8	Quantitative Nonnumencal Relations in Science: Eudoxus, Newton, and Maxwell

Arnold Koslow

Brooklyn College and the Graduate School, C.U.N.Y.

 

 

In an earlier study,¹ I tried to draw the distinction between quantitative and qualitative concepts without reference to or reliance upon numbers, concatenation operations, or ordering relations. Why should one try to do this? There were two very simple needs. The first was the obvious one that would occur to anyone who surveyed the older literature. Certain accounts of scientific method, and at least one influential movement in the theory of measurement, used the quantitative-qualitative distinction freely. Wouldn’t it be nice to have a clear and relatively simple characterization of the distinction? The classic studies of Hempel and Oppenheim (1936), Hempel (1952), and Carnap (1966) provided examples of classificatory, comparative, and quantitative notions. However, there was no account of the distinctions that made it evident that we had been given contrasting classes, where the three types of relations were mutually exclusive and belonged to a restricted but coherent group of relations. It seemed to me to be a small but worthwhile task to draw the distinctions so that these kinds of relations contrasted with each other. The second need was for a characterization of quantitativeness that was independent of numerical assignments. If it was required of all quantitative concepts that they be expressed with the aid of real-valued functions, then one would have to say that the Greeks had no quantitative account of the ratio of magnitudes; for they had no concept of the reals. It would be absurd to think of the Greeks as lacking a quantitative notion of ratios.² In fact Eudoxus developed a theory of a four-place relation R(x,y,z,w): the ratio of x to y is the same as the ratio of z to w, where the variables x,y,z, and w range over magnitudes. As we shall see, the relation R is a quantitative relation and non-numerical, although it takes a good part of Eudoxus’ theory to show that the relation has these features.

There are other historical examples of deliberation over whether a concept is quantitative although nonnumerical. Newton was embroiled in a rather complicated argument with R. Cotes over the concept of one body’s having the same quantity of matter (mass) as another, and Maxwell was probably the first to argue that the notion of “same temperature as” was quantitative. All three cases were discussed without appeal to numbers; all involved issues over quantitativeness. There has to be a way of making sense of these deliberations. However, there is no way of understanding how those issues were resolved by Eudoxus, Maxwell, and Newton, if the only way to understand the concept of quantitativeness requires mappings into the reals.

There were various ancillary theses connected with the division of relations into classificatory, comparative, and quantitative. Carnap suggested that the historical record revealed a predominant use of the quantitative over the comparative, and the comparative over the classificatory, as science became more successful. Further, he suggested that there was an increase in epistemological objectivity as one passed from classificatory to the other kinds of concepts. Whatever one thinks of these claims (and I do not think that any of them are unqualifiedly correct), it still seems important to have an understanding of the distinction between those concepts that are quantitative and those that are qualitative.³

The other source we referred to can be traced to the development of a representational theory of measurement, comprehensively studied in Krantz, Luce, Suppes, and Tversky (1971). It was sometimes argued that the goal of a theory of measurement is to show how one can pass from the qualitative to the quantitative. Thus, Suppes (1957, pp. 265–266) wrote the following:

The primary aim of the theory of measurement, for instance, is to show in a precise fashion how to pass from qualitative observations (“this rod is longer than that one”, “the left pan of the balance is higher than than the right one”) to the quantitative assertions needed in empirical science (“the length of this rod is 7.2 centimeters,” “the mass of this chemical sample is 5.4 grams”). In other words, the theory of measurement should provide an exact analysis of how we may infer quantitative assertions from fundamentally qualitative observations.

Indeed, in Scott and Suppes’ (1958) classic paper, the reason given for using extensional structures—sets together with relations on them of various ranks—is that each of these various relations Rⁿ is supposed to represent a complete set of “yes” or “no” answers to a question asked of every n-termed sequence of objects of the set. This is suppose to convey the idea that it is mainly qualitative empirical data that are under consideration. Not only the data but the axioms of the theory employed are supposed to be qualitative as well. All the more reason then for understanding the difference. We think that the distinction is important despite the fact that the goals for a theory of measurement—whether those set by Hempel and Carnap, or those set by the representational view—are not convincing. The idea of Hempel and Carnap was that there was a special problem about quantitative or metric concepts: How are they “introduced” into science? However, it has now been acknowledged that the problem, posed this way, is badly framed. There is no problem of introducing quantitative terms—no more than there is a problem of how theoretical terms get “introduced” (cf. Putnam, 1962, and Hempel, 1970). This particular goal for quantitative terms presupposes that there is a problem when there is none. On the other hand, the goal sometimes associated with the representational theory of measurement is not persuasive as a general goal. Is it always plausible to think that, by proving a representation theorem, we have shown how to pass from the qualitative to the quantitative?

It seems critical for this task that what we use is a theory that is qualitative. Otherwise there is no passage from the qualitative. However, if our proposed characterization of quantitative relations is correct, then there already are a host of relations and assertions in the theory that will count as quantitative. For example, if the theory has a concatenation operation *, and a two-place relation “xLy” (x is at least as long as y), then any assertion such as [(x* … *x)L(y* … *y) & (y* … *y)L(x* … *x)] for n xs and m ys (i.e., n xs are exactly as long as m ys) expresses a quantitative relation. Along the same lines, the statement that x = (y*z) or that [xL(y*z) & (y*z)Lx] (i.e., that x is the concatenation of y with z, or that x is exactly as long as the concatenation of y with z), counts as an example of a three-place quantitative relation. If the language used to study the quantitativeness of a notion already has quantitative resources in it, then there is no problem in passing from the qualitative to the quantitative for such a theory; we are already there.

There is an additional problem with taking the passage from the qualitative to the quantitative as a general goal. For some concepts, it is a plausible one; for others, it is not. Thus, Suppes (1956) argued persuasively as follows:

Because of the many controversies concerning the nature of probability and its measurement, those most concerned with the general foundations of decision theory have abstained from using any unanalyzed numerical probabilities, and have insisted that quantitative probabilities be inferred from a pattern of qualitative decisions, (p. 88)

Here we have a good example of why a definite restriction to the qualitative is reasonable. Quantitative assertions using the concept of utility and subjective probability are in contention, and the restriction to qualitative patterns represents a minimal, uncontested ground. Carried through, the program would be a real advance, which cuts through and indeed settles a scientific controversy. However, the relevance of these considerations in the case of, say, length or mass, seems remote. Suppose that we paraphrased the previous quotation, using “length” or “mass” rather than “probability”:

Because of the many controversies concerning the nature of length [mass] and its measurement, those most concerned with the general foundations of geometry [mechanics] have abstained from using any unanalyzed numerical lengths [masses], and have insisted that quantitative lengths [masses] be inferred from a pattern of qualitative information.

The plausibility of such a goal for length or mass, in contrast with utility or subjective probability, is extremely low. What are the many controversies about the nature of length (or mass) and their measurement that would render a qualitative theory of those quantities a significant scientific contribution or would encourage us to think that we had cut through those controversies by arguing from an uncontested middle ground?

Of course there have been controversies about the nature and measurement of both length and mass. The history of science is filled with controversies over the nature and measurability of mass: the efforts of Descartes to frame an adequate concept of mass; the extended discussion between Newton and R. Cotes that concerned the variety of concepts that Newton used, and the need to square those concepts with the idea that inertia is a measure of the mass of a body; the controversies raised by the proponents of the electromagnetic world-view over the nature of “mechanical” mass; the differences between relativistic and classical concepts of mass, to mention just a few landmarks. However, the history of science, in my opinion, discloses no case when these issues were resolved by finding a middle ground that was qualitative. For example, I have not found in Mach’s writings any attempt to introduce a qualitative ordering of bodies. Instead he introduced a number of quantitative empirical laws involving the notion of “mutually induced accelerations,” which he thought sufficient to characterize the masses (or mass-ratios) of bodies under consideration.

