Despite the success of Newton's physical theories in explaining and accounting for physical phenomena, critics from the very beginning were skeptical of the absolute structure which Newton assumed as the basis of his theories. On the one hand, Newton's approach was very empirical and, for example, when pressed with gravity he preferred to rest his case with a formula describing how gravity worked. As to the nature of gravity, he did not allow himself to speculate in public. With respect to space and time, however, Newton seemed to be committed to a different approach. At the heart of his system were the concepts of Absolute Space and Absolute Time. These, by Newton's own admission, were not directly experienceable. Arguments which Newton offered in support of the necessity for such concepts were challenged by the critics. It seemed to them that had Newton consistently followed his own empirical methodology he would never have been led to postulate such entities. One never experiences Space in and of itself or Time in and of itself. We experience space only insofar as we experience objects in space and moving with respect to one another. Similarly, we do not experience time in and of itself but only time insofar as we are aware of processes (or events) occurring in succession. Thus, one major line of criticism of the Newtonian approach was to suggest that space and time, after all, are nothing in themselves, but only some sort of relational feature of physical objects and events. Such an approach was adopted by two contemporaries of Newton, Gottfried Wilhelm Leibniz and George Berkeley. Their relational attack was based primarily on metaphysical and philosophical rather than on physical grounds and, for that reason, given the success of the Newtonian approach in the solution of empirical problems, their criticisms did not significantly alter the development of physical theory. Later, in the latter part of the nineteenth century, similar criticisms were made of the Newtonian program by the philosopher and physicist Ernst Mach. Einstein later said that his own thinking about space and time had been influenced by some of Mach's criticisms of Newton. In another one of those ironies in the history of science, Mach, who inspired Einstein to the theory of relativity which ultimately overthrew Newton's view of space and time, himself never accepted the relativistic point of view. He remained an unregenerate absolutist until his death in 1916.
In this chapter we want to both develop the criticisms of Leibniz, Berkeley and Mach in their historical context and also to try and formulate, in a precise way, the difference between the relational and absolutistic theories of space and time. II. Leibniz
A. Life and Times
Leibniz was born in Leipzig on January 1, 1646. As a student, he studied philosophy, mathematics and law. He refused an academic career, preferring to become a man of the world, and he spent the bulk of his adult life as a diplomat in various Courts of Europe. Despite this, he found time to write on a number of subjects, including philosophy, religion, and mathematics. He developed a complex metaphysical view which he spelled out in letters to various friends. In mathematics, he did work on the theory of infinite series and, simultaneously with Newton, invented the differential calculus. Leibniz actually first published his results on the calculus in 1684, three years before Newton published his results in 1687. Initially, each was willing to give appropriate credit to the work of the other, but pressure from various well-meaning and not so well-meaning friends and colleagues pushed those men to a bitter priority dispute over the invention of the calculus. This priority dispute developed into a general philosophical feud which culminated in the Leibniz-Clarke correspondence in which Leibniz attacked and Clarke defended, among other things, the Newtonian views of space and time. The dispute came to an abrupt end upon Leibniz's death in 1716.
B. Leibniz's Metaphysical System
Leibniz's metaphysical system is an extremely complicated structure which attempts to draw together, in one coherent picture of nature, various strands from philosophy, religion, mathematics and science. Leibniz never spelled out the details of this system in any one extended work. Except for a few short pieces, such as the Monadology, Leibniz's system has to be culled from his letters and correspondence, much of which was not published until many years after his death. In this section, we propose to give a brief sketch of Leibniz's complicated system since it serves to provide some basis and background for understanding Leibniz's theory of the nature of space and time.
The key to understanding Leibniz's position is the recognition of the importance of the old Greek distinction between the Appearances and Reality. According to Leibniz, the physical world, which seems to us to be so real, is really only the world of what he calls "phenomena." As such, these phenomena are based upon and grounded in certain real structures which are not themselves experienced by human beings. These real entities Leibniz calls monads or substances. These monads can be thought of as "atoms of substance." They have no spatial parts and they endure forever. There are an infinite number of them and each is self-sufficient unto itself. Perhaps it would help to think of the One of Parmenides. We have here a pluralistic version of the Parmenidean doctrine with one crucial difference. The One of Parmenides was not only simple but unchanging. The monads of Leibniz, on the other hand, are not passive and unchanging but are active beings capable of change. One analogy that Leibniz used to describe them is that they are like souls, simple substances which are capable of experiencing or undergoing change.
Each of these substances has an infinite number of attributes or properties. As they evolve, they are capable of realizing inconsistent or incompatible properties, much as a bar of iron, when heated, is first red, then blue, then white. Nicolas Rescher has suggested that one can think of the monads as computer programs with an infinite number of steps which, once started, goes through an infinite number of states in some predetermined order. Another way to think of them is in terms of mathematical sequences such as 1, 2, 3,... or 2, 4, 6,... Each monad might be considered as the generator of an infinite series. Each contains within itself the rule which determines what sequence the numbers in the series will appear. Once activated, the monads develop according to some inherent plan just as the numbers in an infinite series are arranged in a certain order according to the rule of the series.
Each monad, although self-contained, reflects within itself the development of all other monads with which it is compatible. Thus, the infinite collection of monads can be partitioned into collections which contain monadswhich are compossible with one another. These collections of compossible substances or compossible collections of monads constitutes, for Leibniz, different possible worlds. A possible world, thus, is a collection of monads and their histories. The real world, the world that we live in, is just one among these many possible worlds.
The monads in each possible world, and hence in the actual world, group together in what Leibniz called aggregates. These aggregates are then what is experienced by us human beings in the real world as the objects of everyday experience. Thus, the monads are the underlying reality which gives rise to the appearances (aggregates).
Each possible world and every monad is subject to a number of logical principles. One, which Leibniz says "is sufficient to demonstrate every part of arithmetic and geometry, that is, all mathematical principles..." Leibniz calls the Principle of Contradiction. The Principle of Contradiction is the principle that "a proposition cannot be true and false at the same time..." (Alexander, 1956, 15). The second, which in conjunction with the first, Leibniz thought was sufficient to demonstrate all the principles of natural philosophy, i.e., physics, was called the Principle of Sufficient Reason. This principle is given in various formulations in Leibniz's work, but for our purposes we can treat it as the principle "that nothing happens without a reason why it should be so, rather than otherwise." (Alexander, 1956, 16)
A third principle, which can be derived from the Principle of Sufficient Reason, is the Principle of the Identity of Indiscernibles. This principle, like the Principle of Sufficient Reason, comes in a variety of formulations. Read as an epistemological principle, the principle asserts that two states of affairs, El and E2, which are indistinguishable, are, in fact, only one. That is, unless there is some mark which distinguishes El from E2, then E and E2 are really one and the same state of affairs. A second version of the principle asserts that there cannot exist indistinguishable things. Thus, at the level of monads, each monad is distinguishable from each of the others in some respect or other. Any two monads not so distinguishable would really in fact be only one. Both the Principle of Sufficient Reason and the Principle of the Identity of Indiscernibles play a role in Leibniz's critique of the Newtonian view of space and time.
The logical principles enunciated above hold for all possible worlds. We have yet to say how one of those possible worlds, namely, the actual real world, is singled out. On Leibniz's view, God in His omniscience, is able to see in their totality each and every one of the infinite possible worlds. From this infinite range of possibilities, God creates the real world by actualizing one of the possible worlds. Actualizing one of the possible worlds "starts" the programs and what we call the history of the world is the articulation or the development of the inherent principles of development contained within the monads in this world. How does God choose which world to actualize? According to Leibniz, God does not choose a world at random. God chooses which world to actualize by means of, what Leibniz calls, the Principle of Perfection. God chooses, in effect, the most perfect or best of all possible worlds to actualize. Once God actualizes the most perfect of all possible worlds, God then does not intervene further in the daily day-to-day development of the world. The day-to-day development of world history is governed by the inherent principles of the monads which make up the actualized world. Thus, Leibniz found particularly repugnant the Newtonian idea that God had to tinker with the universe to keep the "clock of the universe" going.
The Principle of Perfection is a maximal principle. It says in effect: actualize that world which maximizes goodness, where goodness is understood in terms of order (the idea that the laws of nature ought to be simple) and variety (the idea that the world ought to be as full of things as possible and ought also to contain the most diverse kinds of things). The idea for such a principle, Rescher suggests, derives from Leibniz's work on what is today called the calculus of variation, of which Leibniz was one of the originators. That God chose to actualize the real world in accordance with the Principle of Perfection leads to certain consequences. Among them is the Principle of Plenitude, from which Leibniz deduces that there can be no vacua. A world in which vacua existed would not be as perfect as one in which they did not because those spaces or times which were empty could have been filled by something or other.
