Space is the boundless, three-dimensional extent in which
objects and events occur and have relative position and direction.
[1] Physical space is often conceived in three
linear dimensions, although modern
physicists usually consider it, with
time, to be part of the boundless four-dimensional
continuum known as
spacetime. In
mathematics one examines 'spaces' with different numbers of dimensions and with different underlying structures. The concept of space is considered to be of fundamental importance to an understanding of the physical
universe although disagreement continues between
philosophers over whether it is itself an entity, a relationship between entities, or part of a
conceptual framework.
Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like the
Timaeus of
Plato, in his reflections on what the Greeks called:
chora /
Khora (i.e. 'space'), or in the
Physics of
Aristotle (Book IV, Delta) in the definition of
topos (i.e. place), or even in the later 'geometrical conception of place' as 'space
qua extension' in the
Discourse on Place (
Qawl fi al-makan) of the 11th century Arab polymath
Ibn al-Haytham (
Alhazen).
[2] Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, particularly during the early development of
classical mechanics. In
Isaac Newton's view, space was absolute - in the sense that it existed permanently and independently of whether there were any matter in the space.
[3]
Other
natural philosophers, notably
Gottfried Leibniz, thought instead that space was a collection of relations between objects, given by their
distance and
direction from one another. In the 18th century, the philosopher and theologian
George Berkely attempted to refute the 'visibility of spatial depth' in his
Essay Towards a New Theory of Vision. Later, the great
metaphysician Immanuel Kant described space and time as elements of a systematic framework that humans use to structure their experience; he referred to 'space' in his
Critique of Pure Reason as being: a subjective 'pure
a priori form of intuition', hence that its existence depends on our human faculties.
In the 19th and 20th centuries mathematicians began to examine
non-Euclidean geometries, in which space can be said to be
curved, rather than
flat. According to
Albert Einstein's theory of general relativity, space around
gravitational fields deviates from Euclidean space.
[4] Experimental
tests of general relativity have confirmed that non-Euclidean space provides a better model for the shape of space.
Philosophy of space
Leibniz and Newton
In the seventeenth century, the
philosophy of space and time emerged as a central issue in
epistemology and
metaphysics. At its heart,
Gottfried Leibniz, the German philosopher-mathematician, and
Isaac Newton, the English physicist-mathematician, set out two opposing theories of what space is. Rather than being an entity that independently exists over and above other matter, Leibniz held that space is no more than the collection of spatial relations between objects in the world: "space is that which results from places taken together".
[5] Unoccupied regions are those that
could have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an idealised
abstraction from the relations between individual entities or their possible locations and therefore could not be
continuous but must be
discrete.
[6] Space could be thought of in a similar way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people.
[7] Leibniz argued that space could not exist independently of objects in the world because that implies a difference between two universes exactly alike except for the location of the material world in each universe. But since there would be no observational way of telling these universes apart then, according to the
identity of indiscernibles, there would be no real difference between them. According to the
principle of sufficient reason, any theory of space that implied that there could be these two possible universes, must therefore be wrong.
[8]
Newton took space to be more than relations between material objects and based his position on
observation and
experimentation. For a
relationist there can be no real difference between
inertial motion, in which the object travels with constant
velocity, and
non-inertial motion, in which the velocity changes with time, since all spatial measurements are relative to other objects and their motions. But Newton argued that since non-inertial motion generates
forces, it must be absolute.
[9] He used the example of
water in a spinning bucket to demonstrate his argument.
Water in a
bucket is hung from a rope and set to spin, starts with a flat surface. After a while, as the bucket continues to spin, the surface of the water becomes concave. If the bucket's spinning is stopped then the surface of the water remains concave as it continues to spin. The concave surface is therefore apparently not the result of relative motion between the bucket and the water.
[10] Instead, Newton argued, it must be a result of non-inertial motion relative to space itself. For several centuries the bucket argument was decisive in showing that space must exist independently of matter.