There have also been some controversies over the nature and measurement of length, area, volume, and angle. However, the ancient struggles to provide a coherent theory of proportion—the comparison or “ratio” of magnitudes—when they might be incommensurable, or the nineteenth and twentieth century controversies over a host of different measures for geometrical and set-theoretical entities—the Riemannian, Borel, and Lebesgue integrals, for example, or debates over the projective account of “metric” concepts—in none of these cases does it seem that the issues were settled or that the controversies were cut through by an appeal to purely qualitative theories.

The goal of passing from the qualitative to the quantitative seem to be well-motivated for concepts like utility and subjective probability, and fails to be well-motivated for concepts like length or mass. Despite these reservations, there is no doubt about the power, fertility, and intellectually exciting character of the particular theories that have emerged within the representational theory of measurement. The difficulties in formulating goals for a theory of measurement may be due to the fact that there are many goals—mathematical ones, those that concern faithfulness to scientific practice, those that concern the inclusion of sophisticated scientific concepts such as relativistic momentum among the mea-sureable. No one goal seems to cover all these concerns—but then, why should it? The present-day theory of measurement may be at least at that level of complexity, where asking for its goals may produce the same problem we encounter when we ask, if we were to ask, for the goals of physics, chemistry, or mathematics. Nevertheless the difference between qualitative and quantitative concepts remains an important one, whatever problems there may be in describing appropriate goals.

The task is to distinguish between quantitative and qualitative relations. It is not our aim to reduce the former to the latter or to systematically replace quantitative relations by qualitative ones. It is not essential to quantitative relations that they be numerical. Some are, and some are not. The conditions for a relation to be quantitative have to do with the way that the relation interacts with a partition of its domain (and, as we shall see, there is only one partition with respect to which a relation can be quantitative, if there are any at all). There are many relations that count as quantitative but not numerical-as we shall see below. Of course there are many numerical relations that are quantitative, although not all of them are. If this is correct, then certain problems do not seem to be well-formed. For example, it seems that many advocates of the representational theory of measurement believe that it is possible to give a qualitative characterization of physical laws (in order, for example, with the aid of numerical representation theorems, to prove dimensional invariance). Although many laws are formulated numerically, it is thought that such equations are only a “shorthand of what is really a complex set of possible qualitative observations.” (Krantz et al., 1971, p. 504) It is hard to see any grounds for pursuing a program of exhibiting physical laws as compendia of qualitative observations. Sometimes the program looks a bit like a refurbished restatement of Machian epistemology, according to which laws are also thought to be summaries of experimental results, provided we thought that the experimental results were qualitative. However, there seems no reason to think that this is generally true. D. Scott and P. Suppes (1958) seem to have thought of a complete set of “yes”-”no” answers recorded by using some n-place relations as counting as qualitative. However, if that is the way things go, then there is a conflict with the present proposal that some relations are already quantitative. In that case, the empirical relational systems as used by the representational school may not be inherently qualitative structures. The distinction that is drawn with the aid of empirical relational systems is not one between qualitative and quantitative structures. Rather it is, I think, a distinction between numerical and nonnumerical structures. If I am correct, the use of empirical relational structures by the representational school has nothing to do with qualitative versus quantitative. Some quantitative relations are numerical; others are not. Furthermore, some qualitative relations are numerical; some are not. Similarly H. Field’s (1980) recent attempt to provide an account of scientific theories that meets nominalist standards is not about the replacement of quantitative relations and laws by nonquantitative or qualitative counterparts. It is, only if certain quantitative relations and laws are described with the aid of numbers, functions, sets or other abstract entities, that they become the target of Field’s nominalistic program. However, if I am correct, there are many quantitative relations that are not numerical. There is, therefore, nothing in Field’s nominalistic program that calls for the replacement of these quantitative terms. On the other hand, there are qualitative relations that are numerical and would have to be replaced in order to meet the Field conditions. Field’s program is, therefore, not to be thought of as requiring the replacement or reduction of quantitative relations and laws by qualitative counterparts. In fact many of the relations that Field used in his account, certainly the congruence relations that he employed, count as quantitative on my account. Consequently I do not think of Field’s program as an attack on the use of quantitative relations in science.

Emil Artin’s generalized discussion (1957) of Descartes’ Coordination Problem showed how the points of various geometries can be characterized by using coordinates from various associated mathematical fields, in such a way that the characteristic relations in each geometry are associated with characteristic algebraic conditions in the associated field. However, even in the special case when the associated field is that of the real numbers and points get “represented” by n-tuples of real numbers, it would be a mistake to think that the geometries in question are purely qualitative theories that are shown to have quantitative representations in the reals. For the geometries in question are certainly not purely qualitative; there are many relations in them that count as quantitative (although of course nonnumerical). Thus, although there are discussions in the literature that look as if they concern the reduction of the quantitative to the qualitative or conversely, I think that, in each case, it is another distinction that is under investigation. In the case of Field’s Program, it is the distinction between expressions that are nominalistically acceptable and those that are not; in the case of Artin’s problem, the distinction concerns the connection between geometric and algebraic theories and their respective relations.

Our task, therefore, is to distinguish between quantitative and qualitative relations. They are, as we shall see, disjoint types of relations. However, from that fact, it does not follow that one type is or ought to be, reducible, definable, or replaceable by the other.

CLASSIFICATORY, COMPARATIVE, AND QUANTITATIVE RELATIONS

The basic idea is to begin with a family of relations on a nonempty set S and to distinguish between those that are classificatory, comparative, and quantitative. The initial qualification is that relations are not described as of one kind or another simpliciter but as comparative or quantitative, or classificatory, relative to some partition of the domain S.

Let S be any nonempty set (the domain). An equivalence relation on S is any binary relation on S that is reflexive, symmetric, and transitive. A partition μ on S is any set of subsets of S (not containing the empty set) such that every member of S is a member of exactly one member of M. To each equivalence relation E, there is a corresponding partition M_E—for any equivalence relation E and any a in S, let E_a = {y|y is in S and E(a, y)}. Let M_E be the set of all subsets of S of the form E_a. It is a partition of S. If M is a partition of S, then let E_M be the binary relation on S such that, for any a and b in S, E_M(a, b), if and only if a and b belong to the same member of M. E_M is an equivalence relation on S.

Let EQ(S) be the set of all equivalence relations on S and Part(S) be the set of all partitions on S. There is a partial order definable for each of these sets: For any two equivalence relations E and E*, let the order be given by the subset relation, and for any two partitions ρ and P*, we shall say that P is a refinement of P* (P ≤ P*), if and only if every member of P is a subset of some member of P*. There is an order-preserving isomorphism of EQ(S) and Part(S)—let ϕ be the mapping that assigns to any equivalence relation E the partition M_E, ϕ is easily seen to be one-to-one, and it is onto, because, for any μ in Part(S), ϕ(E_μ) = μ. Moreover, it is easy to check that, for any equivalence relations E and E* of EQ, E ⊆ E*, if and only if M_E ≤ M_E*). Given the isomorphism, we shall sometimes describe the relativization of the various relations either with respect to equivalence relations or to their associated partitions.

The set EQ(S) of all equivalence relations on S has a decent mathematical structure—it is a complete lattice. The definition of the meet is obviously given by set intersection; the join operation on EQ is nicely defined by Gratzer (1979).

Let Rⁿ be any n–place relation on 5. We think of it either as a function mapping Sⁿ (the n-fold Cartesian product of 5 with itself) to the set {t, f} or equivalently as a subset of Sⁿ.

We shall need the notion of a congruence relation in the characterization of all three kinds of relations. Let E be any equivalence relation on S. Extend E to an equivalence relation on Sⁿ as follows: If p is any member of Sⁿ, then, for any i, 1 ≤ i ≤ n, let p_i be the ith component of p (the ith member of the n–tuple p). For any p and p* in Sⁿ, we shall say that E(p, p*), if and only if E(p_i, p*_i, for all i. We shall say that an equivalence relation E is a congruence of the relation Rⁿ, if and only if the following:

(C) For any p and p* in S, ⁿ if E(p, p*), then Rⁿ(p) if and only if Rⁿ(p*).

