## Space

**-**

## Space

# SPACE

When men began to think about the nature of "space," they thought of it as an all-pervading ether or as some sort of container. Since a thing can move from one part of space to another, it seemed that there was something, a place or a part of space, to be distinguished from the material objects that occupy space. For this reason places might be thought of as different parts of a very subtle jellylike medium within which material bodies are located.

## History of the Concept of Space

Some of the Pythagoreans seem to have identified empty space with air. For more special metaphysical reasons Parmenides and Melissus also denied that there could be truly empty space. They thought that empty space would be nothing at all, and it seemed to them a contradiction to assert that a nothing could exist. On the other hand, there seems to be something wrong with treating space as though it were a material, which, however subtle, would still itself have to be *in* space. Democritus and the atomists clearly distinguished between the atoms and the void that separated them. However, the temptation to think of space as a material entity persisted, and Lucretius, who held that space was infinite, nevertheless wrote of space as though it were a container. Yet he seems to have been clear on the fact that space is unlike a receptacle in that it is a pure void. Since material bodies, in his view, consist of atoms, there must be chinks of empty space even between the atoms in what appear to be continuous bodies.

Plato's views on space have to be gotten mainly from the obscure metaphors of the *Timaeus* ; he, too, appears to have thought of space as a receptacle and of the matter in this receptacle as itself mere empty space, limited by geometrical surfaces. If so, he anticipated the view of René Descartes, where the problem arises of how empty space can be distinguished from nonempty space. Even if, like Lucretius and other atomists, we make a distinction between the atoms and the void, what is this void or empty space? Is it a thing or not a thing?

### aristotle

Aristotle tried to dodge the difficulty by treating the concept of space in terms of place, which he defined as the adjacent boundary of the containing body. For two things to interchange places *exactly,* they would have to be identical in volume and shape. Consider two exactly similar apples that are interchanged in this way. The *places* are not interchanged; rather, the first apple is now at the very same place at which the second apple was and vice versa. We seem, therefore, to be back at the notion of space as a substratum or ether, but it is probable that Aristotle was trying to avoid this and that he meant to define place by reference to the cosmos as a whole. Aristotle thought of the cosmos as a system of concentric spheres, and the outermost sphere of the cosmos would, on his view, define all other places in relation to itself. In the Aristotelian cosmology each of the various "elements" tends toward its own place. Thus, heavy bodies tend toward the center of Earth, and fire goes away from it. This is not, however, for any other reason than that the center of Earth happens to be the center of the universe; the places toward which the elements tend are independent of what particular bodies occupy what places. In more recent times we view these as two different and seemingly irreconcilable ways of thought—the notions of space as a stuff and of space as a system of relations between bodies.

### descartes and leibniz

Descartes held that the essence of matter is extension, and so, on his view, space and stuff are identical, for if the essence of matter is to be extended, then any volume of space must be a portion of matter, and there can be no such thing as a vacuum. This raises the question of how we can distinguish one material object (in the ordinary sense of these words) from another. How, on Descartes's view, can we elucidate such a statement as that one bit of matter has moved relative to another one? In what sense, if matter just *is* extension, can one part of space be more densely occupied by matter than another? Descartes considered these objections but lacked the mathematical concepts necessary to answer them satisfactorily. We shall see that a reply to these objections can be made by denying that space is the same everywhere, and this can be done by introducing the Riemannian concept of a space of variable curvature.

As against Descartes, Gottfried Wilhelm Leibniz held a relational theory of space, whereby space is in no sense a stuff or substance but is merely a system of relations in which indivisible substances, or "monads," stand to one another. Few philosophers have followed Leibniz in his theory of monads, but in a slightly different form the relational theory of space has continued to rival the Cartesian, or "absolute," theory. The issue between the two theories has by no means been decisively settled, at least if we consider not space but space-time. It is still doubtful whether the general theory of relativity can be stated in such a way that it does not require absolute space-time.

### kant

In his *Prolegomena,* Immanuel Kant produced a curious argument in favor of an absolute theory of space. Suppose that the universe consisted of only one human hand. Would it be a left hand or a right hand? According to Kant it must be one or the other, yet if the relational theory is correct it cannot be either. The relations between the parts of a left hand are exactly the same as those between corresponding parts of a right hand, so if there were nothing else to introduce an asymmetry, there could be no distinction between the case of a universe consisting only of a left hand and that of a universe consisting only of a right hand. Kant, however, begged the question; in order to define "left" and "right" we need the notions of clockwise and counterclockwise rotations or of the bodily asymmetry which is expressed by saying that one's heart is on the left side of one's body. If there were only one hand in the world, there would be no way of applying such a concept as left or clockwise. The relationist could therefore quite consistently reply to Kant that if there were only one thing in the universe, a human hand, it could not *meaningfully* be described as either a right one or a left one. (The discovery in physics that parity is not conserved suggests that the universe is not symmetrical with respect to mirror reflection, so there is probably, in tact, something significant in nature analogous to the difference between a left and a right hand.)

Later, in his *Critique of Pure Reason,* Kant argued against both a naive absolute theory of space and a relational view. He held that space is something merely subjective (or "phenomenal") wherein in thought we arrange nonspatial "things-in-themselves." He was led to this view partly by the thought that certain antinomies or contradictions are unavoidable as long as we think of space and time as objectively real. However, since the work of such mathematicians as Karl Theodor Wilhelm Weierstrass, Augustin-Louis Cauchy, Julius Wilhelm Richard Dedekind, and Georg Cantor, we possess concepts of the infinite which should enable us to deal with Kant's antinomies and, indeed, also to resolve the much earlier, yet more subtle, paradoxes of Zeno of Elea.

## Newton's Conception of Space

Isaac Newton held absolute theories of space and time—metaphysical views that are strictly irrelevant to his dynamical theory. What is important in Newtonian dynamics is not the notion of absolute space but that of an inertial system. Consider a system of particles acting on one another with certain forces, such as those of gravitational or electrostatic attraction, together with a system of coordinate axes. This is called an inertial system if the various accelerations of the particles can be resolved in such a way that they all occur in pairs whose members are equal and lie in opposite directions in the same straight line. Finding an inertial system thus comes down to finding the right set of coordinate axes. This notion of an inertial system, not the metaphysical notion of absolute space, is what is essential in Newtonian dynamics, and as Ernst Mach and others were able to show, we can analyze the notion of an inertial system from the point of view of a relational theory of space. Psychologically, no doubt, it was convenient for Newton to think of inertial axes as though they were embedded in some sort of ethereal jelly—absolute space. Nevertheless, much of the charm of this vanishes when we reflect that, as Newton well knew, any system of axes that is moving with uniform velocity relative to some inertial system is also an inertial system. There is reason to suppose, however, that in postulating absolute space Newton may have been partly influenced by theological considerations that go back to Henry More and, through More, to cabalistic doctrines.

We can remove the metaphysical trappings with which Newton clothed his idea of an inertial system if we consider how in mechanics we determine such a system. But even before we consider how we can define an inertial system of axes, it is interesting to consider how it is possible for us to define any system of axes and spatial positions at all. As Émile Borel has remarked, how hard it would be for a fish, however intelligent, which never perceived the shore or the bottom of the sea to develop a system of geometrical concepts. The fish might perceive other fish in the shoal, for example, but the mutual spatial relations of these would be continually shifting in a haphazard manner. It is obviously of great assistance to us to live on the surface of an earth that, if not quite rigid, is rigid to a first order of approximation. Geometry arose after a system of land surveying had been developed by the Egyptians, who every year needed to survey the land boundaries obliterated by the flooding of the Nile. That such systems of surveying were possible depended on certain physical facts, such as the properties of matter (the nonextensibility of chains, for example) and the rectilinear propagation of light. They also depended on certain geodetic facts, such as that the tides, which affect even the solid crust of Earth, were negligible. The snags that arise when we go beyond a certain order of approximation were unknown to the Egyptians, who were therefore able to get started in a fairly simple way.

It might be tempting to say that it was fortunate that the Egyptians were unaware of these snags, but of course in their rudimentary state of knowledge they could not have ascertained these awkward facts anyway. When, however, we consider geodetic measurements over a wide area of the globe we need to be more sophisticated. For example, the exact shape of Earth, which is not quite spherical, needs to be taken into account. Moreover, in determining the relative positions of points that are far apart from one another it is useful to make observations of the heavenly bodies as seen simultaneously from the different points. This involves us at once in chronometry. There is thus a continual feedback from physics and astronomy. Increasingly accurate geodetic measurements result in more accurate astronomy and physics, and more accurate astronomy and physics result in a more accurate geodesy.

Such a geodetic system of references is, however, by no means an inertial one. An inertial system is one in which there are no accelerations of the heavenly bodies except those which can be accounted for by the mutual gravitational attractions of these bodies. It follows, therefore, that the directions of the fixed stars must not be rotating with respect to these axes. In principle we should be able to determine a set of inertial axes from dynamical considerations, even if we lived in a dense cloud, as on Venus, and were unaware of the existence of the fixed stars. This may have influenced Newton to think of space as absolute. However, Newton was not on Venus, and he could see the fixed stars. It is therefore a little surprising that he did not take the less metaphysical course of supposing an inertial system to be determined by the general distribution of matter in the universe. This was the line taken in the nineteenth century by Mach and is referred to (after Albert Einstein) as Mach's principle. It is still a controversial issue in cosmology and general relativity.