Kant
In the eighteenth century the German philosopher
Immanuel Kant developed a theory of
knowledge in which knowledge about space can be both
a priori and
synthetic.
[11] According to Kant, knowledge about space is
synthetic, in that statements about space are not simply true by virtue of the meaning of the words in the statement. In his work, Kant rejected the view that space must be either a substance or relation. Instead he came to the conclusion that space and time are not discovered by humans to be objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences.
[12]
Non-Euclidean geometry
Euclid's
Elements contained five postulates that form the basis for Euclidean geometry. One of these, the
parallel postulate has been the subject of debate among mathematicians for many centuries. It states that on any
plane on which there is a straight line
L1 and a point
P not on
L1, there is only one straight line
L2 on the plane that passes through the point
P and is parallel to the straight line
L1. Until the 19th century, few doubted the truth of the postulate; instead debate centered over whether it was necessary as an axiom, or whether it was a theory that could be derived from the other axioms.
[13] Around 1830 though, the
Hungarian JĂ¡nos Bolyai and the
Russian Nikolai Ivanovich Lobachevsky separately published treatises on a type of geometry that does not include the parallel postulate, called
hyperbolic geometry. In this geometry, an
infinite number of parallel lines pass through the point
P. Consequently the sum of angles in a triangle is less than 180
o and the ratio of a
circle's
circumference to its
diameter is greater than
pi. In the 1850s,
Bernhard Riemann developed an equivalent theory of
elliptical geometry, in which no parallel lines pass through
P. In this geometry, triangles have more than 180
o and circles have a ratio of circumference-to-diameter that is less than pi.
| Type of geometry | Number of parallels | Sum of angles in a triangle | Ratio of circumference to diameter of circle | Measure of curvature |
| Hyperbolic | Infinite | < 180o | > π | < 0 |
| Euclidean | 1 | 180o | π | 0 |
| Elliptical | 0 | > 180o | < π | > 0 |
Gauss and Poincaré
Although there was a prevailing Kantian consensus at the time, once non-Euclidean geometries had been formalised, some began to wonder whether or not physical space is curved.
Carl Friedrich Gauss, a German mathematician, was the first to consider an empirical investigation of the geometrical structure of space. He thought of making a test of the sum of the angles of an enormous stellar triangle and there are reports he actually carried out a test, on a small scale, by
triangulating mountain tops in Germany.
[14]
Henri Poincaré, a French mathematician and physicist of the late 19th century introduced an important insight in which he attempted to demonstrate the futility of any attempt to discover which geometry applies to space by experiment.
[15] He considered the predicament that would face scientists if they were confined to the surface of an imaginary large sphere with particular properties, known as a
sphere-world. In this world, the
temperature is taken to vary in such a way that all objects expand and contract in similar proportions in different places on the sphere. With a suitable falloff in temperature, if the scientists try to use measuring rods to determine the sum of the angles in a triangle, they can be deceived into thinking that they inhabit a plane, rather than a spherical surface.
[16] In fact, the scientists cannot in principle determine whether they inhabit a plane or sphere and, Poincaré argued, the same is true for the debate over whether real space is Euclidean or not. For him, which geometry was used to describe space, was a matter of
convention.
[17] Since
Euclidean geometry is simpler than non-Euclidean geometry, he assumed the former would always be used to describe the 'true' geometry of the world.
[18]
Einstein
In 1905,
Albert Einstein published a paper on a
special theory of relativity, in which he proposed that space and time be combined into a single construct known as
spacetime. In this theory, the
speed of light in a
vacuum is the same for all observers—which has
the result that two events that appear simultaneous to one particular observer will not be simultaneous to another observer if the observers are moving with respect to one another. Moreover, an observer will measure a moving clock to
tick more slowly than one that is stationary with respect to them; and objects are measured
to be shortened in the direction that they are moving with respect to the observer.
Over the following ten years Einstein worked on a
general theory of relativity, which is a theory of how
gravity interacts with spacetime. Instead of viewing gravity as a
force field acting in spacetime, Einstein suggested that it modifies the geometric structure of spacetime itself.
