Leibniz's Theory of Space in the Correspondence with Clarke and the Existence of Vacuums (1) Wolfgang Malzkorn

Introduction It is wellknown that a central issue in the famous debate between Gottfried Wilhelm Leibniz and Samuel Clarke is the nature of space. Leibniz and Clarke, who did not only take a Newtonian standpoint, but was even assisted in designing his answers to Leibniz by Sir Isaac Newton himself, (2) disagree on the ontological status of space rather than on its (geometrical or physical) structure. Closely related to the disagreement on the ontological status of space is a further disagreement on the existence of vacuums in nature: While Leibniz denies it, Clarke asserts it. In this paper I shall focus on Leibniz's position in the debate about these issues. In the first part I shall try to reconstruct the theory of physical space which Leibniz presents in his letters to Clarke. In the second part I shall examine, whether the existence of vacuums is ruled out by that theory of space, as Leibniz seems to imply in one of his letters (see below). To focus exclusively on the correspondence with Clarke is a confinement I am aware of. The theory which I am going to reconstruct differs from Leibniz's ultimate metaphysics of space, (3) but it is particularly interesting for systematic reasons and it also gave rise to a lively discussion in modern philosophy of science. Leibniz's Theory of Space in the Correspondence with Clarke In his letters to Clarke Leibniz trys to "confute the fancy of those who take space to be a substance, or at least an absolute being." (4) To this end, he does not only try to point out the incoherence of Newton's idea of absolute space, but also gives his own positive account of space. As he repeatedly states, the word 'space' denotes, "in terms of possibility, an order of things, which exist at the same time, considered as existing together." (5) Moreover, space is a mere "ideal thing." (6) But what does that mean? In his fifth letter, par. 47, which starts with his explanation of "how men come to form to themselves the notion of space," Leibniz later on gives "a kind of definition":
As has been noticed by several commentators, (7) the general procedure Leibniz adopts here is as follows: First, he defines the relation B is at time t_{2} at the same place as A has been at time t_{1} in terms of "relations of coexistence" of A and B to different physical objects C,E,F,G, etc., which are supposed to have been "fixed" from t1 until t2 (or from t_{2} until t_{1}). Since the relation of being at the same place is an equivalence relation, he can define places as equivalence classes under this relation. (8) Finally, he defines space as the set of all places. Although the general strategy thus seems to be quite clear, some basic questions are still unanswered:
I shall now try to answer these questions, in order to explain the theory in more detail. However, I do not claim that thereby all problems of the theory will be solved. (ad a) In order to explain "how men come to form to themselves the notion of space," Leibniz starts by saying that "[t]hey consider that many things exist at once and they observe in them a certain order of coexistence, according to which the relation of one thing to another is more or less simple. This order, is their situation or distance." However, as the reader is told later, the "situation" of an object is not a relation, but a mere attribut or affection of that object. (9) Thus, one might think that Leibniz thought of distances as the relevant "relations of coexistence" and, presupposing only distances between all physical objects, had in mind the following definition of being at the same place: B is at t_{2} at the same place as A is at t_{1} iff B has at t_{2} the same distances to "fixed" objects C,E,F,G, etc. (which are distinct from A and B) as A has at t_{1}. Of course, this would be an appealing idea, if it worked, since the only structures we would have to presuppose in constructing space are (measurable) distances between physical objects in time, i.e. a distance function on physical objects (and instances of time). Unfortunately, not every distance function suffices to yield an adequate concept of being at the same place. (10) A trivial example for a distance function which is insufficient, is the socalled discrete metric d_{D}. (11) This function does not even determine the unique place of objects A, B at the same time t_{1}, if A has at t_{1} the same distances to distinct "fixed" objects C,E,F,G, etc. as B has (since d_{D}(A,C)=d_{D}(B,C) Ù d_{D}(A,E)=d_{D}(B,E) Ù ... does not imply dD(A,B)=0). Hence, it does not determine the unique place of such objects A, B at different times t_{1},t_{2} at all. Consequently, Leibniz has to presuppose either, that the distances between physical objects define a stronger distance function, or more complicated geometrical "relations of coexistance." An example for a stronger distance function would be a socalled Euclidean metric. (ad b) Even with a Euclidean metric we are in danger to identify different places, if we do not consider a sufficient number of "fixed objects" which satisfy a certain condition, as the following example shows: Let us take three "fixed" objects C,E,F in a Euclidean plane, which are on the same straight line. Let us suppose, the distances between an object A, which is distinct from C,E,F, and C,E,F are the same as the distances between an object B and C,E,F. Then, the following situation may occur: If we say that B is at the same place as A, since A