Mach's principle clearly invites, though it does not compel, a relational theory of space, such as Mach held. The origin of the axes of an inertial system in Newtonian mechanics was naturally taken to be the center of gravity of the solar system, which is nearly, but not quite, at the center of the sun. In fact, it is continually changing its position with reference to the center of the sun. Now that the rotation of the galaxy has been discovered, we have to consider the sun as moving around a distant center. We shall here neglect the possibility that our galaxy is accelerating relative to other galaxies. In any case, once we pass to cosmological considerations on this scale we need to abandon Newtonian theory in favor of the general theory of relativity.

The philosophical significance of the foregoing discussion is as follows: When we look to see how inertial axes are in fact determined we find no need to suppose any absolute space. Because such a space would be unobservable, it could never be of assistance in defining a set of inertial axes. On the other hand, the complexities in the determination of inertial axes are such that it is perhaps psychologically comforting to think of inertial axes, or rather some one preferred set of such axes, as embedded in an absolute space. But Newton could equally have taken up the position, later adopted by Mach, that inertial systems are determined not by absolute space but by the large-scale distribution of matter in the universe.

## Space and Time in the Special Theory of Relativity

We have already noticed the dependence of space measurements on time measurements which sometimes obtains in geodesy. This situation is accentuated in astronomy because of the finite velocity of light. In order to determine the position of a heavenly body we have to make allowance for the fact that we see it in the position it was in some time ago. For example, an observation of a star that is ten light-years away is the observation of it in its position years ago. Indeed, it was the discrepancy between the predicted and observed times at which eclipses of the satellites of Jupiter should occur that led Olaus Rømer to assign a finite, and approximately correct, value to the velocity of light. The correction of position and time on account of the finite velocity of light presupposes in any particular case our knowing what this velocity is, relative to Earth. This would seem to depend not only on the velocity of light relative to absolute space (or to some preferred set of inertial axes) but also on Earth's velocity relative to absolute space (or to the preferred set of inertial axes). The experiment of Albert Abraham Michelson and Edward Williams Morley showed, however, that the velocity of light relative to an observer is independent of the velocity of the observer. This led to the special theory of relativity, which brings space and time into intimate relation with one another. For present purposes it is necessary to recall only that according to the special theory of relativity events that are simultaneous with reference to one inertial set of axes are not simultaneous with reference to another inertial frame. The total set of point-instants can be arranged in a four-dimensional space-time. Observers in different inertial frames will partition this four-dimensional space-time into a "space" and a "time," but they will do so in different ways.

Before proceeding further it is necessary to clear up a certain ambiguity in the word *space.* So far in this entry space has been thought of as a continuant. In this sense of the word *space* it is possible for things to continue to occupy space and to move from one point of space to another and for regions of space to begin or cease to be occupied or to stay occupied or unoccupied. Here space is something that endures through time. On the other hand, there is a different, timeless use of the word *space.* In solid geometry a three-dimensional space is thought of as timeless. Thus, if a geometer said that a sphere had changed into a cube, he would no longer be thinking within the conceptual scheme of solid geometry. In geometry all verbs must be tenseless. In this tenseless way let us conceive of a four-dimensional space-time, three of whose dimensions correspond roughly to the space of our ordinary thought whereas the other corresponds to what we ordinarily call time. What we commonly think of as the state of space at an instant of time is a three-dimensional cross section of this four-dimensional space-time.

Taking one second to be equivalent to 186,300 miles, which is the distance light travels in that time, any physical object, such as a man or a star, would be rather like a four-dimensional worm—its length in a timelike direction would be very much greater than its spacelike cross section. Thinking in terms of space-time, then, two stars that are in uniform velocity with respect to each other and also with respect to our frame of reference will appear as two straight worms, each at a small angle to the other. An observer on either star will regard himself as at rest, so he will take his own world line—the line in space-time along which his star lies—as the time axis. He will take his space axes as (in a certain sense) perpendicular to the time axis. It follows that observers on stars that move relative to one another will slice space-time into spacelike cross sections at different angles. This makes the relativity of simultaneity look very plausible and no longer paradoxical. As Hermann Minkowski observed, the relativity of simultaneity could almost have been predicted from considerations of mathematical elegance even before the experimental observations that led to the special theory of relativity. Indeed, Minkowski showed that the Lorentz transformations of the theory of relativity can be understood as simply a rotation of axes in space-time. (In trying to picture such a rotation of axes it is important to remember that Minkowski space-time is not Euclidean but semi-Euclidean.) In Minkowski's words, "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." We must not forget that space-time is a space in the mathematical sense of the word *space,* not in the sense in which space is a continuant. Thus, certain objectionable locutions are often used in popular expositions. For example, we sometimes hear it said that a light signal is propagated from one part of space-time to another. The correct way to put the matter is to say that the light signal *lies* (tenselessly) along a line between these two parts of space-time. Space-time is not a continuant and is not susceptible of change or of staying the same.

## Euclidean and Non-Euclidean Space

Geometry, as we observed earlier, developed out of experiences of surveying, such as those of the ancient Egyptians. The assumptions underlying the surveying operations were codified by Greek mathematicians, whose interests were mainly theoretical. This codification was developed by Euclid in the form of an axiomatic system. Euclid's presentation of geometry shows a high degree of sophistication, though it falls considerably short of modern standards of rigor. Euclid's geometry was a metrical one. There are, of course, geometries that are more abstract than metrical geometry. The most abstract of all is topology, which deals with those properties of a space that remain unchanged when the space is distorted, as by stretching. Thus, from the point of view of topology a sphere, an ellipsoid, and a parallepiped are identical with one another and are different from a torus. Metrical geometry uses a bigger battery of concepts—not only such notions as those of betweenness and of being longer than (which itself goes beyond topology) but also those of being, say, twice or three and a half times as long as.

Euclid regarded one of his axioms as more doubtful than the others. This is the axiom that is equivalent to the so-called axiom of parallels. It will be more convenient to discuss the axiom of parallels than Euclid's own axiom. The axiom of parallels states that if *C* is a point not on an infinite straight line *AB,* then there is one and only one straight line through *C* and in the plane of *AB* that does not intersect *AB.* Geometers made many efforts to deduce the axiom of parallels from the other, more evident ones. In the seventeenth and eighteenth centuries Gerolamo Saccheri and J. H. Lambert each tried to prove the axiom by means of a reductio ad absurdum proof. By assuming the falsity of the axiom of parallels they hoped to derive a contradiction. They did not succeed; in fact, Saccheri and Lambert proved a number of perfectly valid theorems of non-Euclidean geometry, though they were not bold enough to assert that this was what they were doing.

János Bolyai and N. I. Lobachevski replaced the axiom of parallels with the postulate that *more than one* parallel can be drawn. The type of geometry that results is called hyperbolic. Another way to deny the axiom of parallels is to say that *no* parallel can be drawn. This yields elliptic geometry. (Some adjustments have to be made in the other axioms. For instance, straight lines become finite, and two points do not necessarily determine a straight line.) It is easy to prove (by giving a non-Euclidean geometry an interpretation within Euclidean geometry) that both hyperbolic and elliptic geometries are consistent if Euclidean geometry is. (And all can easily be shown to be consistent if the theory of the real-number continuum is.) A priori, therefore, there is nothing objectionable about non-Euclidean geometries. Unfortunately, many philosophers followed Kant in supposing that they had an intuition that space was Euclidean, and mathematicians had to free themselves from this conservative climate of opinion.

The question then arose whether our actual space is Euclidean or non-Euclidean. In order to give sense to this question we must give a physical interpretation to our geometric notions, such as that of a straight line. One way of defining a straight line is as follows: Suppose that rigid bodies *A, B, C* have surfaces *S _{A} , S_{B} , S_{C} ,* such that when

*A*is applied to

*B,*then

*S*and

_{A}*S*fit; when

_{B}*B*is applied to

*C,*then

*S*and

_{B}*S*fit; and when

_{C}*C*is applied to

*A,*then

*S*and

_{C}*S*fit. Suppose also that

_{A}*S*can all be slid and twisted over one another—that is, that they are not like cogged gears, for example. Then

_{A}, S_{B}, S_{C}*S*are all by definition plane surfaces. The intersection of two planes is a straight line. (In the above we have used the notion of a rigid body, but this can easily be defined without circularity.) With the above definition of a straight line and the like we can make measurements to tell whether the angles of a triangle add up to two right angles. If they make more than two right angles, space is elliptic; if less than two right angles, space is hyperbolic; and if exactly two right angles, space is Euclidean. However, such experiments could not determine the question to any high degree of accuracy. All that this method shows is that, as every schoolchild knows, physical space is

_{A}, S_{B}, S_{C}*approximately*Euclidean.

