 |
| Space in Physical Theories |
Space here means the space of the science of mechanics, which encompasses planetary and celestial (i.e., “outer”) space, but is presupposed by the motion—spatial change—of any bodies whatsoever, from the tiniest particles through human-sized bodies to the whole universe. The investigation of space has been perhaps the most fruitful interaction between physics and philosophy.
Physics endows space with specific properties playing a crucial role in determining the motions of bodies, but, despite being omnipresent, space (prerelativistically) is frustratingly inert—not having even the indirect causal effects of subatomic particles, say. Thus physics ascribes substantive properties to space on the basis of indirect evidence, allowing metaphysical bias to influence understanding, and calling (in part) for philosophical clarification.
One of the main strands of this clarification involves the “absolute-relative” debate. In fact a number of (interconnected) debates go under this title, of which two are focused on in the historical development of mechanics: Of all the motions a body has (relative to different frames of reference), which if any are privileged or “absolute”?
 |
| subatomic particles |
Are such absolute motions determined by the motions of bodies relative to one another, or by motions with respect to space itself: is space a real, substantial entity in addition to bodies? (A third important strand: Are all spatial properties extrinsic—that is, “relative”—or intrinsic?)
Space in Aristotelian Physics
In the European tradition, Eudoxus’s (408–355 BCE) account of the motions of the heavens—which was later significantly extended by Ptolemy (c. 85–165 CE)—is probably the first “physical theory” (in anything like a modern sense) in which space plays a significant role.
According to this theory the Earth is at rest at the center of the universe, surrounded by a series of concentric spheres, interconnected along their axes. The moon, sun, planets, and totality of fixed stars are each located on their own sphere, with the stars farthest out.
 |
| Ptolemy |
The daily apparent motions of the heavens are explained by a daily rotation of the stellar sphere, which carries all the other spheres with it; the “wanderings” of the other bodies through the fixed stars are explained by the additional, slower rotations of the other spheres about their axes. Aristotle (384–322 BCE) provided the philosophical interpretation of this system: a finite, spherical universe with an absolute center (which Aristotle suggests is determined by its position relative to the circumference).
Thus bodies do have an absolute motion, namely relative to the center, which is essential for Aristotle’s mechanics: heavy bodies move naturally toward the center, light bodies away, and the heavenly element, ether, around circularly. (Note that Aristotle denied the existence of space separate from body: no vacuum and no pure extension.)
Although astronomers often took this model instrumentally, Aristotle’s account was the context of debate over the nature of space until the eighteenth century, even after Nicolaus Copernicus (1473–1543) proposed that the earth moved around the sun. Questions seen as important during this period that had bearing on later developments include the possibility of the vacuum and whether God could move the entire universe.
 |
| Aristotle |
Space in Cartesian Physics
In the Early Modern period, René Descartes (1596–1650) is a logical place to start despite numerous important predecessors, especially Galileo Galilei (1564–1642), because of his influence on both physics and its philosophy.
Notable contributions include the development of mechanical explanation, conservation laws, and, with Pierre Gassendi (1592–1655), the correct “law of inertia”: Bodies experiencing no net forces move at constant speeds along straight paths.
According to Descartes, because matter and space have the same essence—“extension”—they are one and the same (Plato’s Timaeus describes a similar view). This identification poses a problem: As a body moves, so does the matter that composes it and hence the space it occupies, but if it does not change with respect to space then it does not move!
 |
| Space in Cartesian Physics |
In his Principles of Philosophy (1644),Descartes’s first solution is to relativize to reference bodies (selected arbitrarily): In thought people identify a relatively moving piece of matter=space as the same body, while they identify as the same spatial region those different pieces of matter=space that bear some fixed relations to the reference bodies.
However, in addition to this “ordinary” concept of motion, Descartes defines motion “properly speaking” as displacement of a body from the bodies in contact with it (in accord with Aristotle’s Physics, Book IV Chapter 4). Why there are dual accounts is a subject of dispute.
According to one interpretation, Descartes took relative motion to be fundamental, but sought to avoid the heretical denial of the earth’s rest; because he wrote only a decade after Galileo was condemned (1633), such concern was real.
 |
| Galileo Galilei |
Descartes claimed that the universe was a plenum in constant agitation, and explained the motions of the planets (including the earth) by postulating a giant vortex of fine matter carrying them around the sun, like leaves in a whirlpool. Hence the earth is in relative motion around the sun and roughly at rest with its surroundings, and so both Copernicus and Aristotle were correct—in the “ordinary” and “proper” senses, respectively.
