Adams on theoretical reduction

MICHAEL A. DAY

A D A M S ON T H E O R E T I C A L R E D U C T I O N

I. I N T R O D U C T I O N

The purpose of this paper is to critically review the work of E. W. Adams on theoretical reduction. Adams' views on reduction are exclusively set forth in his dissertation [1], Axiomatic Foundations of Rigid Body Me- chanics, and an article [2] based upon his dissertation. The reasons for reviewing Adams' work on reduction are the following: (a) Adams' views on reduction are interesting and well developed, but have received little direct attention (except by J. D. Sneed [11]), and sometimes incorrect interpretation, by other philosophers of science, (b) Adams' analysis of reduction is open to certain criticisms, but revisions and extensions in his analysis are possible to make his analysis more acceptable, and (c) Adams' example of the reduction of rigid body mechanics (henceforth abbreviated "RBM") to particle mechanics (henceforth abbreviated "PM") appears to involve a special case of mutual reduction. Also by reviewing Adams' work on reduction, certain aspects of the work done by such "structuralistic" philosophers as J. D. Sneed I11], W. Stegrniiller [12], W. Leinfellner [6], C. U. Moulines [8], and D. Mayr [7] will be highlighted as significant revisions, extensions, and modifications of Adams' work.

I I . ADAMS ON S C I E N T I F I C THEORIES

In discussing the concept of reduction, it is necessary to have a concept or at least a "working" concept of a scientific theory, since reduction is construed as a relation which can exist between scientific theories. Conse- quently, Adams begins by first developing a characterization of scientific theories. Adams holds that his characterization can be applied to a wide variety of theories both inside and outside the natural sciences. However Adams does caution that his characterization should not be expected to cover all the aspects of scientific theories, but that it should be adequate to account for the logical aspects of reduction.

To begin, Adams treats theories as set-theoretical entities rather than as

Erkenntnis 23 (1985) 161-184. 0165~)106/85.10 �9 1985 by D. Reidel Publishing Company

162 MICHAEL A. DAY

linguistic entities. We are first given the scientific theory in a "rough and intuitive form," for example in the discussions and writings of scientists. Based upon this "rough and intuitive form" of the theory, we define a set-theoretical predicate. Now the distinction is drawn between the set of all entities which satisfy this set-theoretical predicate and the set of all entities to which the theory in its "rough and intuitive form" is to be applied. These two sets will consist of ordered n-tuples. On Adams' re- construction, the theory is to be identified as the ordered pair whose first member is the set of all entities satisfying the set-theoretical predicate (this set is called the "characteristic set" of the theory) and whose second member is the set of all entities to which the theory is to be applied (this set is called the set of the "intended interpretations" of the theory).

With this characterization of a scientific theory in mind (i.e., construing a theory as a particular type of ordered pair), Adams says that we can consider a theory, in its "rough and intuitive form," as asserting that its intended interpretations satisfy the set-theoretical predicate associated with that theory, or more formally, as asserting that the set of intended interpretations is a subset of the characteristic set of the theory. Further, a theory is said to be correct if and only if the set of intended interpretations is a subset of the characteristic set of the theory. In most interesting and detailed ways, J. D. Sneed has developed this idea of the central empirical claim (or assertion) of a theory in his book, The Logical Structure of Mathematical Physics ([11], Chapters 2-5). Within his development, Sneed sets forth his concepts of partial possible model, possible model, extensions, constraints, and 0-theoretical functions.

Let us now illustrate Adams' characterization of scientific theories, by considering the theories of PM and RBM as set forth by Adams in his dissertation. With respect to PM, the following definition of a set-theoretical predicate is given, which specifies the characteristic set of PM.

DEFINITION 1. Let F = <P, T, m, S, F>. Then F is a system of classical r-dimensional particle mechanics if and only if F satisfies conditions A1- A6. A1. P is a non-empty finite set. A2. T is an interval of real numbers. A3. S is an r-vector valued function defined over P x T, and for all p in P

and all t in T, d2S(p, t)/dt 2 exists.

A D A M S O N T H E O R E T I C A L R E D U C T I O N 163

A4. A5.

m is a function defined over P taking as values positive real numbers. Fis an r-vector valued function defined over P x T x N(where Nis the set of positive integers), and for all p in P and all t in T, the series

oo

F(p, t, n) n = I

is absolutely convergent. A6. For all p in P and all t in T,

dzS(p, t) oo m(p) dt ~ - ~ F(p, t, n).

I1=1

Let us now consider the set of intended interpretations of PM. For a more complete specification of the intended interpretations of PM, than the one to be given, the reader is referred to either of Adams' works. But the following will suffice for our purposes: F = (P, T, m, S, F ) is an intended interpretation of PM if and only if there exists a complete set of units of measurement/~ and a Cartesian coordinate system C such that the following are satisfied?

P: P is a non-empty set of particles. T: T is an interval of real numbers representing a time interval relative

to the complete set of units p. S: S is a function from P x T taking vectors as values such that S(p, t) is

the vector representing the position of particle p at time t relative to the complete set of units p and the Cartesian coordinate system C. (Note: The reference frame defined by the Cartesian coordinate system C must be inertial.)

m: m is a function from P taking positive real numbers as values such that m(p) is the mass of particle p relative to the complete set of units #.

F: Fis a function from P • T x N (where N is the set of positive integers) taking vectors as values such that F(p, t, n) is the vector representing the n th applied force acting on particle p at time t relative to the complete set of units p and the Cartesian coordinate system C.

In setting forth the definition of the set-theoretical predicate which spec-

164 MICHAEL A. DAY

ifies the characteristic set of RBM, Adams uses a binary matrix operation which is not standard in the literature. Adams defines it as follows: If A and B are two n x r matrices (possibly vectors), then

A x B = A ' B - B*A

where A* and B* are respectively the transpose of A and the transpose of B. Let us now turn to the definition of the set-theoretical predicate which specifies the characteristic set of RBM.

