# Space

- Affine space
- Banach space
- Cauchy space
- Euclidean space
- Function space
- Hardy space
- Hilbert space
- Inner product space
- Kolmogorov space
- Lp space
- Measure space
- Metric space
- Minkowski space
- Normed vector space
- Polish space
- Quotient space
- Sobolev space
- Topological space
- Uniform space
- Vector space

### From Wikipedia, the free encyclopedia that anyone can edit

**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.

Many of the philosophical questions arose in the 17th century, 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.^{[2]} 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, Immanuel Kant described space and time as elements of a systematic framework which humans use to structure their experience.

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.^{[3]} Experimental tests of general relativity have confirmed that non-Euclidean space provides a better model for the shape of space.

## Contents |

## 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 which 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".^{[4]} Unoccupied regions are those which *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.^{[5]} 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.^{[6]} Leibniz argued that space could not exist independently of objects in the world because that would imply that there would be 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 which implied that there could be these two possible universes, must therefore be wrong.^{[7]}

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.^{[8]} 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^{[9]}. 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*.^{[10]} 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.^{[11]}

### Non-Euclidean geometry

Euclid's *Elements* contained five postulates which 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 *L _{1}* and a point

*P*not on

*L*, there is only one straight line

_{1}*L*on the plane which passes through the point

_{2}*P*and is parallel to the straight line

*L*. 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 which could be derived from the other axioms.

_{1}^{[12]}Around 1830 though, the Hungarian János Bolyai and the Russian Nikolai Ivanovich Lobachevsky separately published treatises on a type of geometry which does not include the parallel postulate, called hyperbolic geometry. In this geometry, there are an infinite number of parallel lines which 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 there are no parallel lines which pass through

*P*. In this geometry, triangles have more than 180

^{o}and circles have a ratio of circumference to diameter which 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 | <>o | > π | <> |

Euclidean | 1 | 180^{o} | π | 0 |

Elliptical | 0 | > 180^{o} | < 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, the 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.^{[13]}

Henri Poincaré, a French mathematician and physicist of the late 19th century introduced an important insight which attempted to demonstrate the futility of any attempt to discover by experiment which geometry applies to space.^{[14]} He considered the predicament which 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.^{[15]} 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, it was a matter of convention which geometry was used to describe space.^{[16]} 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.^{[17]}

### 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 which 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.^{[18]} 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, function spaces in general have no close relation to Euclidean space.

## Physics

### Classical mechanics

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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.^{[19]}

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 and has been expanding ever since. The overall shape of space is not known, but space is known to be expanding very rapidly which is evident due to the Cosmic Inflation. Alan Guth whom is known for his Inflationary theory, presented the first ideas in a seminar at Stanford Linear Accelerator Center on January 23, 1980.

## Spatial 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 within science.

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.

## Geography

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

The way in which space is perceived is an area which psychologists first began to study in the middle of the 19th century, and it is now thought by those concerned with such studies to be a distinct branch within 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

- Aether theories
- Cosmology
- General relativity
- Personal space
- Shape of the universe
- Space exploration
- Spatial analysis
- Absolute space and time

## References

**^**Britannica Online Encyclopedia: Space**^**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.**^**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

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