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Space (mathematics)
The modern mathematics treats "space" quite differently from the classical mathematics. The differences are listed below; their origin is explained afterwards.
 
Mathematics uses a wide assortment of spaces. Many of them
are quite far from the ancient geometry.
 
==Short history==
In the ancient mathematics, "space" was a geometric abstraction of the
three-dimensional space observed in the everyday life. Axiomatic method was the main research tool since Euclid (about 300 BC). Coordinate method (analytic geometry) was added by Rene Descartes in 1637. At that time geometric theorems were treated as an absolute objective truth knowable through intuition and reason, similarly to objects of natural science and axioms were treated as obvious implications of definitions.
 
Two equivalence relations between geometric figures were used: congruence and similarity. Translations, rotations and reflections transform a figure into congruent figures; homotheties --- into similar figures. For example, all circles are mutually similar, but ellipses are not similar to circles. A third equivalence relation, introduced by projective geometry (Gaspard Monge, 1795), corresponds to projective transformations. Not only ellipses but also parabolas and hyperbolas turn into circles under appropriate projective transformations; they all are projectively equivalent figures.
 
The relation between the two geometries, Euclidean and projective, shows that mathematical objects are not given to us ''with their structure''. Rather, each mathematical theory describes its objects by ''some'' of their properties, precisely those that are put as axioms at the foundations of the theory.
 
Distances and angles are never mentioned in the axioms of the projective geometry and therefore cannot appear in its theorems. The question "what is the sum of the three angles of a triangle" is meaningful in the Euclidean geometry but meaningless in the projective geometry.
 
A different situation appeared in the 19 century: in some geometries the sum of the three angles of a triangle is well-defined but different from the classical value (180 degrees). The non-Euclidean hyperbolic geometry, introduced by Nikolai Lobachevsky in 1829 and Janos Bolyai in 1832 (and Carl Gauss in 1816, unpublished) stated that the sum depends on the triangle and is always less than 180 degrees. Eugenio Beltrami in 1868 and Felix Klein in 1871 have obtained Euclidean "models" of the non-Euclidean hyperbolic geometry, and thereby completely justified these theories.
 
This discovery forces the abandonment of the pretensions to the absolute truth of Euclidean geometry. It showed that axioms are not "obvious", nor "implications of definitions". Rather, they are hypotheses. To what extent do they correspond to an experimental reality? This important physical problem has nothing anymore to do with mathematics. Even if a "geometry" does not correspond to an experimental reality, its theorems remain no less "mathematical truths".
 
 
Thus, different three-dimensional spaces appeared: Euclidean, hyperbolic and
elliptic. These are symmetric spaces; a symmetric space looks the same
around every point.
 
Much more general, not necessarily symmetric spaces were introduced in
1854 by Riemann, to be used by Albert Einstein in 1916 as a foundation
of his general theory of relativity. An Einstein space looks
differently around different points, because its geometry is
influenced by matter.
 
In 1872 the Erlangen program by Felix Klein proclaimed various kinds
of geometry corresponding to various transformation groups. Thus, new
kinds of symmetric spaces appeared: metric, affine, projective (and
some others).
 
The distinction between Euclidean, hyperbolic and elliptic spaces is
not similar to the distinction between metric, affine and projective
spaces. In the latter case one wonders, which questions apply, in the
former — which answers hold. For example, the question about the sum
of the three angles of a triangle: is it equal to 180 degrees, or
less, or more? In Euclidean space the answer is "equal", in hyperbolic
space — "less"; in elliptic space — "more". However, this question
does not apply to an affine or projective space, since the notion of
angle is not defined in such spaces.
 
The classical Euclidean space is of course three-dimensional. However, the modern theory defines an <math>n</math>–dimensional Euclidean space as an affine space over an <math>n</math>–dimensional inner product space (over the reals); for <math>n=3</math> it is equivalent to the classical theory.
 