—where we write Rⁿ(p) for R Rⁿ(p₁, …, p_n)”. The set of congruences of (on S) also form a complete lattice (Gratzer, 1979, chapter 1, Section 10; Cohn, 1965,).

All three types of relation, classificatory, comparative, and quantitative (on S), will be thought of as relativized to equivalence relations (or their associated partitions), which are also congruences of the relation under study.

Classificatory Relations

Let S be any nonempty set, and E an equivalence relation on S. We shall say that an n–place relation Rⁿ on S is classificatory with respect to E, if and only if the following:

1.   E is a congruence of Rⁿ.

2.   There is a set A = {a₁, …, a_m} of m distinct members of S, no two of which are equivalent under E, such that, for any p in Sⁿ, if Rⁿ(p), then, for any i (1 ≤ i ≤ n), there is a j (1 ≤ j≤ m) such that E(p_i, a_j).

3.   For any p in Sⁿ, if Rⁿ(p), then no two components of P are equivalent (under E).

Two comments are in order. In the older literature, classificatory terms were exemplified by monadic predicates, sorting out elements of a certain kind within a domain. “Red,” “Square,” and so on were the usual examples. There was, as far as I can determine, nothing especially interesting to say about classificatory terms. They were used to highlight the comparative and quantitative terms as the kind of terms that had greater scientific utility. It was noted that classificatory terms could be refined, but, on the whole, there was nothing special that could be said about them beyond what one could say generally about predication and its problems. As far as I can see, the scheme just described generalizes the earlier notion to include a simultaneous sorting of the members of the domain into several disjoint types. However, it is an understatement to say that examples of scientific use of such relations are hard to come by. Its present use is to provide a contrast with the comparative and quantitative relations, but, aside from its being an appropriate generalization of one member of the older trilogy, it hasn’t any scientific or philosophical interest to speak of. One should also mention that the notion of a classificatory term was generalized in another direction by Hempel and Oppenheim (1936). Instead of using one monadic predicate, they considered using several predicates. Thus, one might use the characteristics of having mathematical ability, having physical dexterity, and being emotionally stable, to classify people of some population. A loose analogy was suggested with the idea that, just as a point may be located by reference to several different dimensions, so too objects could be thought of as located in a multidimensional space. Some examples were provided from psychology, but this type of generalization, to my mind, has had no scientifically interesting examples that would make the notion worth studying in any depth.

It should be noted that, given this notion of a classificatory relation, it is possible that a relation can be classificatory with respect to different equivalence relations. For example, if the domain S consisted of objects that were either red or green, with examples of the shades (light red, dark red, light green, dark green), then the relation R(x,y) (either x is red, and y is green, or x is green, and y is red) is easily seen to be classificatory with respect to the equivalence E (same color) as well as the different equivalence E* (same shade of color). E* is a refinement of E, and the set A required by the definition would be different (for E, let A be a set of two members, one of which is red, the other green; for E*, let A* be a set of four members that are light red, dark red, light green, and dark green). As we shall see below, this is in parity with the case of comparative relations for which it always holds that, if a relation is comparative with respect to an equivalence E, it is also comparative with respect to any equivalence E* that refines (is a subset of) E. Furthermore, both classificatory and comparative relations differ from quantitative relations; the latter can be quantitative with respect to only one equivalence relation.

Comparative Relations

Let Rⁿ be a n–place relation on a nonempty set S, and let E be an equivalence relation on S. We shall say that Rⁿ is comparative with respect to E, if and only if the following conditions are satisfied. We first consider the case when n = 2.

1.   E is a congruence of R².

2.   For any a and b in S, either R²(a, b) or R²(b, a) (or both).

3.   R² is transitive.

Now consider the general case. We shall say that (Rⁿ)* is a (two-place) specialization of Rⁿ, if and only if there is some p₀ of Sⁿ, and some i and j, where i ≠ j, and 1 ≤ i, j ≤ n such that, for any ρ in Sⁿ, (Rⁿ)*(p) = Rⁿ(p*), where, for any p in Sⁿ, P* differs at most from p₀ on the ith and jth components, and p* agrees with p on the ith and jth components. The (two-place) specialization of Rⁿ induces a binary relation IR on S according to which, for any a and b in S, IR(a, b), if and only if (Rⁿ)*(p), where the ith and jth components of p are a and b respectively, and any component k of p, other than the ith and jth, is just the kth component of p₀. Finally we shall say that Rⁿ is comparative with respect to the equivalence relation E, if and only if there is a (two-place) specialization of Rⁿ that induces a binary relation on S that is comparative (as in the case previous, for n = 2).

It is worth noting some elementary observations:

1.   If Rⁿ is comparative with respect to an equivalence relation E, then it is also comparative with respect to any equivalence relation E* that refines E (the partition M_E* that is associated with E* is a refinement of the partition M_E associated with E). Thus, in particular, if a relation is comparative with respect to any partition, it is also comparative with respect to the discrete partition on S (the members of the partition consist of sets having exactly one member of S). Consequently, if R is comparative with respect to a nondiscrete partition, there are at least two partitions with respect to which it is comparative.

2.   No relation Rⁿ is both classificatory and comparative. Consider the case where n = 2. If R² is classificatory with respect to E, then, for any p in S², if R²(p), then p₁ and p₂ are not equivalent (under E). R² is comparative; thus, it is reflexive (by the second condition on comparatives). However, that is impossible.⁴

3.   As we noted, the congruences of a relation form a complete lattice. It is possible to use the comparative to identify the maximal congruence. In the case of a two-place comparative, the maximal congruence with respect to which it is comparative is easily seen to be E(a, b): [R²(a, b) & R²(b, a)].⁵

Quantitative Relations

Let Rⁿ be a n-place relation on a nonempty set S, and E an equivalence relation on S. For any p and p* of Sⁿ, and any i (1 ≤ i ≤ n), we shall say that i(p, p*) holds, if and only if p and p* differ at most on their ith components. We shall say that Rⁿ is quantitative with respect to E, if and only if the following:

1.   E is a congruence of Rⁿ.

2.   For any p and p* in Sⁿ, and any i (1 ≤ i ≤ n), if Rⁿ(p), and Rⁿ(p*), and i(p, p*), then E(p_i, p*_i).

This says that, if two n–tuples of objects of S both belong to Rⁿ, and they differ at most in their ith component, then those ith components belong to the same member of the partition. Quantitative relations are, in this intuitive sense, “tight.”

We shall say that a n–place relation Rn on S satisfies the covering condition, if and only if, for any member a of S, there is some p in Sⁿ such that Rⁿ(p) holds, and a is the ith component of p, a = p_i, for some i (1 ≤ i ≤ n).

The following are easily seen:

1.   If Rⁿ is quantitative with respect to the equivalence relation E (on S) and satisfies the covering condition, then E is the only equivalence relation with respect to which Rⁿ is quantitative. For suppose that Rⁿ is quantitative with respect to E and E*. Let a and b be any members of S such that E*(a, b). By the covering condition, there is some p in Sⁿ such that Rⁿ(p), and a is the (say) ith component of p. Let p* be the same as p except for having b as its ith component; then i(p*, p). E* is a congruence of Rⁿ, and Rⁿ(p); thus, it follows that Rⁿ(p*). Therefore, by the second condition on quantitative relations, E(p_i, p*_i). That is, E(a, b). Thus, for any a and b in S, if E*(a, b), then E(a, b). The converse is proved similarly. So E and E* are coextensional.

