Bootstrapping and the Problem of Testing Quantitative Theoretical Hypotheses David Gruenberg

I. Introduction The testing (i.e., the confirmation or disconfirmation) of quantitative theoretical hypotheses, viz., laws or theories involving theoretical functions, has constituted a major problem in contemporary philosophy of science. This problem results from a circularity or infinite regress that can be expressed as follows. (i) In order to confirm a quantitative theoretical hypothesis h belonging to a theory T we need to determine relevant values of the theoretical functions occurring in an instance of h (since one cannot test a quantitative theoretical hypothesis h without knowing the values of the constitutive functions of some instances of h.) (ii) On the other hand, in order to determine the values of theoretical functions in an instance of hypothesis h (of theory T) we need to confirm the laws of theory T including the hypothesis h, because the values of the theoretical functions of a theory T can be determined only by means of the laws of T. Clearly (i) and (ii) give rise to a circle in case the number of the relevant law instances is finite, and to infinite regress in case it is infinite. Now since the values of theoretical functions cannot be measured independently of a theory T, the testing of a hypotheses h containing such functions requires that any theoretical function occurring in h either be eliminated or else that the laws of T be, after all, allowed for the purpose of computing the values of the theoretical function. Both of these ways for solving the abovementioned problemwhich may be called the problem of confirming quantitative theoretical hypotheseshad been tried systematically. Sneed (1979) and Glymour (1980) are important representatives of each of these ways respectively. Sneed considered this problem as constituting a serious difficulty (in particular that the circularity involved is utterly vicious) so that the problem can be solved only by finding a way of eliminating the theoretical functions. Indeed, Sneed reduced the selective confirmation of particular hypotheses belonging to a theory T to the holistic confirmation of the single central empirical claim which expresses the whole empirical content of the theory, but is free of theoretical terms. (The sentence describing this claim is called the RamseySneed sentence of the theory.) Glymour, in contrast, contended that the circularity involved is not vicious and propounded that the proper way of determining the values of theoretical functions consists in allowing the use of the laws of the theory as premises for computation. Glymour called this particular method "bootstrapping". (1) In the course of time criticisms against Glymour to the effect that the use of h itself for the computation of values of h 's theoretical functions has led certain difficulties resulted in Glymour's retreat from his original position, i.e., bootstrapping in the strict sense. (2) This paper aims to show that a general notion of bootstrapping is involved in Sneed's eliminationist structuralism (as well as in Niiniluto's noneliminationist structuralism), and it is this notion which ultimately can solve the problem of theoretical terms, without having recourse to RamseySneed sentences. II. Bootstrapping in Sneed's and Niiniluoto's Structuralist Approach The basic concept of the structuralist approach introduced by Sneed (1979) and Niiniluoto (1981) is the concept of an application (or matrix) of a theory T as the counterpart of a physical system. Such an application is an ntuple of the physical properties, relations or functions to which the theory T refers. An application of T is said to satisfy the laws of that theory in case the components of the application stand in the relation expressed by the laws in question. The components of an application are divided into theoretical and nontheoretical with respect to a theory T. Roughly, those components whose measurement presupposes some successful application of theory T (i.e., an application satisfying the laws T are called Ttheoretical, whereas those which do not presuppose such a condition are called Tnontheoretical. The application of a theory T in so far as it contains Ttheoretical components is called a possible theoretical application whereas the restricted application resulting from the removal of theoretical components from a theoretical application is called a possible nontheoretical application. Relations between different theoretical applications are called constraints, e.g., the condition that the same material particle has the same mass in all different applications of classical particle mechanics involving this particle, is such a constraint. Any scientific theory T at any given time refers to a set J of intended theoretical applications. The theoretical claim of a theory T with set J of theoretical applications is defined as the statement that (i) every application belonging to J satisfy the laws of the theory and (ii) the different applications belonging to J satisfy the constraint of T. The set I of nontheoretical applications resulting from J is called the set of intended nontheoretical applications of T. Then the central empirical claim RS(I ) of a theory T is defined by the statement that (i) every application belonging to I can be enriched to a wider application which satisfies the theory T and (ii) all these wider applications satisfy among themselves the constraint of the theory. (The enrichment of an application means here the adjunction of additional components to the given application.) Furthermore, referring to Sneed (1979) and Niiniluoto (1981), we shall use the following concepts in our subsequent discussions. Given a particular set of theoretical applications {<ni,ti>}i, where ni is a sequence of nontheoretical quantities, and ti a sequence of theoretical ones, we distinguish between (i) the set T(ni,ti) of