Justification, , 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 ( 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) . Let the structure be R = 〈D0, P1, …, Pn〉 (in the case of geometry this structure would be 〈E, Bet, Con〉). The representation theorem says that there is a homomorphism, mapping R onto a subset of a certain mathematical domain D1. More precisely, this theorem says that (1) There is a function f: D0 → D1, and there are formulas A1, …, An definable in language L1 such that for all x1, …, xk ∈ D0, Pi(x1, …, xk) iff Ai(f(x1), …, f(xk)). The second part of the representation theorem (usually called the uniqueness theorem) says that the homomorphism f is unique up to certain scale transformations of the mathematical domain D1. Because I want to keep our considerations possibly most general, I will assume nothing about the nature of these scale transformations. In fact, scale transformations are different in case of Hilbert geometry, Newtonian space-time geometry, and in case of measurement theory for typical physical magnitudes. Therefore we will assume that the set S of all possible scale transformations of mathematical domain D1 is already given. Hence the uniqueness theorem is as follows:

(2) If f satisfies theorem (1), then for every f’: D0 →D1, f’ satisfies (1) iff there is t ∈ S such that f’ = t•f

With these two theorems, Field was able to prove that for certain mathematically stated theories such as Newtonian gravitational theory, it is possible to formulate their nominalistic counterpart, capturing all nominalistically- statable consequences of these theories. My task is to consider the problem more generally, without restricting to any particular theory. So my question is: given that the representation theorem holds, and given any mathematically definable

formulaΨ(x1, …, xk), where x1, …, xk – variables of 0-sort, does there exist a formula Q(x1, …, xk), definable with the help of predicates of 0-sort, and true of exactly the same objects as Ψ? However, it is not difficult to notice that the answer for such a general question should be negative. As an example consider the formula d(x, y) = 2, definable in the analytic version of Euclidean geometry. It is hopeless to try to find its counterpart definable with the help of predicates Bet and Con, because the set of point satisfying this formula depends on a choice of any particular coordinate function (or in other words on a choice of units in which we measure the distance between points). So the above formula does not represent any genuine fact of the matter, but only an artifact, created by the incautious usage of mathematical

4 Throughout this paper bold letters will symbolize relations, whereas italicized letters will stand for predicates, referring to these relations. 5

vocabulary. Therefore we should put one more constrain on our task. We will be seeking nominalistic versions not for all mathematically defined formulas but only for those which are invariant under scale transformation. In expressing this idea we will make use of well-known notion of “empirical meaningfulness”, introduced by Patrick Suppes5. Let A be a formula containing exactly n free variables of 0-sort, and without any bound variables. The definition will be following:

A is empirically meaningful iff for every mapping p: D0 → D0, if there is f: D0 → D1 satisfying the representation theorem, and t ∈ S such that f•p = t•f, then 〈a1, …, ak〉 satisfies A iff 〈p(x1), …, p(xk)〉 satisfies A. The intuitive sense of this definition should be straightforward. A mapping p “represents” certain scale transformation within domain D1 (for example, if t is a scale transformation, changing kilograms into pounds, p would be a function transforming objects having mass equal to x kilograms into objects weighing x pounds). So the condition of empirical meaningfulness amounts to the requirement that the satisfiability of the formula A should not change under any scale transformation. Now we are ready to formulate and prove a very simple theorem, which would provide the answer for our main question.

Th1. Let x1, … xk be variables of 0-sort, and A(x1, …, xk)—a formula definable in L1, fulfilling the definition of empirical meaningfulness. If s is any automorphism of the structure R = 〈D0, P1, …, Pn〉, then A is satisfied by 〈a1, …, ak〉 iff A is satisfied by 〈s(a1), …, s(ak)〉

Before we move to the proof of this theorem, let me first explain its intuitive meaning. An automorphism of the structure R is a transformation of D0 which does not change relations P1, …, Pn. Therefore we can spell out the above theorem as follows: whenever of predicates P1, …, Pn are fixed, so is with the of the formula A: it is simultaneously pinned down. This relationship between predicates P1, …, Pn and the formula A is usually called “implicit (or semantic) definability”. We will see in a moment that this notion lies not very far from the usual notion of definability. But first let’s sketch the proof of the theorem. Let s be an automorphism of R, and let f be any function satisfying the representation theorem. It is easy to notice that the composition of two functions f•s must also satisfy part (1) of the representation theorem. Here is the proof: for all