DEFINITION 2. Let F = (K, T, g, R, H, W, O) . Then F is a system of r-dimensional rigid body mechanics if and only if F satisfies conditions B1-B5.

B1. H i s a function defined over K x T x N (where Nis the set of positive integers), and if H 1 and H 2 are defined over the same domain such that for all k in K, t in T, and n in N,

H(k, t, n) = (Hi (k , t, n), H2(k, t, n)),

then (K, T, g, R, H 1) is a system of classical r-dimensional particle mechanics.

B2. O is a function defined over K x T taking as values r x r orthogonal matrices, and for all k in K and all t in T, d20(k, t)/dt 2 exists.

B3. Wis a function defined over Ktaking as values r x r symmetric positive semi-definite matrices of rank r or r-1.

B4. H 2 takes as values r-vectors, and for all k in K and t in T, the series

(3O

H2(k, t, n) x Hi(k, t, n) n = l

is absolutely convergent. B5. For all k in K and all t in T,

I 1" d20(k, t) =

W(k) dt 2 n = l

O(k, t ) x H2(k, t, n )x Hi(k , t, n).

ADAMS ON THEORETICAL REDUCTION 165

Let us now consider the set of intended interpretations of RBM. Again, for a more complete specification of the intended interpretations of RBM, than the one to be given, the reader is referred to either of Adams' works. But the following will suffice for our purposes: F = (K, T, g, R, H, W, O) is an intended interpretation of RBM if and only if there exists a complete set of units of measurement/~ and a Cartesian coordinate system C such that the following conditions are satisfied. K: K is a non-empty finite set of rigid bodies. T: T is an interval of real numbers representing a time interval relative

to the complete set of units/~. g: g is a function from K taking positive real numbers as values such

that g(k) is the mass of rigid body k relative to the complete set of units/~.

R: R is a function from K • T taking vectors as values such that R(k, t) is the vector representing the position of the center of mass of rigid body k at time t relative to the complete set of units/~ and the Cartesian coordinate system C. (Note: The reference frame defined by the Carte- sian coordinate system C must be inertial.)

H: His a function from K x T x N (where Nis the set of positive integers) taking ordered pairs of vectors as values such that H(k, t, n) = (H 1 (k, t, n), H 2 (k, t, n)) represents the #h-applied force acting on rigid body k at time t in the following manner: H 1 (k, t, n) is the vector such that the magnitude of H 1 (k, t, n) is the magnitude of the nth-applied force acting on rigid body k at time t relative to the complete set of units/~ and the direction of H 1 (k, t, n) relative to the Cartesian coordinate system C is the direction of the #h-applied force acting on rigid body k at time t, and H 2 (k, t, n) is the vector representing the position of the point of application of the nth-applied force acting on rigid body k at time t relative to the complete set of units # and a Cartesian coordinate system whose origin is fixed at the center of mass of k and whose axes remain parallel to the axes of the Cartesian coordinate system C.

W: W is a function from K taking matrices as values such that W(k) is the matrix representing the moment of inertia of rigid body k relative to the set of units/a and a Cartesian coordinate system which rotates with k and whose origin is fixed at the center of mass of k.

O: O is a function from K x Ttaking matrices as values such that O(k, t)

166 MICHAEL A. DAY

is the matrix which represents the "'orientation" at time t of the Carte- sian coordinate system which rotates with rigid body k relative to the Cartesian coordinate system C.

As a final remark on Adams' views on scientific theories, something must be said about the set of intended interpretations of a theory. For Adams, intended interpretations are actual physical systems, or perhaps more properly said, composed of actual physical objects or systems. Secondly, an intended interpretation is a physical system only in the extended sense of being an order n-tuple such that the members of this n- tuple are either actual physical systems (or objects) or sets (relations, functions, etc.) involving actual physical systems (or objects). Lastly, Adams says that the set of intended interpretations of a theory from the natural sciences is

usually defined by means of so called 'coordinating definitions' ... which usually include concepts relating to physical operations or observations, which are more or less vague ... The fact that the intended interpretations of theories in the natural sciences may be defined in terms of vague concepts means that there may be no unequivocal criterion determining whether or not a given n-tuple is an intended interpretation of a given theory. (Adams [1], pp. 14-15)

This remark by Adams that there is no "unequivocal criterion" for determining the intended interpretations reflects an essential factor in the dynamics of theories developed by J. D. Sneed [11] and W. Stegmuller [12]. By allowing the set of intended interpretations to be defined in an inten- sional way (e.g., by paradigm applications) and letting the theory decide in cases of arbitrariness, Sneed and Stegmiiller have set forth a view of the genesis of theories with a definite resemblance to the views of T. S. Kuhn [5].

III . ADAMS ON R E D U C T I O N

We will first consider Adams' analysis of reduction as it is given in his dissertation [1], and then compare this analysis to the analysis given in the article [2] which is based upon his dissertation. In his dissertation, Adams begins his discussion of reduction by "analyzing some of the essential fea- tures of the reduction of thermodynamics to statistical mechanics." Con- cerning this reduction, he says


The reduction of thermodynamics to statistical mechanics involves essentially two aspects: (I) framing a hypothesis about a relation between the fundamental concepts of thermodynamics and those of statisticals mechanics, and (2) showing that with this hypothesis it is possible to deduce the fundamental laws of thermodynamics from the laws of statistical mechanics. (Adams [1], p. 17)

Adams holds that these two aspects of this particular reduction are essential to any reduction, and calls them respectively the "condition of connectability" and the "condition of derivability," and that the so-called hypothesis underlying these two aspects of any reduction is an empirical hypothesis.

Adams' analysis of reduction consists of a formulation of these two conditions in terms of set-theoretical concepts (hence avoiding linguistic concepts) and a specification of what he calls two "informal" conditions of reduction. The following two definitions will be sufficient to set forth in an explicit manner Adams' analysis of reduction. In these definitions, T1 and T2 are theories in the sense of being ordered pairs.

D E F I N I T I O N 3. T2 is reducible to T1 if and only if there exists a reduction relation R from T2 to T1.