instances of the laws of theory T referring to the set {<ni, ti>}i of theoretical applications, (ii) the RamseySneed sentence RS(ni) for the set {<ni>}i of nontheoretical applications corresponding to the set {<ni,ti>}i of theoretical applications, and (iii) the Ramseyelimination equivalent RE(ni) of RS(ni) which , if any , is a sentence extensionally equivalent to RS(ni), referring exclusively to the set {<ni>}i. For a set of particular theoretical applications {<ni,ti>}i if the corresponding sentence RS(ni) has a Ramseyelimination equivalent RE(ni), then the theoretical functions in {<ni,ti>}i are said to be Ramseyeliminable from RS(ni). If this is the case for all sets of possible applications of the theory T the theoretical functions are said to be Ramseyeliminable in the strong sense from theory T, otherwise the theoretical functions are Ramseyeliminable in the weak sense. We have mentioned in the Introduction that the problem of testing quantitative theoretical hypotheses can be solved either by eliminating the theoretical functions of the hypothesis to be tested or by finding a way for computing the values of this function. We shall now consider the first solution as propounded by Sneed. The rationale of this project is that the theoretical functions of a theory T (i.e.,Ttheoretical functions) have no meaning unless the theory's central empirical claim (in which there are no expressions denoting Ttheoretical functions) is true. Therefore, it makes no sense to compute the values of such functions until the theory's central empirical claim is already confirmed. Only subsequent to such a confirmation is the determination of Ttheoretical functions meaningful. However, Sneed qualifies the t functions, even after the confirmation is realized, as mere "calculational devices." This shows that Sneed is instrumentalist in the sense of denying the values of t functions as existing objective physical quantities. (See Sneed, ibid., pp.4445.) Nevertheless we shall argue, first, that this is a selfdefeating view. Indeed, given any theory T the Tnontheoretical functions, according to Sneed himself, are T'theoretical, i.e., theoretical in some theory T'. This is tantamount to saying that the Tnontheoretical functions of a theory T are as much theoryladen as its Ttheoretical functions. Hence, in case the Ttheoretical functions were not denoting objective quantities, the Tnontheoretical functions could not denote objective quantities either and would also be reduced to the status of mere calculational device; but then the elimination of Ttheoretical functions in favor of Tnontheoretical ones via the central empirical claim would fail of its purpose (this purpose being presumably the intention to formulate the content of a theory exclusively in terms of functions whose values are genuine physical quantities). On the other hand, if Sneed could agree to accept that the values of Ttheoretical functions, as well as those of the Tnontheoretical ones, are genuine physical quantities, then there would be again no reason to discriminate between the two categories by eliminating the former in favor of the latter. We have thus constructed a dilemma against eliminationism. Secondly, we shall try to show that the use of a RamseySneed sentence (RSsentence henceforth) is neither necessary nor sufficient for testing theoretical hypotheses and for determining t function values. First, the use of an RSsentence is not necessary because the bootstrapping procedure may perform the task in question as we shall show below. Second, the use of an RS sentence is not sufficient in the sense that any reasoning starting therefrom may not contribute either to the determination of t function values or to the confirmation of theoretical hypotheses, unless the very procedure of bootstrapping is involved in this reasoning. We shall support our argument by the following typical example of an equational theory. (Note that equational theories are themselves typical of quantitative scientific theories.) For this purpose we shall define "x is a T " in analogy to Sneed's "x is an S " defined by (D1) ( 1979, p.11): x is a T if and only if, there exists an n and t such that : (1) x = <n,t > ; (2) n is a nontheoretical and t a theoretical function with a common nonempty domain D and with ranges included in the real numbers; (3) f [n (y ), t (y )] = 0, for all y Œ D where f is a given binary operation on the real numbers. The object x is a theoretical application of theory T. Instead of "x is a T " we shall also say x is a theoretical application of theory T, and from now on we shall use the word 'application' as short for theoretical application, unless otherwise specified. The truth of theory T itself can be defined by the conjunction of the following two conditions. (i) The law of theory T is true or equivalently all intended applications, i.e. all members of set J, are applications of theory T , and (ii) in our case, we stipulate the constraint on t by virtue of which the values of function t for the same argument in different intended applications are equal. Thus, ti(y ) = tj(y ), for any applications i, j ,and any y belonging to the intersection of Di and Dj. The conjunction of these two conditions constitute precisely the socalled theoretical claim of theory T. Hence, we see that the theoretical claim of a theory T is a necessary and sufficient condition of the truth of the theory T. Besides this global theoretical claim concerning the set J of all intended applications we shall use also particular theoretical claims referring to subsets Ji of J. We shall denote this particular theoretical claims by TC (Ji). Then TC(J) denotes the global theoretical claim referring to the set J of all intended applications. In order to show that the use of an RSsentence is not necessary for solving the problems of theoretical terms and of testing theoretical hypotheses, we shall expound the alternative