This page is a sub-page of our page on Mathematical Concepts.


Related KMR-pages:

Oscar Reutersvärd
M.C. Escher

Homology and Cohomology


Other relevant sources of information:

• …


Einstein on the differences between Place and Space:

[Excerpts from the foreword by Albert Einstein to Max Jammer’s monumental work
Concepts of Space – The History of Theories of Space in Physics (Jammer, 1993, (1969, 1954))]

In order to appreciate fully the importance of the investigations such as the present work of Dr. Jammer one should consider the following points. The eyes of the scientist are directed upon those phenomena which are accessible to observation, upon their apperception and conceptual formulation. In the attempt to achieve a conceptual formulation of the confusingly immense body of observational data, the scientist makes use of a whole arsenal of concepts which he imbibed practically with his mother’s milk; and seldom if ever is he aware of the eternally problematic character of his concepts.

He uses this conceptual material, or, speaking more exactly, these conceptual tools of thought, as something obviously, immutably given; something having an objective value of truth which is hardly ever, and in any case not seriously, to be doubted. How could he do otherwise? How would the ascent of a mountain be possible, if the use of hands, legs, and tools had to be sanctioned step by step on the basis of the science of mechanics? And yet in the interests of science it is necessary over and over again to engage in the critique of these fundamental concepts, in order that we may not unconsciously be ruled by them. This becomes evident especially in those situations involving development of ideas in which the consistent use of the traditional fundamental concepts leads to paradoxes difficult to resolve. […]

If two different authors use the words “red,” “hard,” or “disappointed,” no one doubts that they mean approximately the same thing, because these words are connected with elementary experiences in a manner which is difficult to misinterpret. But in the case of words such as “place” or “space,” whose relation with psychological experience is less direct, there exists a far-reaching uncertainty of interpretation. […]

Now as to the concept of space, it seems that this was preceded by the psychologically simpler concept of place. Place is first of all a (small) portion of the earth’s surface identified by a name. The thing whose “place” is being specified is a “material object” or body. Simple analysis shows “place” also to be a group of material objects.

Does the word “place” have a meaning independent of this one, or can one assign such a meaning to it? If one has to give a negative answer to this question, then one is led to the view that space (or place) is a sort of order of material objects and nothing else. If the concept of space is formed and limited in this fashion, then to speak of empty space has no meaning. And because the formation of concepts has always been ruled by instinctive striving for economy, one is led quite naturally to reject the concept of empty space.

It is also possible, however, to think in a different way. Into a certain box we can place a definite number of grains of rice or cherries, etc. It is here a question of a property of the material object “box,” which property must be considered “real” in the same sense as the box itself. One can call this property the “space” of the box. There may be other boxes which in this sense have an equally large “space.” This concept “space” thus achieves a meaning which is freed from any connection with a particular material object.

In this way by a natural extension of “box space” one can arrive at the concept of an independent (absolute) space, unlimited in extent, in which all material objects are contained. Then a material object not situated in space is simply inconceivable; on the other hand, in the framework of this concept formation it is quite conceivable that an empty space might exist.

These two concepts of space may be contrasted as follows: (a) space as a positional quality of the world of material objects; (b) space as container of all material objects. In case (a), space without a material object is inconceivable. In case (b), a material object can only be conceived as existing in space; space then appears as a reality which in a certain sense is superior to the material world. Both space concepts are free creations of the human imagination, means devised for easier comprehension of our sense experience.

These schematic considerations concern the nature of space from the geometric and from the kinematic point of view, respectively. They are in a sense reconciled with each other by Descartes’ introduction of the coordinate system, although this already presupposes the logically more daring space concept (b).

The concept of space was enriched and complicated by Galileo and Newton, in that space must be introduced as the independent cause of the inertial behaviour of bodies if one wishes to give the classical principle of inertia (and therewith the classical law of motion) an exact meaning. To have realized this fully and clearly is in my opinion one of Newton’s greatest achievements.