D E F I N I T I O N 4. Let T1 = (C1, 11) and T2 = (C2, I2>. Then R is a reduction relation from T2 to TI if and only if (1) if iz e 12, then there exists an il e 11 such that i2Ril (condition of connectability), 2 (2) if cl e Cl and c2Rci, then c2 �9 C2 (condition of derivability), (3) if il �9 11, i2 �9 12, and i2Rii, then the physical objects or systems of i2 are numerically the same as or composed out of the physical objects or systems of il (first informal condition), and (4) if il �9 I~, i2 �9 12. and i2Ril, then if the same type of measurement occurs with respect to both il and i2 then the bases (units, zero points, coordinate systems, etc.) of this measurement are the same (second informal condition).

Basic to this analysis is the concept of a reduction relation. A reduction relation relates ordered n-tuples, and must be such that it satisfies conditions (1)-(4) of Definition 4. Therefore, the definition of a relation relating ordered n-tuples and argumentation that conditions (1)-(4) are satisfied by this relation, are essential in establishing a reduction on Adams' analysis. Let us turn to a somewhat informal discussion of the significance of conditions (1)-(4).

168 MICHAEL A. DAY

Condition (1) requires that if T2 is reducible to TI on the basis of R, then for any intended interpretation i2 o f / ' 2 there must exist an intended interpretation il of T~ such that i2 has relation R to i~. Further by condition (3), the physical objects or systems of i2 must be the same as or composed out of the physical objects or systems of il. Roughly, this is requiring that for whatever physical system T2 applies, 7"1 also applies. One would certainly hold that this is a necessary condition for reduction if it is held that a reducing theory is to be at least as "strong" as the reduced theory in application or that the domain of the reducing theory should include the domain of the reduced theory.

Condition (2) requires that if an entity c~ satisfies the set-theoretical predicate associated with T1 and some entity c2 has relation R to c~, then c2 will satisfy the set-theoretical predicate associated with T2. Roughly, this is requiring that if an entity satisfies the "laws of theory TI" and some other entity has relation R to this entity, then this other entity will satisfy the "laws of theory T2." Further, using conditions (1) and (2), it can be proven that if T2 is reducible to TI, then if TI is correct then T2 is correct (correct in Adams' sense). Adams refers to this result as "the correctness consequence" and concerning this result says that "This is intuitively what we should demand of a reduction ..." (Adams [1], p. 22).

Condition (4) requires that if two intended interpretations are related by R, then if the same type of measurement occurs with respect to both then the bases of this type of measurement must be the same for the two intended interpretations. Adams says in support of this condition,

We should regard it as strange, for example, if the reduction relation of rigid body mechanics to particle mechanics were defined so as to hold only between a system of rigid bodies in which masses were measured in pounds and a system of particles in which masses were measured in grams. (Adams [1], p. 36)

Also, a reduction relation R is an empirical relation in the derivative sense that whether two intended interpretations are related by R will depend on empirical relations which exist between the physical objects or systems composing the intended interpretations. This should be expected since a reduction in science is considered as an advance or development that depends essentially on empirical considerations.

Having considered Adams' analysis of reduction as given in his dissertation, a few comparisons and remarks are in order. First of all, Adams' views on reduction are similar to Leinfellner's views [6] on reduction. Lein-

ADAMS ON T H E O R E T I C A L R E D U C T I O N 169

fellner, like Adams, develops his discussion of reduction within a set-theoretical approach. Leinfellner requires, for the reduction of one theory to another, the appropriate deducibility of the formal structures of the "axiomatized kernels" of the two theories and appropriate overlapping of the empirical fields of application of the two theories.

Let us also compare Adams' analysis of reduction to Patrick Suppes' views on reduction. Suppes says

To show in a sharp sense that thermodynamics may be reduced to statistical mechanics, we would need to axiomatize both disciplines by defining appropriate set-theoretical predicates, and then show that given any model T of thermodynamics we may find a model o f statistical mechanics on the basis of which we may construct a model isomorphic to T." (Suppes [14], p. 271)

Concerning models, Suppes says

When a theory is axiomatized by defining a set-theoretical predicate, by a model for the theory we mean simply an entity which satisfies the predicate. (Suppes [14], p. 253)

Therefore, we will temporarily construe Suppes' analysis of reduction as given by the following definition.

DEFINITION 5. Let Tx = (Cx, 11) and T2 = (C2, /2). Then T2 is

reducible to T1 if and only if for any c2 E C2 there exists a cl ~ C1 such that on the basis of c~ we can construct a c3 such that c3 is isomorphic to C2.

It is difficult to see how this analysis of reduction compares with Adams' analysis, since it is not clear what Suppes means by "construct" or even "isomorphic." For example, concerning "isomorphism," he says

A satisfactory general definition of isomorphism for two set-theoretical entities of any kind is difficult if not impossible to formulate. The standard mathematical practice is to formulate a separate definition for each general kind of ordered n-tuple. When the n-tuples are complicated as in the case of models for the theory of particle mechanics, it is sometimes difficult to decide exactly what is to be meant by two isomorphic models ... (Suppes[14], p. 262)

To compare Suppes and Adams, we will have to consider the notion of a representation relation as discussed by Adams. He says

Consider two set-theoretical theories T1 = (C~, 11) and T2 = (C2, 12), and a relation R. If it is the case that an entity x has property C2 if and only if there exists an entity y with property C1 such that xRy, then R is a representation relation between/ '2 and Tt. (Adams [1], pp. 37-38)

170 M I C H A E L A. DAY

Concerning representation relations, Adams says

In one way, the notion of a representation can be regarded as a generalization o f the notion of an isomorphism ... (Adams [1], p. 38)

With these ideas in mind, a plausible way to reconstruct Suppes' analysis of reduction is given in the following two definitions.

DEFINITION 6. T2 is reducible to T1 if and only if there exists a representation relation R from T2 to T1.