2.   If Rⁿ is quantitative with respect to a partition that has at least two members (there are at least two members of S that are not equivalent), then it cannot be comparative with respect to any partition. This is easiest to see in the binary case. For any a and b in S, either R(a, b) or R(b, a), because R is comparative. If R(a, b), then, because R(a, d), and R is quantitative, it follows that E(a, b) (similarly if R(b, a) holds). So every b in S is in the same equivalence class as a. This contradicts the condition that the partition has at least two members.⁶

Although quantitative relations can be numerical, then need not be.⁷ The focus for the remainder of this study will be on three examples of such quantitative but nonnumerical relations that seem to be of the same type but each having a distinct nontrivial scientific interest: (a) sameness of ratio (Eudoxus), (b) sameness of temperature (Maxwell), and (c) sameness of mass (Newton). These are very simple examples as far as quantitative relations go, but their simplicity belies the strategic roles they have played and their conceptual subtlety. There are many other examples of nonnumerical quantitative relations. For example, in empirical relational systems that have some comparative binary relation xLy and some concatenation operation *, then the relations [xLy & yLx] (x is exactly as long as y); [xL(y*y), (y*y)Lx]; (x is exactly twice as long as y); [xL(y*z), (y*z)Lx] (x is exactly as long as the concatenation of y with z) are all quantitative relations with respect to the partition M, where each M_i consists of exactly those elements x,y such that [xLy & yLx]. In Euclidean geometry, the relations P(A, B, C) (A, B, and C are the consecutive sides of a right triangle with hypotenuse C), TΔ(A, B, C) (A, B, and C are the consecutive sides of a triangle that is congruent to the triangle Δ), and SΔ(A, B, C) (A, B, and C are the consecutive sides of a triangle that is similar to the triangle Δ) are all quantitative and non-numerical with respect to the partition of the set S of all line segments with respect to the relation of congruence. In set theory, the relation of a set X being in one-to-one correspondance with the power set P(Y) of the set Y, and the relation by which a set Z is in one-to-one correspondance with the union of two disjoint sets X and Y, are quantitative and nonnumerical relations, where the relevant equivalence relation holds between only those sets that are in one-to-one correspondance with each other. These are all quantitative nonnumerical relations, some patently empirical, others more mathematical in content. There are of course numerical relations that are not quantitative such as the relation of being greater than (on say the reals). The moral is that there is only a partial overlap between quantitative and numerical relations. Being numerical is neither necessary nor sufficient for being quantitative.

EUDOXUS

Let us turn first to Eudoxus’ theory of proportionality of magnitudes.⁸ There is an interesting problem over how “magnitudes” were understood by Euclid and Eudoxus. Are they the lines, plane figures, solids, and angles (as well as other, nongeometric objects), or are they as I. Mueller (1983) suggested, abstractions from geometric objects: lengths (of lines), areas (of plane figures), volumes (of solids), and whatever the property is in the case of angles? Either view has its problems, exegetical as well as substantive. It might seem plausible to think of lengths, areas, and volumes as features, attributes, or properties of things rather than the things themselves. If that is so, then it is hard to understand what could be meant by Euclid’s assumption, for example, that magnitudes have parts (Mueller, 1983, p. 123). It does seem easier to think of magnitudes as objects, abstracted in such a way that only geometric properties serve to individuate them—provided that we have a grasp of what it is to be a geometric property.

However “magnitude” is understood, there is no question that, for Eudoxus, the ratio of two magnitudes is not a magnitude. Consequently the sameness of two ratios is not settled by the sameness or difference of magnitudes. In effect Eudoxus developed a theory of a four-place relation R(x, y, z, w) with x, y, z, and w ranging over magnitudes. It was thought of as expressing the relation that the ratio of x to y is the same as the ratio of z to w and was defined this way. R(x, y, z, w), if and only if (for every m and n): (i) (mx > ny → mz > nw), and (ii) (mx ≃ ny → mz ≃ nw), and (iii) (mx < ny → mz < nw). The use of terms that refer to natural numbers is, as Mueller noted, an expository device used by Mueller but not by Euclid. Euclid assumed that there are magnitudes that are multiples of others, and, as Mueller observed, Euclid consistently used the relation “x is the same multiple of y as z is of w” (Mueller, 1983, p. 121). Thus, there is, in Euclid’s account, no essential reference to numbers.⁹

Eudoxus’ explanation of the sameness of ratios does not presuppose that there is any object or magnitude that is the ratio of two magnitudes. There is an explanation that he offered of the idea that x and y have a ratio that is expressed by saying that some multiple of x exceeds y and some multiple of y exceeds x (Mueller, 1983, p. 144)

We have then a four-place relation R, defined over magnitudes or objects abstracted or individuated in some special way, and the question is whether it is quantitative, for it is not numerical.¹⁰ It is fairly straightforward to check that R is quantitative with respect to the equivalence relation ≃. The first condition to be checked is that ≃ is a congruence of R. Consider one case: If x ≃ x*, and R(x, y, z, w), then R(x*, y, z, w). This holds, because mx > ny → mz > nw, and mx* ≃ mx. Therefore, mx* > ny → mz > nw. Thus, clause (i) of the definition of R holds—similarly for clauses (ii) and (iii). Consequently R(x*, y, z, w). Similar remarks hold for y and y*. The case for z ≃ z* and for w ≃ w* requires a slight modification in the original definition of R. Instead of using conditionals, the revision uses biconditionals. This makes no difference to the Eudoxian theory of proportion, because that theory also requires that R(x, y, z, w) → R(z, w, x, y) for all x, y, z, and w.

In order for R to count as quantitative, it only remains to check that the second condition holds: Let p and p* be any quadruples of magnitudes. If R(p) and R(p*) where p and p* differ only in the first component (say x is the first component of p, and x* the first of p*), then x ≃ x*—similarly for the other components. Suppose, for example, that R(x, y, z, w) and R(x*, y, z, w). Then, by (i), for all m and n, (mx > ny ↔ mz > nw), and (mx* > ny ↔ mz > nw). Consequently (mx > ny ↔ mx* > ny), for all multiples (mx), (mx*), and (ny) of x, x*, and y respectively. Therefore, x > y ↔ x* > y. By clauses (ii) and (iii) of the definition of R, it follows that x ≃ y ↔ x* ≃ y, and x < y ↔ x* < y. Thus, from the definition of R, we have R(x, y, x*, y). However, it is also a feature of Eudoxus’ theory that, if R(x, z, y, z), then x ≃ y. In particular then, x ≃ x*. Similar arguments hold for R(x, y, z, w) & R(x, y*, z, w), implying that y ≃ y* (as well as the cases for z and z* and w and w*). Thus, R(x, y, z, w) is a quantitative relation on the set of magnitudes, with respect to the equivalence relation.¹¹

Our aim here was the limited one of showing that a relation central to Eudoxus’ theory of proportions is quantitative and not numerical. We have taken it for granted that Eudoxus’ theory is a landmark in the history of mathematics. However, one should not think of its influence as limited to the mathematical. The influence upon Newton was a powerful one (McGuire & Tamny, 1983), and no less a physicist than C. G. Stokes was concerned about the use of proportions and incommensurables in his definition of mass, and explored the connection between the algebraic (numerical) and the Euclidean (Eudoxian) theory of proportions.¹²

MAXWELL

The second example of a scientifically significant quantitative relation we wish to consider is the notion of one body’s having the same temperature as another. In the second chapter of his Theory of heat, thermometry, or the theory of temperature (1871), Maxwell defined the temperature of a body as a (thermal) state of the body related to its power to communicate heat to other bodies. If two bodies are thermally connected, and one gains and the other loses heat, then the one that “gives out” heat is said to have the higher temperature. The corollary he drew is that, when two bodies are “in thermal contact,” and neither loses nor gains heat, then they have the same temperature. These remarks are followed by what Maxwell called the Law of Equal Temperatures: Bodies whose temperatures are equal to that of the same body have themselves equal temperatures.

The law of equal temperatures (LET) might appear to be a special case of the Euclidean claim that things equal to the same thing are equal to each other. Maxwell, however, was quite explicit on the matter. The law is not a truism, not a special case of the Euclidean dictum, and, we might add, not a theorem of the theory of identity. The truth that, if A and C are in thermal equilibrium, and B and C are in thermal equilibrium, then so too are A and B, is nothing less, he said, than “the foundation of the whole science of thermometry (Maxwell, 1871, p. 33).