In contrast to Leibniz and Huygens, it was clear to Newton that the space concept (a) was not sufficient to serve as the foundation for the inertia principle and the law of motion. He came to this decision even though he actively shared the uneasiness which was the cause of the opposition of the other two: space is not only introduced as an independent thing apart from material objects, but is also assigned an absolute role in the whole causal structure of the theory. This role is absolute in the sense that space (as an inertial system) acts on all material objects, while these do not in turn exert any reaction on space.

The fruitfulness of Newton’s system silenced these scruples for several centuries. Space of type (b) was generally accepted by scientists in the precise form of the inertial system, encompassing time as well. Today one would say about that memorable discussion: Newton’s decision was, in the contemporary state of science, the only possible one, and particularly the only fruitful one.

But the subsequent development of the problems, proceeding in a roundabout way which no one then could possibly foresee, has shown that the resistance of Leibniz and Huygens, intuitively well founded but supported by inadequate arguments, was actually justified.

It required a severe struggle to arrive at the concept of independent and absolute space, indispensable for the development of theory. It has required no less strenuous exertions subsequently to overcome this concept – a process which is probably by no means as yet completed.

Dr Jammer’s book is greatly concerned with the investigation of the status of the concept of space in ancient times and in the Middle Ages. On the basis of his studies, he is inclined towards the view that the modern concept of space of type (b), that is, space as container of all material objects, was not developed until after the Renaissance. It seems to me that the atomic theory of the ancients, with its atoms existing separately from each other, necessarily presupposed a space of type (b), while the more influential Aristotelian school tried to get along without the concept of independent (absolute) space. […]

The victory over the concept of absolute space or over that of the inertial system became possible only because the concept of the material object was gradually replaced as the fundamental concept by that of the field. Under the influence of the ideas of Faraday and Maxwell the notion developed that the whole of physical reality could perhaps be represented as a field whose components depend on four space-time parameters. If the laws of this field are in general covariant, that is, are not dependent on a particular choice of coordinate system, then the introduction of an independent (absolute) space is no longer necessary. That which constitutes the spatial character of reality is then simply the four-dimensionality of the field. There is then no “empty” space, that is, there is no space without a field. Dr Jammer’s presentation also deals with the memorable way in which the difficulties of this problem were overcome, at least to a great extent. Up to the present time, no one has found any method of avoiding the inertial system other than that by way of the field theory.

Princeton, New Jersey

Albert Einstein


Jammer on the differences between Place and Space:
[Quoted from (Jammer, 1993 (1969, 1954))]

(p.9): Space as a subject of philosophical inquiry appears very early in Greek philosophy. According to Aristotle, numbers were accredited with a kind of spatiality by the Pythagoreans: “The Pythagoreans, too, asserted the existence of the void and declared that it enters into the heavens out of the limitless breath – regarding the heavens as breathing the very vacancy – which vacancy ‘distinguishes’ natural objects, as constituting a kind of separation and division between things next to each other, its prime seat being in numbers, since it is this void that delimits their nature.” […]

The concept of space is still confounded with that of matter. A J. Burnett says: “The Pythagoreans, or some of them, certainly identified ‘air’ with the void. This is the beginning, but no more than the beginning, of the conception of abstract space or extension.” Only later is this confusion cleared up by Xutus and Philolaus. In Simplicius we find that Archytas, the Pythagorean, already had a clear understanding of this abstract notion, since, as related by Eudemus, he asked whether it would be possible at the end of the world to stretch out one’s hand or not. Unfortunately, Archytas’ work on the nature of space is lost except for a few fragments to be found in Simplicius’ Commentaries, according to which Archytas distinguished between place (topos), or space, and matter.

Space differs from matter and is independent of it. Every body occupies some place, and cannot exist unless its place exists. “Since what is moved is moved into a certain place and doing and suffering are motions, it is plain that place, in what is done and suffered exists, is the first of things. Since everything which is moved is moved into a certain place, it is plain that the place where the thing moving or being moved shall be, must exist first. Perhaps it is the first of all beings, since everything that exists is in a place and cannot exist without a place. If place has existence in itself and is independent of bodies, then, as Archytas seems to mean, place determines the volume of bodies.”