The second interpretation claims that Descartes took proper motion more seriously, as the correct, “true” sense of motion in physics; in particular, his laws of collision are blatantly contradictory if taken to concern relative motion, but not if they concern proper motion, because it is “absolute” in the sense of being privileged over all other relative motions. (As Christiaan Huygens [1629–1695] realized, Descartes should have changed the laws to make them consistently describe relative motions, not relied on his absolute notion.)
Space in Newtonian Physics
Although Descartes’s views were influential, Isaac Newton’s (1643–1727) physics and philosophy (arguably his epistemology as well as his metaphysics) were infinitely more successful. In his Principia (1687) and in an unpublished essay, De Gravitatione (undated), he attacks Descartes’s views concerning space and motion and lays out his own.
 |
| Space in Newtonian Physics |
Newton claims that space is three-dimensional and Euclidean, persists through time, and is neither a substance such as mind or matter (because it has no causal powers—the law of inertia holds because space does not act on bodies) nor a property of substances (because in a vacuum there is space but no substance): Space is outside of the categories of traditional metaphysics.
He takes it to be a pseudosubstance, causally inert, but metaphysically necessary for the existence of anything, including God, because everything exists somewhere. Commentators often stretch metaphysical categories, and count Newton’s “absolute space” as a nonmaterial, nonmental substance, regions of which may be occupied by other substances: they rather inaccurately ascribe “substantivalism” to Newton.
Newton famously argues against the Cartesian view of space using the example of a bucket of water, though it is only one of a series of arguments he gives. If bucket and water, initially at rest, are set spinning about their axis, initially the water will remain at rest, and hence be in motion relative to its contiguous surroundings (the side of the bucket); the water will be rotating properly speaking. Later, friction with the sides of the bucket will have set the water rotating at the same rate as the bucket, and so it will be at “proper” rest, according to Descartes.
 |
| Isaac Newton |
In the first instance, because it is not yet rotating, the surface of the water will be flat, whereas in the second it will be concave (just like tea stirred in a cup). By Descartes’s and Newton’s (and most of their contemporaries’) explicit principles, it follows that only in the second case is the water “truly,” physically rotating.
And so in the experiment the water has physical motion if and only if it has no motion properly speaking. Cartesian “ordinary” motion fares no better: The water spins at a unique height in the bucket, indicating a unique rate of rotation, while it moves at different rates relative to different reference bodies.
Newton concludes that because true motion is neither kind of Cartesian motion, it must be the only other option on the table: motion relative to absolute space (which he calls “absolute motion,” though it was seen above that proper motion too is “absolute” in the sense of being privileged).
 |
| Christiaan Huygens |
Leibniz’s Relationist Response
Gottfried Leibniz’s (1646–1716) position is complex: He argued persuasively against substantivalism, but was motivated by idiosyncratic metaphysics; and he gave a sophisticated account of “relationism”—space is not a substance, and all spatial properties and motions are determined by relations—but it conflicted with his theory of collisions (the so-called “Newtonian” or “classical” theory of elastic collisions).
At the end of his life Leibniz arguably held:
- that every body possesses a unique quantity of “living force” or “vis viva”, measured by mass x speed2 (basically kinetic energy), and hence a unique speed;
- that living force and pure Cartesian extension are “form” and “substance” in an updated Aristotelian metaphysics;
- that force entails the laws of mechanics (living force and momentum are conserved in elastic collisions);
- to avoid being an occult power, the actual force must have no detectable effects, so the laws must satisfy the “equivalence of hypotheses” and hold in all frames (Leibniz was mistaken to think this was true of his laws); and
- that space is not only merely relative, so bodies and their relations exhaust all spatial facts, but also ideal (not a “well-founded phenomenon” in his terms), arguing in part that because no two things can literally stand in the same relation to a third, only a mental identification allows two things to stand in the same relative place one after the other.
 |
| Gottfried Leibniz |
Thus Leibniz opposes both Descartes and Newton: Against Descartes he rejects the claims that space is matter (space is ideal, whereas matter is well-founded) and that “proper” motion is privileged (Descartes also held that vis viva was mass ¥ speed); against Newton he rejects the view that space is absolute.