To make measurements that could settle the question to any high degree of accuracy we should have to make them on an astronomical scale. On this scale, however, it is not physically possible to define straight lines by means of the application of rigid bodies to one another. An obvious suggestion is that we should define a straight line as the path of a light ray in empty space. One test of the geometry of space might then come from observations of stellar parallax. On the assumption that space is Euclidean, the directions of a not very distant star observed from two diametrically opposite points on Earth's journey round the sun will be at a small but observable angle. If space is hyperbolic, this angle, which is called the parallax, will be somewhat greater. If space is elliptic, the parallax will be less or even negative. If we knew the distance of the star, we could compare the observed parallax with the theoretical parallax, on various assumptions about the geometry. But we cannot know the distances of the stars except from parallax measurements. However, if space were markedly non-Euclidean, we might get some hint of this because the distribution of stars in space, calculated from parallax observations on Euclidean assumptions, would be an improbable one. Indeed, at the beginning of the twentieth century Karl Schwarzschild made a statistical analysis of parallaxes of stars and was able to assign an upper limit to the extent to which physical space deviates from the Euclidean.

A good indication that space, on the scale of the solar system at least, is very nearly Euclidean is the fact that geometrical calculations based on Euclidean assumptions are used to make those predictions of the positions of the planets that have so strongly confirmed Newtonian mechanics. This consideration points an important moral, which is that it is impossible to test geometry apart from physics; we must regard geometry as a part of physics. In 1903, Jules Henri Poincaré remarked that Euclidean geometry would never be given up no matter what the observational evidence was; he thought that the greater simplicity of Euclidean, as against non-Euclidean, geometry would ensure our always adopting some physical hypothesis, such as that light does not always travel in straight lines, to account for our observations. We shall not consider whether—and if so, in what sense—non-Euclidean geometry is necessarily less simple than Euclidean geometry. Let us concede this point to Poincaré. What he failed to notice was that the greater simplicity of the geometry might be bought at the expense of the greater complexity of the physics. The total theory, geometry plus physics, might be made more simple even though the geometrical part of it was more complicated. It is ironical that not many years after Poincaré made his remark about the relations between geometry and physics he was proved wrong by the adoption of Einstein's general theory of relativity, in which overall theoretical simplicity is achieved by means of a rather complicated space-time geometry.

In three-dimensional Euclidean space let us have three mutually perpendicular axes, *Ox* _{1}, *Ox* _{2}, *Ox* _{3}. Let *P* be the point with coordinates (*x* _{1} , *x* _{2} , *x* _{3}), and let *Q* be a nearby point with coordinates (*x* _{1} + *dx* _{1}, *x* _{2} + *dx* _{2} , *x* _{3} + *dx* _{3}). Then if *ds* is the distance *PQ,* the Pythagorean theorem*ds* ^{2} = *dx* _{1}^{2} + *dx* _{2}^{2} + *dx* _{3}^{2}

holds. In a "curved," or non-Euclidean, region of space this Pythagorean equation has to be replaced by a more general one. But before considering this let us move to four dimensions, so that we have an additional axis, *Ox* _{4}. This four-dimensional space would be Euclidean if*ds* ^{2} = *dx* _{1}^{2} + *dx* _{2}^{2} + *dx* _{3}^{2} + *dx* _{4}^{2}.

In the general case*ds* ^{2} = *g* _{11}*dx* _{1}^{2} + *g* _{22}*dx* _{2}^{2} + *g* _{33}*dx* _{3}^{2} + *g* _{44}*dx* _{4}^{2} + 2*g* _{12}*dx* _{1}*dx* _{2} + 2*g* _{13}*dx* _{1}*dx* _{3} + 2*g* _{14}*dx* _{1}*dx* _{4} + 2*g* _{23}*dx* _{2}*dx* _{3} + 2*g* _{24}*dx* _{2}*dx* _{4} + 2*g* _{34}*dx* _{3}*dx* _{4}.

The *g* 's are not necessarily constants but may be functions of *x* _{1}, *x* _{2}, *x* _{3}, *x* _{4}. That it is impossible to choose a coordinate system such that for a certain region *g* _{12}, *g* _{13}, *g* _{14}, *g* _{23}, *g* _{24}, *g* _{34} are all zero is what is meant by saying that the region of space in question is curved. That a region of space is curved can therefore in principle always be ascertained by making physical measurements in that region—for instance, by testing whether the Pythagorean theorem holds. There is, therefore, nothing obscure or metaphysical about the concept of curvature of space. The space-time of special relativity, it is worth mentioning, is semi-Euclidean and of zero curvature. In it we have*g* _{11} = *g* _{22} = *g* _{33} = − 1, *g* _{44} = + 1,

and *g* _{12}, *g* _{13}, *g* _{14}, *g* _{23}, *g* _{24}, *g* _{34} are all zero.

According to the general theory of relativity, space-time is curved in the neighborhood of matter. (More precisely, it has a curvature over and above the very small curvature that, for cosmological reasons, is postulated for empty space.) A light wave or any free body, such as a space satellite, is assumed in the general theory to lie along a geodesic in space-time. A geodesic is either the longest or the shortest distance between two points. In Euclidean plane geometry it is the shortest, whereas in the geometry of space-time it happens to be the longest. Owing to the appreciable curvature of space-time near any heavy body, a light ray that passes near the sun should appear to us to be slightly bent—that is, there should be an apparent displacement of the direction of a star whose light passes very near the sun. During an eclipse of the sun it is possible to observe stars very near to the sun's disk, since the glare of the sun is blacked out by the moon. In the solar eclipse of 1919, Sir Arthur Stanley Eddington and his colleagues carried out such an observation that gave results in good quantitative accord with the predictions of relativity. In a similar way, also, the general theory of relativity accounted for the anomalous motion of the perihelion of Mercury, the one planetary phenomenon that had defied Newtonian dynamics. In other cases the predictions of Newtonian theory and of general relativity are identical, and general relativity is, on the whole, important only in cosmology (unlike the special theory, which has countless verifications and is an indispensable tool of theoretical physics).

## Is Space Absolute or Relative?

The theory of relativity certainly forces us to reject an absolute theory of space, if by this is meant one in which space is taken as quite separate from time. Observers in relative motion to one another will take their space and time axes at different angles to one another; they will, so to speak, slice space-time at different angles. The *special* theory of relativity, at least, is quite consistent with either an absolute or a relational philosophical account of space-time, for the fact that space-time can be sliced at different angles does not imply that it is not something on its own account.

It might be thought that the *general* theory of relativity forces us to a relational theory of space-time, on the grounds that according to it the curvature of any portion of space-time is produced by the matter in it. But if anything the reverse would seem to be the case. If we accept a relational theory of space-time, we have to suppose that the inertia of any given portion of matter is determined wholly by the total matter in the universe. Consider a rotating body. If we suppose it to be fixed and everything else rotating, then we must say that some distant bodies are moving with transitional velocities greater than that of light, contrary to the assumptions of relativity. Hence, it is hard to avoid the conclusion that the inertia of a body is partly determined by the local metrical field, not by the total mass in the universe. But if we think of the local metrical field as efficacious in this way, we are back to an absolute theory of space-time. Furthermore, most forms of general relativity predict that there would be a curvature (and hence a structure) of space-time even if there were a total absence of matter. Indeed, relativistic cosmology often gives a picture of matter as consisting simply of regions of special curvature of space-time. (Whether this curvature is the cause of the existence of matter or whether the occurrence of matter produces the curvature of space-time is unclear in the general theory itself.) The variations of curvature of space-time enable us to rebut the objection to Descartes's theory that it cannot differentiate between more and less densely occupied regions of space.

Nevertheless, there are difficulties about accepting such a neo-Cartesianism. We must remember that quantum mechanics is essentially a particle physics, and it is not easy to see how to harmonize it with the field theory of general relativity. One day we may know whether a particle theory will have absorbed a geometrical field theory or vice versa. Until this issue is decided we cannot decide the question whether space (or space-time) is absolute or relational—in other words, whether particles are to be thought of as singularities (perhaps like the ends of J. A. Wheeler's "wormholes" in a multiply connected space) or whether space-time is to be understood as a system of relations between particles. This issue can be put neatly if we accept W. V. Quine's criterion of ontological commitment. Should our scientific theory quantify over point-instants of space-time, or should we, on the other hand, quantify over material particles, classes of them, classes of classes of them, and so on? The latter involves a commitment to particle physics, but if a unified field theory is successful, our ontology may consist simply of point-instants, classes of them, classes of classes of them, and so on, and physical objects will be definable in terms of all of these. So far neither Descartes nor Leibniz has won an enduring victory.

** See also ** Aristotle; Atomism; Cantor, Georg; Cartesianism; Descartes, René; Eddington, Arthur Stanley; Einstein, Albert; Geometry; Kant, Immanuel; Lambert, Johann Heinrich; Leibniz, Gottfried Wilhelm; Leucippus and Democritus; Logical Paradoxes; Lucretius; Mach, Ernst; Melissus of Samos; More, Henry; Newton, Isaac; Parmenides of Elea; Philosophy of Physics; Plato; Poincaré, Jules Henri; Pythagoras and Pythagoreanism; Quantum Mechanics; Quine, Willard Van Orman; Relativity Theory; Time; Zeno of Elea.