A characteristic property of space is that all things are in it, but it is never in something else; its surroundings are the infinite void itself. Apart from this metaphysical property, space has the physical property of setting frontiers or limits to bodies in it and of preventing these bodies from becoming infinitely large or small. It is also owing to this constraining power of space that the universe as a whole occupies a finite space. To Archytas, space is therefore not some pure extension, lacking all qualities or force, but is rather a kind of primordeal atmosphere, endowed with pressure and tension and bounded by the infinite void.

The function of the void, or of space, in the atomism of Democritus is too well known to need any elaboration here. But it is of interest to note that according to Democritus infinity of space is not only inherent in the concept itself, but may be deduced from the infinite number of atoms in existence, since these, although indivisible, have a certain magnitude and extension, even if they are not perceptible to our senses. Democritus himself seems not to have attributed weight to the atoms but to have assumed that as a result of constant collisions among themselves they were in motion in infinite space. It was only later, when the explanation of the cause of their motions was sought, that his disciples introduced weight as the cause of the “up and down” movements (Epicurus). If Aristotle says that Democritus’ atoms differed in weight according to their size, one has to assume – in modern words – that it was not gravitational force but “force of impact” that was implied. This point is of some importance for our point of view, since it shows that in the first atomistic conception of physical reality space was conceived as an empty extension without any influence on the motion of matter.

However, there still remains one question to be asked: Was space conceived by the atomists of antiquity as an unbounded extension, permeated by all bodies and permeating all bodies, or was it only the sum total of all the diastemata, the intervals that separate atom from atom and body from body, assuring their discreteness and possibility of motion? The stress laid time and again by the atomists on the existence of the void was directed against the school of Parmenides and Melissus, according to whom the universe was as compact plenum, one continuous unchanging whole. “Nor is there anything empty,” said Melissus, “for the empty is nothing and that which is nothing cannot be.”

Against such argument Leucippus and Democritus maintained the existence of the void as a logical conclusion of the assumption of the atomic structure of reality. But here the void or the empty means clearly unoccupied space. The universe is the full and the empty. Space, in this sense, is complementary to matter and is bounded by matter; matter and space are mutually exclusive. This interpretation gains additional weight if we note that the term “the empty” (kenon) was used often as synonymous with the word “space”; the term “the empty” obviously implies only the unoccupied space. Additional evidence is furnished by Leucippos’ explicit use of the adjective “porous” (manon) for the description of the structure of space, which indicates that he had in mind the intervals between particles of matter and not unbounded space.

Although Epicurus’ recurrent description of the universe as “body and void” seems also to confirm this interpretation, we find in Lucretius, who bases himself on Epicurus, a different view. In general, Lucretius’ complete and coherent scheme of atomistic natural philosophy is the best representation of Epicurean views. As far as the problem of space is concerned, Lucretius emphasises in the first book of De rerum natura the maxim: “All nature then, as it exists, by itself, is founded on two things: there are bodies and there is void in which these bodies are placed and through which they move about.” Here we find, in contrast to the early Greek atomism, a clear and explicit expression of the idea that bodies are placed in the void, in space. With Lucretius, therefore, space becomes an infinite receptacle for bodies.


Lucretius adduces a further argument for the infinitude of space which reveals an important physical aspect of the atomistic theory: If space were not infinite, he claims, all matter would have sunk in the course of past eternity in a mass to the bottom of space and nothing would exist any more. This remark shows clearly that Lucretius, in the wake of Epicurus, conceived space as endowed with an objectively distinguished direction, the vertical. It is in this direction in which the atoms are racing through space in parallel lines. According to Epicurus and Lucretius, space, though homogeneous, is not isotropic.