The real world, so conceived, can be thought of in one of two different ways. On the one hand, it is a collection of monads unfolding, as it were, as they develop in accordance with their pre-programmed instructions. On the other hand, the real world is the system of phenomenal appearances that constitutes what we would ordinarily take to be the physical world. These are not two separate collections of things but rather the same collection but viewed from different perspectives. The real world as a set of aggregated monads, i.e., appearances, represents the point of view of human cognizers. What we, from our limited perspective, see as world history is the unfolding of the monads.
Each monad, recall, is an individual self-contained substance. Yet the
fact that the world is a cosmos, a coherent whole, indicates that they all unfold together in an orderly fashion. Since they are self-contained, they cannot interact nor causally influence one another. To explain the order and coherence of their development, Leibniz invokes a principle which he calls "pre-established harmony." The idea here is that each monad, although completely independent from all the others, is preset in such a way that it unfolds in a coherent way with the unfolding of all the others. That such a harmony should exist is part of the idea that the world is orderly, i.e., subject to laws. Thus, the pre-established harmony that is evidenced in the real world is a result of the fact that, of all the possible worlds, God chose to actualize that world which was most perfect.
The monads themselves are atemporal and do not exist in space. Space and time, for Leibniz, are part of the phenomenal reality. We turn now to a more detailed consideration of Leibniz's views on space and time.
C. Leibniz's Relational Theory of Space and Time
The connection between Leibniz's metaphysical position and his views on physical theory is suggestive rather than deductive. The metaphysical principles serve as a constraint on what can count as a legitimate physical theory, but there was no question in Leibniz's mind that those principles were compatible with more than one physical theory-. In the light of this, one can see that Leibniz's objections to the Newtonian view are based on a number of considerations, some physical, some metaphysical, some epistemological and some theological.
The basic outline of Leibniz's relational view of space and time can be developed more or less independently of his metaphysical position. Certain specific features, such as the continuity of space and time, and the homogeneity of space, do rely for their support on an appeal to the metaphysics. Leibniz's relational theory is spelled out in greatest detail in The Leibniz-Clarke Correspondence (1716) and in a short paper "Metaphysical Foundations of Mathematics" (1715) (see Alexander [1956] for the (1716); Wiener [1951] for the (1715) )
The basic idea is that space and time are nothing in themselves, but rather are constituted by relationships between existents. Leibniz puts it as follows: "As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is; that I hold it to be an order of coexistences, as time is an order of successions. For space denotes, in terms of possibility, an order of things which exist at the same time, considered as existing together; without inquiring into their manner of existing. And when many things are seen together, one perceives that order of things among themselves [our italics] (Alexander, 1951, 25)." Space, Leibniz holds, is an "order of coexistences," and time is an "order of successions." How are we to understand what this means? One way is as follows. Suppose we imagine that we have a collection of things that exist. We might, for reasons that will become clear later, call them events or objects. These events or objects can be arranged or related to one another in various ways. What we call "time" and "space" are just two (special) ways in which these existent things relate to one another. We can make this concrete with the following illustration. Consider some collection of events, e.g., "The birth of Ronald Reagan (el),'' "The election of Ronald Reagan as President of the United States (e2)," "The defeat of Jimmy Carter in his bid for a second term as President of the United States (e3) ," The election of Jimmy Carter as President of the United States (e4) . " These events can be arranged or ordered in various ways . El, e2 and e4, for instance are not contemporaneous. We can arrange them according to which came "earlier" and which "later. " The resulting order is what we call time. Of course, time is not constituted by the ordering in terms of "earlier" and "later" of just these three events, but of all events. Leibniz did not stop to consider whether or not this ordering must be unique. He certainly took it to be unique, in the sense that any two observers arranging the events in order of earlier to later would come up with the same sequence, but modern theories of time order suggest this need not be so.
Have we done away with the notion of time as an independent existent? It might seem that we have cheated. We eliminated time in favor of an ordering of events with respect to "earlier" and "later, " but are not these themselves temporal notions? Doesn't 'a is earlier than b' just mean that a occurred at some time before the time at which b occurred? If this is correct, that we understand the notions of "earlier" and "later" in terms of their relative positions on a temporal scale, then, of course, we have not gained anything over Newton and the Absolutists, since the concept of time reappears in an essential way in our analysis. To succeed, a relational theory of the nature of time must explain time in terms of non-temporal concepts. In order for Leibniz's attempt to succeed, he must provide an account of the notion of "earlier" and "later" which does not explain these concepts in terms of temporal position. Leibniz has such an account, which many modern day commentators think has a great deal of merit and anticipates moves made by 20th century science. It is called the causal theory of time. We will return to it in the next section. First, however, let us consider what a relational theory of space would look like, on Leibniz ' s view. Consider, again, our small set of events {el, e2, e3, e4}. Not all of these events can be ordered on a scale of earlier to later. The events e2 and e3 are co-existing or contemporaneous. If we consider all events, there will be a large number of such events, all contemporaneous with el and e2. Consider, e.g., (e5): a brushfire in California on Jan. 22, 1981 (which we will take to be the day on which el and e2 officially occurred), (e6): a traffic accident in Boise, (e7): a murder in New York City, etc. We could define what we mean by saying that all these events occurred at the same time in the following way: consider any other event e: if e coexists with any one of the events e2, e3, e5, and e7, then it coexists with all of them. (Recall that we are assuming, with Leibniz, that there is a unique ordering of all events, an assumption which Einstein's theory of relativity forces us to abandon.)
The sequence of events by earlier and later thus turns out to be a sequence of bundles of events. Is there some way in which the events or objects within a given bundle can be ordered? Space, for Leibniz, just is the order of these coexisting objects. In the 1715 paper, Leibniz suggests that spatial positions can be relatively determined by considering how many coexisting objects or events can be interposed between two given coexistents. Consider the events e5, e6, and e7. The event e6 occurs at a place closer to e5 than does e7 in virtue of the fact that every event which occurs at a place between e5 (in California) and e6 (in Idaho) occurs at a place between e5 (in California) and el (in New York) but there are other events, occurring in places like Toledo, for example, which occur at places between where e6 occurs and where el occurs but not between where e5 occurs and where e6 occurs (Wiener, 1951, 202). This is the gist of Leibniz's idea, but he does not develop it in any great detail. Spelling out the details of such a construction turns out to be quite complicated and we will not pursue the details here (cf. van Fraassen, 1970, Chapter 4). Leibniz gives no details as to exactly how the relative situations of the various coexistents are to be empirically determined and to this extent his account is incomplete. Modern analyses suggest that the problem of arranging a spatial "picture" of a collection of contemporaneous existents requires more than can be provided by an ordering relation alone. In addition, some account of measure or magnitude is needed (cf. Grunbaum, 1973, Chapter 22). We shall say some more about this below (pp 8-26 ff.).
Setting aside the difficulties in providing a complete account of spatial ordering, we have developed a picture of what it means to say that time is the order of successive existents and space is the order of coexistents. Space and time are not, however, completely analogous. The order which constitutes time, for example, is (or appears to us to be) one-dimentional, whereas the ordering of coexistent objects is three-dimensional. A second, more troublesome disanalogy is that once the succession of coexistents is fixed, they remain fixed in their relative order, but the coexistent objects whose relative positions constitutes space are free to move around with respect to one another. Does space then change every time the objects constituting it do? If so, then the very idea of motion becomes unintelligible, since something moves if it moves with respect to something else at rest, e.g., if it moves from one position to another. But, if there is nothing to the idea of spatial position but the relative situation of bodies, then if all bodies are constantly in "motion" the idea of a fixed place becomes difficult to define. And without fixed places, it is hard to see how the idea of motion is intelligible. Leibniz was aware of this problem and was trying to deal with it in the passage cited above where he says that "space denotes, in terms of possibility, an order of things..." Space, then, is not the actual ordering of bodies at any given instant, since this is subject to constant change, and, space, we want to say, in some sense remains the same through these changes. Rather, Leibniz seems to be suggesting that space be identified with a possible ordering. This is reinforced by later remarks in his Fifth reply to Clarke, paragraph 47. There what he suggests is the following.