It is remarkable that Maxwell stressed that LET was not a logical truth but a law that is the basis for all of thermometry. It is a law that supports the simplest use of thermometers: If a piece of iron is plunged into water and found to be in thermal equilibrium with it, and then, without altering its temperature, it is plunged into oil and is in thermal equilibrium with it, then if the oil and water were in thermal contact, they would also be in thermal equilibrium. As for the “foundational” claim—I believe that it just comes to this: If LET were not true, then the use of thermometers would be very limited. We would have no basis for thinking that two bodies that have the same temperature as the thermometer also have the same temperature. We would be in a situation comparable to relating things by sameness of color; where it is not generally true that two things having the same color as a third, have the same color as each other. However, Maxwell, as well as Mach, thought that LET was true and had empirical content. At the very least, it was not regarded as a logical truism.¹³

Now it follows from the Law of Equal Temperatures that the relation E(A, B) (A has the same temperature as B, or A and B are in thermal equilibrium upon thermal contact) is a quantitative relation. The reason is that, because E is symmetric (from its characterization: upon thermal contact, neither of A and B gives up heat to the other), then given LET, E is also transitive. To see that E is quantitative, a partition is needed. For each body A, let E_A be {C| E(A, C)}. The collection of sets of the form E_A for all A in S is a partition of S, and E is clearly quantitative with respect to that partition. Of course, if it were assumed that E is an equivalence relation, the matter of its being quantitative would be obvious.

Thus, the relation “A has the same temperature as B” is a quantitative relation, and it is the Law of Equal Temperatures, or as it is currently known, the Zeroth Law of Thermodynamics, that insures that the relation is quantitative. We also think that E(A, B) and the Law of Equal Temperatures are nonnumerical—at least in the accounts provided by Maxwell and Mach (1868, 1911), as well as Helmholtz (1977). For it is clear that, even if a thermometer is used to determine that two bodies are in thermal equilibrium, neither one gaining nor losing heat, it is only the mark on the thermometer, not its particular value that is used to determine whether E(A, B) holds. It comes as no surprise that, in Maxwell’s orderly exposition of the subject, that it is only after the discussion of the Law of Equal Temperatures that there is an account of the different kinds of thermometers and the construction of scales. None of the latter is needed to express and to determine the truth of LET. Thus, although it is a simple quantitative relation, E(A, B) is nonnumerical and has a significant role to play in the expression of the Zeroth Law of Thermodynamics.

The law itself is rather striking. It does look, at first sight, to be like a truism, deserving passing attention at most. However, one can understand why Mach might have called special attention to the law that bodies that behave to a third as equal masses will also behave that way to each other. On his account of mass, the claim is far from obvious. For he characterized the mass-ratio of two bodies i and j, as the negative inverse ratio of their accelerations, ϕ_ij,ϕ_ji, which, under certain circumstances, the bodies induce in each other (ϕ_ij is the acceleration that the ith body induces on the jth, and Mach required that m_ijm_jk = m_jk, or that ϕ_ijϕ_jkϕ_ik = ϕ_jiϕ_kjϕ_ki.¹⁴ However, this relation among induced accelerations is hardly obvious, although it yields the law of equal masses. Thus, on Mach’s construal of relative masses, the law of equal masses is not transparently true. However, Maxwell seems to have been unaware of Mach’s early discussion of this law (1868), and there is no mention of it in the appropriate place in his Matter and motion (1924).¹⁵

Why did Maxwell isolate LET as especially significant? Unfortunately there are no hints in The theory of heat itself. There is an original draft of the work, but the second chapter, curiously enough, is missing from it.¹⁶ In October of 1878, the last year of his life, Maxwell lectured on thermodynamics. Those lectures might have shed some light on the matter. There is a bound volume of notes taken by A. Fleming, who, when he presented them to the Cavendish Laboratory some fifty two years later, said that he “took careful notes and wrote them out with some additions after each lecture. …” However, Fleming has to be mistaken. The presented volume cannot be a record of Maxwell’s last thoughts about the theory of heat.¹⁷

Why did Maxwell single out LET? What called it to his attention? Speculation here has to be brief; the facts are few and indecisive at best. It is possible that Maxwell, as well as Mach and Helmholtz, knew of other relations for which a Zeroth-style law failed, and this might have raised the question of when such a type of law holds. For example, when R(A, B) is the relation of just matching in color, a “Law of Equal Colors” fails. It is hard to think that Maxwell was not aware of such examples, because he knew Helmholtz’s Physiological optics; he carried out experiments on the eye’s sensitivity to light; he probably knew of Fechner’s work; and, in some of his scientific studies on colors, Maxwell used the distinction between optical and chromatic similarity, which is related to threshold phenomena. However, this is speculative. He made no reference, as far as I am aware, to the failure of a Zeroth style law for the chromatic matching of colors.

We shall have to reserve for another occasion, the seminal way in which Helmholtz thought about these issues.¹⁸ He thought of the cases of sameness of temperature, of mass, and other quantities, as special cases of a more general relation of “alikeness.” Alikeness relations are denoted by “ =,” and Helmholtz required that, for any alikeness relation, if A = B, and A = C, then B = C. There are two features of his account that are worth special notice. The first is connected with some issues of realism. There is an ambiguity in Helmholtz’s description of the alikeness relation. On the one hand, “A = B” is supposed to be a relation among physical objects or magnitudes. So one thinks of the special cases of “A = B” when A and B are bodies that are in thermal equilibrium or have equal mass. On the other hand, Helmholtz sometimes described the alikeness relations as holding between properties or attributes of objects. The view he held, although I cannot argue the case here, is that a specific alikeness relation R holds between A and B, and there are certain attributes of A and B that cause or issue in A’s being related to B by R. There is no guarantee that there will be such an attribute, and it seems to me that what is needed to express this idea is a family of attributes. However, it is evident that Helmholtz thought that such attributes and their causal role are a part of the story about alikeness relations. Although Helmholtz referred to attributes, he also thought of the relevant attributes as capacities of the objects for producing the effect of being alike. It is not clear whether it has to be properties, or whether it could be certain states of these objects that result in their being alike. Clearly Maxwell thought of temperature as a state “with reference to” a power to communicate heat to other bodies. It seems that something causal is being suggested for the outcome of A and B as being alike (if they are alike). One of the reasons for going causal is that, if it should turn out that, whenever A and B are alike (with respect to some relation R), they are also alike by some other relation S, then it is the attribute that resulted in A and B as being alike (by R) that also causes them to be alike (by S). At least that seems to be the program in which attributes are supposed to have a special role. Briefly if, whenever objects (or properties) are alike by one relation, they are also alike by another relation, then the explanation will appeal to attributes that caused the first kind of outcome. Although states of objects might do as well, Helmholtz called attention to the kind of simple problem for which realism suggests the kind of solution that he thought was needed: a causal one, or at the very least, a kind of empirical dependency.¹⁹

Helmholtz was also explicit about the Zero-type laws for alikeness relations. Roughly speaking, for each alikeness relation R, we have the corresponding Zeroth Law: (Z_R); If R(A, B), and R(A, C), then R(B, C). Each (Z_R) has empirical content. However, Helmholtz also remarked that, if (Z_R) were false, then R would not be an alikeness relation. Thus, he thought that the schema (Z_R) had no objective content but was a necessary condition for being an alikeness relation. However, the particular instances of (Z_R) are supposed to have empirical content. Helmholtz’s account of the measurement of (say) length requires that the relation ”A has the same length as B” is an alikeness relation; thus, it follows that, if the Zeroth law of lengths (Z_L) were false, then, according to his theory, length would not be measurable. Consequently the Zeroth law becomes crucial on Helmholtz’s theory of measurement.²⁰

NEWTON AND THE SAMENESS OF MASS

The third example of a nonnumerical quantitative relation belongs to a little-known episode that occurred between the publication of the first and the second editions of Newton’s (1687/1713) Principia. The problem was raised by Roger Cotes, the editor of the second edition, in correspondence. What made it a serious rather than a minor problem for Newton was the combination of assertions of the first edition against the background of thoughts that Newton held ever since his undergraduate days at Cambridge, and had expressed in unpublished studies and notes.

