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Science Without Numbers: A Defence of Nominalism

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Why review an out-of-print book? Because some are just too good to pass up. I am currently working through the "heavy" section of "Science Without Numbers", wherein Hartry Field shows how to do Newtonian gravitational physics without abstracta. This book is a brilliant and *highly* unorthodox defence of mathematical nominalism, the view that there are no abstract mathematical objects. While subtle in execution, the basic ideas behind Field's approach are straightforward (though probably his gifts of exposition make them seem simpler than they are).

The traditional problem of the nominalist, or would-be nominalist, is to account for the truth of mathematical statements, apparently about abstract objects like numbers and sets, without supposing that there really are any such objects. Not so for Field; he sidesteps that issue entirely by showing that it makes no difference, so far as *applications* of mathematics are concerned, whether mathematical theories are true or false--what is really necessary is that they be consistent and satisfy a few other weak conditions. If consistency (plus these other conditions) are satisfied, then the mathematical theory will be *conservative* whether or not it is true. This means that any inferences made from nominalistic claims--claims that don't entail that there are abstracta--to other nominalistic claims with the aid of the theory are such that the conclusion is formally entailed by the nominalistic premises *alone*. So, if your concern is to make inferences about the concrete world from premises about the concrete world, you can use any math you like (all good mathematical theories are conservative, as Field demonstrates), because the ultimate *conclusions* you draw will in fact be entailed by nominalistic claims alone. Thus, whether the math used for the inferences is true or not, your conclusions will always be right--so long as your nominalistic premises are. All you need to do in order to *use* mathematics, then, is to take care that your *physical theory* is formulated nominalistically. This can always be done, at least provided that your theory could be expressed in (possibly higher-order) predicate calculus, because of a logical theorem which allows us to re-axiomatize the nominalistic consequences of *any* such theory from purely nominalistic axioms. One might however wonder whether such an axiomatization would be usable--whether there are any *interesting* nominalistic physical theories. It turns out that there are; Euclidean geometry as axiomatized by Hilbert, for instance, is a nominalistic theory. The heart of Field's book, which I'm now reading, is to show that there are interesting nominalistic formulations of typical scientific theories, like Newtonian physics. (He says that his approach here could be easily extended to classical electrodynamics with special relativity. I believe him.) Given such a formulation, abstract mathematics becomes merely a useful tool, making essentially nominalistic inferences easier--as natural scientists have always thought. If there are good reasons for adopting Platonism in mathematics, Field has at any rate shown that the use of mathematics in science (and everyday life) is not one of them.