We first come to our idea of space by seeing bodies move with respect to one another. This is basically an epistemological point, i.e., a point about how we acquire our beliefs about and knowledge concerning space. Leibniz then goes on to argue that space is to be identified with the set of possible places (relative positions) which bodies can occupy with respect to one another. This is a slightly different point, an ontological point, about the nature of space rather than our knowledge about it. The basic point can be illustrated by considering a swarm of bees. They are constantly moving with respect to one another but form a more or less compact, though fluid, unity. Leibniz suggests we understand motion, and, hence, relative position and space in the following way. Choose one bee and, at an instant, imagine all the other bees to be frozen in position. Now follow the course of this one bee with respect to all the others considered fixed. Then we can say that this one bee is moving with respect to the others because his relative position with respect to the (presumed fixed) positions of the others is constantly changing. We can now define different positions to be different insofar as their relative position to this (presumed fixed) set of bees is different. Of course, in actual fact, none of the bees is at rest with respect to any of the others. But, the same argument can be applied to any other bee. Follow it and assume the rest (including our original bee) at rest. In this way, one can understand what it means to say that any given bee is moving, even when all the reference bees are also, in fact, moving. That this device works depends on our defining space in terms of possible positions rather than actual ones, but if one allows this, then one can understand both motion and space from a relationalist point of view. We should point out that some commentators are disturbed by this fundamental appeal to "possible positions." They argue, in effect, that understanding possible positions relies implicitly on our prior understanding of actual positions, and that Leibniz is, in effect, smuggling in a reference to an absolute spatial framework at this time (see, e.g., Sklar, 1974, p. 171).
In the 1715 paper, Leibniz gives an account of how a relationalist might handle a metrical concept such as "length." He says: "Quantity or magnitude is that determination of things which can be known in things only through their immediate contemporaneous togetherness (or through their simultaneous observation). For example, it is impossible to know what the foot and yard are if there is not available an actually given object applied as a standard to compare different objects. What "a foot" is can, therefore, not be explained completely by a definition, i.e., by one which does not contain a determination of the same sort. For we may always say that a foot consists of 12 inches, but the same question arises again concerning the inch, and we have made no progress. Also we cannot say whether the concept of an inch is logically prior to that of a foot, for the choice of a fundamental unit lies entirely within our will (Wiener, 1951, 202-3)." Leibniz is making two basic points in this passage, one of which his Absolutist opponent could and would agree with, the other which he would not agree with. The point conceded by the Absolutist is that contained in the last sentence, namely that "the choice of a fundamental unit lies entirely within our will." An Absolutist, such as Newton, could agree with this as long as it is understood that with respect to lengths that can be measured by us, it is a matter of convention (i.e., within our will) which unit we choose to use. The Absolutist, however, then wants to go on to say something stronger. Suppose we find ourselves in possession of a standard foot rule and use it to determine that some object (say a rod R) at point A is, indeed, one foot long. We now transport the object, without destroying it (perhaps, we should assume it is made of some highly indestructible material), to a point B which is some distance away. We now ask: has the length of the rod changed? The simple method of telling whether or not it has is to transport the standard foot rule (or some appropriate surrogate) to B and make a second measurement of the rod R. Suppose that done and the result is that, at B, the rod R measures one foot long. Are we done? The relationalist, following Leibniz's view, says: yes, we are done, the length of the rod at A equals the length of the rod at B (= one foot). The Absolutist says: not so fast. True, the apparent, measurable, relative length of the rod at A and B appears to be the same, but what about its Absolute length? At A, the rod occupied a certain stretch of Absolute Space. At B, the rod occupied a certain other stretch of Absolute Space. The Absolute length of the rod at A is equal to the Absolute length of the rod at B (and, hence, its Absolute length has not changed) if and only if the two stretches of Absolute Space are the same length. The fact that the standard foot rule gave the same measurement in both places proves nothing, since, for all we know, the Absolute length of the standard foot rule changed when it was moved from A to B. So, despite the results of the measurement, it remains an open question for the Absolutist whether or not the length of R (or the standard foot rule, for that matter) changed from A to B. The Leibnizian view suggests that this is absurd. In effect, the Absolutist contends that objects have intrinsic lengths determined by how much Absolute Space they occupy, regardless of the results of any measurement. The relationalist, however, wants to argue that lengths, like velocities, are only relative. Just as one cannot speak (so says the relationalist) of the velocity of an object A without specifying with respect to what frame of reference that velocity is being measured, so, one cannot speak of the length of an object A without specifying some usable standard with respect to which the length of A is being measured.
Because of Leibniz's peculiar metaphysical convictions, he was persuaded, for this reason, to call lengths, and space and time determinations in general, ideal as opposed to real. But, we must not be misled here. Leibniz, and the Relationalists, should not be construed as arguing that space, time, lengths, and velocities, etc. are subjective as opposed to objective. Such determinations are as objective as anything in the sense that different observers, using the same standards and in the same (or equivalent) frames of reference, will come up with identical measurements.
Leibniz uses the analogy of a family tree to illustrate the sense in which he takes space to be set of relations but nothing in itself. Consider what a family tree is. It is a representation of the familial relationships which exist between the members of a family. An individual's "place" within the family is defined by his (or her) relationship to the other members of the family. The relationships are often represented as a tree, thusly,
- - t {] ~ deg.~]
!~ g~O,na~p~L 9
J~r~e5 ! I ;~on~S S~ ~
r ~ ~
L~es ~ I
~9 8.~
The structure appears to be "real," but, of course, only the individuals are real, the interconnections don't exist in and of themselves. Take the fact that Jones Sr. is the father of Jones Jr. Two real individuals are related by the "is the father of" relation. Suppose now that Jones Sr. and Jones Jr. did not exist (not: they did exist but are now dead; rather: than never existed at all). If Jones Sr. and Jones Jr. had never existed (actually it suffices in this case that Jones Jr. has never existed) then the relation that Jones Sr. is the father of Jones Jr. would not exist either. In a similar fashion, if we think away the individuals in the tree (imagine that they never existed), we, at the same time, are thinking away the relations that connect them as well. If there were no individuals in the Jones family, then the relationships which connect them would not exist either. The genealogical relationships are ideal, they don't exist in and of themselves. According to Leibniz, space is an ideal set of relations in much the same way. The objects and events which are related to one another are real but the spatial relationships are ideal. If one "thought away" all the objects
* *
then nothing, not even empty space, would remain behind. ( We are ignoring here certain subtleties involving the relationship between Leibniz's metaphysics and his physics. For a further discussion, see Rescher, 1980, Chapter X).
D. The Causal Theory of Time
"Time," says Leibniz, "is the order of non-contemporaneous things." The basic idea, as we saw above, is to define temporal order by ranking events from earlier to later. In order to avoid a circular argument, one must show how the notions of "earlier" and "later" can be understood in non-temporal terms. Leibniz's solution, which was the forerunner of many similar attempts, was to try to define the earlier/later relation by means of causal connections. Such a theory is known as a causal theory of time. Versions of the causal theory of time were developed in the next 200 years by Kant, among others, and defended in the 20th century by e.g., Hans Reichenbach, Adolf Grunbaum and Bas van Fraassen (Reichenbach (1956), Grunbaum (1973), van Fraassen (1970)).
As van Fraassen remarks, Leibniz's theory is fundamentally a theory of temporal order and must be supplemented by a theory of temporal measure or duration (corresponding to the spatial measure or length). Leibniz does say that "Duration is the quantity of time " (Wiener, 1951, 202) and that "Quantity or magnitude is that determination of things which can be known in things only through their immediate contemporaneous togetherness (or through their simultaneous observation) (Wiener, 1951, 202). From these remarks, we may infer that Leibniz would say about duration something similar to what he says about length, namely, that the choice of a fundamental unit of time is a matter of convention and that the duration of some process is not an intrinsic property of the process in question but only a relational property of the process as compared to some standard process, such as the motion of the heavenly sphere or the "regular" swing of a pendulum (or, if he had lived in the 20th century, the "steady" ticking of an atomic clock) (cf. Chapter 5 above). But, this aspect of Leibniz's theory of time is not developed by him. We turn to Leibniz's version of the causal theory of temporal order. The key points are made as follows: When one of two non-contemporaneous elements contains the ground for the other, the former is regarded as the antecedent, and the latter as the consequent. My earlier state of existence contains the ground for the existence of the later. And since, because of the connection of all things, the earlier state in me contains also the earlier state of the other thing, it also contains the ground of the later state of the other thing, and is therefore prior to it. All existing elements may thus be ordered either by the relation of contemporaneity (co-existence) or by that of being before or after in time (succession) (Wiener 1951, 201-2)~ The basic idea underlying the causal theory is easy to grasp, but filling in the details in such a way to make the theory work turns out to be a complicated and controversial problem. The basic point is this. We start with a fundamental conviction that certain events are the causes of other events. If A is the cause of B, then, in Leibniz's terminology, A is the ground for B. The concept of causal connection that is being employed is taken to be a primitive, unanalysable notion. The fundamental intuition is that A causes B when A produces B or when the occurrence of A is a condition which leads to the production of A. Thus, we are inclined to say that striking a match (under normal conditions) causes it to light, heating water causes it to boil, and what we mean is that the one is a productive factor in bringing about the other. Leibniz's own example suggests he was thinking of a cause as a condition, in the sense that Jones's being 10 years old is a condition or ground for his being 11 years old. These examples are only meant to illustrate the notion of causal connection and not to explain it. With this notion more or less firmly in hand, we can proceed to define a temporal order in the following way. If A is indeed the cause of B then A must precede B, since B, in some sense, arises out of or because of A. Unstruck matches do not, in general, light by themselves. They must first be struck, then they light. The striking of the match is an antecedent physical condition of which the lighting of the match just struck is the consequence. In general, if A causes B then, not only are A and B non-contemporaneous but A invariably precedes B.