DEFINITION 7. Let T1 = (C1, 11) and T2 = (C2, 12). Then R is a representation relation from Tz to T1 if and only i f ( l ) if c2 e C2, then there exists a cl e C1 such that c2Rc,, and (2) if Cl ~ C1 and c2Rcl, then c2 e C2.

From this formulation of Suppes' analysis of reduction, we can see that Adams' analysis differs from Suppes' analysis in requiring a reduction relation instead of a representation relation. These two types of relations essentially disagree with what might be called the "applied aspect" (intended interpretations) of scientific theories. Suppes' analysis of reduction only considers what might be called the "formal aspects" of theories, while neglecting the "applied aspects". For example, Suppes can not derive from his analysis the "correctness consequence" as Adams can on his analysis.

Further, neither analysis entails the other. While both analyses agree on their second conditions, their first conditions are not related as such (or related by entailment). Adams' first condition is concerned with intended interpretations, where Suppes' first condition is concerned with the entities that satisfy the associated set-theoretical predicates; and there is no guar- antee that the intended interpretations of the theories satisfy their respec- tive set-theoretical predicates. Even if we assume the correctness of the theories, these two analyses are still not related by entailment. Therefore, these two analyses will (in all probability) disagree in many cases on what is to be classified as a reduction. For example, Adams considers two physical theories which have the same formal structure (i.e., the theory of vi- brating strings and the theory of electric currents on a transmission line). On his analysis of reduction, neither theory is reducible to the other; but on Suppes' analysis there is a reduction in this case, as a matter of fact each theory would be reducible to the other. 3


It certainly appears to be the case that Adams' analysis is more adequate than Suppes' analysis of reduction since it is reasonable to assume that "applied aspects" of theories must be considered in any adequate analysis of reduction. But this should not be construed as implying that Adams' analysis is fully adequate; as is well-known, his analysis is open to serious criticisms.

Let us now consider, Adams' analysis of reduction as it is given in the article which is based upon his dissertation. In this analysis, Adams is clearly only giving necessary conditions for reduction. The following definition and requirement will be sufficient to set forth this analysis.

D E F I N I T I O N 8. T2 is reducible to T1 if and only if there exists a reduction relation R from T2 to T1.

R E Q U I R E M E N T 1. Let TI = (Cx, 11) and T2 = (C2, I2). If R is a reduction relation from T2 to T1, then (1) if i2 E I2, then there exists an i,

11 such that i2Ril, and (2) if cl ~ C1 and c2Rcl, then c2 e C2.

This analysis gives necessary conditions for reduction in the sense of only giving two necessary conditions for reduction relations in Requirement 1. Further, these two conditions are the same as the first two conditions for reduction relations as given in Adams' dissertation.

Concerning these two conditions (which he calls conditions A and B), Adams says

As has been pointed out, Conditions A and B do not define the concept of reduction. A complete analysis of this notion would undoubtedly formulate considerably more restrictive conditions than ours on the concept. In fact, our conditions are so weak that for any two correct theories it is possible to construct a trivial relation 'reducing' one to the other satisfying Conditions A and B. (Adams [2], p. 262)

To see how such a trivial reduction can be constructed for any two correct theories, just consider the following definition of the relation R: Let T1 = ( C 1 , 11) and T2 = ( C 2 , 12) where T1 and T2 are correct theories, then c2Rcl if and only if cl e C1 and c2 e C2. Such a relation R can easily be shown to satisfy the two previous conditions for reduction relations. 4

Clearly in his article, Adams is only formulating necessary conditions for reduction; and the question might arise whether our previous account, involving necessary and sufficient conditions, of Adams' analysis of re-

172 MICHAEL A. DAY

duction as set forth in his dissertation is a correct interpretation. This question is legitimate since it is not exactly clear that Adams is giving necessary and sufficient conditions for reduction in his dissertation. In defense of our interpretation of Adams, we will say that it certainly makes Adams' analysis more interesting if we construe him as giving necessary and sufficient conditions for reduction. Further, Adams never denies in his dissertation that he is giving necessary and sufficient conditions for reduction. Finally, Adams' analysis in his dissertation is stronger than the one given in his article. For example, the so-called trivial reductions considered above can not be constructed on the analysis given in the dissertation since R-related intended interpretations must consist of or be composed of the same physical objects or systems, s Further, this analysis does not even allow reductions to be constructed on specialized representation relations (e.g., Adams' example of the relation "kinematically represents" for PM and RBM) even if we assume certain truths about the empirical world. For these reasons, we have construed Adams as giving necessary and sufficient conditions for reduction in his dissertation. Also, in what follows, we will take the analysis given in his dissertation to be Adams' analysis of reduction. Let us now turn to Adams' discussion of the reduction of RBM to PM.

IV. ADAMS ON THE R E D U C T I O N OF R I G I D BODY M E C H A N I C S TO

P A R T I C L E M E C H A N I C S

To establish the reduction of RBM to PM on his analysis of reduction, Adams must first define a relation R, and then argue that this relation is a reduction relation from RBM to PM (i.e., it satisfies conditions (1)-(4) of Definition 4). Since it would be cumbersome to fully define the relation R presented by Adams and since such a definition is not necessary for our purposes, we will only present some of Adams' conditions upon R in the following definitional sketch.

DEFINITION 9. Let F = (K, T', g, R, H, W, O) and A = (P, T, m, S, F) . Then F is R-re la ted to A if and only if conditions C1-C6 are satisfied. C1. K is partition of P. C2. For all p, q �9 k where k �9 K and t, t ' �9 T,

S (p , t) - S(q , t) = S(p , t ' ) - S (q , t').


C3. T'---T. C4. For all k ~ K,

g(k ) = ~ m(p). pek

C5. For a l l k ~ K a n d t ~ T ,

1 R(k, t) g(k) ~ m(p)S(p, t).

p~k

C6. Further conditions involving m, S, F, R, H, O and W. Let us now consider whether R as given in Definition 9 satisfies condi-

tions (1) and (2) of Definition 4 with respect to RBM and PM. We will not concern ourselves with the question whether R satisfies conditions (3) and (4) of Definition 4 since such a discussion would involve certain tech- nicalities which are not directly related to our purposes. Now Adams has shown in his dissertation that R satisfies condition (2) of Definition 4, i.e., if an entity satisfies the set-theoretical predicate associated with PM and another entity has relation R to this entity, then this other entity satisfies the set-theoretical predicate associated with RBM. Roughly put in an alternative way, Adams has shown that the laws of rigid body mechanics follow from the laws of particle mechanics (given certain assumptions of course).