In his Correspondence (1715–1716) with Samuel Clarke (1675–1729), Leibniz gives relativity arguments against Newton. He argues that because two systems differing only in their absolute positions or velocities cannot be told apart, they must not differ at all: that absolute locations and velocities, and absolute space itself, are unreal.
While this argument impressed later empiricists, Leibniz himself argued from the theological “principle of sufficient reason.” Leibniz claims that his relationism avoids Newton’s arguments, but this is highly doubtful: He only hints at (in his Specimen of Dynamics, 1695) a relational account of rotating bodies (such as the bucket), and fails to see that Newton’s arguments disprove the relativity of his own mechanics.
 |
| Samuel Clarke |
Modern Arguments
In his Science of Mechanics (1893) Ernst Mach (1836–1916) criticized Newton for making a non sequitur: Rotation relative to the bucket fails to explain the curvature of the water, but it does not follow that the water must be rotating relative to absolute space—could not the curvature show motion relative to some other body? Mach’s reading fails to understand how Newton refuted Descartes, and ignores his attack on relative (“ordinary”) motion, but asks a reasonable question as a non-Cartesian relationist.
Mach proposed that sufficiently massive bodies act to cause distant bodies to move in constant, linear relative motion unless acted on by forces: in particular, that the fixed stars determine which motions are inertial, not absolute space. (Newton considered this idea, but dismissed it because it involved action at a distance—a questionable argument given his theory of gravity.)
Mach’s arguments were influential on contemporary physicists who were developing the idea of an “inertial frame”: a frame in which Newton’s laws hold, and in particular in which bodies experiencing no net forces move inertially.
 |
| Modern Arguments |
In practice, physicists have since taken inertial frames to be sufficient, and viewed absolute space (if at all) as an early formulation of that idea, though whether this approach amounts to relationism is debatable.
Mach was also a hero of empiricist philosophers, however; beginning in the 1960s, a reappraisal of Newton led to a defense of absolute space (simultaneous with a general philosophical turn from strict empiricism toward realism).
In the late twentieth and early twenty-first centuries,Newton is often taken to argue abductively that his theory gives the best explanation of the bucket; better than Descartes’s, and by extension, because he offered no real theory, better than Mach’s.
 |
| Ernst Mach |
It is (arguably) no non sequitur to infer that motion is absolute because Newtonian mechanics in absolute space explains better than any relational theory. Note that although Newton might have endorsed this argument as a response to Mach, it is weaker than Newton’s demonstration of the inconsistency of Descartes.
A major innovation has been to transfer the arguments into the context of (nonrelativistic) spacetime. One can then distinguish (a) “Newtonian spacetime” with a preferred standard of rest—geometrically speaking, a “rigging” that picks out stationary trajectories—from (b) “Galilean spacetime” with only a preferred standard of constant motion—no rigging but an “affine connection” that picks out nonaccelerated, inertial trajectories.
In (a) both velocity and acceleration are well defined and hence “absolute,” but in (b) only acceleration is, avoiding (part of) Leibniz’s relativity argument. Thus, plausibly, Newtonian mechanics—which distinguishes different states of absolute acceleration but not velocity—in Galilean spacetime offers the best mechanical explanations.
 |
| Newtonian spacetime |
Modern substantivalists infer first from the need for well-defined accelerations in Newtonian mechanics, to spacetimes with “absolute structures” such as a connection, and then further to the substantiality of those spacetimes, particularly of Galilean spacetime. Several relationists have responded by arguing that acceleration can be understood without substantial spacetime: that Newtonian mechanics has a relational interpretation.
Other relationists have attempted to construct a theory that does explain this as well as Newton: Most attempts rely on the fact that if it is postulated that the total angular momentum of a system (such as the whole universe) is zero, then Newtonian mechanics determines a well-defined evolution for the relative state of the system.
Relativity
How does the absolute-relative debate change in relativity theory? Consider the special theory of relativity (henceforth “STR”). First distinguish “relativity” from relationism. Broadly speaking, a theory is relativistic if it admits no unmeasureable quantities.
 |
| Relativity |
Then Newtonian mechanics in absolute space or Newtonian spacetime is not relativistic, because it admits absolute velocity, whereas Newtonian mechanics in Galilean spacetime and electromagnetism in Minkowski spacetime are relativistic.
