Space (mathematics)

The modern mathematics treats "space" quite differently from the classical mathematics. The differences are listed below; their origin and meaning are 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
relationships between points, lines etc. are determined by their nature	relationships between points, lines etc. are essential; their nature is not
mathematical objects are given to us with their structure	each mathematical theory describes its objects by some of their properties
geometry corresponds to an experimental reality	geometry is a mathematical truth
all geometric properties of the space follow from the axioms	axioms of a space need not determine all geometric properties
geometry is an autonomous and living science	classical geometry is a universal language of mathematics
the space is three-dimensional	different concepts of dimension apply to different kind of spaces

History

Before the golden age of geometry

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 René 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 forced 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".

A Euclidean model of a non-Euclidean geometry is a clever choice of some objects existing in Euclidean space and some relations between these objects that satisfy all axioms (therefore, all theorems) of the non-Euclidean geometry. These Euclidean objects and relations "play" the non-Euclidean geometry like contemporary actors playing an ancient performance! Relations between the actors only mimic relations between the characters in the play. Likewise, the chosen relations between the chosen objects of the Euclidean model only mimic the non-Euclidean relations. It shows that relations between objects are essential in mathematics, while the nature of the objects is not.

The golden age and afterwards: dramatic change

According to Nikolas Bourbaki, the period between 1795 ("Geometrie descriptive" of Monge) and 1872 (the "Erlangen programme" of Klein) can be called the golden age of geometry. Analytic geometry made a great progress and succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups. Since that time new theorems of classical geometry interest amateurs rather that professional mathematicians.

However, it does not mean that the heritage of the classical geometry was lost. Quite the contrary! According to Nikolas Bourbaki, "passed over in its role as an autonomous and living science, classical geometry is thus transfigured into a universal language of contemporary mathematics".

According to the famous inaugural lecture given by Bernhard Riemann in 1854, every mathematical object described by $n$ real-valued parameters may be treated as a point of the $n$-dimensional space of all such objects. Nowadays mathematicians follow this idea routinely and find it extremely suggestive to use the terminology of classical geometry nearly everywhere.

In order to fully appreciate the generality of this approach one should note that mathematics is "a pure theory of forms, which has as its purpose, not the combination of quantities, or of their images, the numbers, but objects of thought" (Hermann Hankel, 1867).

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 generally, 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 $n$–dimensional Euclidean space as an affine space over an $n$–dimensional inner product space (over the reals); for $n=3$ 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.

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.

Space	Is richer than	Stipulates
Projective	Topological space.	Straight lines.
Affine	Projective space.	Parallel lines.
Linear	Affine space.	Origin. Vectors.
Linear topological	Linear space. Topological space.
Metric	Topological space.	Distances.
Normed	Linear topological space. Metric space.
Inner product	Normed space.	Angles.
Riemann	Metric space.	Tangent spaces with inner product
Euclidean	Affine space. Riemann space.	Angles.

A finer classification uses answers to some (applicable) questions.

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 Andrei Kolmogorov's approach to probability theory.

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.

Space	Special cases	Properties
Measurable	standard	Isomorphic to a Polish space with the Borel σ-algebra.
Measure	standard	Isomorphic mod 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 Nikolas Bourbaki) that often (but not always) are more geometric than other structures.