The major difficulty, according to Adams, is whether R satisfies condition (1) of Definition 4, i.e., whether for any intended interpretation i2 of RBM there is an intended interpretation il of PM such that i2 is R-related to il.6 Now since the intended interpretations of RBM are septuples whose first members are sets of rigid bodies and the intended interpretations of PM are quintuples whose first members are sets of particles, the question whether R satisfies condition (1) depends whether it is empirically true that each rigid body is composed of finitely many particles which are fixed with respect to their mutual positions, that the mass of each rigid body is equal to the sum of the masses of the particles that compose it, etc. Further, Adams holds that whether the reduction of RBM to PM is successful will also depend upon the empirical truth of these assumptions. But at the end of the article based upon his dissertation, Adams leaves us with a some-

174 MICHAEL A. DAY

what disturbing passage concerning the truth of these empirical assumptions.

The empirical question here raised is a very difficult one, and involves in addition the problem of clarifying the rather vague notion of a particle. It may be observed that the molecular theory lends support to the hypothesis that rigid bodies are composed of entities small enough to approximate the point-particles required in the derivation of the laws of RBM, and the theory of solids indicates that these molecules remain relatively fixed within rigid bodies. However, the facts that molecules only approximate point-particles, and that they are not perfectly rigidly fixed within the bodies they compose, shows that the deduction of the laws of RBM from those of PM depends on an hypothesis which, taken exactly, is false. The necessary revisions are, however, complicated, and are, in any case, beyond the scope of this paper. (Adams [2], p. 264)

From this passage, it seems that Adams still holds that RBM is reducible to PM, but it is clear that for him the relation R given in Definition 9 will not establish the reduction of RBM to PM. Further, he has not even hinted at what revisions are necessary to make the reduction go through. More will be said concerning this problem in Section VI of this paper; let us now turn to an evaluation of Adams' account of theoretical reduction.

V. GENERAL EVALUATION OF ADAMS' ACCOUNT OF REDUCTION

In evaluating Adams' account of reduction, three main criticisms and certain replies to these criticisms will be considered. The three criticisms are given in the following.

(1)

(2)

(3)

Most traditional cases of reduction do not satisfy Adams' analysis of reduction. Adams has not considered in detail what might be called the "semantical nature" of reduction relations. Adams has not considered the possibility of different types of reduction.

The second and third criticisms focus on the incompleteness of Adams' account of reduction, and if valid require at least extensions of his account. The first criticism goes to the heart of Adams' account, and if valid calls for a revision of his account. Let us therefore consider the first criticism.

It should be obvious that most examples which have been classified as reductions will not satisfy Adams' analysis of reduction, for example the

A D A M S ON T H E O R E T I C A L R E D U C T I O N 175

reduction of Galilean mechanics to Newtonian mechanics. This can be seen by just considering the "correctness consequence" of Adams' analysis, namely if the reducing theory is correct then the reduced theory is correct. But this is not the case for most reductions. For if Newtonian mechanics is correct, then Galilean mechanics is incorrect. The trouble in Adams' analysis is coming from his second condition for reduction relations (i.e., condition (2) of Definition 4); this condition is just too strong. Consider the case of the reduction of Galilean mechanics to Newtonian mechanics. Fundamental to Galilean mechanics is the thesis that near the surface of the earth, bodies in free fall have a constant acceleration provided such things as air resistance are neglected. According to Newtonian mechanics, this is not strictly the case, for there is a small increase in acceleration as a body in free fall gets closer to the earth. One might say that if Newtonian mechanics is correct, then Galilean mechanics is correct "given allowable approximations."

With this idea in mind, we could reformulate Adams' second condition for reduction relations as follows:

(2') if cl e C1 and c2RCl, then "given allowable approximations" C 2 ~ C 2.

Even though such a revision in Adams' analysis allows us to avoid the criticism under consideration, we are now confronted with the problem of specifying what is to be meant by an "allowable approximation. ''7 The recent work done by such "structuralistic" philosophers as C. U. Moulines [8] and D. Mayr [7] deals with the difficult problems introduced when one attempts to explicate the notions of approximative reduction and/or intertheoretic approximation.

Let us now consider the criticism of Adams' account of reduction which is based upon the "semantical nature" of reduction relations. To under- stand the significance of this criticism, we must briefly consider what other philosophers (Sklar [10], Schaffner [9], Causey [3], etc.) have said concerning certain aspects of the phenomenon of reduction. It is held that in many cases of reduction, the two theories under consideration are such that certain terms appear in the theory to be reduced (henceforth called the "secondary theory") that do not appear in the theory which is to do the reducing (henceforth called the "primary theory"). For example, the term

176 M I C H A E L A. D A Y

"temperature" occurs in thermodynamics, but does not occur in statistical mechanics. Now according to these philosophers if the secondary theory is to be reducible to the primary theory, some connections must be made between the terms which appear in the secondary theory and the terms which appear in the primary theory. Further, these connections will consist of statements in which these terms essentially occur. The question arises, What is the semantical status of these connecting statements?

One might defend the view that these connecting statements are nomological statements which assert the correlation of certain properties. But such a view has been criticized by certain philosophers (e.g., Sklar, Schaff- ner, and Causey) who defend the view that such statements must be construed as synthetic identifications involving types of things and attributes (or properties), s Some of the reasons given for this view are that scientists consider such statements as identifications in many cases and that only by such identifications can reduction accomplish linguistic and ontological simplifications. Also, it is held that if such statements are interpreted as nomological correlations then they are in need of explanation, and hence the reduction has not fully accomplished one of its goals - the explanation of the secondary theory by the primary theory. But if we interpret the connecting statements as identifications, no explanation of them is needed. All that is required is a justification for each identification.