## Bibliography

An excellent, mainly historical, account of the philosophy of space is given in Max Jammer, *Concepts of Space* (Cambridge, MA: Harvard University Press, 1954; rev. ed., New York, 1960). Ch. 1, "The Concept of Space in Antiquity," is particularly valuable as a guide to the very scattered and obscure references to space in Greek philosophy. Also useful is John Burnet, *Early Greek Philosophy,* 3rd ed. (London: A. and C. Black, 1920). For Descartes, see his *Principles of Philosophy,* Part II, Secs. 4–21. For Leibniz, see especially his correspondence with Clarke, third paper, Secs. 3–6, and fifth paper, Secs. 32–124. The Leibniz-Clarke correspondence has been edited, with introduction and notes, by H. G. Alexander (Manchester, U.K., 1956). See also Bertrand Russell's *The Philosophy of Leibniz* (London, 1900). Newton's metaphysical views on space are to be found in the Scholium to the definitions of the *Principia* (*Mathematical Principles of Natural Philosophy,* translated by Florian Cajori, Berkeley: University of California Press, 1934). For Kant's example of the left hand and the right hand, see his *Prolegomena to Any Future Metaphysics,* translated by P. G. Lucas (Manchester, U.K., 1953), Sec. 13. Kant's most characteristic doctrines about space are to be found in his *Critique of Pure Reason,* in Secs. 2–7 of the "Transcendental Aesthetic" and in the "First Antinomy." In N. Kemp Smith's translation (London, 1929) these passages will be found on pp. 67–82 and 396–402. A criticism of Kant, Zeno, and other philosophers is to be found in Bertrand Russell's *Our Knowledge of the External World,* Lecture VI, "The Problem of Infinity Considered Historically" (London: Allen and Unwin, 1914). See also Adolf Grünbaum, "A Consistent Conception of the Extended Linear Continuum as an Aggregate of Unextended Elements," in *Philosophy of Science* 19 (1952): 288–306. For Mach's criticism of Newton, see especially Secs. 2–6 of his *Science of Mechanics,* translated by J. T. McCormack, 6th ed. (La Salle, IL: Open Court, 1960).

On a fairly elementary level, and although somewhat out of date in places, Émile Borel, *Space and Time* (New York: Dover, 1960)—a translation, by Angelo S. Rappoport and John Dougall, of the French edition published in 1922—can be recommended. So can the more difficult *Philosophy of Space and Time,* by Hans Reichenbach (New York, 1958), and *Philosophical Problems of Space and Time,* by Adolf Grünbaum (New York: Knopf, 1963). See also Grünbaum's paper "Geometry, Chronometry and Empiricism," in *Minnesota Studies in the Philosophy of Science,* edited by Herbert Feigl and Grover Maxwell, Vol. III (Minneapolis: University of Minnesota Press, 1962). A criticism of Grünbaum's views is given by Hilary Putnam in his paper "An Examination of Grünbaum's Philosophy of Geometry," in *Philosophy of Science, the Delaware Seminar,* edited by Bernard Baumrin, Vol. II (1962–1963; published New York: Interscience, 1963). Chs. 8 and 9 of Ernest Nagel's *Structure of Science* (New York: Harcourt Brace, 1961) are very useful. An interesting dialogue by A. S. Eddington, "What Is Geometry?," is a prologue to his *Space, Time and Gravitation* (Cambridge, U.K.: Cambridge University Press, 1920). On relativity, see Hermann Minkowski, "Space and Time," in *The Principle of Relativity,* by Albert Einstein, et al. (London: Methuen, 1923); Hans Reichenbach, "The Philosophical Significance of Relativity," and H. P. Robertson, "Geometry as a Branch of Physics," both in *Albert Einstein: Philosopher-Scientist,* edited by P. A. Schilpp, 2nd ed. (New York: Tudor, 1951); and Adolf Grünbaum, "The Philosophical Retention of Absolute Space in Einstein's General Theory of Relativity," in *Philosophical Review* 66 (1957): 525–534 (a revised version appears in *Problems of Space and Time,* edited by J. J. C. Smart, New York: Macmillan, 1963). See also J. A. Wheeler, "Curved Empty Space-Time as the Building Material of the Physical World: An Assessment," in *Logic, Methodology, and Philosophy of Science* edited by Ernest Nagel, Patrick Suppes, and Alfred Tarski (Stanford, CA: Stanford University Press, 1962), pp. 361–374.

For a discussion of the asymmetry between clockwise and counterclockwise rotations in relation to the nonconservation of parity, which has some relevance to Kant's problem of the left and right hands, see the brilliant popular exposition by O. R. Frisch, "Parity Is Not Conserved, a New Twist to Physics?," in *Universities Quarterly* 11 (1957): 235–244, and the article by Philip Morrison, "Overthrow of Parity," in *Scientific American* 196 (April 1957). For Poincaré's views, see *Science and Hypothesis* (New York: Dover, 1952), especially pp. 72–73. In connection with the sharpening of the issue between absolute and relational theories of space and time into an issue of ontology, see W. V. Quine, *Word and Object* (Cambridge, MA: Technology Press of the Massachusetts Institute of Technology, 1960), especially Sec. 52, "Geometrical Objects." A book of readings on space and time is *Problems of Space and Time,* edited by J. J. C. Smart (New York: Macmillan, 1964).

I should like to thank Professor B. C. Rennie, who read an earlier draft of this entry and made helpful comments.

*J. J. C. Smart (1967)*

## Space

# SPACE

The term space, derived from the Latin *spatium.* has a variety of meanings both in ordinary language and in philosophical and scientific usage. While corresponding etymologically to the Greek στάδιον, it has a much wider meaning than the latter; thus it can signify distance or length, place, temporal duration or interval, or other types of dimensionality. In the Greek language there is no term corresponding to this broad signification of the Latin and modern term. It is difficult, then, to speak about the historical development of the concept of space. Rather one must speak about various concepts and problems that are more or less connected with the meanings attributed to the word in modern usage. This article proposes to do so by a philosophical analysis of the concept.

**Philosophical Analysis.** A philosophical analysis of the concept of space distinguishes a threefold aspect or usage of the concept, viz, the psychological, the mathematical, and the physical.

**Psychological Aspect.** According to the investigations of contemporary psychology, the representation of space is gradually formed in the consciousness of the child through a complex experience that begins from the very early months of his life. The senses principally involved in this experience present a whole field wherein single objects are perceptible; this forms, as it were, a unitary and permanent picture wherein individual objects appear and disappear. The unitary and permanent picture gradually becomes distinct and separate from the individual objects and, notably in the imagination, is established as something independent and existing in itself. Thus, by successive integrations, there is formed visual, tactile, imaginary, etc., space.

By reason of its very origin, the space pertinent to perception and imagination has an essential unity, an absolute center of reference, and privileged directions of up and down, length, width, and depth. With the extension of infantile experience, the space relevant to perception, initially limited to present perception, is progressively broadened and identified first with the space of familiar surroundings, places of travel and play, and the region about the horizon of one's own experience. Finally, by way of information gleaned from stories and schooling, it is identified with terrestrial space, wherein the earth, with its complex of mutually similar objects, constitutes the privileged and absolute platform on which bodies exist and physical phenomena develop and about which the celestial sphere moves with the stars.

This geocentric representation, which has formed the space of humanity for thousands of years, has the same characteristics of absoluteness and independence of individual bodies, together with unity, center, and privileged directions, as the primitive representation in the child. When the Copernican revolution substituted the heliocentric system for the geocentric, it did not make essential modifications in the representation of space. A more decisive and revolutionary step was taken shortly afterward: this resulted from the recognition of the sun's equality with other stars, the recognition of equivalence in inertial systems, and the definitive renunciation of privileged centers and direction—all associated with the beginnings of modern mechanics, especially the work of Galileo galilei. Did these developments imply a renunciation of all systems of extrinsic reference for the position and movement of bodies or of all stable platforms for the description of physical phenomena? In the 17th century an affirmative answer would have demanded an audacity of which a genial poet might possibly have been capable, but not a methodical and reflective scientist such as Isaac Newton. A man of his times, Newton restricted himself to the path that seemed safer and less risky for him. For the egocentric, geocentric, and heliocentric platform, he substituted the notion of absolute space, existing by itself, always conformed to itself, and immobile, infinite, and eternal, without being dependent upon particular terrestrial and heavenly bodies.

From the psychological point of view, Newton's absolute space constitutes the projection of imaginary space into the world of physical reality, while maintaining the same characteristics of unity, absoluteness, and necessity. In fact, although the fantasy can imagine that it is suppressing all existing bodies, it cannot suppress imaginary space. This is a matter however, of psychological necessity deriving from the fact that the imagination is an organic faculty and, therefore essentially spatial. However, as a spiritual faculty, the intellect—which has being as such as its object—can discern (that is, judge, and not merely imagine) that the world and space itself would not exist if matter were totally annihilated. The intellect, then, can recognize that imaginary space is fictitious entity lacking reality in itself, independent of material bodies. This realization opens the door to a more rational consideration of space through the sciences of mathematics and physics.