This notion of a causal connection between A and B defining an ordering of A and B is still viable. If A and B are causally connected then whichever is the cause precedes the other which is the effect. This result still stands even in the light of modifications due to Einstein's theory of relativity. If every pair of non-contemporaneous events was causally connected then the problem of defining a temporal order would be child's play. Unfortunately, things are not so simple. Suppose, for example, that, while Jones is striking the match (A), Smith is whistling a tune (C). Just before the match lights (B)l Smith stops whistling. Under these conditions, the events B and C are not contemporaneous but they cannot be ordered by the causal argument because B and C are not causally connected. In general, if one chooses any two arbitrary non-contemporaneous events el and e2, they will not be causally connected either. In order to order the events B and C in time, one must use the fact, established by some other method, that A and C are contemporaneous. Given that any pairs of events can be established as either contemporaneous or not, then a temporal order of all events can be defined. If this procedure worked, then we would have succeeded in defining time and temporal relations in terms of the non-temporal relation of causal connectibility.
This possibility has exerted a powerful influence on many thinkers from Leibniz to the present day. Einstein's theory of relativity is compatible with Leibniz's theory with the addition of one slight wrinkle. For events which are causally connected, the story in the theory of relativity is exactly as Leibniz has it. For events which are not causally connected, Einstein's theory introduces a curious twist. The time order of such events is not unique. In the example above, some observers will see event C occur before event B (as Smith and Jones do) whereas others will see event B occur before event C and the theory of relativity insists that both observers are right. All observers, however, will see A occur before B. The explanation of this effect is discussed in Chapter 11 below.
Unfortunately, sketching out what the theory is supposed to accomplish turns out to be simpler than producing a theory which actually does it. The problem is with the notion of "cause" itself. Leibniz himself had no problem with the notion of causality because he thought of it as a 'productive' force ultimately resting on the analogy of a person being able to effect the movement of his body by an effort of will. In the 1700's and subsequently, this concept of 'cause' came under increasing attack especially from empiricists such as David Hume and his followers. Hume professed not to understand what this sense of causal connection was. He thought that 'A caused B' meant no more (and no less) than that events of type A were constantly associated with events of type B and that A was earlier than B. The notion of a productive connection existing between A and B has disappeared. But, now the causal theory of time order is in trouble, for in order to determine that A is the cause of B and not vice versa, we must first say which event came first in time. The temporal order of A and B has been smuggled back in to establish which of the pair is the cause of the other and, hence, the causal order of A and B cannot be used to define their temporal order.
Most contemporary philosophers are more likely to side with Hume rather than Leibniz. This places a heavy burden on those of them who would like to defend the causal theory of time- Hans Reichenbach, one such 20th century follower of Hume, tried to determine causal order independently of temporal order by a method he called the "mark method." Briefly, the idea is to provide a non-temporal criterion of causal precedence by considering that small variations in the cause lead to variations in the effect but not vice versa. Thus, suppose I am examining a solitary footprint in the sand of some secluded beach. I reason that this footprint was caused by someone's foot. When I find the matching foot, I will have found the cause. How do I know that the foot is the cause of the footprint and not the other way around? The mark method tells me as follows: If I mark the footprint, by drawing a pattern in it, by gouging, for example, a similar pattern does not appear on the foot in question. If, however, I were to mark the foot in question then footprints made by that foot would bear the mark. Thus, the foot causes the footprint and not the other way around. The problem is that this method tacitly employs concepts of temporal order. In order to work, the marks employed must be irreversible, i.e., they cannot be erased once made. But, when we try to explain the notion of "irreversible," we wind up using concepts of temporal order. A process going from state A to state B is irreversible just in case once an object in state A is transformed to state B it cannot later be brought back to state A. But "later" is a concept of temporal order. These criticisms have been raised against Reichenbach's method by Grunbaum (1973, Chapter 7), van Fraassen (1970, chapter VI) and others. Neither sees the objections as fatal and both have offered alternative constructions of a causal theory of time which themselves have come under fire. The possibility of a viable causal theory of time remains an open question. Should one not be forthcoming then we may have to come to grips with the possibility that temporal order is a basic unanalyzable feature of the world or of human experience. E. Clarke's Criticisms of Leibniz's View
Clarke, Newton's spokesman in the debate with Leibniz over Absolute Space and Absolute Time, had three basic criticisms of Leibniz's approach.
(l) Clarke argued that, on Leibniz's view, if the entire universe were to be moved say, ten feet to the left of its present position then it would be in the same place as it is now. But, Clarke argues, this is an "express contradiction" and, hence, the Leibnizian view is absurd (Alexander, 1956, 31). Why is such a supposition contradictory? Presumably because if the entire universe were first in position A and then moved 10 feet to the left to a new position B, then A and B could not be the same place since they are 10 feet apart. Leibniz had no problem in disposing of this objection (Alexander, 1956, 38). Clarke's point is question begging in that it assumes that the notion of Absolute Space makes sense, and, hence, that A and B are different. But, this is just what the Relationalist is denying. Clarke's point reduces to arguing that Leibniz is wrong because he is wrong. Hardly telling.
(2) Clarke's second argument is more telling and Leibniz does not really satisfactorily answer it. Clarke asks, in effect, how Leibniz is prepared to deal with the experimental evidence (i.e., the bucket experiment) which "proves" that there is a difference between absolute and merely relative motion and, hence, by implication, between absolute and merely relative space and time. A consistent relationalist would have to deny that the experimental result does enable one to distinguish between absolute and relative motion. Leibniz does not take this tack. He agrees that there is a distinction between absolute and relative motion but denies that this entails that there is a distinction between absolute and relative space (Alexander, 1956, 74). Clarke professes that surely this is untenable and he presses Leibniz for a fuller explanation (Alexander, 1956, 105). Unfortunately, Leibniz died before he could formulate a satisfactory answer and there is no way of knowing if, indeed, he had one.
Leibniz's position seems to be that absolute motions can be distinguished from purely relative motion by considering the causes (forces) which act to set the bodies in motion. If A and B are moving in relation to one another, then if an impressed force on A set A in motion, then it is A (rather than B) which is undergoing absolute motion. As far as the relative motions of A and B are concerned, there is no distinguishing them. When we consider the forces acting on A and B, then, Leibniz suggest, absolute motions can be distinguished from relative motions. In effect, what Leibniz is saying is that kinematically speaking, all motions are purely relative, but dynamically speaking, some motions can be distinguished from others. Newton took this difference between kinematic and dynamic effects to be evidence for the existence of Absolute Space. Leibniz, rather inconsistently, agreed that the effects were different but arbitrarily denied that they had the implications that Newton tried to draw from them. If Leibniz had been a consistent relationalist he could have denied that consideration of forces does make a difference. He would have argued that, dynamically speaking, the motions of A and B were indistinguishable as well. He did not, however, make this move. It was left to Ernst Mach in the late 1800's to argue for the dynamical equivalence as well as the kinematic equivalence of two observers in relative motion to one another (see pp 8-40 ff. below).
(3) Clarke's third basic criticism of Leibniz was also not adequately answered by Leibniz in the Leibniz-Clarke correspondence. Leibniz had argued that space was a system of relations. Clarke urged that space had a quantity as well, but relations could have no quantity and that, therefore, a theory which held space (or time) to be merely relational could not be adequate (Alexander, 1956, 32). Leibniz responds by arguing that a relation "also has its quantity". Leibniz says: As for the objection that space and time are quantities, or rather things endowed with quantity; and that situation and order are not so: I answer, that order also has its quantity; there is in it, that which goes before, and that which follows; there is distance or interval. Relative things have their quantity as well as absolute ones. For instance, ratios or proportions in mathematics, have their quantity, and are measured by logarithms; and yet they are relations. And therefore though time and space consist in relations, yet they have their quantity (Alexander, 1956, 75). This is pretty obscure stuff and Clarke, perhaps, can be forgiven for thinking that Leibniz was just evading the issue.