The significance of the second criticism of Adams' account of reduction is that he has not taken a clear position with respect to the semantical status of these connecting statements. This is certainly the case, but certain questions arise: Where within Adams' account does this semantical issue appear? What if anything does Adams say about this issue? Can Adams' account be extended to take a position with respect to this issue?

First of all, this semantical issue will be an issue concerning what might be called the "semantical nature" of reduction relations. The "semantical nature" of a reduction relation R is to be, roughly speaking, the nature of the relationships which must exist for it to be true that i2 is R-related to il where i2 and il are respectively intended interpretations of the theories T2 and T1 (here R is assumed to be a reduction relation from T2 to /'1). Now by specifying the "semantical nature" of reduction relations, one could take a definite stand on the semantical issue concerning the connecting statements between the so-called primary and secondary theories.

Secondly, Adams does say something about the "semantical nature" of


reduction relations in the sense that whether two intended interpretations are related by a reduction relation will depend on empirical relations which exist between the physical objects or systems composing the intended interpretations. Also, condition (3) of Definition 4 for reduction relations requires that R-related intended interpretations must be such that the physical objects or systems of these intended interpretations are numerically the same or composed out of one another. But Adams has not taken a complete position with respect to the "semantical nature" of reduction relations since he has not said anything about the correlation and/or identification of the attributes (or properties) instantiated by the physical objects of systems which compose intended interpretations which are related by reduction relations. But it does seem reasonable to assume that Adams' account of reduction can be extended so that it can take a complete position on this semantical issue.

Let us now turn to the third criticism concerning Adams' account of reduction - that he has not considered the possibility of different types of reduction. This criticism is certainly valid. First of all, there do seem to be different types of reduction. For example, we speak of microreductions. Also, it seems reasonable to assume that there are reductions in which the secondary theory deductively follows from the primary theory given certain assumptions (deductive reductions), which would contrast with the case where the secondary theory deductively follows from the primary theory only given certain allowable approximations (approximative reductions). Now Adams' account of reduction is incomplete since it does not consider these different types of reduction. But Adams' account could be extended to cover these different types of reduction, for example by placing different requirements on reduction relations. If we wanted to dis- tinguish deductive reductions from approximative reductions, we could do this by leaving condition (2) of Definition 4 the same for reduction relations which characterize deductive reductions, and change condition (2) of Definition 4 to (2'), as previously given, for reduction relations which characterize approximative reductions.

VI. THE R E D U C T I O N OF P A R T I C L E M E C H A N I C S TO R I G I D BODY

M E C H A N I C S

In this section, an argument will be presented for the view that the reduc-

178 MICHAEL A. DAY

tion of RBM to PM is a special case of reduction. We are going to accept that Adams has presented a plausible case for the view that RBM is reducible to PM. Now we will argue that PM is reducible to RBM, and hence that RBM and PM are mutually reducible. Since reduction is gen- erally an asymmetric relation between theories, the reduction of RBM to PM must be a special case of reduction. Also, it appears that from another angle we are involved with a special case of reduction, since the reduction of RBM to PM is most likely a deductive reduction and such reductions seldom appear in science.

In arguing for the reduction of PM to RBM, we will accept Adams' formulations of these two theories and his analysis of reduction. Even though his analysis of reduction is defective, assuming it for the sake of arguing that PM is reducible to RBM should not affect the validity of the argument since Adams' analysis is too strong if anything. 9 With this in mind consider the following definition of the relation R'.

DEFINITION 10. Let F = (K, T', g, R, H, W, O ) and A = (P, T, m, S, F ) . Then A is R'-related to F if and only if conditions D1-D6 are satisfied.

D1. D2. D3. D4. D5.

D6.

P = K . T = T '. m = g . S = R . H is a function such that there exist functions H 1 and H 2 having the same domain as H such that for all x belonging to the domain of H, H(x) = (H' (x ) , H2(x)) and F = H ~.

If F is an intended interpretation of RBM and A is an intended interpretation of PM, then if the same type of measurement occurs with respect to both F and d then the bases (units, zero points, coordinate systems, etc.) of this measurement are the same.

From this Definition of R', we immediately see that R ' satisfies condition (2) of Definition 4 for reduction relations. This simply follows from con-


dition B1 of Definition 2. Also, it is obvious that R ' satisfies conditions

(3) and (4) of Definition 4. This can be easily seen by noting conditions D1 and D6 of Definition 1.

The difficult aspect in arguing for this reduction is establishing that R'

satisfies condition (1) of Definition 4, i.e., to show that for any intended interpretation i2 of PM, there exists an intended interpretation it of RBM such that i2 is R'-related to it. This will involve us in a discussion of what is a particle. Adams distinguishes two types of particles as can be seen in the following passage.

The first type of particle is any physical object which has mass and location but whose size, for the purposes of some given problem, is negligibly small. We shall call this type of particle a mass poin t . . . An alternative interpretation of the notion of a particle embraces any physical object at all which has mass and a definite center of mass. This interpretation is not common, but is admissible in a certain sense because it can be shown that if the theory of particle mechanics is correct for the set of mass point interpretations, then it is also correct when the class of intended interpretations is widened to include particles of the second type. (Adams [1], pp. 72-73)

From this passage, it appears clear that Adams is holding that the intended interpretations of PM are composed of physical objects which have size. Therefore, there appear to be no reasons why it will not be the case that

for any intended interpretation i2 of PM that there will exist an intended interpretation il of RBM such that i2 is R'-related to il. The main reason one would have for saying that R' does not satisfy condition (1) of Defi- nition 4 is that particles are extensionless point masses where rigid bodies are not. But Adams does not hold this view. Therefore on Adams ' analysis,

PM is reducible to RBM. One might reply at this point and say that Adams is just mistaken here,

for the intended interpretations of PM should be so defined as to include extensionless point masses. Now, the intended interpretations of PM con- taining extensionless point masses can not be R'-related to intended interpretations of RBM since there will not be intended interpretations of RBM which contain these extensionless point masses. Hence, PM is not reducible to RBM. The only difficulty with this reply is that for Adams intended interpretations can only contain actual physical systems; and since there are no extensionless point masses, all intended interpretations of PM can still be R'-related to intended interpretations of RBM.