**Mathematical Aspect.** Mathemathical space, which is the proper object of classical geometry and its more recent generalizations, can be defined as pure extension. Absolutely considered in its essence, abstracting from all the concrete conditions of its realization, as well as from every relation, either of dependence or of content, to real bodies. As such, it is absolutely possible extension.

Mathematical space received a systematic treatment, admirable for its logical rigor and completeness, in the *Elements* of Euclid about the year 300 b.c. Transmitted through Arabian culture to medieval Europe, this treatise still constitutes the nucleus of geometric knowledge for the person of ordinary culture. At the basis of this teaching there is a natural, uncritical, and directly realistic conception taking fundamental notions (point, line, surface, etc.) from spontaneous intuition. It proceeds from principles held as evident and necessary truths, and it claims that all its propositions contain absolute truth. Euclidean space is three-dimensional, homogeneous, and isotropic, as well as infinite, or rather, indefinite. In it, parallel straight lines are equi-distant; there is the possibility of similar figures, and the form of geometric things is independent of their position and extension. In short, Euclidean space is straight, having no curvature.

All the properties of Euclidean space can be deduced by introducing a postulate, namely, Euclid's fifth postulate, which states in substance that through a point external to a straight line one and only line can be drawn parallel to that line. The evidence and necessity of this postulate, and equivalent statements that have been substituted for it, have been disputed since the time of Euclid's first commentators. In fact, it constitutes an extrapolation from immediate intuition, lacking logical justification. The critique of Euclidean geometry, through the work of many mathematicians, such as K. F. Gauss, J. Bolyai (1802–60), N. I. Lobachevskĭ, and G. F. B. Riemann, has led to the conclusion that Euclid's postulate does not have an intrinsic and exclusive logical necessity, since it is possible to substitute opposite postulates for it. These lead to concepts of curved, hyperbolic, or elliptical spaces having properties different from those of Euclidean space, yet logically coherent and free from internal contradictions. This development has led to a generalization of the concept of mathematical space, further elaborated by the consideration of nonisotropic and differential spaces, of hyperspaces having more than three dimensions, and of functional spaces having infinite dimensions, etc.

The evolution of geometry, then, has shown the need for distinguishing the question of absolutely possible mathematical spaces from that of a really existing physical space. Rene descartes and classical physicists deemed these questions to be identical.

**Physical Aspect.** By physical space is meant the first, fundamental, dimensional quantity that, along with time, enables one to describe the emotion of bodies, define velocity as the derivative of space with respect to time, and, consequently, determine the position of the body at the term of its motion. Two questions are posited in reference to physical space: (1) Does physical space haven an existence distinct from the bodies that fill it (or can there be space without matter)? And (2) What are the properties of physical space?

As regards the first question, the Greek atomists, in opposition to parmenides, claimed that the void exists together with the plenum and that this is necessary for the multiplicity and motion of atoms. This teaching was restored by Pierre gassendi during the 17th century, in opposition to Descartes; the latter, identifying the essence of bodies with their extension, denied all possibility of empty space. Aristotle, and scholastic philosophers generally, deny the possibility of an absolute void; as a mere nonentity, this cannot exist in itself. Aristotelians explain the multiplicity and movement of bodies in terms of real divisibility of matter, itself made intelligible by the concept of potency. There would seem to be no validity Isaac Newton's arguments in favor of an absolute space independent of sensible bodies or fixed stars, since the position and movement of bodies can be explained in reference to the complex of sensible bodies or fixed stars, without recourse to a system of uncontrollable and purely imaginary reference. Physical space, then, is identical with the complex of extended bodies constituting the universe—not only directly sensible bodies, that is, solids, liquids, and gases, but also ether and the fields of modern physics.

As regards properties, man has always asked whether physical space is finite or infinite. According to the Ptolemaic conception, it was easy to deem physical space to be finite and limited by the outermost celestial sphere; according to the Copernican conception, however, one was led to think of space as open and infinite. More recently, equilibrium considerations in celestial mechanics and deductions from the theory of general relativity lead scientists to conceive of physical space as finite. In recent times a question has also arisen as to whether physical space is Euclidean or non-Euclidean. There is no absolute answer to this question, since all physical measurements for ascertaining the geometry of space are approximate and imprecise; one can speak only of a greater or lesser approximation to a geometric space. Within the limits of attainable measures and terrestrial experiments, physical space constitutes a good approximation to Euclidean space. Where astronomical distances are involved, however, it is more exact to say (in keeping with the theory of general relativity) that physical space is what Riemann has described it to be—elliptical, curved in its totality, and enclosed within itself, as a spherical hypersurface. Finally, as regards the relations between space and time, the theory of special relativity rejects the absolute separation of these categories and claims the existence of a sole spatial-temporal physical reality, the space-time continuum, wherein spatial and temporal relations among various bodies and events depend upon the state of reciprocal movement.

See Also:continuum; place; time; motion

**Bibliography:** m. j. adler, ed., *The Great Ideas: A Syntopicon of Great Books of the Western World*, 2 v. (Chicago 1952) 2:811–25. c. b. garnett, *The Kantian Philosophy of Space* (New York 1939). m. jammer, *Concepts of Space* (Cambridge, Mass.1954). g. bachelard, *L'Experience de l'espace dans la physique contemporaine* (Paris 1937). l. p. eisenhart, *Reimannian Geometry* (Princeton 1949). f. gonseth, *La Geometrie et le probleme de l'espace,* 6 v. (Neuchatel 1945–55). f. soccorsi, *Quaestiones scientificae cum philosophia coniunctae: De geometriis et spatiis non euclideis* (Rome 1960). a. einstein, *Relativity*, tr. b. w. lawson (15th ed. London 1953). a. s. eddington, *Space, Time and Gravitation* (Cambridge, Eng. 1920). h. margenau, *The Nature of Physical Reality* (New York 1950). d. nys, *La Notion de'espace* (Brussels 1922). f. selvaggi, *Cosmologia* (Rome 1962); *Scienza e metodologia* (Rome 1962).

[f. selvaggi]

## Space

# Space

Space, as referred to by astronomers, is the region beyond the atmosphere of the Earth. It is also called outer space. Generally, space is defined as the three-dimensional extension in which all things exist and move. Humans intuitively feel that they live in an unchanging space. In this space, the height of a tree or the length of a table is the same for everybody. German–American physicist Albert Einstein’s (1879– 1955) *special theory of relativity* tells that this intuitive feeling is really an illusion. Neither space nor time is the same for two people moving relative to each other. Only a combination of *space and time*, called space-time, is unchanged for everyone.

Einstein’s general theory of relativity tells that the force of gravity is a result of a warping of this space-time by heavy objects, such as planets. According to the big bang theory of the origin of the universe, the expansion of the universe began from infinitely curved space-time. Scientists still do not know whether this expansion will continue indefinitely or whether the universe will collapse again in a big crunch. Meanwhile, astronomers are learning more and more about outer space from terrestrial and orbiting telescopes, space probes sent to other planets in the solar system, and other scientific observations. This is just the beginning of the exploration of the unimaginably vast void, beyond the Earth’s outer atmosphere, in which a journey to the nearest star would take 3,000 years at a million miles an hour.

The difference in the perception of space and time, predicted by the special theory of relativity, can be observed only at very high velocities close to that of light. A person driving past at 50 mph (80 km/h) will appear only a hundred million millionth of an inch thinner as another person stands watching on the sidewalk. By themselves, three-dimensional space and one-dimensional time are different for different people. Taken together, however, they form a four-dimensional space-time in which distances are the same for all observers. One can understand this idea by using a two-dimensional analogy. Suppose the definition of south and east is not the same for all people. Bill travels from city A to city B by going ten miles along his south and then five miles along his east. Sallly travels from A to B by going two miles along her south and 11 miles along her east. Both people, however, move the same distance of 11.2 miles south-east from city A to B. In the same way, if one thinks of space as south and time as east, space-time is something like south-east.

The general theory of relativity tells one that gravity is the result of the curving of this four-dimensional space-time by objects with large mass. A flat stretched rubber membrane will sag if a heavy iron ball is placed on it. If one now places another ball on the membrane, the second ball will roll towards the first. This can be interpreted in two ways; as a consequence of the curvature of the membrane, or as the result of an attractive force exerted by the first ball on the second one. Similarly, the curvature of space-time is another way of interpreting the attraction of gravity. An extremely massive object can curve space-time around so much that not even light can escape from its attractive force. Such objects, called *black holes*, could very well exist in the universe. Astronomers believe, for example, that the disk found in 1994 by the Hubble Space Telescope, at the center of the elliptical galaxy M87 near the center of the Virgo cluster, is material falling into a supermassive black hole estimated to have a mass three billion times the mass of the sun.