Plato, who, according to Aristotle, was not satisfied, as his predecessors were, with the mere statement of the existence of space, but “attempted to tell us what it is,” develops his theory of space mainly in Timaeus. The upshot of the rather obscure exposition of this dialogue, as interpreted by Aristotle, and in modern times by E. Zeller, is that matter – at least in one sense of the word – has to be identified with empty space. Although “Platonic matter” was sometimes held to be a kind of body lacking all quality (Stoics, Plutarch, Hegel) or to be the mere possibility of corporeality (Chalcidius, Neoplatonists), critical analysis seems to show that Plato intended to identify the world of physical bodies with the world of geometric forms. A physical body is merely a part of space limited by geometric surfaces containing nothing but empty space. With Plato, physics becomes geometry, just as with the Pythagoreans it became arithmetic. Stereometric similarity becomes the ordering principle in the formation of macroscopic bodies. […]

Physical coherence, or, if one likes, chemical affinity, is the outcome of stereometric formation in empty space, which itself is the undifferentiated material substrate, the raw material for the Demiurgus. […] Geometric structure is the final cause of what has been called “selective gravitation” where like attracts like. […] As much as matter is reduced to space, physics is reduced to geometry.

This identification of space and matter, or, in the words of later pseudo-Platonic teachings, of tridimensionality and matter, had a great influence on physical thought during the Middle Ages. For although Aristotle’s Organon was the standard text in logic, Plato’s Timaeus was succeeded by Aristotle’s Physics only in the middle of the twelfth century. It is perhaps not wrong to assume that the obscure and vague language of the Timaeus contributed to preventing the concept of space from becoming a subject of strict mathematical research. Greek mathematics disregards the geometry of space. Plato himself, for whom solid bodies and their geometry were of fundamental importance in the formulation of his philosophy, lamented the neglecting of this branch of mathematics. In the Republic he apologizes for failing to discuss solid geometry when listing the essential subjects for instruction. […]

Aristotle’s theory of space is expounded chiefly in his Categories and, what is of greater relevance for our purpose, in his Physics. In the Categories Aristotle begins his short discussion with the remark that quantity is either discrete or continuous. “Space,” belonging to the category of quantity, is a continuous quantity. “For the parts of a solid occupy a certain space, and these have a common boundary; it follows that the parts of space also, which are occupied by the parts of the solid, have the same common boundary as the parts of the solid. Thus, not only time but space also, is a continuous quantity, for its parts have a common boundary.” “Space” here is conceived as the sum total of all the places occupied by bodies, and “place” (topos), conversely, is conceived as that part of space whose limits coincide with the limits of the occupying body.

In the Physics Aristotle uses exclusively the term “place” (topos), so that strictly speaking the Physics does not advance a theory of space at all, but only a theory of place or a theory of positions in space. However, since the Platonic and the Democritian conceptions of space are unacceptable to the Aristotelian system of thought, and since the notion of empty space is incompatible with his physics, Aristotle develops only a theory of positions in space, with the exclusion of the rejected concept of general space.

For our purpose, Aristotle’s theory of places is of greatest pertinence not only because of its important implications of physics, but also because it was the most decisive stage for the further development of space theories. In our treatment we shall adhere as much as possible to Aristotle’s original terminology and use the term “place.”

In Book IV of the Physics Aristotle develops on an axiomatic basis a deductive theory of the characteristics of place. Place is an accidence, having real existence, but not independent existence in the sense of a substantial being. Aristotle’s four primary assumptions regarding our concept are as follows: “(1) That the place of a thing is no part or factor of the thing itself, but is that which embraces it; (2) that the immediate or “proper” place of a thing is neither smaller nor greater than the thing itself; (3) that the place where the thing is can be quitted by it, and is therefore separable from it; and lastly (4) that any and every place implies and involves the correlatives of ‘above’ and ‘below,’ and that all the elemental substances have a natural tendency to move towards their own special places, or to rest in them when there – such movement being ‘upward’ or ‘downward,’ and such rest ‘above’ or below,’ ”