The relativity of STR is thus of a specific kind: So that no body can be said to be at rest, all must agree on the speed of light. Thus the difference between the relativity of STR and Newtonian mechanics lies in whether one takes account of electromagnetic phenomena, a difference that has no immediate bearing on whether space is absolute or relative.
Indeed, Minkowski spacetime has an affine connection, so acceleration is as absolute as in Galilean spacetime, thus the same question of whether a connection provides evidence for the substantiality of spacetime arises.
 |
| Minkowski spacetime |
In the general theory of relativity (“GTR”) spacetime is not a fixed background but is acted on by matter and has more robust causal powers. For example, rapidly rotating bodies (e.g., a black hole) can produce gravitational waves with the power to stretch and squeeze bodies as they pass through them.
The causal powers of spacetime are a serious problem for relational interpretations of GTR: If a gravitational wave knocks a person down it would be odd to say that person’s body merely moved relative to the ground “as if ” a wave were present.
Thus only a strictly relational theory in agreement with the evidence for GTR will suffice for the relationist. However, the causal nature of the spacetime of GTR makes it metaphysically different from Newton’s, and so hardly vindicates prerelativistic substantivalism either.
 |
| theory of relativity |
GTR is sometimes mistakenly claimed to be relational. First, the action of matter on space means that the affine connection, and hence inertial motion, is dependent on the distribution of matter in distant regions (in the causal past), as Mach claimed.
However, the distribution does not determine inertial motion, because the connection also depends on the geometry of spacetime in the causal past: for instance, even if there is no matter at all, the connection in a region is not fixed.
Thus Mach’s relationism is not vindicated by GTR. Second, the theory is “generally covariant”: Its equations take the same form in every frame. Thus, unlike Newtonian mechanics, one cannot define absolute acceleration as acceleration in some privileged class of frames, and any relative frame will do for formulating the theory.
 |
| inertial motion |
But these points do not settle the absolute-relative debate: GTR has an affine connection, and every body still has an absolute acceleration. Further, Newtonian mechanics can be formulated generally covariantly too, so arguably general covariance shows nothing.
It is true, however, that, unlike Newtonian mechanics and STR, the dynamic nature of spacetime in GTR makes general covariance necessary: Intuitively, how can spacetime have privileged frames if spacetime is not independently given?
More or less equivalently, GTR is “diffeomorphism invariant”: If all the dynamical quantities in a model (the distribution of matter and the geometry of spacetime) are continuously rearranged over the points of spacetime then the result is still a model. (Spacetime theories with static geometries are also “diffeomorphism invariant” in the weaker sense that the diffeomorphism of a model is also a model if the dynamical quantities and the nondynamical geometry are permuted.)
 |
| diffeomorphism invariant |
Diffeomorphism invariance drives the “hole argument” against substantivalism, because it entails a kind of indeterminism if the spacetime of GTR has a substantival interpretation.
That antisubstantivalists and some substantivalists avoid such indeterminism by claiming that distinct diffeomorphic models represent the same physical world, which is to say that the physical content of the models is captured by the relation between matter and the geometry of space, because this is what the models have in common.
 |
| geometry of space |
If so, GTR is a theory of the relations between dynamical quantities, which is what the prerelativistic relationists sought, though in terms of different dynamic quantities, namely relative distances. Thus it can be argued that GTR is as sympathetic to prerelativistic relationism as to prerelativistic substantivalism (note that physicists tend to emphasize the relational nature of GTR far more than philosophers).
Quantum Gravity
What of the absolute-relative debate in a sought-after quantum theory of gravity, such as string theory? First, the interpretation of diffeomorphism invariance is important for certain approaches to quantizing general relativity, which some argue gives physical import to the philosophical debate concerning the hole argument.
 |
| Quantum Gravity |
Second, space is likely to be an “effective” notion, which does not appear as a fundamental element of the theory, but only phenomenologically in particular circumstances. If so, then neither relationism nor substantivalism will be correct interpretations of quantum gravity, and the debate may seem doomed. However, quantum gravity could shed new light on the matter in the following sense.
One could ask what quantities count as observables in the effective context: If the theory can be given completely in terms of observables relating bodies then effective space could be said to be relational, whereas if the theory contains observables concerning points of space, then it seems that effective space is substantial.
 |
| fundamental element |