noneliminationist bootstrap procedure in our example. We shall thus attempt to test the theory T described in the example. The shortest way of confirming theory T, i.e., the law of the theory, consists in a set {<n1,t1> , <n2, t2>} of two intended theoretical applications where the domains of t1 and t2 contain a common member y such that t1(y ) = t2(y ). The bootstrap procedure consists in using as an assumption the theoretical claim referring to this set of applications. We obtain then the following piece of reasoning: 1. TC( { <n1, t1> , < n2, t2> } ) (Assumption of bootstrapping) 2. <n1, t1> is a T (From 1, by definition of TC) 3. <n2, t2> is a T (From 1, by definition of TC) 4. t1= t2= t (From 1, by definition of TC, t being the common value of the functions t1 and t2) 5. f (n1, t ) = 0 (From 2, 4) 6. f (n2, t ) = 0 (From 3,4) 7. t = g (n1) (From 5, computation of t ) 8. f [(n2, g (n1)] = 0 (From 6,7). This reasoning can be used for two purposes. First, it affords the computation of tfunction values in so far as these values are in fact determined by TC. Second, it serves to derive from TC the last line which is a relation between nfunction values expressing a prediction about nfunction values, hence an empirically testable one. Therefore, the bootstrapping procedure in principle constitutes a satisfactory solution to the problem of testing quantitative theoretical hypothesis, so that the use of RSsentences is not necessary for that purpose. In order to show the insufficiency of the use of RSsentences let us rework our abovementioned bootstrap reasoning via replacing the theoretical claim at the top by the corresponding RSsentence: $ t 1$ t 2[ f (n1, t1) = 0 & f (n2 , t 2) = 0 & t 1 = t 2 ] (Assumption, RSsentence corresponding to TC in the above example) f (n1 , t*) =0 & f (n2 , t*) = 0 (From 1, taking t* to be the common value of the instances of t 1 and t 2) 3. f (n1 , t*) = 0 (From 2) f (n2 , t*) = 0 (From 2) t* = g (n1) (From 3, computation of t*) f (n2, g (n1)) = 0 (From 4 and 5). Now it is important to note that the RSsentence on the top entails only the last line, viz., a prediction about n function values, but it does not entail the intermediary lines which involve t*values. Therefore, these intermediary lines cannot be interpreted as formulating a computation of t function values, even not of the value of t*function, occurring in the reasoning. Thus, the use of the RSsentence cannot justify the computation of tfunction values. However, the reasoning in question involves a subreasoning which indeed constitutes a formally correct justification of the computation of t*function values, viz., the one starting with the second line, i.e., the existential instantiation of the first line. But, of course, the second line which is of the form T(ni, t*), is an exact counterpart of T(ni, ti). Hence, in order to justify the computation of t*function values we do not need at all the RSsentence, which should be then deleted from the reasoning. But, then, the resulting reasoning starting from T(ni, t*) is a form of bootstrapping. One may object by remarking that there still exists a possibility of testing the central empirical claim RS(I ) of a theory T directly (i.e., without computing t function values). This refers to cases in which the t functions are Ramseyeliminable in the strong sense from the theory. Indeed, in such a case RS(I ) can be tested by means of testing its Ramseyelimination equivalent RE(ni) which depends exclusively on the determination of n function values via measurements. However, this seems to be the case only for artificial examples, and not for genuine physical theories. For example, Sneed himself acknowledges that ". . . it is not likely that the mass function can be Ramsey eliminated from any claim of classical particle mechanics" (1979, p. 152). Besides, the force functionthough Ramseyeliminable from RSsentences concerning applications without constraint and special lawsit is not Ramseyeliminable in case of applications with constraints and special laws (ibid., p. 152). However, in the general case of Ramseynoneliminability it is very difficult, if possible at all, to test RS(I ) directly, as Sneed himself points out: "Of course, even if we knew all the values of [nfunctions involved in the given applications referred to by the RSsentence] we might simply not be clever enough mathematicians to discover whether the requisite tfunction exists" (ibid., p. 51). We can say, then, that the use of RSsentences is insufficient in computing tfunction values. III. Conclusion We conclude that structuralist eliminationism does not succeed in computing the values of theoretical functions. On the contrary, bootstrapping can be used successfully in accomplishing this task, and, as a matter of fact, is implicitly present in structuralism (both in its eliminationist and noneliminationist versions). 
Notes (1) Glymour pointed out that "Sneed and Stegmueller came upon a central feature of theory testing [viz., circular confirmation, or in other words bootstrapping] and, very much like their predecessors [Reichenbach and Carnap, past it up because of a commitment to a particular and unnecessary conception of justification [i.e., linear or foundational justification]" (1980, p.109). (2) We elsewhere, in an unpublished paper, argue that bootstrapping in the strict sense can be salvaged from various cuounterexamples raised against it. Bibliography Glymour, C. (1980), Theory and Evidence. Princeton, New Jersey: Princeton University Press. Niiniluoto, I. (1981), "The Growth of Theories: Comments of the Structuralist Approach," in: Theory Change, Ancient Axiomatics and Galileo's Methodology: Proceedings of the 1978 Pisa Congress on History and Philosophy of Science (ed. by J. Hintikka, D. Gruender and E. Agazzi). Dordrecht: Reidel. Sneed, J. D. (1979), The Logical Structure of Mathematical Physics, second edition, revised. Dordrecht: Reidel. 