What's Hot in Mathematical Philosophy? #2 (The Reasoner)

Each month MCMP (well, me ...) contributes a column to The Reasoner magazine, edited by Jon Williamson at the University of Kent. The most recent edition (August 2011) has just been published.
Here is the current column (the first column is available at The Reasoner website too):

What’s hot in mathematical philosophy? (#2)

Instrumentalist nominalism: nominalism is the view that there are no numbers, sets, sequences of symbols, computer programs, languages, formal systems, symmetry groups, wavefunctions, manifolds, tensor fields, Hilbert spaces, etc., and is usually motivated by a sceptical argument: in order to know that there are exactly two continuous automorphisms of the complex field to itself, the mind must be in ‘causal contact’ with these, which is impossible. On the other hand, Quine and Putnam argued that our best scientific theories, being mathematicized, are inconsistent with nominalism and that the use of mathematics in such theories is indispensable in some sense. For example, the electromagnetic field is a function from spacetime to a vector space. How can we even formulate Maxwell’s equations without appealing to the electromagnetic field, current densities, and so on?

Standard responses have been to engage in nominalization programmes: either eliminate mathematicalia or reconstrue them as nominalistically benign. The magnetic field might be eliminated and replaced by certain intrinsic spatio-temporal relations (Field) or perhaps reconstrued as a ‘possible sentence token’ (Chihara). But there are a number of difficulties with these programmes, connected to the awkwardness of the resulting reconstructions and the logical and metaphysical resources to which they appeal (for a survey, see Burgess & Rosen 1997, A Subject with No Object).

The last decade has seen the emergence of a more radical strategy responding to the indispensability arguments: instrumentalism. Contemporary instrumentalist nominalists would like to combine realism about science with anti-realism about mathematics, while insisting that there is no need to nominalize our best scientific theories. Given an explanatory and predictively successful scientific theory T, inconsistent with nominalism, the realist says,
$T$ is a good approximation to the truth,
while the instrumentalist says,
the concrete things behave as if $T$ (or, $T$ is nominalistically adequate),
while maintaining that no intrinsic description of the concrete things need be given to replace/reconstrue $T$. So, compasses, computers and constellations behave as if there is an electromagnetic field, even though there isn’t. The approach is conceptually similar to van Fraassen’s constructive empiricism: replace ‘empirically adequate’ by ‘nominalistically adequate’ to get instrumentalist nominalism.

Instrumentalist nominalism was criticized by John Burgess (‘Why I am Not a Nominalist’, Notre Dame J. Formal Logic 1983) and further discussed in Burgess & Rosen 1997. Over the last decade or so, several authors have proposed similar views, occasionally called ‘fictionalism’, often incorporating ideas from the literature on the semantics and pragmatics of fictional discourse. Examples are Joseph Melia (‘Weaseling Away the Indispensability Argument’, Mind 2000), Gideon Rosen (‘Nominalism, Naturalism, Epistemic Relativism’, Nous 2001), Stephen Yablo (‘Go Figure: A Path Through Fictionalism’, Midwest Studies in Philosophy 2001), Mary Leng (‘Revolutionary Fictionalism: A Call to Arms’, Philosophia Mathematica 2005), and some recent work by Richard Pettigrew. Building on her previous work, Leng has recently published a monograph defending instrumentalism (Mathematics and Reality, OUP 2010), reviewed by Burgess in Philosophia Mathematica (Vol. 18, 2010) and by Chris Pincock in Metascience (forthcoming). Criticisms of instrumentalism include recent articles by Stathis Psillos (‘Scientific Realism: Between Platonism and Nominalism’, Philosophy of Science 2010), Mark Colyvan (‘There is No Easy Road Nominalism’, Mind 2010) and myself (‘Nominalistic Adequacy’, Proceedings of the Aristotelian Society 2011). Whether viable or not, instrumentalist nominalism has become a major topic in contemporary philosophy of mathematics.