The relativity of space and time and the curvature of space-time do not affect human’s daily lives. The high velocities and huge concentrations of matter, needed to manifest the effects of relativity, are found only in outer space on the scale of planets, stars, and galaxies. Earth’s own Milky Way galaxy is a mere speck, 100,000 light years across, in an observable universe that spans over ten billion light-years (where one light-year is the distance that light travels in a vacuum in one year). Though astronomers have studied this outer space with telescopes for hundreds of years, the modern space age began only in 1957 when the Soviet Union put the first artificial satellite, *Sputnik 1*, into orbit around the Earth.

At present, there are hundreds of satellites in orbit gathering information from distant stars, free of the distorting effect of the Earth’s atmosphere. Even though no manned spacecraft has landed on other worlds since the Apollo Moon landings, several space probes, such as the *Voyager 2* and the *Magellan*, have sent back photographs and information from the moon and from other planets in the solar system. Other spacecraft, such as New Horizons, are on their way to particular destinations in the solar system. *New Horizons* is traveling toward the dwarf planet system Pluto-Charon to investigate this little explored region of space. There are many questions to be answered and much to be achieved in the exploration of space. The Hubble telescope, repaired in space in 1993, has sent back data that has raised new questions about the age, origin, and nature of the universe.

Along with Hubble, NASA has also launched the Compton Gamma Ray Observatory, the Chandra

### KEY TERMS

**Big Bang theory** —The currently accepted theory that the universe began in an explosion.

**Black hole** —An object, believed to be formed by a very massive star collapsing on itself, which exerts such a strong gravitational attraction that nothing can escape from it.

**General theory of relativity** —Albert Einstein’s theory of physics, according to which gravity is the result of the curvature of space-time.

**Light year** —The distance traveled by light in one year (within a vacuum) at the speed of about 186,000 miles per second.

**Space probe** —An unmanned spacecraft that orbits or lands on the Moon, planet, or other celestial body to gather information that is relayed back to Earth.

**Space station** —A manned artificial satellite in orbit about the earth, intended as a base for space observation and exploration.

**Space-time** —Space and time combined as one unified concept.

**Special relativity** —The part of Einstein’s theory of relativity that deals only with non-accelerating (inertial) reference frames.

X-ray Observatory, and the Spitzer Space Telescope, as part of its Great Observatories program, to explore the far reaches of the universe at various wavelengths of the electromagnetic spectrum. The fate of Hubble is unsure as of October 2006. If nothing is done to repair it, it will re-enter the Earth’s atmosphere sometime after 2010. Currently, the James Webb Space Telescope (JWST) is partially replacing Hubble in 2013. It will operate at the infrared region of the electromagnetic spectrum, with a much better ability to see into those parts of the universe. However, it does not operate in the visible part of the spectrum, as does Hubble.

Countries such as Japan, the countries of the European Space Agency (such as England, Germany, and Italy), China, Russia, and India, are also sending probes into outer space to investigate the solar system and launching telescopes into space to discover more about the solar system, galaxies, and the universe in general. As they do these investigations, humans learn more about life on the Earth and how life developed on Earth and, possibly, in other places within the immense expanse of outer space called the universe.

*See also* Cosmology; Relativity, general; Relativity, special.

## Resources

### BOOKS

Eckart, Andreas. *The Black Hole at the Center of the Milky Way*. London, UK: Imperial College Press, 2005.

Harland, David Michael. *The Big Bang: A View from the 21st Century*. London and New York: Springer, 2003.

Kundt, Wolfgang. *Astrophysics: A New Approach*. Berlin and New York: Springer, 2005.

Mallary, Michael. *Our Improbable Universe: A Physicist Considers How We Got Here*. New York: Thunder’s Mouth Press, 2004.

Mark, Hans, ed. *Encyclopedia of Space Science & Technology*. New York: John Wiley & Sons, 2003.

Raine, Derek J. Black Holes: *An Introduction*. London, UK: Imperial College Press, 2005.

Zelik, Michael. *Astronomy: The Evolving Universe.* Cambridge and New York: Cambridge University Press, 2002.

Sreela Datta

## Space

# SPACE

To the question, "Where are you in this moment?" a pilot would answer, "At longitude *x*, latitude *y*, altitude *z*." But if one asks, "Where do you live?", the answer may instead evoke neighborly relations weaved through the years, a climate, old stones, the freshness of water. Depending on who is asked about what, the *where* question can be answered by *space* determinations or by the memories of a concrete *place.* Space and place are two different ways of conceiving the "where" or, using the Latin word for "where" as a *terminus technicus,* two answers to the *ubi* question.

## Place and Space

Place is an order of beings vis-à-vis the body. This order (*kosmos* in Greek) always mirrors the great cosmos. This vis-à-vis or mirroring is the essence of what has been called *proportionality* (Illich and Rieger 1996). According to Albert Einstein, the concept of space disembedded itself from the "simpler concept of place" and "achieve[d] a meaning which is freed from any connection with a particular material object" (Einstein 1993, p. xv). Yet Einstein insisted that space is a free creation of imagination, a "means devised for easier comprehension of our sense experience" (Einstein 1993, p. xv). In pure space, however, the body would be out of place and in a state of perceptual deprivation.

The focus here is on the radical monopoly that space determinations exert on the *ubi* question. Wheels and motors seem to belong to space as feet do to places. And just as the radical monopoly of motorized transportation on human mobility leaves some freedom to walk, space determinations leave remnants of placeness to linger in perception and memory. Ethics, then, can only be rebuilt by a recovery of placeness.

## Origins of Space

A general conception of space is conspicuously absent from ancient mathematics, physics, and astronomy. The Greek language, so rich in locational terms, had no word for space (Bochner 1998). *Topos* meant place, and when Plato in the *Timaeus* (360 b.c.e.) located the demiurge in an uncreated *ubi* in which one can have no perception because it does not *exist,* he called it *chôra,* fallow land, the temporary void between the fullness of the wild and cultivation. According to Plato, the demiurge's chôra could only be conceived "by a kind of spurious reason," "as in a dream," in a state in which "we are unable to cast off sleep and determine the truth about it" (passage Timaeus 52). In hindsight, one may say that this was a first intuition of the antinomy between place and what is has come to be called "space." In the fourteenth century, Nicolas d'Oresme imagined an incorporeal void beyond the last heavenly sphere, but still insisted that, in contrast, all real places are full and material. Space, still only a pure logical possibility, became a *possibile realis* between the times of d'Oresme and Galileo (Funkenstein 1986, p. 62).

Following the canons of Antiquity and medieval cartography, a chart summarized bodily scouting and measuring gestures. Pilgrims followed *itineraria;* sailors, charts of ports; and surveyors consigned ritually performed acts of mensuration on marmor or brass plates. These were not maps in the modern sense, because they did not postulate a disembodied *eye* contemplating a land or a sea from above. The first maps in the modern sense were contemporary with early experimentations in central perspective and, like these, construed an abstract eye contemplating a distant grid in which particulars could be relatively situated. In 1574, Peter Ramus wrote a *lytle booke* in which he exposed a *calculus of reality* where all topics were divided in mental spaces that immobilized objects in their definitions precluding the understanding of knowledge as an act (Pickstock 1998). Cartesian coordinates and projective geometry gave the first mathematical justification to the idea of an immaterial vessel, unlimited in extent, in which all material objects are contained.

## Non-Euclidean Space

Had space been invented, as Einstein contended, or discovered? In the eighteenth century, Immanuel Kant announced that space was an *a priori* of perception. For him, Euclidean geometry and its axioms were the mathematical expression of an entity—space—that cannot be perceived, but, like time, underlies all perceptions. The first attempt to contradict Euclidean geometry was published in Russian in 1829 by Nicolay Lobachevsky (1792–1856), whose ideas were rooted in an opposition to Kant. For him, space was an *a posteriori* concept. He sought to prove this by demonstrating that axioms different from Euclid's can generate different spaces. In light of Lobachevsky's—and then Georg Riemann's (1826–1866)—non-Euclidean geometries, Euclidean geometry appears ex post facto as just another axiomatic construct. There is no a priori space experience, no *natural,* or *universal* space. Space is not an empirical fact but a construct, an arbitrary frame that *carpenters* the modern imagination (Heelan 1983).

Einstein occupies an axial and simultaneously ambiguous position in the history of this understanding. In order to express alterations of classical physics that seemed offensive to common sense, he adopted a mathematically constructed *manifold* (coordinate space) in which the space coordinates of one coordinate system depend on *both* the time and space coordinates of another relatively moving system. On the one hand, like Lobachevsky and Riemann (1854), Einstein insisted on the constructed character of space: Different axioms generate different spaces. On the other, he not only came to consider his construct as ruling the unreachable realms of the universe, but reduced earthly human experience to a particular case of it. In Einstein's space, time can become extension; mass, energy; gravity, a geometric curvature; and reality, a distant shore, indifferent to ethics. This view of space has reigned over the modern imagination for a century. Yet the idea that the realm of everyday experience is a particular case of this general construct has not raised fundamental ethical questions.