[19] According to the general theory, time
goes more slowly at places with lower gravitational potentials and rays of light bend in the presence of a gravitational field. Scientists have studied the behaviour of
binary pulsars, confirming the predictions of Einstein's theories and Non-Euclidean geometry is usually used to describe spacetime.
Mathematics
In modern mathematics
spaces are defined as
sets with some added structure. They are frequently described as different types of
manifolds, which are spaces that locally approximate to Euclidean space, and where the properties are defined largely on local connectedness of points that lie on the manifold. There are however, many diverse mathematical objects that are called spaces. For example,
vector spaces such as
function spaces may have infinite numbers of independent dimensions and a notion of distance very different to Euclidean space, and
topological spaces replace the concept of distance with a more abstract idea of nearness.
Physics
Classical mechanics
Space is one of the few
fundamental quantities in
physics, meaning that it cannot be defined via other quantities because nothing more fundamental is known at the present. On the other hand, it can be related to other fundamental quantities. Thus, similar to other fundamental quantities (like
time and
mass), space can be explored via
measurement and experiment.
Astronomy
Astronomy is the science involved with the observation, explanation and measuring of objects in
outer space.
Relativity
Before
Einstein's work on relativistic physics, time and space were viewed as independent dimensions. Einstein's discoveries showed that due to relativity of motion our space and time can be mathematically combined into one object —
spacetime. It turns out that distances in
space or in
time separately are not invariant with respect to Lorentz coordinate transformations, but distances in Minkowski space-time along
space-time intervals are—which justifies the name.
In addition, time and space dimensions should not be viewed as exactly equivalent in Minkowski space-time. One can freely move in space but not in time. Thus, time and space coordinates are treated differently both in
special relativity (where time is sometimes considered an
imaginary coordinate) and in
general relativity (where different signs are assigned to time and space components of
spacetime metric).
Furthermore, in
Einstein's general theory of relativity, it is postulated that space-time is geometrically distorted-
curved -near to gravitationally significant masses.
[20]
Experiments are ongoing to attempt to directly measure
gravitational waves. This is essentially solutions to the equations of general relativity, which describe moving ripples of spacetime. Indirect evidence for this has been found in the motions of the
Hulse-Taylor binary system.
Cosmology
Relativity theory leads to the
cosmological question of what shape the universe is, and where space came from. It appears that space was created in the
Big Bang, 13.7 billion years ago and has been expanding ever since. The overall shape of space is not known, but space is known to be expanding very rapidly due to the
Cosmic Inflation.
Spatial measurement
Main article:
Measurement
The measurement of
physical space has long been important. Although earlier societies had developed measuring systems, the
International System of Units, (SI), is now the most common system of units used in the measuring of space, and is almost universally used.
Currently, the standard space interval, called a standard meter or simply
meter, is defined as the
distance traveled by light in a vacuum during a time interval of exactly 1/299,792,458 of a second. This definition coupled with present definition of the
second is based on the
special theory of relativity in which the
speed of light plays the role of a fundamental constant of nature.
Geographical space
Geography is the branch of science concerned with identifying and describing the
Earth, utilizing spatial awareness to try and understand why things exist in specific locations.
Cartography is the mapping of spaces to allow better navigation, for visualization purposes and to act as a locational device.
Geostatistics apply statistical concepts to collected spatial data to create an estimate for unobserved phenomena.
Geographical space is often considered as land, and can have a relation to
ownership usage (in which space is seen as
property or territory). While some cultures assert the rights of the individual in terms of ownership, other cultures will identify with a communal approach to land ownership, while still other cultures such as
Australian Aboriginals, rather than asserting ownership rights to land, invert the relationship and consider that they are in fact owned by the land.
Spatial planning is a method of regulating the use of space at land-level, with decisions made at regional, national and international levels. Space can also impact on human and cultural behavior, being an important factor in
architecture, where it will impact on the design of buildings and structures, and on
farming.
