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Field's Program: a Defence

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Field's Program: a Defence Justification, Truth, and Belief http://www.jtb-forum.pl December 2001 TOMASZ BIGAJ Field’s Program: A Defense It has been over twenty years since Hartry Field’s famous book Science without Numbers was published, yet it still gives rise to controversies and discussions. In 1998 April’s issue of Analysis an interesting article appeared (Melia 1998), in which the author formulates several objections to the program of nominalization of science, not discussed extensively elsewhere. As it happened, I have worked extensively on Field’s program for some time, trying to overcome difficulties created by his approach and to generalize his particular results. I presented some outcomes of my investigations in my book (Bigaj 1997). Here I would like to concentrate on particular arguments against Field’s attempt, put forward in the aforementioned article. Because I happen to be a proponent (naturally, not without certain reservations) of Field’s general approach, I feel obliged to point out certain weaknesses of Melia’s arguments. Answering to Melia’s criticism I also want to report some of my earlier results to the effect that it is possible to strengthen substantially Field’s method of nominalization by showing that it can be applied to a wider range of physical theories without any additional proofs. I am assuming that the reader is familiar with the basic ideas of Field’s approach (in Melia’s article one can find a very clear and accurate1 outline of this approach). 1. A nominalistic interpretation of irrational ratios between distances Melia in his article argues that Field’s program of eliminating reference to mathematical objects in science is incomplete. Usually critics of Field’s approach point to the fact that he had failed to show that all (or at least most of) interesting scientific theories, like for example quantum theory, can be nominalized along his line2. Field claims that he succeeded in nominalizing Newtonian gravitational theory, but this theory is still far away from modern physics. However, Melia goes even further in his criticism. He maintains that even in the simple case of Euclidean geometry Field’s way of nominalization is incomplete, because it does not give any method of interpreting sentences stating that the distance between certain points equals to some irrational number (e.g. the number e, the base of natural logarithms). In that way he creates the challenge for the nominalist, who would like 1 Maybe I should mention that I found in Melia’s paper one obvious mistake: the formula in the last paragraph on page 64 expresses not the fact that the space is 3-dimensional, but rather that it is 2- dimensional (it says that there are at least three non-collinear points). However, this mistake has no consequences in further analysis. 2 Cf. for example Malament 1982. 2 to employ Field’s method. In what follows, I am going to meet this challenge and to show that, contrary to Melia’s claim, it is in general possible to define nominalistically the platonist formula d(x, y) = αd(w, z), where α is an irrational number. But I am aware of the fact that this does not solve the general problem, for one can still raise the questions, whether this or that notion is nominalistically definable. Therefore as a next step I will consider the possibility of proving generally that under certain circumstances every notion definable with the help of a language of a given mathematical theory can be also defined purely nominalistically. If we could prove this, we would undermine every possible objection of that sort. Let me first start with a very simple case. Melia mentions in a footnote on page 69 that for some irrational numbers, like for example √2, it is relatively easy to find a nominalistic counterpart of the above formula. I think that it would be instructive to see how to define nominalistically every formula of the type d(x, y) = √nd(z, w), where n is a natural number (with the possibility of extending this definition for the square root of any rational number). The definition is going to be inductive. Our primitive terms are usual, Hilbert-style predicates: 3-argument predicate x Bet yz (meaning intuitively “x lies on a straight line between x and z”) and 4-argument predicate xy Con zw (meaning “x is equally distant from y as z from w”). With the help of these notions we can define 3-argument predicate Right, where x Right yz means intuitively that the angle ∠(y, x, z) is right. The definition is as follows: x Right yz ≡ ∃w (x Bet wz ∧ wy Con yz ∧ xw Con xz) and the inductive definition is: d(x, y) = √2 d(z, w) ≡ ∃u (w Right zu ∧ zw Con wu ∧ zu Con xy) (intuitively: the segment xy is equal to the diagonal of a square, whose side is the segment zw) d(x, y) = √n d(z, w) ≡ ∃u [d(u, w) = √(n-1) d(z,w) ∧ w Right zu ∧ zu Con xy], for any natural number n. Obviously this construction is not general enough to serve our purpose, which is to find a way to express nominalistically all formulas of the type d(x, y) = α d (z,w), for any irrational number α. So let us move to this task. We will assume that each irrational number is represented by a Cauchy sequence {an} of rational numbers (for example, in case of the number e, the sequence is given by an = (1 + 1/n)n ). We will need an auxiliary definition of a given sequence of points. For any points x, y, w, z, the n-th element of this sequence (symbolized by xn) is given with the help of the following expression: (xn Bet xy ∨ y Bet xxn) ∧ d(x, xn) = an d(z, w) Hence xn is a point which lies on the halfline with the endpoint x and containing y, and such that the distance between x and xn equals an times the distance between z and w. It is obvious that because an is a rational number, the last expression has its natural nominalistic counterpart. 3 Let α be an irrational limit of the sequence {an} and let xn be a point defined for certain x, y, w, z by the above formula. Then our nominalistic interpretation is as follows: d(x, y) = α d(z, w) ≡ ∀ u ∀v {u Bet xy ∧ y Bet xv → ∃xn ∀xm [ xm > xn → xm Bet uv]}, where xm > xn, when m>n. It should be quite clear that the above definition is a nominalistic version of a usual δ-ε definition of the limit of sequence. The definition says that no matter how small interval including point y we choose (this interval is defined by points u, v), 3 every point from the set {xn} except finite numbers of points will be found within this interval. This definition is applicable to the case of e number, as well as to every other irrational number. Therefore, one of the main Melia’s objections is met. 2. A more general approach to the problem of incompleteness of Field’s analysis For reasons presented earlier, it would be much better for nominalists if they had at their disposal a general theorem, assuring that it is always possible to find a nominalistic counterpart of any mathematically definable formula. In the following I will try to show that under certain assumptions such a theorem can be in fact proved. First let me specify some of these assumptions. To start we will presuppose that any mathematically formulated scientific theory can be expressed in a two- sorted language (of either first or second order), where the first sort consists of non-mathematical and the second of mathematical vocabulary. Let L symbolize the language of a theory T, L0—its non-mathematical vocabulary (including variables, ranging over non-mathematical domain, predicate symbols, and possibly constants and function symbols), L1—its mathematical vocabulary. We will assume that every predicate in L0 can take only variables of 0-sort (it means that it is meaningless to apply physical notions to mathematical objects). However, with the mathematical vocabulary it will be different. Beside predicate and function symbols, which can create well-formed formulas only with 1-sort variables (those symbols can be interpret as expressing purely mathematical notions) we will need to have some mixed notions, connecting physical and mathematical objects. According to common practice, we introduce to our language function symbols, which take arguments in a physical domain and values in mathematical domains (these can be thought of as measurement functions for certain physical magnitudes, like mass or temperature). Therefore in L1 there will be at least one function symbol Φ, such that its variables belong entirely to L0 vocabulary. For the sake of simplicity, let us assume that there is only one such a function symbol, and that it is 3 Speaking of sets of points in this context shouldn’t worry a nominalist, for we can interpret them as certain regions of space. See Field 1980, pp. 36-40, for an extensive discussion. 4 a one-argument function. This should not diminish generality of resulting conclusions. So much for the preliminary assumptions. The crucial assumption, on which the whole Field’s program depends, is the so-called representation theorem. Because the notion of representability in this context is widely known, let me only roughly sketch its meaning. We assume that a certain structure is defined on the physical domain D0. This structure consists of relationships defined on D0 (usually 4 this structure is given axiomatically) .
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