## Ethics in Space

The subsumption of the neighborhood where one lives into the same category as distant galaxies transforms neighbors into disembodied particularities. This loss of the sense of immediate reality invites a moral suicide. Hence, ethics in the early-twenty-first century requires an epistemological distinction that evokes that of d'Oresme in the fourteenth: Contrary to outer space, the perceptual milieu is a place of fullness. According to its oldest etymology, *ethos* means a place's gait. Space recognizes no gait, no body, no concreteness, and, accordingly, no ethics. The *ubi* question must thus be ethically restated.

Body historians and phenomenologists provide tracks toward an ethical recovery of placeness in the space age. Barbara Duden (1996) argues that one can only raise fundamental ethical questions related to pregnancy by relocating the body in its historical places. For their part, phenomenologists, those philosophers who cling to the *primacy of perception* in spite of tantalizing science-borne and technogenic *certainties,* restore some proportionality between body and place. For Gaston Bachelard (1884–1962), for instance, there is no individual body immersed in the apathetic void of space, but an experience of *mutual seizure* of the body and its natural *ubi.* Maurice Merleau-Ponty (1908–1964) further articulates the complementarity of these two sides of reality. These can be steps toward a recovery of the sense of the vis-à-vis without which there is no immediate reality, and hence no ethics.

JEAN ROBERT

SEE ALSO *Cyberspace;
Einstein, Albert;
Foucault, Michel.*

## BIBLIOGRAPHY

Bachelard, Gaston. (1983). *Water and Dreams. An Essay on the Imagination of Matter.* ed. and trans. Edith R. Farrell. Dallas: Dallas Institute of the Humanities and Culture. Originally published in French in 1956.

Bochner, Salomon. "Space." (1998). In *Raum und Geschichte,* Kurseinheit 4, comp. Ludolf Kuchenbuch and Uta Kleine. Hagen: Fernuniversität.

Coxeter, H. S. (1965). *Non-Euclidean Geometry.* Toronto: University of Toronto Press.

Duden, Barbara. (1996). *The Woman Beneath the Skin*. Cambridge, MA: Harvard University Press. Originally published in 1991.

Einstein, Albert. (1993). "Foreword." In *Concepts of Space: The History of Theories of Space in Physics,* by Max Jammer. New York: Dover. Einstein's text was originally published in 1954.

Funkenstein, Amos. (1986). *Theology and the Scientific Imagination from the Middle Ages to the Seventeenth Century.* Berkeley: University of California Press.

Heelan, Patrick A. (1983). *Space-Perception and the Philosophy of Science.* Berkeley: University of California Press.

Illich, Ivan, and Rieger, Matthias. (1996). "The Wisdom of Leopold Kohr." In *Fourteenth Yearly E. F. Schumacher Conference.* New Haven, Connecticut, October 1994. Reprinted in French as "La sagesse de Leopold Kohr," in *Ivan Illich, La Perte des sens,* trans. Pierre-Emmanuel Dauzat. Paris: Fayard. The French version is the most recent authorized version; the English version is not generally available.

Lobachevsky, Nikolay Ivanovich. (1886). *Geometrische Untersuchungen zur Theorie der Parallellinien.* Kasan, Russia: Kasan University. Originally published in Berlin in 1840.

Merleau-Ponty, Maurice. (1964). *The Primacy of Perception.* Chicago: Northwestern University Press.

Pickstock, Catherine. (1998). *After Writing: On the Liturgical Consummation of Philosophy.* Malden, MA: Blackwell.

Ramus, Peter. (1966). *Logike.* Leeds: The Scholar Press. Originally published in 1574.

Riemann, Bernhard. (1876). "*Über die Hypothesen, welche der Geometrie zu Grunde liegen*" [On the hypotheses that are the base of geometry.]. In *Gesammelte mathematische Werke und wissenschaftlicher Nachlaß* [Collected mathematical works and legacy], ed. Richard Dedekind and Heinrich Weber. Leipzig, Germany: Teubner. Originally published in 1854.

## Space

# Space

Space is the three-dimensional extension in which all things exist and move. We intuitively feel that we live in an unchanging space. In this space, the height of a **tree** or the length of a table is exactly the same for everybody. Einstein's *special theory of relativity* tells us that this intuitive feeling is really an illusion. Neither space nor time is the same for two people moving relative to each other. Only a combination of space and time, called *space-time*, is unchanged for everyone. Einstein's general theory of relativity tells us that the **force** of gravity is a result of a warping of this space-time by heavy objects, such as planets. According to the big bang theory of the origin of the universe, the expansion of the universe began from infinitely curved space-time. We still do not know whether this expansion will continue indefinitely or whether the universe will collapse again in a big crunch. Meanwhile, astronomers are **learning** more and more about outer space from terrestrial and orbiting telescopes, space probes sent to other planets in the **solar system** , and other scientific observations. This is just the beginning of the exploration of the unimaginably vast void, beyond the earth's outer atmosphere, in which a journey to the nearest **star** would take 3,000 years at a million miles an hour.

The difference in the **perception** of space and time, predicted by the special theory of relativity, can be observed only at very high velocities close to that of **light** . A man driving past at 50 MPH (80 km/h) will appear only a hundred million millionth of an inch thinner as you stand watching on the sidewalk. By themselves, three-dimensional space and one-dimensional time are different for different people. Taken together, however, they form a four-dimensional space-time in which distances are same for all observers. We can understand this idea by using a two-dimensional analogy. Let us suppose your definition of south and east is not the same as mine. I travel from city A to city B by going ten miles along my south and then five miles along my east. You travel from A to B by going two miles along your south and 11 miles along your east. Both of us, however, move exactly the same **distance** of 11.2 miles southeast from city A to B. In the same way, if we think of space as south and time as east, space-time is something like south-east.

The general theory of relativity tells us that gravity is the result of the curving of this four-dimensional space-time by objects with large **mass** . A flat stretched rubber **membrane** will sag if a heavy **iron** ball is placed on it. If you now place another ball on the membrane, the second ball will roll towards the first. This can be interpreted in two ways; as a consequence of the curvature of the membrane, or as the result of an attractive force exerted by the first ball on the second one. Similarly, the curvature of space-time is another way of interpreting the attraction of gravity. An extremely massive object can curve space-time around so much that not even light can escape from its attractive force. Such objects, called *black holes*, could very well exist in the universe. Astronomers believe that the disk found in 1994 by the Hubble **telescope** , at the center of the elliptical **galaxy** M87 near the center of the Virgo cluster, is material falling into a supermassive **black hole** estimated to have a mass three billion times the mass of the **Sun** .

The relativity of space and time and the curvature of space-time do not affect our daily lives. The high velocities and huge concentrations of **matter** , needed to manifest the effects of relativity, are found only in outer space on the scale of planets, stars, and galaxies. Our own **Milky Way** galaxy is a mere speck, 100,000 light years across, in a universe that spans ten billion light years. Though astronomers have studied this outer space with telescopes for hundreds of years, the modern space age began only in 1957 when the Soviet Union put the first artificial **satellite** , *Sputnik 1*, into **orbit** around the **earth** . At present, there are hundreds of satellites in orbit gathering information from distant stars, free of the distorting effect of the earth's atmosphere. Even though no manned spacecraft has landed on other worlds since the Apollo **moon** landings, several space probes, such as the *Voyager 2* and the *Magellan*, have sent back photographs and information from the moon and from other planets in the solar system. There are many questions to be answered and much to be achieved in the exploration of space. The Hubble telescope, repaired in space in 1993, has sent back data that has raised new questions about the age, origin, and nature of the universe.

See also Cosmology; Relativity, general; Relativity, special.

## Resources

### books

Burrows, William E. *Exploring Space.* New York: Random House, 1990.

Ellis, G.F.R., and R.M. Williams. *Flat and Curved Space-Times.* Oxford: Clarendon Press, 2000.

*Introduction to Astronomy and Astrophysics.* 4th ed. New York: Harcourt Brace, 1997.

Krauss, Lawrence M. *Fear of Physics.* New York: BasicBooks, 1993.

Mark, Hans, Maureen Salkin, and Ahmed Yousef, eds. *Encyclopedia of Space Science & Technology.* New York: John Wiley & Sons, 2001.

Smolin, Lee. *The Life of the Cosmos.* Oxford: Oxford University Press, 1999.

Thorne, Kip S. *Black Holes and Time Warps.* New York: W. W. Norton & Company, 1994.

### periodicals

Bruning, David. "A Galaxy of News." *Astronomy* (June 1995): 40.

Malin, M.C., and K.S. Edgett. "Evidence for Recent Groundwater Seepage and Surface Runoff on Mars." *Science* no. 288 (2000): 2330-2335.

Sreela Datta

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Big bang theory**—The currently accepted theory that the universe began in an explosion.

**Black hole**—An object, formed by a very massive star collapsing on itself, which exerts such a strong gravitational attraction that nothing can escape from it.

**General theory of relativity**—Albert Einstein's theory of physics, according to which gravity is the result of the curvature of space-time.

**Light year**—The distance traveled by light in one year at the speed of 186,000 miles per second.

**Space probe**—An unmanned spacecraft that orbits or lands on the Moon or another planet to gather information that is relayed back to Earth.

**Space station**—A manned artificial satellite in orbit about the earth, intended as a base for space observation and exploration.