Again, one might reply that if there were extensionless point masses, then PM would not be reducible to RBM on Adams ' analysis. But it is

180 MICHAEL A. DAY

not clear that even this is correct. Since even if we assume the existence of extensionless point masses, it still might be the case that we can argue for the reduction of PM to RBM while holding on to the basic views of Adams' account. Remember Adams grants that there is some imprecision in specifying the intended interpretations of PM and RBM. Therefore, it could be argued that extensionless point masses would be included in the intended interpretations of RBM, for extensionless point masses would just be rigid bodies with zero spatial dimension.

One must be careful at this point since if extensionless point masses are admitted into the intended interpretations of RBM, RBM becomes an incorrect theory. This is because according to Definition 2 the rank of the moment of inertia matrix must be r or r-1 (for intended interpretations r = 3); but for extensionless point masses this matrix has rank zero. To get around this difficulty, we would have to allow the rank of the moment of inertia matrix to be zero, and hence change Adams' formulation of RBM. The reason Adams imposes a restriction on the rank of this matrix is so that the equation of angular motion will give a deterministic solution. Hence if we omit this restriction, we are faced with indeterminacy with respect to angular motion. But this indeterminacy should not concern us since extensionless point masses are not capable of angular motion. There- fore, by allowing the rank of this matrix to be zero, we can include systems of extensionless point masses in the set of intended interpretations of RBM while keeping RBM a correct theory. Further, this extension of RBM will give the same empirical results as PM with respect to the physical systems in question.

One might finally object that such an extension of RBM is not allowable since rigid bodies must have size and extensionless point masses do not have size. Maybe this is correct. But is it not also the case that particles must have size and hence that the intended interpretations of PM should not include these extensionless point masses? Also, it must be remembered that these so-called extensionless point masses do not exist; hence there are no intended interpretations of PM that are not R'-related to intended interpretations of RBM under Adams' analysis. As a matter of fact, it appears that we can make a better case that PM is reducible to RBM than that RBM is reducible to PM under Adams' analysis. Remember that Adams was confronted with the problem that RBM is reducible to PM only if rigid bodies are composed of finitely many particles which remain

A D A M S ON T H E O R E T I C A L R E D U C T I O N 181

fixed with respect to their mutual positions. However in arguing for the reduction of PM to RBM, we are not confronted with such an analogous problem.

At this point, one might contend that since RBM and PM are most likely mutually reducible on Adams' analysis, something must be incorrect with his analysis since reduction is an asymmetric relationship between theories. For example, thermodynamics is reducible to statistical mechanics, but statistical mechanics is not reducible to thermodynamics. But it is not clear that reduction must in all cases be an asymmetric relation. As a matter of fact, there are reasons to expect that the relationship between PM and RBM is one of mutual reducibility; for PM and RBM are both branches of Newtonian mechanics, and it seems to be an arbitrary decision whether to start with the notion of a particle and then define what a rigid body is to be, or vice versa. One might well construe the argument for the mutual reducibility of PM and RBM based upon Adams' analysis, as giving support to his analysis.

In concluding this section, we must consider a possible inadequacy in Adams' analysis of scientific theories. Adams holds that the intended interpretations must consist of actual physical systems. This might be an oversimplication on Adams' part, since one might contend that scientific theories are not directly concerned with actual physical systems but are such that they characterize the behavior of "idealized replicas" of actual physical systems, and hence are in some sense indirectly about physical systems. This is a view set forth by Fredrick Suppe ([13], pp. 45-46). With such a view in mind, one might hold that within Adams' analysis we must make the distinction between the "intended scope" of a theory (which would consist of actual physical systems) and the intended interpretations (which would consist of "idealized replicas" of these actual physical systems). How such a distinction would affect Adams' analysis of reduction will not be considered here. However we must be aware of this possible oversimplification when holding that the intended interpretations of a theory consist of actual physical systems.

As a final comment, this oversimplification (assuming now that it is one) is perhaps partially reflected in the problem which was raised in the last part of Section IV of this paper. Remember that Adams remarked that the deduction of the laws of RBM from the laws of PM depended upon a hypothesis, which taken exactly, is false (i.e., that each rigid body is com-

182 MICHAEL A. DAY

posed of finitely many particles which are fixed with respect to their mutual positions). This leads to the result that the relation R as defined in Defi- nition 9 is not a reduction relation from RBM to PM. Adams concludes by saying that the necessary revisions are complicated and are, in any case, beyond the scope of his article. But perhaps Adams is somewhat misguided at this point.

To illustrate this, let us adopt Fredrick Suppe's approach to scientific theories and consider an actual physical object (e.g., a rotating top) which could be in the intended scopes of both PM and RBM. Now assuming PM is such that the bodies within its intended scope are idealized as extensionless point masses, it becomes impossible for PM to consider the rotation of the top unless the top is composed of other bodies which also occur within the intended scope of PM. Hence, it would be reasonable to suppose that the reduction of RBM to PM depends upon the empirical truth of the hypothesis that the top is composed of other bodies. But is this the proper way of construing the case at hand?

Another way to view this case is that PM can consider the rotation of the top even if the top is not composed of other bodies. Because PM now idealizes the top not as an extensionless point mass but as a set of extensionless point masses which satisfy certain constraints. The important thing to show is that the so-called behavior of this idealized set of mass points can successfully represent the behavior of the top. Therefore, the question of the reduction of RBM to PM should be considered as a question whether rigid bodies (e.g., the top) can be successfully represented by sets of extensionless mass points, and not as a question whether rigid bodies are composed of other bodies.