In the Third Corollary to Proposition Six of the Third Book of the Principia (first edition), Newton asserted that not all spaces are equally full (of matter) or that there is a vacuum. The target was of course Descartes, and the argument seemed simple: If all spaces are equally full of matter, then any two spaces of equal volume are completely filled. Consequently they have the same quantity of matter. However, Newton thought that, by his experiments with pendulums, he had shown that the weight of a body is proportional to its quantity of matter. Consequently any two spaces of equal volume have the same weight. Now any sphere, of gold or any other substance, will displace a volume of air that thas the same volume as the sphere. So the weight of any body is the same as the (same) volume of air that it displaces. However, by Archimedes’ law (the buoyant force on a body in a fluid is equal to the weight of the fluid that the body displaces), there will not be any net force on a body, because its weight and the buoyant force are equal. Thus, no body will descend in air. Now that is patently false. The conclusion that Newton wished to draw was that the first assumption was false: Not all spaces are equally full; there is a vacuum.

Cotes had no objection to the conclusion. However, he did raise serious questions about the argument that Newton used. It seemed to Cotes that the relation of having the same quantity of matter was used ambiguously by Newton. One way of smoothing over the ambiguity required an assumption about pri-migenial particles for which there was no evidence. Even worse, as Cotes stressed, Newton (1952/1704) was already on record (in the Optics) as considering the possible falsity of that assumption. In the end, Newton conceded the point, and instead of the categorical conclusion he had in the first edition—that there was a vacuum—he now settled for a conditional conclusion: There is a vacuum, if all the solid particles of all bodies have the same density (and cannot be rarefied without pores). There was a struggle in the correspondence over the relation of sameness of quantity of matter that is worth reviewing briefly.

Let us denote, by “M(A, B),” the condition that body A has the same quantity of matter as body B. There is no question that both Newton and Cotes understood the condition M(A, B) to be a quantitative relation. The difficulty, as Cotes saw it, arose, because Newton linked M(A, B) to two other quantitative relations: I(A, B) (A and B have the same inertia); and V(A, B) (the volume that body A completely fills is the same as the volume that body B completely fills). The argument for a vacuum would then look like this:

1.   M(A, B), if and only if V(A, B), and

2.   M(A, B), if and only if I(A, B). Consequently

3.   V(A, B), if and only if I(A, B). The volumes that the bodies A and B can completely fill are the same, if and only if their inertias are the same.

Newton then used the results of his pendulum experiments, according to which the weight of a body is proportional to its inertia. Consequently W(A, B), if and only if I(A, B). Therefore, V(A, B), if and only if W(A, B). It is at this stage that Newton considers any body A that completely fills a volume (i.e., pulverize it, or condense it until it cannot be condensed further, or “discount” the pores). That volume is the same as the volume of air that it displaces. Consequently the body and the air that it displaces have the same weight. However, by Archimedes’ law, the net force on the body is then zero. Consequently a body will not descend in air, and Newton added, that goes for any body. Thus, Descartes was wrong. For, if one assumed that all spaces are equally filled with matter, then (1) holds. For, if A and B have equal volumes, then they have the same quantity of matter, because all volumes are completely filled with matter according to Descartes. This way of connecting (1) with Descartes is plausible given the Cartesian identification of extension (volume) with matter. On that view, any two bodies are completely filled with one kind of matter (there is only one kind of matter for Descartes, although there are different kinds of bodies) so that any two equal volumes have to contain the same quantity of matter.

Cotes thought that there was a difficulty with (3). Why should one think that two globes A and B of equal volume that are completely filled with matter will have the same inertia? The quantity of matter might be different. Is it impossible, he asked Newton, that God could give them different inertia?²¹ Cotes continued with a diagnosis: The notion of quantity of matter does double duty in (1) and (2). The second assertion was, he thought, what Newton meant all along. Thus, he pressed Newton not to think of the quantity of matter as proportional to the space that the body can perfectly fill without void interstices.

Newton’s reply is a defense of the original argument: “This happens only if the quantity of matter is proportional to its gravity, and in addition matter is impenetrable, so that matter always has the same density in completely filled spaces.”²² Thus, Newton seems to have thought that, from the proportionality of the quantity of matter to weight (this would be shown by Proposition 6, Book III, whose Third Corollary was under discussion), and the impenetrability of matter, it followed that matter in completely filled spaces always has the same density.

However, Cotes remained unconvinced. He repeated his view that Newton always estimated the quantity of matter by inertia, and that the objection had not not been obviated. He then suggests this as a possible emendation: “This holds only if the magnitude (bulk) or extension of matter in completely filled spaces is always proportional to the quantity of matter and to the force of inertia, and also thus to gravity: For by this proposition [Theorem VI, Book III] it is shown that the force of inertia and the quantity of matter are proportional to gravity.”²³

The effect of Cotes’ suggestion is to isolate the claim that M(A, B), if and only if V(A, B), and to render the conclusion that there is a vacuum as conditional upon it. He ignored Newton’s attempt to rest the conclusion on the impenetrability of matter. In his response,²⁴ Newton said that he will prevent the cavils (sic) of those who think that there are two kinds of matter. He then spelled out in greater detail just how the impenetrability of matter leads to the conclusion that poreless bodies of the same volume will have the same quantity of matter. The idea is that, once the pores of a body are contracted, the solid (primigenial) bodies in it will all abut on each other (by their impenetrability), and the body so reduced completely fills its space. Newton then thought that, if the volumes of two such bodies are equal, then so too are their quantities of matter. That is, M(A, B), if and only if V(A, B). However, such a deduction requires that the densities of all the solid particles are the same (and nonzero). This is the point that Cotes pressed in his reply.²⁵ He informed Newton that he was still not satisfied on this issue— unless Newton was prepared to make the argument for the existence of a vacuum conditional upon this: that the primigenial particles, out of which the world is supposedly constituted, are all equally dense. However, he pointed out, this is an assumption that has no known support. For, he said, he did not see how it can be proved a priori or inferred from experiments. We are back again to what Newton had earlier described as the cavils of those who think that there might be two sorts of matter. So a concession has to be added, according to Cotes; there is no known support for it. Moreover, Cotes added, things may be worse: Newton seems to have thought that the denial was possible (Optics, Query 31).²⁶

The result of this series of proposals and counterproposals is Newton’s conditional assertion of the existence of a vacuum: “If all the solid particles of all bodies are of the same density, and cannot be rarefied without pores, then a void, space, or vacuum must be granted. By bodies of the same density, I mean those whose inertias are in the proportion of their bulks” [emphasis added]. (1947, p. 414)

The issue is now resolved with an emended Corollary 3 and an added Corollary 4, by reverting to a conditional proof of the existence of a vacuum. The additional condition requires that matter is impenetrable and that the solid particles of all bodies have the same density. However, the explanation is added, as we have seen, that bodies have the same density, if and only if the ratio of their inertias to their volumes is the same. That, Newton said, is what sameness of density means. It is this particular explanation of density that recalls the thorny issues that have been raised, from Mach onward, about the circularity of Newton’s definition of mass. “Mass,”it is said, is defined with the aid of density, but “density” is defined with the aid of “mass.” However, I think that, strictly speaking, there is no circularity. In the first “Definition” of Principia, the one concerned with mass or the quantity of matter, (1) a “measure” is given for the quantity of matter, using density and volume. (2) In the addendum (Corollary 4) of the second edition, the sameness of density is explained using inertia and volume. It would be a mistake to think that (1) provides a definition of a previously unexplained term, “quantity of matter.” The concept of the quantity of matter was one that was familiar to Descartes, Newton, and their contemporaries. I do not think it is a case of defining an unfamiliar phrase; it is more a matter of stating a truth about something familiar. Even familiar terms, however, can be defined, but I think that Newton believed that there was no difference between himself and Descartes over the extension of the relation “M(A, B).” For Newton’s argument against Descartes begins with the observation that two bodies that completely fill spaces of the same volume have the same quantity of matter. The argument would be completely off the mark, if there were disagreement over this initial assumption. Newton seems to have thought of quantity of matter in such a way that the argument will hold against Descartes’ view of matter as completely filling any space. If this is correct, then it would explain why he did not define the quantity of matter of a body as its inertia, despite Cotes’ pressure in that direction. It is evident that that would have begged the issue against Descartes, because Descartes would not have allowed quantity of matter to be defined as anything other than a geometric notion such as the content or volume of the appropriate region of space. Inertia was clearly not a geometric concept (as they thought of such concepts), and it would have been patent to all that inertia would not have been countenanced as belonging to physical theory by Descartes or any of his strict followers. However, Newton did think of the inertia of a body as a measure of its quantity of matter. The use of the concept of inertia points to a deep difference between Newton and Descartes over what concepts may be employed in physical explanations, but it is not a difference over how quantity of matter was understood.