It is this last assumption that makes space a carrier of qualitative differences and furnishes thereby the metaphysical foundation of the mechanics of “natural” motion. Starting from these assumptions, Aristotle proceeds by a lucid process of logical elimination to his famous definition of “place” as the adjacent boundary of the containing body. By this definition the concept became immune to all the criticisms that were designed to show the logical inconsistency of former definitions, as, for instance, Zeno’s famous epicheirema (Everything is in place; this means that it is in something; but if place is something, then place itself is in something, etc.). In fact, this “nest of superimposed places” is mentioned as an argument against the existence of a kind of dimensional entity – distinct from the body that has shifted away when the encircled content is taken out and changed again and again, while the encircling continent remains unchanged.

Further, this “replacement” of the content of a vessel by another content reveals that place is something different from its changing contents and so proves the reality of space. Of great importance from our point of view is a passage in Aristotle’s Physics in which space is likened (using a modern expression) to a field of force: “Moreover the trends of the physical elements (fire, earth, and the rest) show not only that locality or place is a reality but also that it exerts an active influence; for fire and earth are borne, the one upwards and the other downwards, if unimpeded, each towards its own ‘place,’ and these terms – ‘up’ and ‘down’ I mean, and the rest of the six dimensional directions – indicate subdivisions or distinct classes of positions or places in general.”

The dynamical field structure, inherent in space, is conditioned by the geometric structure of space as a whole. Space, as defined by Aristotle, namely, as the inner boundary of the containing receptacle, is, so to speak, a reference system which generally is of very limited scope. The place of the sailor is i the boat, the boat itself is in the river, and the river is in the river bed. This last receptacle is at rest relative to the earth and therefore also to the universe as a whole, according to contemporary cosmology. For astronomy, with its moving spheres, the reference system has to generalized still further, leading to the finite space of the universe limited by the interior boundary of the outermost sphere, which itself is not contained in any further receptacle.

This universal space, of spherical symmetry, has as its center the center of the earth, to which heavy bodies move under the dynamic influence intrinsic to space. It is natural for us, who have read Mach and Einstein, to raise the question whether the geometric aspect of this dynamical “field structure” depends on the distribution of matter in space or is completely independent of mass. Aristotle anticipated this question and tried to show that the dynamics of natural motion depends on spatial conditions only.


(p.25): As we have tried to show in this chapter, space was conceived by classical Greek philosophy and science at first as something inhomogeneous because of its local geometric variance (as with Plato), and later as something anisotropic owing to directional differentiation in the substratum (Aristotle). It is perhaps not too conjectural to assume that these doctrines concerning the nature of space account for the failure of mathematics, especially geometry, to deal with space as a subject of scientific inquiry. Perhaps this is the reason why Greek geometry was so much confined to the plane.

It may be objected that “space” according to Aristotle is “the adjacent boundary of the containing body” and so by its very definition is only of a two-dimensional character. But this objection ignores a clear passage in the Physics and another passage in De caelo. As Euclid’s Elements show, the science of solid geometry was developed only to a small extent and mostly confined to the mensuration of solid bodies, which is at least one reason why even the termini technici of solid geometry, compared with those of plane geometry, were so little standardized. The idea of coördinates in the plane seems to go back to pre-Greek sources, the ancient Egyptian hieroglyphic symbol for “district” (hesp) being a grid (plane rectangular coordinate system). It would therefore be only natural to expect some reference to spatial coordinates in Greek mathematics. But in the whole history of Greek mathematics no such reference is found. […]

The use of a three-dimensional coördinate system, and in particular of a rectangular spatial coördinate system, was not thought reasonable until the seventeenth century (Descartes, Frans van Schooten, Lahire, and Jean Bernoulli), when the concept of space had undergone a radical change. Undoubtedly Greek mathematics dealt with three-dimensional objects; Euclid himself, as related by Proclus, saw perhaps in the construction and investigation of the Platonic bodies the final aim of the Elements. Yet space, as adopted in mechanics or in astronomy, had never been geometrized in Greek science. For how could Euclidean space, with its homogeneous and infinite lines and planes, possibly fit into the finite and anisotropic Aristotelian universe?