VII . C O N C L U D I N G REMARKS

Having completed our discussion of Adams' work on reduction, some general remarks are in order. First of all, Adams' account of reduction can not be straightforwardly accepted since it is open to some serious criticisms. But there are certain modifications and extensions within his account which can be made to make it more acceptable. Secondly, the inadequacies of Adams' account do not appear to be a result of his basic "structuralistic" approach to scientific theories and the problem of reduction. Furthermore, this approach has been successful in setting forth in a


rigorous manner two scientific theories (namely PM and RBM) and some of the important relationships between these two theories. Finally, Adams' example of a reduction (i.e., the reduction of RBM to PM) appears to be a special case of mutual reduction.

NOTES

1 In specifying the intended intrpretations of PM and RBM, we will find that it is useful to speak of a complete set o f units of measurement. By this, we simply mean a set of units of measurement such that there is a unit of measurement for each type of physical quantity of interest. A common method for specifying a complete set of units is by specifying a unit of mass, length, and time (also, possibly electric charge and temperature), and specifying the rules for determining all other relevant units from these so-called "fundamental units." Fur- ther, we will speak of a particular quantity, vector, etc., relative to a complete set of units o f measurement. What is meant here should be clear from the given context. For example, when speaking of the mass of a body relative to a complete set of units o f measurement, we will mean the numerical value of the mass of that body as given in terms of the unit of mass associated with that complete set of units of measurement. 2 Actually, Adams formulated this first condition in his dissertation in terms of a bicondi- tional, hence it should be "i2 ~ 12 if and only if there exists an il E It such that i2Ril." But Adams does say that it would be admissible to weaken this condition as it is given in Defi- nition 4; and since Adams formulates this condition as such in the article based upon his dissertation, we will so weaken this first condition to allow for a closer correspondence between his dissertation and article. 3 Schaffner [9] considers what he calls four "paradigms" of reduction and places Adams under the "Suppes paradigm." This appears to be a mistake. Adams should most likely be placed under what Schaffner calls the "Nagel-Woodger-Quine paradigm," especially in view of Adams' analysis being based to a large extent on his so-called conditions of "connectability" and "derivability." 4 Actually, this last remark is not exactly correct. The qualification must be added that 11 is not identical to the empty set. 5 Trivial reductions can still be constructed on Adams' analysis of reduction (as we have given it) for correct theories where one of the correct theories "deals" with all the physical systems dealt with by the other theory. This possibility of constructing such trivial reductions is most likely a reflection of the incompleteness of Adams' analysis with what is called the "semantical nature" of reduction relations as discussed in Section V. One possible way to get around such trivial reductions is to require the truth of i2Ri~ to depend in an essential way on the correlation and/or identification of properties of the physical systems composing the intended interpretations il and i2. 6 Strictly speaking, R does not satisfy condition (1). To see this, just note that for intended interpretations condition C1 of Definition 9 requires a rigid body to be identical to a set (abstract entity) of particles. Adams footnotes this difficulty. Since the modifications to correct for it are rather straight-forward (but cumbersome) and this difficulty should not lead to confusion, it is best forgot. 7 Other analyses of reduction suffer from similar problems involving the concept of approximation (e.g., Hempel [4]). 8 Also, these connecting statements could in some cases be analytic identifications.

184 MICHAEL A. DAY

9 One might object that Adams' analysis of reduction can not simply be too strong since it allows the trivial reductions mentioned in Note 5. This is correct but the forthcoming argument for the reduction of PM to RBM should still be valid since the reduction relation defined in Definition 10 depends in an essential way on the correlation and/or identification of properties.

REFERENCES

[1] Adams, E. W.: 1955, Axiomatic Foundations of Rigid Body Mechanics, unpublished Ph.D. dissertation, Stanford University.

[2] Adams, E. W.: 1959, 'The Foundations of Rigid Body Mechanics and the Derivation of its Laws from Those of Particle Mechanics,' in The Axiomatic Method (ed. by Hen- kin, Suppes, Tarski), North-Holland, Amsterdam, pp. 250-265.

[3] Causey, R. L.: 1972, 'Attribute-Identities in Microreductions,' The Journal of Philos- ophy 49, 407-422.

[4] Hempel, C. G.: 1969, 'Reduction: Linguistic and Ontological Issues,' in Philosophy, Science and Method: Essays in Honor of Ernest Nagel (ed. by Morgenbesser, Suppes, and White), St. Martins, New York, pp. 179-199.

[5] Kuhn, T. S.; 1970, The Structure of Scientific Revolutions (2nd ed.), University of Chi- cago Press, Chicago.

[6] Leinfellner, W.: 1974, 'A New Epitheoretical Analysis of Social Theories' in Develop- ments in the Methodology of Social Science (ed. by Leinfellner and Kochler), D. Reidel, Dordrecht, pp. 3-43.

[7] Mayr, D.: 1981, 'Investigations of the Concept of Reduction II,' Erkennmis 16, 109- 129.

[8] Moulines, C. U.: 1980, 'Intertheoretic Approximation: The Kepler-Newton Case,' Synthese 45, 387-412.

[9] Schaffner, K. F.: 1967, 'Approaches to Reduction,' Philosophy of Science 24, 137-147. [10] Sklar, L.: 1964, Inter-Theoretic Reduction in Natural Science, unpublished Ph.D. dis-

sertation, Princeton University. [I 1] Sneed, J. D.: 1971, The Logical Structure of Mathematical Physics, D. Reidel, Dor-

drecht, 2nd edition, 1979. [12] Stegmiiller, W.: 1976, The Structure and Dynamics of Theories, Springer-Verlag, New

York-Heidelberg-Berlin (German edition 1973). [13] Suppe, F.: 1974, 'Theories and Phenomena,' in Developments in the Methodology of

Social Science (ed. by Leinfellner and Koehler), D. Reidel, Dordrecht, pp. 45-91. [14] Suppes, P. C.: 1975, Introduction to Logic, Van Nostrand, New York.

Manuscript submitted 6 August 1983 Final version received 25 April 1984

Department of Physics Thiel College Greenville, PA 16125 U.S.A.

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Adams on theoretical reduction