investigations often cannot. What I mean here by “understanding” is simply having a theoretical account of how the system is constituted that enables us to solve problems, make predictions, and explain why the phenomena in question behave in the way they do. Although much of my discussion will focus on the role of mathematics in biology, specifically population genetics, I also want to look at some examples from physics to highlight the ways in which understanding via mathematical abstraction in these two fields can be strikingly similar. Many philosophical accounts link understanding with explanation. While some (Hempel 1965, van Fraassen 1980) claim that understanding is a pragmatic notion and hence epistemically irrelevant to assessing the merit of a theory, most others see understanding as an important goal of science but define the notion in a variety of different ways. One option is to link understanding with unification (Friedman 1974; Kitcher 1989), where the explanation that best unifies the phenomena is the one that produces the greatest understanding. Salmon (1984), on the other hand, sees knowledge of causal mechanisms as the 123 de Regt Txt•.indd 123 9/8/09 11:27:03 AM 124 feature that furnishes understanding of the physical world. What both of these accounts have in common is that they privilege a particular type of explanation as the vehicle for producing understanding. De Regt and Dieks (2005) criticize these monolithic accounts of understanding and instead define it in terms of a criterion of intelligibility that involves having a theory where one can recognize qualitatively characteristic consequences without performing exact calculations . This is achieved using a variety of conceptual tools that are relevant to the problem at hand. Their point is that in some cases causal knowledge will be relevant to understanding, and in some cases it will not—the tool kit contains a number of different resources that, depending on the context, will produce understanding . An important feature of their account is that its pragmatic quality in no way compromises the epistemic relevance of understanding for scientific investigation. My own sympathies lie with the type of contextual analysis provided by de Regt and Dieks, but I would put the point in slightly more negative terms. While I agree that what it means to “understand” depends on contextual factors such as the nature of the problem and the resources available to solve it, I want to claim that it is neither possible nor desirable to formulate a “theory” of understanding. That does not mean that we cannot explain what it is to understand why a phenomenon behaves as it does or what we mean when we say someone understands a mathematical concept.1 Rather, there is no canonical account of what it means to understand. I would suggest the same holds for explanation. Very often in science we desire explanations that are causal, but that is not to say that this is the only form of explanation that is acceptable, or that we can give a “theory” of explanation that centers on causal relations. On the contrary, whether something has been successfully explained will depend on the question and the stock of available answers, answers that come from our background theories. We simply cannot specify in advance what qualifies as an explanation or what form it will take. However, I do think explanation and understanding are linked in the intuitive sense that one often accompanies the other; that is, we demonstrate our understanding by being able to offer explanations of the object /concept in question and the success of our explanations is typically a function of how well we understand what we are trying to explain. My position is a minimalist one insofar as neither understanding nor explanation is capable of being codified into a philosophical theory.2 Bound up with the question on how mathematical abstraction enables us to understand physical phenomena is the role played by models...

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