One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. Its not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics. From the rather remarkable but seemingly uncontroversial fact that mathematics is indispensable to science, some philosophers have drawn serious metaphysical conclusions. In particular, Quine (1976; 1980a; 1980b; 1981a; 1981c) and Putnam (1979a; 1979b) have argued that the indispensability of mathematics to empirical science gives us good reason to believe in the existence of mathematical entities. According to this line of argument, reference to (or quantification over) mathematical entities such as sets, numbers, functions and such is indispensable to our best scientific theories, and so we ought to be committed to the existence of these mathematical entities. Field (1980) presents a case for denying the second premise of the Quine-Putnam argument. That is, he suggests that despite appearances mathematics is not indispensable to science. There are two parts to Fields project. The first is to argue that mathematical theories dont have to be true to be useful in applications, they need merely to be conservative. (This is, roughly, that if a mathematical theory is added to a nominalist scientific theory, no nominalist consequences follow that wouldnt follow from the nominalist scientific theory alone.) This explains why mathematics can be used in science but it does not explain why it is used. The latter is due to the fact that mathematics makes calculation and statement of various theories much simpler. Thus, for Field, the utility of mathematics is merely pragmatic — mathematics is not indispensable after all. The second part of Fields program is to demonstrate that our best scientific theories can be suitably nominalised. That is, he attempts to show that we could do without quantification over mathematical entities and that what we would be left with would be reasonably attractive theories. To this end he is content to nominalise a large fragment of Newtonian gravitational theory. Although this is a far cry from showing that all our current best scientific theories can be nominalised, it is certainly not trivial. The hope is that once one sees how the elimination of reference to mathematical entities can be achieved for a typical physical theory, it will seem plausible that the project could be completed for the rest of science. Read More: plato.stanford.edu/entries/mathphil-indis/
Posted on: Mon, 24 Mar 2014 13:41:21 +0000