The basic problem is that specifying a given order of points (objects, events) is not sufficient to determine a unique quantitative measure (length, duration) for them. A simple example in one dimension will illustrate the difficulty. Suppose we consider three points A, B, and C, lying along a line such that B is between A and C. This order is compatible with the following two pictures (and many more as well).
I. A B C
II. A B C
The order of the points in I and II is exactly the same but the quantitative measure of "space" between A and C is clearly different. The same order is compatible with any measurable quantity of "space" between the extremities. Leibniz tries to give some account of the metrical properties of order in the 1715 paper: In each of both orders -- in that of time as that of space -- we can speak of a propinquity or remoteness of the elements according to whether fewer or more connecting links are required to discern their mutual order. Two points, then, are nearer to one another when the points between them and the structure arising out of them with the utmost definiteness, present something relatively simpler. Such a structure which unites the points between the two points is the simplest, i.e., the shortest and also the most uniform, path from one to the other; in this case, therefore, the straight line is the shortest one between two neighboring points (Wiener, 1951, 202).
There are two related difficulties with this approach. Both arise from Leibniz's failure to realize that a metrical structure cannot be generated from a topological structure alone. The first difficulty concerns Leibniz's characterization of the "nearness" relation. The second difficulty concerns his characterization of "straightness."
The problem with "nearness" is this. Consider our three points A, B, and C again. According to Leibniz, B is "nearer" to A than C just in case there are 'fewer' points between A and B than there are between A and C. Intuitively, this seems right, but our earlier consideration of Zeno's paradoxes should have cautioned us that our intuitions, as often as not, are liable to lead us astray. They do so here, if we are persuaded by Leibniz's diagram. The fact is that we can only talk about the 'number' of intervening points between A and B or between A and C as being different if we assume that space is discrete, if, that is, we assume that there is a definite finite number of spatial positions between A and B and a greater number (assuming any two spatial points are at a fixed distance from one another) between A and C. Then, of course, to find the distance between any two points A and B, we find the smallest number of points linking A to B and use that count as a natural measure. We can imagine such a space (in two dimensions), the generalization to three dimensions is straightforward) as follows:
B~
Fl9 ~.~
Imagine the lattice extended infinitely in all directions, then, it is clear that any two points in the lattice are separated by a finite number of other points, and by determining the shortest path, one has a natural definition of "nearness."
While we are at it, we should point out that on such a view, the straight line from A to B just is the shortest path (in terms of intervening connecting points) which connects them. Thus, if space or time were composed of discrete points (or instants) then the definition of measure by means of order might be possible.
However, there are two difficulties. The first is that the view that space and time are discrete leads to paradoxical results (cf. Chapter 3 above). The second is that Leibniz's own view was that space and time were continuous, that is, infinitely divisible (The modern concept of continuity, recall, is a product of the l9th century, 150 years after Leibniz's work). But if space is infinitely divisible (or worse, continuous, in the modern sense) then the 'number' of points between any two points A and B is not finite and the "natural" method of counting intervening links and, thereby, determining "shortest" paths, fails. The link between the order structure of space and time and its metrical structure is broken.
The task of defining metrical concepts and characterizing the metrical structure of continuous space and time requires an additional account, not derivable from considerations of order alone. The details of the construction are beyond the scope of this book. (cf. van Fraassen, 1970, Chapter III and Grunbaum 1973). Insofar as metrical concepts cannot be derived from considerations of order alone, Leibniz's account is incomplete. This does not detract, however, from the essentially correct insights of Leibniz which are today being vindicated, in modified form, by the success of Einstein's theories of relativity.
F. The Characterization of Space and Time According to Leibniz
In order to facilitate a comparison of Leibniz's view on space and time with that of Aristotle and Newton, it will be useful to construct a table of properties of space and time for the Leibnizian view which parallels that already constructed for the Aristotelian and Newtonian view.
lS. Space is homogeneous. Different positions in space are not, merely in virtue of their difference in position, distinguishable. It is clear why a relationalist like Leibniz should hold such a view. If two positions did differ intrinsically, then if the entire universe were moved 10 feet to the left or rotated 30deg., such a change would be discernible and, hence, distinguishable. But, Leibniz argues, such differences are not distinguishable and are not really differences. Therefore, positions in space cannot be intrinsically distinguishable. Space must be homogeneous.
In the 1715 paper, Leibniz gives a different argument for the same conclusion based on the fact that space is boundless. Since it is boundless, and since space, as such, has parts which are themselves spaces, it must be "universally uniform," i.e., homogeneous (Wiener, 1951, 206; cf. 204 for a discussion of Leibniz's concept of homogeneity).
lT. Time is inhomogeneous. One would think that Leibniz would produce a similar argument with respect to time. But, he does not. Instead he says I admit, however, that there is this difference between instants and points -- one point of the universe has no advantage of priority over another, while a preceding instant always has the advantage of priority, not merely in time but in nature, over following instants (Loemker, 1956, 664). Leibniz's point seems to be that the anisotropy of time introduces an inhomogeneity. We should pause to reflect here on why this should be so for Leibniz but not for Aristotle or Newton. Both take time to be anisotropic and yet hold that it is homogeneous as well. The difference may lie in the fact that for Leibniz, time is not only anisotropic but is constituted by the succession of events which makes up the history of the universe. For Aristotle and Newton, the succession of such events, at best, provides a measure of the passage of time, but is not, itself, time. Time extends infinitely into the past, for Aristotle and Newton. For Aristotle the physical universe does so as well: both time and the physical universe being coeternal. Newton, however, was a creationist. Although time (Absolute Time) has always existed, the physical world was created by God at some particular time. Thus, it makes sense for Clarke and Newton to suppose that God might have created the world earlier or later than he actually did. There was nothing special about the moment of creation, as such, for Newton. Thus, any instant is just like any other instant and time is homogeneous. Leibniz, of course, rejects this supposition as absurd (Alexander, 1956, 27).
For Leibniz, time is constituted by the succession of events which is the result, recall, of the "unfolding" of the actualized monads according to their pre-established harmony. It follows that each instant is a unique reflection of the state of the developing monads, not merely in virtue of its temporal position, but in virtue of the fact that it is just that instant which is associated with that stage in the development of the universe. Perhaps this accounts for Leibniz's view that time is inhomogeneous.
2S. Space is infinite. In the 1715 paper, Leibniz makes it clear that he takes space to be infinite. As noted above, Leibniz sees this property as connected with homogeneity. If space were finite, it could not be homogeneous because some places would be singled out as special, namely, the center and the boundaries.
2T. Time is infinite. Leibniz was, like Newton, a creationist. The world was created by God at some finite time in the past. Since the creation of time and the creation of the world are contemporaneous, the actual elapsed time from the beginning to the present is also only finite. What makes time infinite is that it extends (potentially) forever into the future. The development of the history of the world, recall, is like writing down an infinite series of numbers. The series has a definite beginning, but, once started, it never ends. There is no last term in the series just as there is no last movement of time.
3S,T. Space and time are both continuous. The continuity of space and time follow for Leibniz, from his view that motion is continuous. This follows, in turn, from his metaphysical principle of continuity which holds that there are no gaps or 'jumps' in nature (cf. Rescher, 1979, Chapter X).
4S. Space is isotropic. That space is isotropic follows from its being infinite and homogeneous.
4T. Time is anisotropic. The anisotropic character of time is a reflection of our lived experience which is always from past to future. Both Aristotle and Newton, as well as all the other theorists we have so far considered in this book, agreed that time is anisotropic. What distinguishes Leibniz's view from theirs is that, whereas, for them, the anisotropic character of time is a brute fact, Leibniz's causal theory of time is an attempt to explain the anisotropy of time.
5S, 5T. Space and time are object dependent. The object (or event) dependence of space and time, for Leibniz, should by now be obvious. Space and time are nothing in themselves. They are merely structural features of objects and events. No objects or events, no space and time.
6S, 6T. The mind dependence of space and time. Would space and time exist, for Leibniz, if there were no minds to perceive the objects and events which ground them? Recall that space and time are features of the developing monads as perceived by us conscious beings. Suppose, if we may, that the best of all possible worlds had turned out to be one in which consciousness did not exist. God would have actualized that world instead of our world. Would that world be a world in space and time? One can go two ways with this, depending upon how one reads Leibniz. If one takes Leibniz to be saying that space and time are characteristics of the world solely in virtue of the fact that there are conscious beings in the world who perceive things in the way they do, then if the actual world had had no conscious beings, it would not have been a spatio-temporal world either. If, however, one reads Leibniz as holding that space and time are relational properties of objects and events as such, insofar as they are aggregates of monads, then such a world could be spatio-temporal. The incompleteness and unclarity of Leibniz's views on this do not allow a clear cut resolution of the issue (see Rescher, 1979, Chapter X).
