# Field, Hartry (1956–)

# FIELD, HARTRY

*(1956–)*

Hartry H. Field was born in Boston. He received his BA in Mathematics at the University of Wisconsin (1967) and his Ph.D. at Harvard (1972) working under Hilary Putnam and Richard Boyd. He has taught at Princeton, USC, CUNY Graduate Center, and NYU, where he is currently Silver Professor of Philosophy. Field is the recipient of, among other awards, a Guggenheim Foundation Fellowship (1979–1980) and the Lakatos prize (1986) for his book *Science without Numbers* (1980). He was elected in 2003 to the American Academy of Arts and Sciences.

Field has made significant contributions in a number of areas. He is best known for his work in philosophy of mathematics and on a variety of issues connected with realism and with the notion of truth. In philosophy of mathematics, Field has defended a version of fictionalism: a view according to which mathematics, which he takes at face value as asserting the existence of numbers, pure sets, and so on, is literally false and cannot be interpreted via a nonliteral reading in such a way that it works out true. Field sees the central argument in favor of realism about mathematics to be its indispensability for formulating and making use of scientific theories, and he proposes to answer this argument by giving an account of the use of mathematics in the sciences that does not require that the mathematics be true: If T is a nominalistic physical theory (roughly, one that makes no mention of mathematical entities), and M is a mathematical theory used to derive consequences from T (an example of such a theory might be a version of set theory that allows one to treat the objects of T as urelements and that allows the vocabulary of T to appear in the comprehension axioms) then M is said to be conservative over T if any such consequences, if entirely stated in the vocabulary of T, are already (semantic) consequences of T—that is, true in any model of T.

Field points out that people have always expected mathematics to be conservative over physical theories, and that in fact there is good reason to believe it is. The importance of this observation is the following. Suppose P is a physical theory that, like most such theories, is not nominalistic. It may be possible to find a nominalistic theory N, from which one can derive P via definitions and mathematics. It will then follow that P and mathematics are jointly conservative over N. This at least suggests that N captures all the physical content of P, and that the mathematics (together with P itself) is simply a convenient device for drawing out the consequences of N. Following (and significantly extending) techniques familiar to decision theorists and others under the title of "measurement theory," Field succeeded in constructing a natural nominalistic N for the case where P is a form of Newtonian gravitation theory.

Field's project of extending this result to all of physics has stimulated widespread interest in a number of issues. To name just one, Newtonian gravitation, and any theory remotely like it, requires an N that quantifies over sets of points, which may be identified with regions of space; the sense of consequence in which anything about N provable in P + mathematics is already a consequence of N is second-order consequence, thought of as the complete logic of the part-whole relation. This raises interesting questions, both about the extent to which first-order approximations to Field's result are available or convincing, and about whether one can speak about second-order consequence while continuing to be a fictionalist about mathematics. Indeed, the latter question arises for first-order consequence, despite that it is coextensive with a syntactic notion—because a fictionalist about mathematics ought to be a fictionalist about, for example, the claim that a given theory is syntactically consistent. Field has responded to this question with an interesting theory of (purely) logical necessity as a sui generis kind of necessity, one that is not explained in terms of models or possible worlds.

Field's earliest work on truth, the essay "Tarski's Theory of Truth" (1972), appeared at a time when Putnam and others were trying to argue for a form of scientific realism that stressed, as against, for example, Thomas Kuhn, the continuity of reference across changing scientific theories. Integral to this view was a conception of reference that made it a nontrivial question how use of the word "water" brings it about that "water" refers to the particular chemical compound it does, and thereby a nontrivial question, what brings it about that "Water tastes good," as uttered by an American, is true (beyond the fact that it does taste good). This conception, which sometimes goes (as do many other views) under the name "correspondence theory" (of reference or of truth), contrasts with the "deflationist" idea according to which "'water' refers to water (in English)" is nothing more than a straightforward consequence of a natural definition of "refers in English." In this paper and later related essays, Field forcefully articulated what has turned out to be the most persuasive argument in favor of the need for a correspondence theory: namely, that human success in interacting with the world using language requires a systematic explanation of a kind a deflationist is unable to supply.

It turns out that deflationists have some at least initially plausible responses to this argument, and in fact Field has been increasingly sympathetic to deflationism. One topic he has addressed is what the theory of meaning looks like from a deflationist perspective, given that deflationism needs to sever the apparently intimate connections between meaning and reference. Another has been what a deflationist (or anyone else—but the problem is particularly pressing for deflationists) is to make of situations where it seems correct to say that "there is no fact of the matter"; these include not only areas where philosophers have traditionally debated about realism, but also in borderline cases involving vague expressions like "bald." Field has presented an appealing picture in which one both abandons excluded middle, and introduces a "determinately" operator into the language. The "determinately" operator is not given a semantics; it is rather understood both through its connections with degrees of belief, and through its relations to a natural non-truth-functional conditional. Field shows that such a language allows one consistently (despite the presence of the "determinately" operator) to introduce a truth predicate T such that the Tarski sentences (written using the new conditional) work out to be theorems; in fact "T(A)" is everywhere substitutable for "A."

** See also ** Mathematics, Foundations of.

## Bibliography

The fictionalist approach to mathematics was first set out in Field's *Science without Numbers: A Defence of Nominalism* (Princeton, NJ: Princeton University Press, 1980). Many of Field's papers can be found in the collections *Realism, Mathematics, and Modality* (Oxford: Blackwell, 1989; rev. ed. 1991) and *Truth and the Absence of Fact* (New York: Oxford University Press, 2001).

For the recent work on vagueness and the liar, see "A Revenge-Immune Solution to the Semantic Paradoxes," in *Journal of Philosophical Logic* 72 (2003): 139–177; and "No Fact of the Matter," in *Australian Journal of Philosophy* 81 (2003): 457–480. Some recommended papers not in the collections, and on topics not mentioned in this entry, are "Logic, Meaning and Conceptual Role," in *Journal of Philosophy* 74 (1977): 379–409; "A Note on Jeffrey Conditionalization," in *Philosophy of Science* 45 (1978): 361–367; "The A Prioricity of Logic" *Proceedings of the Aristotelian Society* 96 (1996): 359–379; and "Causation in a Physical World," in *The Oxford Handbook of Metaphysics*, edited by M. Loux and D. Zimmerman, 435–460 (Oxford: Oxford University Press, 2003; reprinted in *The Philosopher's Annual* 26 (2003), edited by P. Grim, K. Baynes, and G. Mar). See also "Tarski's Theory of Truth," *Journal of Philosophy* 69 (13) (July 1972): 347–375.

*Stephen Leeds (2005)*

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**Field, Hartry (1956–)**