x1, …, xk ∈ D0, Pi(x1, …, xk) ≡ Pi(s(x1), …, s(xk)) (from the definition of an automorphism) ≡ Ai(f(s(x1)), …, f(s(xk)) (from the representation theorem) ≡ Ai(f•s(x1), …, f•s(xk)) (from the definition of the composition of functions), where Ai—a formula mentioned in the representation theorem. Therefore, from the uniqueness theorem we know that there must be t ∈ S such that f•s = t•f. But in that way we obtained the antecedent of the definition of empirical meaningfulness. Thus by an appeal to this definition we see that A must be preserved under the automorphism s, which completes the proof. Having accomplished this, we should now ask whether we are in a position to answer our main question. In fact we are very close. First notice that if the formula A has a normal definition in terms of predicates P1, …, Pn, i.e. if there is a formula

5 See (Suppes 1959) and also my article (Bigaj 1999). 6

consisting only of P1, …, Pn and true of exactly the same objects as A, then A is implicitly definable by P1, …, Pn. However, is it so that the opposite implication also holds? Under certain circumstances, the answer is yes. We can namely appeal to Beth’s theorem stating that for first-order languages implicit (semantic) definability is equivalent to explicit (syntactic) definability6. Therefore, for first- order theories we have achieved our goal: every mathematical notion, having any importance for empirical science, can be defined purely nominalistically, with the help of certain primitive qualitative predicates. However, we cannot make use of Beth’s theorem in the case of second order language. So in principle it is possible that each mathematical notion of our theory is semantically dependent upon non-mathematical vocabulary, but our vocabulary L0 isn’t powerful enough to express all of these notions. Nevertheless I don’t think that this should worry much a nominalist. First of all, it seems that a great portion of our knowledge can be expressed without resorting to any language stronger that first order. And secondly, even if we had to use second order logic in expressing certain mathematical theory, we could always appeal to the above theorem, arguing that ontologically there are no facts of the matter over and above purely physical ones, although it may be useful to describe some of them with the help of certain non-physical notions (we could express this intuition in a more fancy way saying that mathematically expressible facts of the matters are supervenient on purely physical states of affairs). However, the method of nominalizing physical theories proposed above has obvious limits. Namely, it applies only to those theories for which the representation theorem holds, i.e. roughly speaking to the theories which employ the classical notion of physical magnitude. It is obvious that our method does not extend in a straightforward way for such theories as quantum mechanics. Hence the main moral which can be drawn from the above theorem is that the fact that Field succeeded in nominalizing Newtonian gravitational theory was not a sheer luck, for this theory is a classical one. But an open question remains, how far can we go along this line.

3. Ontological costs of Field’s program

It should be obvious that we must pay a certain price for eliminating mathematical vocabulary from our scientific theories. First of all, nominalistic versions of usual theories are in general much harder to handle: the definitions are long and obscure, the theorems are likely to be complicated and not easy to comprehend. But this is all right. After all, the same difficulty would occur when someone tried to reformulate all mathematics in the language of set theory. So, as in cases of other reductions, what a nominalist needs is a proof that such a reformulation is possible, and not that it is useful or elegant. However, Melia argues that Field must pay an even higher price—in fact, too high to be outweighed by an advantage of elimination of mathematical objects