7S, 7T. The mutability of space and time. Time, for Leibniz, is not mutable. Instants, for example, cannot change their places with respect to one another. The same is true for space. Remember that space is not the order of the actual relative positions of objects, which is subject to change (recall the swarm of bees analogy) but rather the possible order of such positions. As such, even though the relative positions of the objects constituting space are mutable, space itself is not.
8S, 8T. Space and time are ideal, not real. By calling space and time ideal and not real, Leibniz meant to deny that they were substances in the sense that either Descartes or Newton thought that they were. He also wanted to deny that they were substances in his own sense, i.e., monads. Space and time are ideal in much the same sense that the relationships in a family tree are ideal and not real. They are nothing in themselves and would not exist if what they related did not exist.
9S, 9T. Neither space nor time is causally active. This is a direct consequence of Leibniz's view that space and time are nothing in themselves and that they are not substances. Only substances (i.e., monads) can be causally active.
lOS, lOT. There are no vacua in space or time. That space is a plenum follows from metaphysical considerations. A universe in which empty spaces existed would be less perfect than one in which they don't, since God seeks to maximize both the amount and the variety of material in the universe (Alexander, 1956, 16 and 43-45). One would expect a similar argument for time. But, in this respect, Leibniz is somewhat inconsistent. In the Leibniz-Clarke correspondence, he does argue that there are no "empty times" (i.e., times during which no processes take place) there is no vacuum..., (if I may so speak), in times, any more than in places (Alexander, 1956, 90). Rescher, however, points out that in Leibniz's polemic against John Locke, The New Essays on Human Understanding, written in the early 1700's but not published in Leibniz's lifetime, Leibniz seems to admit the possibility of empty times on the grounds that if someone argued that such existed, there would be no way to refute him, since no measurable distinction could be made between a universe in which such a void existed and one in which it did not (Rescher, 1979, 92). Rescher points out, quite correctly, that in accordance with the principle of the identity of indiscernables, Leibniz should have denied that any real difference exists in the two cases considered. Someone who claimed a void existed in time would, thus, be refuted by pointing out that what he claims is no different from the claim of someone who denies voids can exist in time.
With this, we bring our discussion of Leibniz to an end. His relational critique of Newton, while incomplete, was, at times, quite perceptive and it is for this reason that he is acknowledged to be a forerunner of contemporary ideas in physical theory. III. Berkeley
A. Life and Times
George Berkeley was a younger contemporary of both Newton and Leibniz. He was born in Ireland in 1685. He studied mathematics and philosophy at Trinity College in Dublin, and was a Fellow at Trinity until 1724. He became bishop of Cloyne in 1734 and died in Oxford in 1753. He was interested in and wrote on a wide variety of topics, including a critique of the calculus, a theory of vision, a treatise on motion, a theory of human knowledge, a book of Christian apologetics, and, late in life, a treatise on the beneficial medicinal aspects of tar-water for treating human diseases.
Philosophically, Berkeley is a curious figure. He presents himself as a common-sense philosopher and, as such, is very critical of what he considers to be the metaphysical excesses of Newton and Leibniz. His own philosophical position is that of radical empiricism, a doctrine that can be encapsulated in Berkeley's slogan "Esse est percipi," or "To be is to be perceived." What he meant by this was that the only existents and the only things we can only have knowledge of are minds and what they perceive. These entities he took to be, not material objects, but sensory impressions of material objects or, as he called them, ideas. Anything which could not be understood in terms of sensory impressions alone was rejected by Berkeley, as meaningless. In this vein, he attacked not only Newton's concepts of Absolute space, Absolute time and gravity, but also Leibniz's concept of causality. Such abstract, general ideas, as Berkeley called them, are meaningless precisely because they are not derived from sensory impressions, i.e., they have no perceptual content.
Berkeley's general philosophical views led him to formulate a position about the nature of science which foreshadows what, in the 20th century, came to be known as positivism. Theoretical concepts which cannot be cashed out in terms of observable processes or properties are held to be empty, devoid of empirical content and lacking in explanatory power. Thus, the Newtonian concept of "gravitational attraction" is attacked by Berkeley as being meaningless and non-explanatory. That the planets move in the ways that they do is not denied by Berkeley. Nor does he deny that Kepler's laws or Newton's law of gravity can be used to correctly describe that motion. These are empirical facts. But, to add to this description the claim that these planetary motions are explained by a mutual attraction between the sun and the planets is to add nothing, on Berkeley's view, to our original descriptive account. We are capable of observing bodies moving with respect to one another, but we do not observe them attracting one another. We infer that they do so, and such inferences are ruled out by Berkeley as illegitimate. According to Berkeley, the job of physics is not to explain by causes but merely to formulate rules which describe how bodies move or how, in general, processes evolve and so on. Causes are ruled out of science. One "explains" a particular motion by showing how a description of that motion can be deduced from the general laws (rules) of motion, much as one proves a theorem in geometry by deducing it from the axioms. In physics, though, there are no axioms, only general empirical rules (Berkeley, De Motu, paragraph 71). Although he anticipated some of the doctrines of the 20th century positivists, Berkeley did not go as far as they did. He did believe in causes, for example, and he dabbled in metaphysics and theology. According to Berkeley, only spirits or minds could affect causal changes, but these were beyond the province of the study of natural science. Latter day positivists desired to rid the world of all metaphysics but Berkeley was content to rid only natural science of it.
To this end, Berkeley proposed to sharply distinguish physics from metaphysics (Berkeley, De Motu, paragraph 72). This can be read as an indirect attack on Leibniz and his freewheeling approach to science combining empirical research, metaphysical speculation and theology. Likewise, Newton comes under attack for failing to observe his own methodological principles (e.g., "I feign no hypotheses") with strict enough regularity. In addition to these methodological comments, Berkeley also put forward specific criticisms of Newtonian doctrines, criticisms which are relationalist in tone. We turn to those comments now.
B. Berkeley's Critique of the Newtonian View of Space and Time
Berkeley did not, himself, put forward a comprehensive general doctrine of space and time. To have done so would have been out of line with his general philosophical approach. As we remarked, Berkeley prided himself on his common sense point of view. From the point of view of the common man, Berkeley argued, we are all aware of time to the extent that we know how to answer questions such as "What time is it?" or to the extent that we are capable of being on time for appointments. Our awareness and understanding of time, for Berkeley, is grounded in such common run of the mill experiences. The difficulties begin when we try to ask about time in the abstract (What is it, really?) "taken exclusive of all those particular actions and ideas that diversify the day" (Berkeley, 1963, page 113_~97).
Berkeley makes similar remarks with respect to space and motion, and always sides with the man in the street. With respect to motion, for example, Berkeley argues that nothing moves except insofar as it moves with respect to something else. That is, all motion is relative motion (Berkeley, 1965, 121 SS112). But, at this point Berkeley adds a peculiar "common sensical" twist. If we examine the way we ordinarily use the word "motion," we discover that we are not, in general, inclined to say that B is moving with respect to A just because A is moving with respect to B For, however, some may define relative motion, so as to term that body moved which changes its distance from some other body, whether the force or action causing that change were impressed on it or no, yet as relative motion is that which is perceived by sense, and regarded in the ordinary affairs of life, it follows that every man of common sense knows what it is as well as the best philosopher. Now, I ask anyone whether, in his sense of motion as he walks along the streets, the stones he passes over may be said to move, because they change distance with his feet? (Berkeley, 1963, 121).Normally, Berkeley is claiming, we don't say that the stones in the street or the stores or other fixed objects are moving as we move past them.
Berkeley uses this asymmetry in our ordinary speech patterns concerning motion to deny Newton's inference from the bucket experiment. Newton argued, recall, that the concave shape of the water could not be due to its relative motion with respect to the bucket because in stages 2 and 4 the relative motion is tile same but the shape of the water surface is flat in the one case and concave in the other (cf. the discussion in Chapter 7, pp 7-23, ff. above). In accordance with the above reasoning, Berkeley denies that the water in stage 2 has any motion at all (Berkeley, 1963, 122, SS 114), In the De Motu, (1721) Berkeley has more to say about the bucket. His view there is a direct anticipation of Mach's l9th century critique of Newton.
In a similar vein, Berkeley denies that the concept of Absolute space can play an effective role in science. First, it is an abstract idea and ruled out on that account. Second, since it is unobservable, it cannot serve as a frame of reference in terms of which we might determine relative motions. What philosophers and physicists take to be Absolute space is, according to Berkeley, an illegitimate conception of "the Corporeal World as finite, [with] the utmost unmoved walls or shell thereof" serving as an absolute framework (Berkeley, 1963, 122, ~ 114). In De Motu, Berkeley suggests that this "shell" can just as well be taken to be "the heavens of the fixed stars, considered at rest... [and]... motion and rest marked out by such relative space can conveniently be substituted in place of the absolutes, which cannot be distinguished from them by any mark" (Berkeley, De Motu, 49, ~ 64).
