Zoom Talk: Bill D'Alessandro und Silvia Jonas (MCMP)
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Mathematics in Science
Bill D'Alessandro: Modeling in Mathematics: Understanding, Explanation, Counterfactuals
Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two cases: Cramér’s random model of the prime numbers and the dyadic model of the integers. These cases show that mathematicians, like empirical scientists, rely on simple (and often distorted or unrealistic) models to gain understanding of complex phenomena. There are also morals here for some much-discussed theses about scientific understanding. Two issues in particular are worth highlighting. First, modeling practices in mathematics seem to confirm that one can gain understanding without obtaining an explanation (contra [de Regt 2009], [Khalifa 2012], [Strevens 2013], [Trout 2007]). Second, these cases cast doubt on the idea that unrealistic models confer understanding by imparting counterfactual knowledge (contra [Bokulich 2011], [Grimm 2011], [Hindriks 2013], [Lipton 2009], [Rice 2016], [Saatsi forthcoming]).
Silvia Jonas: Realism, Pluralism, and Indispensability
Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics: it accurately reflects mathematical practice, matches crucial mathematical facts, and offers elegant solutions to a number of philosophical objections against mathematical realism. However, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most one mathematical universe. Science may thus lose its guiding role in the debate about mathematical realism.