**Space-time**—Space and time combined as one unified concept.

**Special relativity**—The part of Einstein's theory of relativity that deals only with nonaccelerating (inertial) reference frames.

## Space

# Space

Space is the three-dimensional extension in which all things exist and move. Intuitively, it feels that we live in an unchanging space. In this space, the height of a tree or the length of a table is exactly the same for everybody. Einstein's special theory of relativity explains that this intuitive feeling is really an illusion. Neither space nor time is the same for two people moving relative to each other. Only a combination of space and time, called space-time, is unchanged for everyone. Einstein's general theory of relativity states that the force of **gravity** is a result of a warping of this space-time by heavy objects, such as planets. According to the **Big Bang theory** of the origin of the universe, the expansion of the universe began from infinitely curved space-time. Scientists still do not know whether this expansion will continue indefinitely, or whether the universe will collapse again in a Big Crunch. Meanwhile, astronomers are continually learning about outer space from terrestrial and orbiting telescopes, space probes sent to other planets in the **solar system** , and other scientific observations. This is just the beginning of the exploration of the unimaginably vast void, beyond Earth's outer atmosphere, in which a journey to the nearest star would take 3,000 years traveling at a million miles per hour.

The difference in the perception of space and time, predicted by the special theory of relativity, can be observed only at very high velocities close to that of light. A man driving past at 50 mph (80 kph) will appear only a hundred million millionths of an inch thinner as you stand watching on the sidewalk. By themselves, three-dimensional space and one-dimensional time are different for different people. Taken together, however, they form a four-dimensional space-time in which distances are the same for all observers. We can understand this idea by using a two-dimensional analogy. Suppose that a man's definition of south and east is not the same as a woman's. The woman travels from city A to city B by going 10 miles along her south and then 5 miles along the man's east. The man travels from A to B by going 2 miles along his south and 11 miles along the woman's east. Both, however, move exactly the same distance of 11.2 miles south-east from city A to B. In the same way, if we think of space as south and time as east, space-time is something like south-east.

The general theory of relativity states that gravity is the result of the curving of this four-dimensional space-time by objects with large mass. A flat stretched rubber membrane will sag if a heavy **iron** ball is placed on it. If you now place another ball on the membrane, the second ball will roll towards the first. This can be interpreted in two ways: as a consequence of the curvature of the membrane, or as the result of an attractive force exerted by the first ball on the second one. Similarly, the curvature of space-time is another way of interpreting the attraction of gravity. An extremely massive object can curve space-time around so much that not even light can escape from its attractive force. Such objects, called black holes, probably exist in the universe. Astronomers believe that the disk found in 1994 by the Hubble **telescope** , at the center of the elliptical galaxy M87 near the center of the Virgo cluster, is material falling into a supermassive black hole estimated to have a mass three billion times the mass of the **Sun** .

The relativity of space and time and the curvature of space-time do not affect our daily lives. The high velocities and huge concentrations of matter, needed to manifest the effects of relativity, are found only in outer space on the scale of planets, stars, and galaxies. Our own Milky Way galaxy is a mere speck, 100,000 light years across, in a universe that spans ten billion light years. Though astronomers have studied this outer space with telescopes for hundreds of years, the modern space age began only in 1957 when the Soviet Union put the first artificial **satellite** , *Sputnik 1*, into orbit around the earth. At present, there are hundreds of satellites in orbit gathering information from distant stars, free of the distorting effect of the earth's atmosphere. Even though no manned

spacecraft has landed on other worlds since the Apollo **Moon** landings, several space probes, such as the *Voyager 2* and the *Magellan*, have sent back photographs and information from the Moon and from other planets in the solar system. There are many questions to be answered and much to be achieved in the exploration of space. The Hubble telescope, repaired in space in 1993 and 2002, has sent back data that has raised new questions about the age, origin, and nature of the universe. The launch of a United States astronaut to the Russian *Mir* space station in March 1995, the docking of the United States **space shuttle** *Atlantis* with *Mir*, and the **international space station** currently under construction have opened up exciting possibilities for space exploration.

** See also ** Astronomy; Celestial sphere: The apparent movements of the Sun, Moon, planets, and stars; History of manned space exploration; Physics; Relativity theory; Solar system; Space and planetary geology; Space physiology

## space

space
/ spās/
•
n.
1.
a continuous area or expanse that is free, available, or unoccupied:
*a table took up much of the space* |
*we shall all be living together in a small space*
*he backed out of the parking space.*
∎
an area of land that is not occupied by buildings:
*she had a love of open spaces.*
∎
an empty area left between one-, two-, or three-dimensional points or objects:
*the space between a wall and a utility pipe.*
∎
a blank between printed, typed, or written words, characters, numbers, etc.
∎
Mus.
each of the four gaps between the five lines of a staff.
∎
an interval of time (often used to suggest that the time is short, considering what has happened or been achieved in it):
*both their cars were stolen in the space of three days.*
∎
pages in a newspaper, or time between television or radio programs, available for advertising.
∎
(also commercial space)
an area rented or sold as business premises.
∎
the amount of paper used or needed to write about a subject:

*there is no space to give further details.*∎ the freedom and scope to live, think, and develop in a way that suits one:

*a teenager needing her own space.*∎ Telecommunications one of two possible states of a signal in certain systems. The opposite of mark

^{1 }(sense 2). 2. the dimensions of height, depth, and width within which all things exist and move:

*the work gives the sense of a journey in space and time.*∎ (also outer space) the physical universe beyond the earth's atmosphere. ∎ the near vacuum extending between the planets and stars, containing small amounts of gas and dust. ∎ Math. a mathematical concept generally regarded as a set of points having some specified structure. • v. 1. [tr.] (usu. be spaced) position (two or more items) at a distance from one another:

*the houses are*∎ (in printing or writing) put blanks between (words, letters, or lines): [as n.] (spacing)

**spaced out**.*the default setting is single line spacing.*2. (usu. be spaced out or space out) inf. be or become distracted, euphoric, or disoriented, esp. from taking drugs; cease to be aware of one's surroundings:

*I was so tired that I began to feel totally spaced out*

*I kind of space out for a few minutes.*PHRASES: watch this space inf. further developments are expected and more information will be given later.DERIVATIVES: spac·er n.

## Space

# Space

The term space has two general meanings. First, it refers to the three-dimensional extension in which all things exist and move. We sometimes speak about outer space as everything that exists outside our own solar system. But the term space in astronomy and in everyday conversation can also refer to everything that makes up the universe, including our own solar system and Earth.

Mathematicians also speak about space in an abstract sense and try to determine properties that can be attributed to it. Although they most commonly refer to three-dimensional space, no mathematical reason exists not to study two-dimensional, one-dimensional, four-dimensional, or even n-dimensional (an unlimited number of dimensions) space.

## Space-time continuum

One of the most important scientific discoveries of the twentieth century had to do with the nature of space. Traditionally, both scientists and nonscientists thought of space and time as being two different and generally unrelated phenomena. A person might describe where he or she is in terms of three-dimensional space: at the corner of Lithia Way and East Main Street in Ashland, for example. Or he or she might say what time it is: 4:00 P.M. on April 14.

What the great German-born American physicist Albert Einstein (1879–1955) showed was that space and time are really part of the same way of describing the universe. Instead of talking about space or time, one needed to talk about one's place on the space-time continuum. That is, we move about in four dimensions, the three physical dimensions with which we are familiar and a fourth dimension—the dimension of time.

Einstein's conception of space-time dramatically altered the way scientists thought about many aspects of the physical world. For example, it suggested a new way of defining gravity. Instead of being a force between two objects, Einstein said, gravity must be thought of in terms of irregularities in the space-time continuum of the universe. As objects pass through these irregularities, they exhibit behaviors that correspond almost exactly to the effects that we once knew as gravitational attraction.

[*See also* **Big bang theory; Cosmology; Relativity; Time** ]

## space

**space**
•**abase**, ace, apace, backspace, base, bass, brace, case, chase, dace, efface, embrace, encase, enchase, enlace, face, grace, interlace, interspace, in-your-face, lace, mace, misplace, outface, outpace, pace, place, plaice, race, space, Thrace, trace, upper case
•airbase • freebase • wheelbase
•database • steeplechase • paperchase
•paleface • typeface • whiteface
•boldface • coalface • interface
•staircase • briefcase • slipcase
•packing case • doorcase • showcase
•notecase • pillowcase • suitcase
•bookcase • nutcase • marketplace
•anyplace • everyplace • showplace
•shoelace • bootlace • someplace
•Lovelace • fireplace • commonplace
•workplace • birthplace • tenace
•airspace • aerospace • hyperspace
•carapace • workspace • ratrace
•millrace • Fuentes • rosace

## space

**space** Boundless three-dimensional expanse in which objects are located. Relativity states space and time are aspects of one thing, known as space-time. Also, space has a non-Euclidean geometry wherever there is a gravitational field. In another sense, space, sometimes referred to as outer space, is taken to mean the rest of the Universe beyond the Earth's atmosphere.

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#### NEARBY TERMS

**Space**