This criticism, that the fixed stars can be used as a frame of reference in place of Newton's Absolute space, and, thus, that all spaces and motions are relational was suggested again in the 1870's by Ernst Mach. But, Berkeley, unlike Mach, was not a physicist. He did not develop a complete dynamical theory in terms of which his criticisms of Newton, however sound, could be worked out in detail. Thus, Berkeley's telling criticisms of Newton, like those of Leibniz, fell, for the most part, stillborn from the press. The success of Newtonian physics plus the lack of a viable alternative meant that despite these contemporary critics, Newtonian science grew and prospered for the next 200 years. In this respect, the fate of Newton's theory is analogous to that of Aristotle's, which survived despite piecemeal attacks until a viable alternative appeared in the 1600's. Part of the staying power of the Newtonian physics was due to the fact that, as the critics rightly urged, Absolute Space and Absolute Time did not play any role in any mechanical determinations. It was only in the latter part of the 1800's that cracks began to appear in the Newtonian edifice due in part to challenges from electromagnetic theory (See Chapter 10 below). The time was ripe for a re-examination of the foundations of mechanics. IV. Ernst Mach
A. Life and Times
Ernst Mach was born in Moravia in 1838 and died in Munich in 1916. He was trained as a physicist and, although he disclaimed the title "philosopher," he made a number of important contributions to the philosophy of science. In a book, The Analysis of Sensations, subtitled "The relation of the Physical to the Psychical," he developed a phenomenalistic approach which was, in many respect, akin to that developed by Berkeley. His most important work was The Science of Mechanics: A Critical and Historical Account of its Development, which, as its subtitle suggests, was a historical critique of the development and foundations of mechanics. Mach was a great believer in the historical approach to understanding science. He argued that a complete understanding of science could only be attained through a historical analysis of the development of scientific concepts. In keeping with this view, he wrote historical and critical analyses of the concept of energy, of the theory of heat and of optics. Insofar as the approach of this book is that a critical understanding of the concepts of space and time can only be achieved through a historical critical analysis of the development of these concepts, it is Machian in spirit.
A number of writers have spotted Berkeleyan themes in Mach (Popper, 1953; Whitrow, 1953; Suchting, 1967). They certainly shared some common methodological themes, although Mach was more "positivistic" than Berkeley. Mach, unlike Berkeley, rejected all metaphysics and was himself an antireligious thinker. Some have argued that, in addition, Mach's substantial criticisms of Newton are also anticipated by Berkeley (e.g., Popper, 1953). There can be no doubt that both men express themselves in a similar way, but a careful reading of Berkeley shows that any claim of significant substantial anticipation by Berkeley is somewhat of an exaggeration (cf. Suchting, 1967).
Among the common methodological themes shared by Berkeley and Mach, we may take note of four. (l) Both Berkeley and Mach are phenomenalists. They argue that the data of science (and all other knowledge) is ultimately the sensory impressions (phenomena) of knowing subjects. (2) According to Mach, the basic function of science is that it provides a method for collecting data into convenient formulae thus achieving an "economy of thought" which enables knowing subjects to make their way about in the world more efficiently. Scientific laws and theories, thus, are nothing more than rules for codifying observational data and using it for predictive purposes. (3) A related point is that the aim of science is to provide comprehensive descriptions but not explanations. (4) Mach rejected, in principle, the appeal to unobservable entities and processes in science. In his critique of Newton's concepts of Absolute Space and Absolute Time, this attitude stood Mach in good stead, but, with respect to the reality of physical atoms (still a controversial issue in the early l900's), it led him astray.
Einstein, in his obituary notice for Mach in 1916, and in other writings, credited Mach's critique of Newton as being an important influence on his own thinking which led to the theory of relativity. Even earlier, Mach was being seen as a forerunner of the then new theory of relativity. But, Mach, despite his criticism of Newton, was no relativist. In the preface to his Principles of Physical Optics, written 1913, he writes I gather from the publications which have reached me, and especially from my correspondence, that I am gradually becoming regarded as the forerunner of relativity. I am able even now to picture approximately what new expositions and interpretations many of the ideas expressed in my book on Mechanics will receive in the future from the point of view of relativity.
It was to be expected that philosophers and physicists should carry on a crusade against me, for, as I have repeatedly observed, I was merely an unprejudiced rambler, endowed with original ideas, in varied fields of knowledge. I must, however, as assuredly disclaim to be a forerunner of the relativists as I withhold fromthe atomistic belief of the present day (Mach, 1916, vii-viii). Mach goes on to suggest that the theory of relativity will "prove to be [no] more than a transitory inspiration in the history of science..." The young revolutionary, attacking the foundations of classical mechanics had grown into an aged conservative unable to follow through on the implications of his earlier insights. In the preface to the seventh edition of the Mechanics (1912), Mach had written, I myself -- seventy-four years old, and struck down by a grave malady -- shall not cause any more revolutions (Mach, 1960,
xxvi ii ) .
On these sentiments, Reichenbach comments, No one who has read Mach's excellent critical analysis of classical mechanics without preconceived notions could doubt that the young Mach, the gifted critic of Newton, would have become a convinced supporter of Einstein's theory. It frequently happens that the author of a revolutionary idea shuts himself off from those who attempt to develop his idea further. It seems that Mach exhausted his revolutionary strivings on this one achievement, and became insensitive to the imaginative daring of others; apparently the liberating effect of a fruitful idea consists, for its creator, more in the intensity with which he has produced it than in its content. How much more pronounced must this attitude be when age and illness prevent one from sharing the enthusiasm of other thinkers (Reichenbach, 1959, 19-20). We might say of Mach what he himself said of Newton: He has done enough if he has discovered truths on which future generations can build... [even though]... he was... not perfectly clear himself concerning the import... of his principles (Mach, 1960, 305, 3deg.4)It is to these discoveries that we now turn.
B. Mach's Critique of Newton
Absolute Space and Absolute Time, according to Mach, are "monstrous conceptions" and, despite his reservations about the relativistic implications of his relational view of space and time, he did not change his mind about the illegitimacy of these Newtonian concepts (Mach, 1960, viii).
Mach's view is that physics must work with what is given in observation. Absolute motions or absolute intervals of time are never observed. What are observed are always relative motions or relative time intervals. The concept of Absolute Time, Mach argues, is an "idle metaphysical speculation" (Mach, 1960, 273). We measure time by comparing one process with another taken as a standard. We take one process, e.g., the diurnal motion of the stars, as a standard in terms of which we measure the durations of other processes, but "time" in and of itself does not play a role in our determinations. We are free to choose another process as the standard if we so desire. If we do, then we may measure the duration of successive days and determine with respect to the new standard, e.g., an atomic clock, that the lengths of successive days (determined by successive reappearances of the same configuration of stars) are not constant. Have we "discovered" that the sidereal clock which was our original standard was not keeping the right time? By no means. We are free to say either that the rotation of the earth as measured by the atomic clock is irregular or that the atomic clock as measured by the rotation of the earth is erratic. Which we choose to say is determined, in part, by which resulting physical theories of the world are easier to deal with.
With respect to space and motion, the point is the same. Absolute space does not play any role in our empirical determination that some object A is in motion or has moved such a distance. The only motions or movements which are detectable are relative motions or movements, i.e., A's moving with respect to some background of objects which is chosen as a convenient standard. For Mach, as for Berkeley, the fixed stars serve as well as anything.
In general, Mach was critical of thought experiments (or interpretations of the results of actual experiments) which required one to assume conditions in the Universe to be other than they are. Such interpretations miss the point, according to Mach, "that the system of the world is only given once to us..." (Mach, 1960, 279). By this, Mach meant that the physicist is given a set of data and his task is to provide mathematical models of these data as best he can. If two different models each organize the data in some particular pattern or other, then, on Mach's view, there is no fact of the matter about which interpretation is the correct one.
Consider, for example, the difference between the Ptolemaic (geocentric) view of the Universe and the Copernican (heliocentric) view. Each model accounts for the observational data. Each represents an interpretation of that data. We are free to choose whichever one we wish. But, even having chosen the Copernican model (or some variation of it), the Ptolemaic model remains as an acceptable although, perhaps, cumbersome alternative. Thus the life and death issue of whether the earth really moves is reduced, by Mach, to the question of which model we are prepared to adopt. On the Ptolemaic model, the earth is not moving; on the Copernican model it does move. But, the observational facts are the same on both models -- the earth moves relatively with respect to the stars. How we want to account for that relative motion is our problem, according to Mach, but nothing about whether the earth really moves follows no matter what choice we make.