6 See for example Chang and Keisler 1973, pp. 87-8. 7

from scientific discourse. For example, in order to properly account for mathematically stated relations between masses of physical objects, a nominalist must postulate the existence of additional massive bodies, different from these objects. Indeed this seems to be very strange: we would certainly oppose the claim that for a body x being twice as heavy as a body y, there must exist a third body, disjoint from y and equally massive to y. After all, what would happen if all things in the universe except x and y disappeared? Would x cease to be twice as massive as y? Certainly it seems to be absurd. Fortunately, it is not as bad as it might appear. We should remember that all nominalistic definitions of that sort are in fact conditional definitions. They apply only when certain assumptions are met: in our case, when there are sufficiently many objects of distinctive masses. We know that in order to prove the representation theorem for the mass-function, we must postulate the existence of infinitely (even uncountably) many different physical objects. With the definition of the phrase “being twice as much heavy as” it is somehow better: all we need is the existence of one extra body. So the correct version of the nominalistic interpretation of our notion should be following: If there exists such z that z is disjoint from, and as heavy as y, then x is twice as heavy as y if and only if x is equally massive to the sum of y and z. Therefore, when there is no such z, our definition is no longer applicable—it doesn’t decide whether x is or is not two times heavier than y. Thus the nominalist is not committed to the unacceptable claim that when there are, for example, only two distinct bodies x and y in the universe, the sentence “x is two times heavier than y” is always false, and the fact of the matter expressed in it does not exist. Rather, in such an universe, we don’t have tools to properly express facts about these objects and their masses. But doesn’t that mean a victory of the platonist, anyway? For he seems to be able to express the mass-relations between considered objects, resorting to an endless domain of abstract objects, e.g. numbers. Not so fast. In fact I will argue the platonist and the nominalist are both in the same position with respect to this problem. The platonist can formulate the sentence “The mass of x equals 2 times the mass of y”, but does this phrase have any empirical significance? In order to verify it empirically, we would have to resort to a certain measurement procedure, which requires some extra massive bodies, unavailable in our sparse universe. So both platonist and nominalist are incapable of checking, what the measurement-independent mass-ratio facts really are. Nevertheless, unless they are radical empiricists, they both can equally claim that these facts of the matter are really there, and they can even try to express them in inevitably unverifiable, lacking empirical content sentences—the platonist in the sentence written above, and the nominalist in the equally hopeless counterfactual expression: “If there were the third body z, weighing as much as y, then x would be equally massive as the sum of y and z”. They both can grasp the intuitive meaning of what measurement-independent mass ratios are, without proposing any method of verifying them in our extremely narrow universe. In the remaining two paragraphs of the paper I will shortly refer to Melia’s other arguments. He claims also that the synthetic version of Euclidean geometry 8

leaves us with exactly the same difficulties, which Field ascribed to the analytical formulation. Namely, in expressing relations between different distances, we must appeal to certain objects (points) which are extrinsic, causally irrelevant and arbitrary—hence they possess the same unwanted features as numbers and other abstract objects. I think that in a sense Melia is right, but he probably overestimates the role of these auxiliary entities. Remember that the existence of certain objects is only a precondition of applicability of nominalistic definitions for certain notions. For example: if we want to express the idea that the distance between x and y is twice as long as the distance between w and z, by saying that there must be a point u exactly in the middle of x and y, then do we really need u to be causally relevant for the described fact? I think no more than when referring to certain distance as 20 meters long, we assume implicitly the existence of the standard meter in Sevres (forgetting, for the sake of argument, that it no longer serves as a standard of length unit in physics). Of course the existence of such an object is causally irrelevant to this fact, and it is also arbitrary, for there could be many different standards of meter in many different places on earth. What is more important, is that both meters and points are, in a sense, empirical objects—that we can physically perform certain procedures with the help of them, which is impossible in the case of mathematical objects. I also think that an attempt to explain the difference between usage of mathematical objects and usage of other theoretical entities, like electrons, which is proposed at the end of Melia’s paper, is misguided. The claim that we use mathematical objects only in describing empirical facts, but not in explaining them, is not justified. For it is a perfectly acceptable explanation of, let’s say, the fact that the acceleration of a body x with the mass 5 equals 5 (in some units), that the applied force was equal to 25 (in appropriate units). Numbers in fact appear in almost all scientific contexts, and therefore the Quine-Putnam indispensability argument is so powerful. For now I cannot see any compelling way of rejecting this argument except Field’s nominalization program.7

Tomasz Bigaj Warsaw University, Poland

References

Bigaj, T. 1999. “Formalne i nieformalne kryteria sensu w naukach empirycznych” (“Formal and informal criteria of meaning in empirical sciences”), in M. Heller, J. Maczka, J. Urbaniec (eds.) Sensy i nonsensy w nauce i filozofii (Senses and Nonsenses in Science and Philosophy), pp. 117-125, OBI - Biblos, Kraków – Tarnów Bigaj, T. 1997. Matematyka a swiat realny (Mathematics and the Real World), WFiS, Warsaw Chang, C.C. and Keisler, H.J. 1973. Model Theory, Amsterdam, London: North Holland – American Elsevier Field, H. 1980. Science without Numbers, Princeton: Press Malament, D. 1982. Review of Science without Numbers, Journal of Philosophy 79: 523-34 Melia, J. 1998. “Field’s program: some interferences”, Analysis 58: 63-71

7 I would like to thank Joe Mourad for reading an earlier draft of this paper and for his useful corrections and comments. 9

Suppes, P. 1959. “Measurement, empirical meaningfulness and three-valued logic”, in: C.W. Churchman, P. Ratoosh (eds.), Measurement: Definitions and Theories, New York: John Wiley & Sons Inc., 129-43.

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