What seems to be involved in Newton’s argument for the vacuum can be thought of as the use of a term such as “M(A, B)” used in two claims that connect that notion with (1) equal volumes of bodies A and B that completely fill their spaces, and (2) equal inertias of A and B. (1) and (2) are both taken as “measures” of the same quantity of matter, and having it both ways, was, as Cotes clearly saw, not justifiable on any grounds that Newton had yet provided.

There is another way, other than definitional, of thinking of the two measures that Newton wished to retain. The notion M(A, B) is a quantitative relation on bodies that completely fill their spaces or for which interstitial pores are discounted. On our account of quantitative relations, some partition is needed. If the collection of bodies that completely fill their spaces is partitioned by the equivalence relation V(A, B), let that partition be denoted by M_v. Then it is easy to see that, if M(A, A) holds for all A, then, for all A and B, M(A, B), if and only if V(A, B), if and only if the relation M(A, B) is quantitative with respect to the partition M_v. Thus, the requirement that “same quantity of matter” be quantitative with respect to the partition induced by sameness of volume leads to the conclusion that, for all A and B, M(A, B), if and only if V(A, B)—the first premise of Newton’s original argument for the vacuum. This biconditional is not a definition by contemporary standards, and it is not clear that it would be a definition by eighteenth century standards. Those standards are probably closer to the ones found in Euclid than they are to those of present logical theory. In any case, the biconditional between sameness of mass and sameness of volume is the one that Newton attributed to Descartes, and, if Cotes had not made things difficult to do so, Newton would have adopted it for the Primigenial particles of bodies and ultimately for all bodies that are rendered poreless.

Similar remarks hold for the relation between mass and inertia. Newton did think that inertia was a measure of the quantity of matter of a body. However, there is no evidence that he thought of this as a matter of definition. Suppose now that the set of bodies is partitioned by the equivalence relation I(A, B), and let M, denote the corresponding partition. Then we have the result that, if M(A, A) holds for all A, then the relation M(A, B) is quantitative with respect to the partition M,, if and only if, for all A and B, M(A, B), if and only if I(A, B). This is the second premise of Newton’s original argument for the existence of a vacuum. On this nondefinitional reconstruction, we can say that Newton wished to assert that M(A, B) was quantitative with respect to the relation of sameness of volume and also quantitative with respect to the relation of sameness of inertia. According to this gloss, to say that inertia is a measure of the quantity of matter is to say that the relation “A has the same quantity of matter as B” is quantitative with respect to the partition induced by the relation of having the same inertia. Although the claims that the relation M(A, B) is quantitative with respect to M_v and that M(A, B) is quantitative with respect to M₁ are weaker than their corresponding definitional counterparts, Newton’s argument against Descartes still goes through, if it is assumed, as he did, that μ is quantitative with respect to each of the partitions M_v and M₁. For, in that case, we know that μ can be quantitative with respect to only one partition, if it is quantitative with respect to any. Consequently, if μ is quantitative with respect to M_v as well as M₁, it follows that, for all A and B, I(A, B), if and only if V(A, B). This is the third step of Newton’s original argument for the vacuum. Furthermore, from here onward, the original Newtonian argument can be resumed by using the pendulum experiments to support the connection between sameness of inertia and sameness of weight. Cotes’ objection, on this reconstruction, is that Newton had all along assumed that M(A, B) is quantitative with respect to M₁ and that, in his discussion of the existence of the vacuum, Newton also seems to have assumed that μ is also quantitative with respect to M_v. However, there are no known grounds, either a priori or empirical, to show that M(A, B) is quantitative with respect to the partition M_v.

The question “What is the measure of the quantity of matter?” is not, we have suggested, the question of how quantity of matter is to be defined. It is the question of what partition it is, with respect to which the relation “A and B have the same quantity of matter” is quantitative. On the definitional view, the conflict between Newton and Cotes resembles some of the difficulties familiar from the work of Carnap, in which two stipulations or definitions of the meaning of a term yield an empirical consequence (in this case, that bodies have the same inertia, if and only if they have the same volume). On our reconstruction using quantitativeness with respect to a partition, the partitions with respect to which a relation is quantitative have to be coextensional, and that would, in Newton’s argument, force sameness of inertia to be coextensional with sameness of volume (for bodies that are poreless).

There is a further question about M(A, B), whose resolution is far from clear. We have seen how the quantitativeness of M(A, B) figured in a significant way in the discussion of whether there is a vacuum. That it was quantitative was clear; less clear was the appropriate partition. What we have called the choice of a partition is what Newton refered to as an empirical measure or an estimate of the sameness of the quantity of matter. The best known discussion of Newton about empirical estimates or measures of certain quantities occurs in his discussions of relative space and time in the Principia. In those passages, he said that, because the parts of space and time are not sensible, we need to have sensible measures of them. These are the relational notions he mentioned, such as hours, days, months, and years used to estimate or measure absolute duration. Similar remarks hold for places of absolute space. We think that there is a parity of matter with space and time in the Principia. The quantity of matter, no less than regions of absolute space and durations of absolute time, also fail to be sensible quantities and require sensible, empirical measures as well. Much of the dispute between Cotes and Newton centers around the several measures that Newton employed in his argument against Descartes’ plenum. Newton thought that the volume of bodies without pores was a natural measure of their quantity of matter. Anyone who thought differently was described as someone who was caviling. Certainly the objection by Cotes seems to have caught Newton short, but there is no mistaking the determination with which Newton persisted in his view. In fact there is reason to believe that, ever since his undergraduate days, he believed it possible that bodies might just be “determined quantities of extension which omnipresent God endows with certain conditions.”²⁷ That would certainly make it plausible that the quantities of matter of poreless bodies is just the volume or extent of some region of space. The comparison of the mass of two (poreless) bodies would be a comparison of the associated regions of space. Although it was speculation on Newton’s part that bodies be thought of in this way, it was a central part of his early unpublished attack on Descartes’ physics, and it seems clear that he thought that it was true. He certainly needed no reminder from Cotes, although he did receive one, that although the masses of bodies were to each other as their volumes, it was still a question of evidence that was needed. Otherwise its use would have violated the canons of scientific reasoning that Newton used to distinguish his own scientific work from that of Descartes and others. The picture that results is one in which nonsensible quantities have their sensible measures. However, in the case of mass or quantity of matter, just as in the case of absolute space and time, the measures were neither a definition nor a replacement for what they measured. What they measured in the case of the quantity of matter seems to have been a completely filled part of space.

ACKNOWLEDGMENTS

This research was supported by a grant from the CUNY Research Foundation.

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¹A. Koslow, “Quantity and quality: Some aspects of measurement.”

²This point was forcefully made and cogently stated by E. Adams in “The aim and purpose of measurement.”

³The program for showing that significant parts of physical theories can be formulated using only qualitative relations stands Carnap’s view on its head.

⁴The case for n ≠ 2 is proved similarly by noting that any (two-place) specialization of Rⁿ will be reflexive in the ith and jth places of the specialization.