Ownership of space is not restricted to land. Ownership of
airspace and of
waters is decided internationally. Other forms of ownership have been recently asserted to other spaces — for example to the
radio bands of the electromagnetic
spectrum or to
cyberspace.
Public space is a term used to define areas of land as collectively owned by the community, and managed in their name by delegated bodies; such spaces are open to all. While
private property is the land culturally owned by an individual or company, for their own use and pleasure.
Abstract space is a term used in
geography to refer to a hypothetical space characterized by complete homogeneity. When modeling activity or behavior, it is a conceptual tool used to limit
extraneous variables such as terrain.
In psychology
Psychologists first began to study the way space is perceived in the middle of the 19th century. Those now concerned with such studies regard it as a distinct branch of
psychology. Psychologists analyzing the perception of space are concerned with how recognition of an object's physical appearance or its interactions are perceived.
Other, more specialized topics studied include
amodal perception and
object permanence. The
perception of surroundings is important due to its necessary relevance to survival, especially with regards to
hunting and
self preservation as well as simply one's idea of
personal space.
Several space-related
phobias have been identified, including
agoraphobia (the fear of open spaces),
astrophobia (the fear of celestial space) and
claustrophobia (the fear of enclosed spaces).
See also
References
- ^ Britannica Online Encyclopedia: Space
- ^ Refer to Plato's Timaeus in the Loeb Classical Library, Harvard University, and to his reflections on: Chora / Khora. See also Aristotle's Physics, Book IV, Chapter 5, on the definition of topos. Concerning Ibn al-Haytham's 11th century conception of 'geometrical place' as 'spatial extension', which is akin to Descartes' and Leibniz's 17th century notions of extensio and analysis situs, and his own mathematical refutation of Aristotle's definition of topos in natural philosophy, refer to: Nader El-Bizri, 'In Defence of the Sovereignty of Philosophy: al-Baghdadi's Critique of Ibn al-Haytham's Geometrisation of Place', Arabic Sciences and Philosophy: A Historical Journal (Cambridge University Press), Vol.17 (2007), pp. 57-80.
- ^ French and Ebison, Classical Mechanics, p. 1
- ^ Carnap, R. An introduction to the Philosophy of Science
- ^ Leibniz, Fifth letter to Samuel Clarke
- ^ Vailati, E, Leibniz & Clarke: A Study of Their Correspondence p. 115
- ^ Sklar, L, Philosophy of Physics, p. 20
- ^ Sklar, L, Philosophy of Physics, p. 21
- ^ Sklar, L, Philosophy of Physics, p. 22
- ^ Newton's bucket
- ^ Carnap, R, An introduction to the philosophy of science, p. 177-178
- ^ Lucas, John Randolph. Space, Time and Causality. p. 149. ISBN 0198750579.
- ^ Carnap, R, An introduction to the philosophy of science, p. 126
- ^ Carnap, R, An introduction to the philosophy of science, p. 134-136
- ^ Jammer, M, Concepts of Space, p. 165
- ^ A medium with a variable index of refraction could also be used to bend the path of light and again deceive the scientists if they attempt to use light to map out their geometry
- ^ Carnap, R, An introduction to the philosophy of science, p. 148
- ^ Sklar, L, Philosophy of Physics, p. 57
- ^ Sklar, L, Philsosophy of Physics, p. 43
- ^ chapters 8 and 9- John A. Wheeler "A Journey Into Gravity and Spacetime" Scientific American ISBN 0-7167-6034-7
Our brains take their input from the rest of our bodies. What our bodies are like and how they function in the world thus structures the very concepts we can use to think. We cannot think just anything - only what our embodied brains permit."
Rafael NĂºĂ±ez and George Lakoff have been able to give an elaborate first answer to the questions: How can advanced mathematics arise from the physical brain and body? Given the very limited mathematical capacity of human brains at birth, how can advanced mathematical ideas be built up using the basic mechanisms of conceptual structure: image-schemas, frames, metaphors, and conceptual blends?