has the same distances to C,E,F, as B has, we obviously identify different places. However, this would not be possible, if C,E,F were not on the same straight line. But, it would have been possible as well, if we had considered only two "fixed" objects (e.g. C and E). (12) Whereas in a Euclidean plane we need three "fixed" objects which are not on the same straight line, in the projective plane P2 (which is a simplification of the unit sphere S2) even three "fixed" objects, which are not on the same line, are insufficient. This is exemplified by the following picture: Consequently, the number of "fixed" objects, which is needed to gain an adequate concept of being at the same place solely by means of distances, depends on the distance function which is defined by the actual distances between physical objects. (ad c) The word 'fixed', however, cannot simply mean 'not in motion', since motion for Leibniz is always relative motion. There may be cases in which we suppose that objects C,E,F,G, etc. continue to be "fixed" during a certain time period, although the distances of C,E,F,G, etc. to some object W change. Bertrand Russell who assumed that Leibniz understood that fact and therefore confused his theory with a reference to absolute motion, wrote in his book on Leibniz:
However, as Edward J. Khamara has pointed out, for Leibniz an object C may change its distances to other objects and still be "fixed" if the cause for the change of distances does not lie in C. (14) For example, a man who stands at a traffic light and is passed by a car, continues to be "fixed" although his distances to the car change constantly. Thus, Leibniz's theory of space needs to be supplied by an adequate causal theory of motion which I cannot discuss here. (I do not even claim that there is such a theory.) But, though such a theory is a necessary supplement to Leibniz's theory of space, it is not part of the core of that theory, as Khamara has plausibly argued. (15) Leibniz's Theory of Space and the Existence of Vacuums So far we have seen, that for Leibniz space is an abstraction from distances between physical objects in time, which is made up by the human mind and thus only ideal. To be precise, it is a set of equivalence classes of ordered pairs consisting of physical objects and instants of time (not just a set of equivalence classes of physical objects, since an object which moves from one place to another would then be a member of different equivalence classes). Each equivalence class, called place, contains at least one pair. Therefore, only that can be considered to be a place, where a physical object has been at some time t. Without physical objects and instants of time there are no places and thus no space. However, it is not necessary, that a place contains an ordered pair áA,tñ (where A is a physical object) for every instant of time t. There may have been times which are not represented in the ordered pairs of a place, meaning that there have been no objects at that place at those times or, in other words, that the place was void at those times. And, if there may have been or may be void places at a time t, there may as well have been or be collections of such places at t. Hence, the existence of vacuums is not generally ruled out by Leibniz's theory of space. To the contrary, the theory even provides us with a way to define exactly the notion of a void place and the notion of a vacuum: A place P is void at a time t iff there is no physical object A such that áA,tñ Î P; a space V is a vacuum at a time t iff every place PÎV is void at t. To put the matter in a slightly different way, like Khamara did, we may say: Leibniz's general denial of the existence of vacuums does not belong to the core of his theory of space. (16) But, unlike Khamara, I do not believe that Leibniz's reasons for denying the existence of vacuums were only theological. There are some passages in the correspondence with Clarke in which Leibniz distinguishes different kinds of vacuums, and there is at least one passage in which Leibniz seems to deny the existence of a certain kind of vacuums referring to his theory of space as a reason:
Although we already know that the existence of vacuums is not generally ruled out by Leibniz's theory of space, we can take this passage as a reason to ask whether certain kinds of vacuums are ruled out by that theory. Leibniz himself distinguishes two kinds of vacuums: (a) vacuums within the world, and (b) extramundane vacuums. (18) Unfortunately, Leibniz does not tell us what the difference is between vacuums within the world and extramundane vacuums (at least not in the correspondence with Clarke). But, on the basis of what has been said in the first part of this paper, I am able to explain that difference, thus reducing it to a distinction between void places within the world and void extramundane places: A void place P is within the world, iff for all pairs áA,tñÎP the object A is a proper part of the universe. As we have seen above, there is no problem with the existence of such places. However, a void place P is extramundane, iff there is a pair áA,tñÎP such that A is either the whole universe or a physical object which is neither identical with nor part of the universe. The latter alternative is ruled out by the definition of the universe: All physical objects belong to the universe, so that there can be no physical objects which are neither identical with nor parts of the universe. The former alternative, however, is ruled out by the definition of the universe in combination with the definition of the concept of being at the same place: (19) There can be no "fixed" physical objects distinct from the universe which can serve as a reference frame for determining the place of the universe at the time t.Therefore, we can conclude that although the existence of vacuums is not ruled out in general by Leibniz's theory of space, there is a certain kind of vacuums whose existence is ruled out by that theory, namely extramundane vacuums. This is probably what Immanuel Kant had in mind when he wrote in the observation on the antithesis of the first antinomy of pure reason:

Notes (1) I would like to thank Eric Watkins and Rainer Noske for valuable comments and suggestions. (2) Cf. A. Koyré/I.B.Cohen (1962) (3) For an account of Leibniz's ultimate metaphysics of space as well as its development see, for example, M.Gueroult (1946), A.T.Winterbourne (1982), and G.A.Hartz/J.A.Cover (1988). (4) Leibniz's third letter, par. 5. In quoting from the LeibnizClarke Correspondence I follow the text of H.G.Alexanders edition (see Bibliography). (5) Leibniz's third letter, par. 4. Cf. his fourth letter, par. 41, and his fifth letter, par. 29. (6) Leibniz's fifth letter, par. 33; see also par. 104. (7) See, for example, A.T.Winterbourne (1982), p.203, K.L.Manders (1982), p.578ff., E.J.Khamara (1993), p.474 and passim, and R.Athur (1994), p.237. (8) This interpretation is confirmed by the following passage from the same letter and paragraph: "I have here done much like Euclid, who not being able to make his readers well understand what ratio is absolutely in the sense of geometricians; defines what are the same ratios. Thus, in like manner, in order to explain what place is, I have been content to define what is the same place." (9) Cf. A.T.Winterbourne (1982), p.203 (10) I will assume here that a distance function d assigns a nonnegative real number to every pair of objects and satisfies the following conditions: (i) d(x,y)=0 iff x=y, (ii) d(x,y)=d(y,x), and (iii) d(x,y)£d(x,z)+d(z,y). (11) dD is defined as follows: dD(x,y):=0, if x=y, and dD(x,y):=1, if x¹y. (12) Cf. E.J.Khamara (1993), p.479.Generally, if one wants to construct ndimensional Euclidean space in the way described above, one needs n+1 "fixed" objects which do not lie within the same n1dimensional subspace. Yet, what it means for objects C,E,F,G, etc. to lie in a mdimensional space can be defined in terms of distances between C,E,F,G, etc. Hence, an analysis of the distances between all physical objects could in principle provide one with the knowledge of how many dimensions space has, which one at least needs to know in order to postulate a sufficient number of "fixed" objects when defining the concept of having the same place. (13) B.Russell (1951), p.121; see also pp.847. (14) Cf. E.J.Khamara (1993), p.474 (15) Op. cit., p.481 (16) See E.J.Khamara (1993), p.480.Yet, Leibniz may have had a different opinion; in par. 62 of the his fifth letter he says: "I don't say that matter and space are the same thing. I only say, there is no space, where there is no matter." For Leibniz's denial of the existence of vacuums see also his fourth letter, par. 4 and 78, and his fifth letter, par. 3335. (17) Leibniz's fifth letter, par. 29 (italics added by the author). (18) Cf. Leibniz's fifth letter, par. 33 (19) It is important to consider this alternative since according to Clarke and the Newtonians the universe can move through (absolute) space and, therefore, have been at places which are void later on; see the passage from par. 29 of Leibniz's fifth paper quoted above. (20) Critique of Pure Reason A431/B459 Bibliography Athur, R. (1994): Space and Relativity in Newton and Leibniz; in: British Journal for the Philosophy of Science 45 (1994) 219240 Earman, J. (1989): Leibniz and the Absolute vs. Relational Dispute; in: Leibnizian Inquiries. A Group of Essays, ed. by N.Rescher; LanhamNew YorkLondon 1989, 922 Grant, E. (1981): Much Ado about Nothing. Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution; Cambridge 1981 Gueroult, M. (1946): Space, Point, and Void in Leibniz's Philosophy, transl. by R.Ariew; in: Leibniz: Critical and Interpretative Essays, ed. by M.Hooker; Manchester 1982, 284301 Hartz, G.A./Cover, J.A. (1988): Space and Time in Leibnizian Metaphysic; in: Noûs 22 (1988) 493519 Jammer, M. (1969): Concepts of Space. The History of Theories of Space in Physics, 2nd ed.; Cambridge, Mass. 1969 Kant, I.: Critique of Pure Reason, transl. by N.Kemp Smith, 2nd ed.; London 1933 Khamara, E.J. (1993): Leibniz' Theory of Space: A Reconstruction; in: The Philosophical Quaterly 43 (1993) 472488 Koyré, A./Cohen, I.B. (1962): Newton and the LeibnizClarke Correspondence; in: Archives Internationales d'Historie des Sciences 15 (1962) 63126 Manders, K.L. (1982): On the SpaceTime Ontology of Physical Theories; in: Philosophy of Science 49 (1982) 575590 Mundi, B. (1983): Relational Theories of Euclidean Space and Minkowski Spacetime; in: Philosophy of Science 50 (1983) 205226 Newman, A. (1989): A Metaphysical Introduction to a Relational Theory of Space; in: The Philoso phical Quaterly 39 (1989) 200220 Russell, B. (1951): A Critical Exposition of the Philosophy of Leibniz, 3rd ed.; London 1951 Ryan, P. J. (1986): Euclidean and NonEuclidean Geometry. An Analytic Approach; Cambridge University Press 1986 The LeibnizClarke Correspondence, ed. by H.G.Alexander; Manchester 1956 Winterbourne, A.T. (1982): On the Metaphysics of Leibnizian Space and Time; in: Studies in the History and Philosophy of Science 13 (1982) 201214 