⁵The case for a comparative Rⁿ (with respect to E), of greater than two places is complicated by the fact that there may be several two-place specializations available. Each of those induces a binary relation (say IR_i) on S that is comparative with respect to the congruence relation given by the symmetrization [IR_i(a,b) & IR_i(b, a)], which contains E. The maximal congruence with respect to which Rⁿ is comparative is also a congruence of all the induced binary relations IR_i. The join of all these congruences exists, but it is difficult to obtain a clear connection between it and the maximal congruence of Rⁿ. (If the join of all the maximal congruences of all IR_i, is a congruence of Rⁿ, then the two are identical.)

⁶The case for n greater than 2 is essentially the same. Take any (two-place) specialization of Rⁿ. “Suppose that it is the ith and jth places that have not been fixed. Then for any a and b in S, either (Rⁿ)*(p) or (Rⁿ)*(p^{^}), where the ith component of p is a, the jth component is b, and the ith component of p^{^} is b, and its jth component is a, and all the other components are the respective ones from po used to obtain the specialization. Since the (two-place) specialization is comparative, we have (Rⁿ)*(p’) where the ith and jth components of p’ are both a, and the kth component (k different from i and from j) is the kth component of p₀. Moreover, because any two-place specialization of a quantitative relation is quantitative, it follows that a and b are equivalent (under E), for any a and b in S. This contradicts the assumption that the partition has at least two members.

⁷Even if a relation R is quantitative and nonnumerical, there still will be cases where R can be represented homomorphically by an appropriate relation defined over a numerical relational system. Such representation theorems are proved in Koslow (1982). However, the fact that a relation R can be represented numerically is not what makes it quantitative; certain ordering relations can be represented in numerical extensional systems.

⁸I am indebted here to I. Mueller’s exceptional study (1983), especially chapter 3.

⁹Cf. Mueller’s discussion of the constructivity of Euclid’s notion of the multiplication of magnitudes (p. 121 et seq.). It is possible to express the multiplication of magnitudes geometrically, cf. D. Hubert (1968) for the case of multiplication of line segments in a purely geometrical way, without the use of terms that refer to or range over numbers.

¹⁰We take for granted that the point of studying such a relation is the construction of a general theory of proportionality of magnitudes that covers incommensurable as well as commensurable magnitudes.

¹¹R(x, y, z, w) is a condition for sameness of ratios. It is quantitative, as the previous argument shows. There are other relations that figure in the full theory of proportions, such as G(x, y, z, w): the ratio of x to y is greater than the ratio of z to w (for some multiple mx of x, and ny of y, mx > ny, but not (mz > nw).G is easily seen to be comparative.

¹²These interesting but undated studies can be found in Stokes’ collected papers, Cambridge University Library. The definition of mass is given in PA1091 (Definition of Mass, Third Law of Motion), and PA1071 (Definition of Proportion in Euclid & Algebra).

¹³Notice that, if temperature were thought of as a function mapping bodies to the reals, then LET would be a consequence of the theory of identity. For A and ? would have the same temperature, if and only if T(A) = T(B), and it is a consequence then that, if T(A) = T(B), and T(A) = T(C), then T(B) = T(C). This construal of temperature is incompatible with Maxwell’s view of the nontriviality of LET. It is worth noting that Mach, in his Theory of heat, which appeared after Maxwell’s (1871) work, agreed with the thermal example and had earlier (1868) already considered a Zeroth style law for mass in “Uber die Definitionen der Masse,” Reportorium fŭr physikalische Technik, Bd IV, 1868, 355 et seq., and in History and root of the principle of the conservation of energy (trans, and annotated by P. ?. B. Jourdain, Open Court, Chicago, 1911, p. 83), he wrote that “Only experience can teach us that two bodies which behave to a third as equal masses will also behave to one another as equal masses.” A similar moral can be drawn from the Law of Equal Masses (LEM), as one might call it: If it is not a theorem of the theory of identity, then mass cannot be construed as a function from bodies (say) to the reals. Otherwise LEM is just a special case of a theorem of the identity relation.

¹⁴For a fuller discussion, cf. Koslow (1968) and the literature cited there.

¹⁵A question has been raised by an anonymous referee, whether J. Black (1803) has priority over Maxwell and Mach for the recognition of the law of equal temperatures (Lectures on the elements of chemistry, 1803). Mach seems not to have thought so, because no credit is given to him despite Mach’s extensive citation of Black’s work. The relevant passage is reprinted in Magie’s Sourcebook of physics (Harvard), in which Black wrote: “All bodies communicating freely with each other, and exposed to no inequality of external action, acquire the same temperature, as indicated by a thermometer. All acquire the same temperature as the surrounding medium.” At best this seems to be a special case of LET, when all the bodies are simultaneously in thermal equilibrium. LET permits the ascription of the same temperature to bodies, although they are not in thermal equilibrium with each other. The passage could also be read less generously as indicating only that bodies (under certain conditions) that are in thermal equilibrium, have the same temperature.

¹⁶Through the tenth edition (1894) with additions by Rayleigh, the relevant sections on the Law of Equal Temperatures remains the same as in the first edition. The draft manuscript is in Add.7655/IV, l (Box 2), and contains only the Preface, parts of chapters 3–8, 18, and 22. One suggestion has been that the relevant chapters were used for other writing or scrap. So far there is no trace of those missing pages.

¹⁷Fleming’s remarks occur in J. C. Maxwell, A commemoration volume, 1831–1931, Cambridge Press, 1931, and the bound volume is Add.8082 (Cambridge University Library). Unfortunately the reader should be advised that these notes turn out to be a verbatim copy of large sections of a book by R. E. Baynes, who wrote a more mathematical supplement for his students at Oxford, based upon the fourth edition of Maxwell’s work. He sent a copy of it to Maxwell on 27 June, 1878. Why Fleming copied Baynes’ book, commas and all, is puzzling. One can only hope that the lecture notes still exist somewhere.

¹⁸They can be found in “Numbering and measuring” and “The facts of perception”, in Helmholtz (1921, 1977).

¹⁹Thus, it is a mistake to say, as did P. Hertz in his admirable notes to the Helmholtz article on measurement and numbering, that Russell and Frege showed how there is always such a property underlying any alikeness relation. If “alikeness” is an equivalence relation, Hertz noted that there always is a property P such that, if A and B both have P, then they will be related by the equivalence and conversely. Even if we granted these observations of Hertz, they still miss Helmholtz’s point. Although A and B will be alike, if and only if they both have the property P, Helmholtz wanted a property P that causally results in A and B being alike. The Russell-Frege construction would be a logically sufficient but not necessarily a causal condition for A and B being alike.

²⁰Despite the many references to Helmholtz in the literature of the representation theory of measurement, Helmholtz’s theory differs in several key ways: the mathematical analog used is not the Dedekind-Peano version of arithmetic but Grassmann’s, and there are no ordering relations assumed. On the contrary, special ordering relations are introduced by Helmholtz using a whole-part relation on the objects.

²¹ Letter of Cotes to Newton, Feb. 16, 17, in Hall and Tilling (1975),

²²cf. Cotes to Newton, Feb. 23, 17, ibid., in which Cotes quotes Newton’s response to his opening objection: “Hoc ita se habebit si modo materia sit gravitati suae proportionalis & insuper impenetrabilis adeoq: ejusdem semper densitatis in spatii plenus.”

²³ “Hoc ita se habebit si modo magnitudo vel extensio materiae in spatiis plenis, sit semper proportionalis materiae quantitati & vi Inertiae atq: adeo si gravitatis: nam per hanc Propositionem constitit quod vis inertiae & quantitas materiae sit ut ejusdem gravitas.” Parenthetical reference added.

²⁴Newton to Cotes, Feb. 26 17, ibid.

²⁵Cotes to Newton, Feb. 28, 17, ibid.

²⁶Newton did not think that it was a contradiction to think otherwise. That is the point of his remarks about the possibility of particles having different density, forces, and different laws of nature.

²⁷cf. Newton’s “De Gravitatione et Aequipondio Fluidorum” which contains this suggestion about bodies being at any time some region of space that God focuses upon and makes the apparent source of various sensible effects upon us. The text and a translation can be found in Hall and Hall (1962, p. 140).