Euclidean axioms leave no freedom, they determine uniquely all
geometric properties of the space. More exactly: all three-dimensional
Euclidean spaces are mutually isomorphic. In this sense we have "the"
three-dimensional Euclidean space. Three-dimensional symmetric
hyperbolic (or elliptic) spaces differ by a single parameter, the
curvature. The definition of a Riemann space leaves a huge freedom,
more than a finite number of numeric parameters. On the other hand,
all affine (or projective) spaces are mutually isomorphic, provided
that they are three-dimensional (or n-dimensional for a given n) and
over the reals (or another given field of scalars).
 
== Modern approach==
Nowadays mathematics uses a wide assortment of spaces. Many of them
are quite far from the ancient geometry. Here is a rough and
incomplete classification according to the applicable questions
(rather than answers). We start with a basic class.
 
{| class="wikitable"
|-
! Space
! Stipulates
|-
| Topological
| Convergence, continuity. Open sets, closed sets.
|}
 
Straight lines are defined in projective spaces. In addition,
all questions applicable to topological spaces apply also to projective
spaces, since each projective space (over the reals) "downgrades" to the corresponding
topological space. Such relations between classes of spaces are shown
below.


==Differences==
{| class="wikitable"
{| class="wikitable"
|-
|-
! Space
! Classic
! Is richer than
! Modern
! Stipulates
|-
| Projective
| Topological space.
| Straight lines.
|-
| Affine
| Projective space.
| Parallel lines.
|-
| Linear
| Affine space.
| Origin. Vectors.
|-
|-
| Linear topological
| a single space
| Linear space. Topological space.
| many spaces of various kinds
|-
|-
| Metric
| axioms are obvious implications of definitions
| Topological space.
| axioms are conventional
| Distances.
|-
|-
| Normed
| theorems are absolute objective truth
| Linear topological space. Metric space.
| theorems are implications of the corresponding axioms
|-
| Inner product
| Normed space.
| Angles.
|-
| Euclidean
| Affine space. Metric space.
| Angles.
|}
|}
A finer classification uses answers to some (applicable) questions.
{| class="wikitable"
|-
! Space
! Special cases
! Properties
|-
| Linear
| three-dimensional
| Basis of 3 vectors.
|-
|
| finite-dimensional
| A finite basis.
|-
| Metric
| complete
| All Cauchy sequences converge.
|-
| Topological
| compact
| Every open covering has a finite sub-covering.
|-
|
| connected
| Only trivial open-and-closed sets.
|-
| Normed
| Banach
| Complete.
|-
| Inner product
| Hilbert
| Complete.
|}
Waiving distances and angles while retaining volumes (of geometric bodies) one moves toward measure theory and the corresponding spaces listed below. Besides the volume, a measure generalizes area, length, mass (or charge) distribution, and also probability distribution, according to Andrey Kolmogorov's approach to probability theory.
{| class="wikitable"
|-
! Space
! Stipulates
|-
| Measurable
| Measurable sets and functions.
|-
| Measure
| Measures and integrals.
|}
Measure space is richer than measurable space. Also, Euclidean space is richer than measure space.
{| class="wikitable"
|-
! Space
! Special cases
! Properties
|-
| Measurable
| standard
| Isomorphic to a Polish space with the Borel σ-algebra.
|-
| Measure
| standard
| Isomorphic <i>mod</i> 0 to a Polish space with a finite Borel measure.
|-
|
| σ-finite
| The whole space is a countable union of sets of finite measure.
|-
|
| finite
| The whole space is of finite measure.
|-
|
| Probability
| The whole space is of measure 1.
|}
These spaces are less geometric. In particular, the idea of dimension, applicable to topological spaces, therefore to all spaces listed in the previous tables, does not apply to measure spaces. Manifolds are much more geometric, but they are not called spaces. In fact, "spaces" are just mathematical structures (as defined by Nikola Bourbaki) that often (but not always) are more geometric than other structures.

Revision as of 10:18, 19 July 2009

The modern mathematics treats "space" quite differently from the classical mathematics. The differences are listed below; their origin is explained afterwards.

Differences

Classic Modern
a single space many spaces of various kinds
axioms are obvious implications of definitions axioms are conventional
theorems are absolute objective truth theorems are implications of the corresponding axioms
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