(p.54): The first major contribution to the clarification of the concept of absolute space was made by Philoponus, or John the Grammarian, as he is often called (fl. c. A.D. 575). Philoponus is well known as the forerunner of the so called “impetus theory” in mechanics, which was the subject of profound investigation during the fourteenth century and which became in its later development the main point of departure for Galileo’s formulation of the basis of modern dynamics. We shall have occation to see how Philoponus’ revision of the Aristotelian conception of space is intrinsically connected with his impetus theory. He begins by pointing out an inner inconsistency in Aristotle’s theory of space. To Aristotle, place is the adjacent boundary of the containing body, provided this containing body itself is not in motion. If, for example, we hold a stone in a current of water, the constantly changing envelope of water clearly is not the “place” of the stone; otherwise the motionless stone would change its place continuously, which is self-contradictory. The stone’s place must therefore be the inner surface of the first immobile containing body, as, for instance, the river bed.

Philoponus now asks what actually is the place of the sublunary world of matter, subjected to generation and decay. According to Aristotle it is the concave surface of the first celestial sphere, the orbit of the moon. But, says Philoponus, this surface itself is constantly rotating, and therefore not immobile; on the contrary, a certain part of this surface successively touches other parts of the contained matter, even if these parts themselves happen not to be moving. To ascribe the place of our changing world to one of the highest spheres is of no avail, since all of them are in rotational motion. Philoponus rejects the argument that rotation about a fixed axis or fixed point is not local motion, since the sphere as a whole always occupies, so to say, the same place. Philoponus concentrates upon a fixed part of the rotating sphere and shows how this part occupies different places in the course of time. Hence he concludes that Aristotle’s definition of “place” leads to a cul-de-sac and must be rejected. The definition not only makes it impossible to determine the place of the sublunar world, but provides no answer to the question of the place or the space in which the outermost sphere is moving, for moving it certainly is.


(p.56): Inconsistencies of this kind proved to Philoponus that a new definition of “place” or space was necessary. According to him, the nature of space is to be sought in the tridimensional incorporeal volume extended in length, width, and depth, different altogether from the material body that is immersed in it. “Space is not the limiting surface of the surrounding body … it is a certain interval, measurable in three dimensions, incorporeal in its very nature and different from the body contained in it; it is pure dimensionality void of all corporeality; indeed, as far as matter is concerned, space and the void are identical.”


(p.57): It is clear that this rather abstract notion of space is incompatible with Aristotle’s dynamics, for Philoponus conceives space as pure dimensionality, lacking all qualitative differentiation. Space can no longer be conceived as the efficient cause of motion, compelling the body to move to its “natural place.” It is ridiculous to pretend that space, as such, possesses an inherent power. If every body tends to its natural place, it is not because it seeks to reach a certain surface; the reason is rather that it tends to the place which was assigned to it by the Demiurgus.”


(p.98): Although Newton cannot, as we have already remarked, be regarded as a positivist in the modern sense of the word, yet he drew a clear line of demarcation between science on the one hand and metaphysics on the other. The famous “Hypotheses non fingo,” although originally expressed only with relation to an explanation of gravitation, became his motto for the exclusion of the occult, metaphysical, or transcendental religious entities. His aim was not to abolish metaphysics, but to keep it distinct from physical investigation. It is well known that Newton, himself a religious man, never denied the existence of beings and entities that transcend human experience; he contended only that their existence had no relevance to scientific explanation: In its mundus discorsi, science has no place for them. Intimately acquainted with the problems of religion and metaphysics, Newton managed to keep them in a separate compartment of his mind, but for one exception, namely, his theory of space. Space thus occupies a unique place in his teachings.


Leave a Reply

Your email address will not be published. Required fields are marked *