Mary Leng
I am a Lecturer in Philosophy at the University of York. Prior to coming to York in 2012, I was a lecturer at the University of Liverpool (2006-2011) and a Research Fellow at St John's College, Cambridge (2002-2006). I have held visiting positions at the University of California at Irvine, and the Peter Wall Institute for Advanced Studies at the University of British Columbia. I did my PhD at the University of Toronto (1996-2001) and also held a post doctoral fellowship there (2001-2002). I studied Mathematics and Philosophy as an Undergraduate at Balliol College, Oxford (1993-1996).
Supervisors: James Robert Brown and Ian Hacking
Supervisors: James Robert Brown and Ian Hacking
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According to recent explanatory versions of the Quine-Putnam indispensability argument for mathematical realism (QPIA), we have reason to believe in mathematical objects since mathematics plays an indispensable explanatory role in some of our best explanations of empirical phenomena. Some nominalists respond to this challenge by rejecting the indispensability of mathematics to these explanations. However, I agree with Christopher Pincock that in at least some cases (the honeycomb example being one), mathematics does do genuine explanatory work. I also agree with Pincock that mathematical explanations explain by picking out structural features of physical systems. Where I disagree, however, is the assumption that structural explanations that make use of mathematical theories to explain physical phenomena require those theories to be true. So, I will argue, mathematical explanation doesn't require mathematical truth. As Pincock explains, the original QPIA depends on a naturalist commitment to look to our best empirical scientific theories to determine our ontological commitments. In Quine's view, our best scientific theory – that being, as Pincock puts it, " the regimented theory that maximizes the theoretical virtues " – quantifies over platonic mathematical entities, and is thus ontologically committed to such entities. So if naturalism requires us to believe our best scientific theory – at least " as a going concern " (Quine, 1975, p. 72) – then it looks as though naturalism requires us to believe in mathematical objects. According to Pincock, there are two points of attack for nominalists in the naturalist tradition. " They may revise or deny Quine's test for the ontological commitments of a scientific theory " , or " question the application of the Quinean theoretical virtues " , arguing that the best theory does not quantify over abstract objects. The latter approach is that pursued by Hartry Field (1980), who wishes to show that we can formulate nominalistically acceptable versions of our ordinary 'platonistic' (or mathematically-stated) scientific theories, and that these versions are preferable to platonic interpretations. In particular, Field argues that his nominalistic version of Newtonian gravitational theory is preferable to the standard platonistic alternative in that it is able to provide intrinsic explanations of physical phenomena (that appeal only to causally relevant features), rather than the extrinsic explanations provided by their platonistic counterparts. If in explaining the behavior of a physical system, one formulates one's explanation in terms of relations between physical things and numbers, then the explanation is what I would call an extrinsic one. It is extrinsic because the role of the numbers is simply to serve as labels for some of the features of the physical system: there is no pretence that the properties of the numbers influence the physical system whose behavior is being explained. (Field 1985: 192-3) Field's contention is that explanations that are formulated in terms of the relation between physical and mathematical objects cannot be fundamental. The mathematical objects posited in these explanations are simply serving to enable us to represent, or index, relevant features of the physical system, and it is these features that are doing all the genuine explanatory work. Nominalistically-stated alternatives to ordinary platonistic scientific theories are preferable because their explanations, appealing only to physical features of physical systems, pick out the genuinely explanatorily relevant features.