In the case of Newton's bucket experiment, however, something else seems to be going on. Newton and his followers argue that the results of the experiment show that the bucket is really rotating with respect to some Absolute Space. But, what, Mach asks, do we observe? What we observe is that the bucket rotates with respect to the fixed stars. But, who is to say that the bucket is really rotating with respect to the stars at rest or that the stars are really rotating with respect to the bucket? The parallel with the Ptolemaic/Copernican controversy is clear. Neither party is in a position to say which is really rotating with respect to the other, which is the same as to say that both are rotating with respect to each other. Mach says Try to fix Newton's bucket and rotate the heavens of fixed stars and then prove the absence of centrifugal [inertial] forces (Mach, 1960, 279).
The point is that the inertial effects which produce the receding of the water up the edge of the spinning bucket are observed when there is a relative rotation of the bucket with respect to the fixed stars. It is the job of physics to deal with this phenomenon and not speculate on what might be the case in a situation which cannot be realized, i.e., fixing the bucket and rotating the heavens.
By the time Mach was writing in the late 1860's, there were a number of phenomena other than the diurnal revolution of the heavens to indicate that the earth was rotating with respect to the fixed stars. These effects were inertial effects like that produced by Newton's spinning bucket. They were generally taken to establish the reality of the Copernican doctrine that the earth really rotates on its axis. Mach discusses three such results, although, of course, he challenges the interpretation usually put on them. (l) The Earth is an oblate spheroid. Accurate measurements of the Earth's diameter indicated that the Earth was not a perfect sphere but that it was flattened at the poles and bulging at the equator. The explanation given was that the Earth, spinning on its axis, "forced" material away from the axis of rotation just as the spinning water in Newton's bucket was forced away from the axis of its rotation. The water in Newton's bucket was constrained by the solid walls of the bucket, so the net effect was to force the edge of the water up the sides of the bucket thus deforming the surface of the water. The material of the Earth is, of course, under no constraint. In addition, the Earth is a solid not a liquid. The net effect is to create a bulging at the equator where the tendency to recede from the axis of rotation is the greatest. Such a flattened sphere is called an "oblate spheroid."
(2) The value of g (the force of gravity) is least at the equator and greatest at the poles. Independent confirmation of the fact that the Earth is not a perfect sphere is indicated by sensitive measurements of the pull of gravity at various latitudes. If the Earth is an oblate spheroid, flattened at the poles, one would expect that the pull of gravity should be a maximum at the poles and diminish to a minimum at the equator. Recall that the formula for the gravitational attraction between two bodies is
G is a universal constant. F, the force that an object, of mass m, on the surface feels from the Earth is equal to mg (Newton's Second Law), where g is the acceleration of the object due to gravity. The mass of the Earth is mE and r is the distance from the center of the Earth to the object in question. A simple substitution
(~ rnF ~
YY) 9
yields, after cancelling the 'm''s from both sides
g -
~ ?
But GmE is a constant, so we are left with the result that g is inversely proportional to the squared distance of an object to the center of the Earth. Consider two identical objects which, when at the same place, weigh the same. Let one be sitting at the north pole and one be sitting on the equator. The one at the north pole will be closer to the center of the Earth and, hence, in accordance with the above result, will feel a greater gravitational attraction than the one sitting on the equator. The one sitting at the pole will weigh more than the one sitting on the equator. Sensitive weighing experiments verify this result and corroborate the oblate shape of the Earth. Again, this result was taken to be evidence for the real rotation of the Earth.
(3) Foucault's pendulum. A more spectacular effect of the rotation of the Earth with respect to the stars is afforded by a Foucault pendulum. Such a pendulum is constructed by suspending a bob from a support in such a way that the attached end of the pendulum is able to rotate freely. If we imagine such a pendulum constructed at the north pole, then the support, rigidly fixed to the Earth will rotate with the Earth. The pendulum, however, we assumed to be freely swinging. A pendulum, once set in motion, will oscillate in a fixed plane with respect to the fixed stars if it is a non-rotating frame of reference.
If we assume that the pendulum is set in motion oscillating in a plane fixed by the stars A and B and the pole P (the plane of the paper), then, since it is not rigidly attached to the support, it will continue to rotate in that plane as the Earth rotates around its axis. To an observer standing on the rotating Earth, however, the support will seem to be at rest, and the plane of the pendulum will rotate. This effect is readily detectable by letting the pendulum bob release a trail of sand and noticing the shifting patterns of the trail as the Earth rotates. The pendulum, of course, need not be constructed at the north pole, the effect is detectable anywhere. There is, in fact, such a pendulum at the United Nations building in New York. A circle of dominoes which is systematically toppled over by the rotating pendulum bobs indicates the motion.
Again the standard analysis of such results was that they were due to the real rotation of the Earth with respect to the stars assumed at rest in Absolute Space. Mach insisted, however, that all that was observable was that such effects were produced when there was relative rotation of some body with respect to the fixed stars. It was illegitimate, Mach claimed, to conclude from such relative rotations that either the Earth or the fixed stars were really rotating.
In this regard, Mach considers a thought experiment suggested by C. Neumann in 1870 (Mach, 1960, 340). Neumann's argument, as cited by Mach goes as follows: If a heavenly body be conceived rotating about its axis and consequently subject to centrifugal [inertial] forces and therefore oblate, nothing, so far as we can judge, can possibly be altered in its condition by the removal of all the remaining heavenly bodies. The body in question will continue to rotate and will continue to remain oblate. But if the motion be relative only, then the case of rotation will not be distinguishable from that of rest. All the parts of the heavenly body are at rest with respect to one another, and the oblateness would also disappear with the disappearance of the rest of the universe. (Mach, 1960, 340, our italics) The emphasized sections of the preceding quotation are the inferential steps to which Mach objected. Specifically, he raised two criticisms against Neumann's view. First, in thinking away the universe, Neumann has made a "meaningless assumption." Second, neither Neumann nor anyone else is competent to say what would happen if the universe were not present. In particular, Mach challenges Neumann's contention that the centrifugal forces which produce the oblateness are a result of the rotation of the body about its axis simpliciter with no regard for the presence or absence of the rest of the universe. When experimenting in thought, it is permissible to modify unimportant circumstances in order to bring out new features in a given case; but it is not to be antecedently assumed that the universe is without influence on the phenomenon here in question (Mach, 1960, 341). In fact, Mach was of the opinion that the universe had a great deal to do with the phenomenon in question. Recall that the problem that Newton had in considering the fixed stars (or the distant mass of the universe) as having an effect on producing inertial effects in the bucket was that he could not see how the influence of the stars could be transmitted to the water in the bucket. The only other relative motion was that of the water with respect to bucket but no noticeable inertial effects are produced by that relative rotation. But, Mach argues No one is competent to say how the experiment would turn out if the sides of the vessel increased in thickness and mass till they were ultimately several... [miles]... thick. The one experiment only lies before us, and our business is to bring it into accord with the other facts known to us, and not with the arbitrary fictions of our imagination (Mach, 1960, 284). For all we know, Mach is suggesting, were the walls of the bucket made that thick then the inertial effects would be produced when the water and the bucket were rotating relatively to one another.
Even if the walls of Newton's bucket were made several miles thick, the mass of the bucket although near the water would be miniscule compared to the total mass of the universe. Perhaps, Mach thought, there might be some compensating factor for the immense distances which was producing inertial effects in the water when it rotated with respect to the fixed stars. As a matter of fact, Mach set for himself the task of determining what that law might be, much as Newton had set for himself the task of determining what the law of gravity was. Newton's worry about how to causally account for the effect would not, of course, worry Mach who saw the task as science of science as purely descriptive and not explanatory.
The net effect of Mach's critique is to produce a full relational theory of space, time and motion. The one class of motions that had resisted relational treatment had been accelerations which appeared, even to Leibniz, to produce Absolute effects. But, for Mach, accelerations (and, therefore, rotations) turn out to be purely relational as well. That objects accelerating with respect to the fixed stars (by rotating or otherwise) produce dynamical or inertial effects is the given. The mistake, Mach urged, was to think that such effects could be used to determine which object, e.g., the Earth or the fixed stars (considered as the rest of the universe), was really accelerating. The observed effects are produced when such objects rotate relatively to one another. It is idle speculation to try to infer anything absolute from the data. The scientific job that lies ahead is to try to determine what the force law is which correctly describes the production of these inertial effects. As the nature or cause of inertia, Mach would surely echo Newton: Hypotheses non fingo.
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