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
the space is the universe of geometry	spaces are just mathematical structures, they occur in various branches of mathematics

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 than professional mathematicians.

However, it does not mean that the heritage of the classical geometry was lost. Quite the contrary! According to 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 parametrized by $n$ real numbers 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).

Functions are important mathematical objects. Usually they form infinite-dimensional spaces, as noted already by Riemann and elaborated in the 20 century by functional analysis.

An object parametrized by $n$ complex numbers may be treated as a point of a complex $n$-dimensional space. However, the same object is also parametrized by $2n$ real numbers (real parts and imaginary parts of the complex numbers), thus, a point of a real $2n$-dimensional space. The complex dimension differs from the real dimension. This is only the tip of the iceberg. The "algebraic" concept of dimension applies to linear spaces. The "topological" concept of dimension applies to topological spaces. There is also Hausdorff dimension for metric spaces; this one can be non-integer (especially for fractals). Some kinds of spaces (for instance, measure spaces) admit no concept of dimension at all.

The original space investigated by Euclid is now called "the three-dimensional Euclidean space". Its axiomatization, started by Euclid 23 centuries ago, was finalized in the 20 century by David Hilbert, Alfred Tarski and George Birkhoff. This approach describes the space via undefined primitives (such as "point", "between", "congruent") constrained by a number of axioms. Such a definition "from scratch" is now of little use, since it hides the standing of this space among other spaces. The modern approach defines the three-dimensional Euclidean space more algebraically, via linear spaces and quadratic forms, namely, as an affine space whose difference space is a three-dimensional inner product space.

Also a three-dimensional projective space is now defined non-classically, as the space of all one-dimensional subspaces (that is, straight lines through the origin) of a four-dimensional linear space.

A space consists now of selected mathematical objects (for instance, functions on another space, or subspaces of another space, or just elements of a set) treated as points, and selected relationships between these points. It shows that spaces are just mathematical structures. One may expect that the structures called "spaces" are more geometric than others, but this is not always true. For example, a differentiable manifold (called also smooth manifold) is much more geometric than a measurable space, but no one calls it "differentiable space" (nor "smooth space").

Taxonomy of spaces

Three taxonomic ranks

Spaces are classified on three levels. Given that each mathematical theory describes its objects by some of their properties, the first question to ask is: which properties?

For example, the upper-level classification distinguishes between Euclidean and projective spaces, since the distance between two points is defined in Euclidean spaces but undefined in projective spaces.

Another example. The question "what is the sum of the three angles of a triangle" makes sense in a Euclidean space but not in a projective space; this is an upper-level distinction. In a non-Euclidean space the question makes sense but is answered differently, which is not an upper-level distinction.

Also the distinction between a Euclidean plane and a Euclidean 3-dimensional space is not an upper-level distinction; the question "what is the dimension" makes sense in both cases.

In terms of Bourbaki the upper-level classification is basically the classification by "typical characterization" (or "typification").

On the second level of classification one takes into account answers to especially important questions (among the questions that make sense according to the first level). For example, this level distinguishes between Euclidean and non-Euclidean spaces; between finite-dimensional and infinite-dimensional spaces; between compact and non-compact spaces, etc.

In terms of Bourbaki the second-level classification is the classification by "species".

On the third level of classification, roughly speaking, one takes into account answers to all possible questions (that make sense according to the first level). For example, this level distinguishes between spaces of different dimension, but does not distinguish between a plane of a three-dimensional Euclidean space, treated as a two-dimensional Euclidean space, and the set of all pairs of real numbers, also treated as a two-dimensional Euclidean space. Likewise it does not distinguish between different Euclidean models of the same non-Euclidean space.

More formally, the third level classifies spaces up to isomorphism. An isomorphism between two spaces is defined as a one-to-one correspondence between the points of the first space and the points of the second space, that preserves all relations between the points, stipulated by the given "typification". Mutually isomorphic spaces are thought of as copies of a single space.

Two relations between species

Topological notions (continuity, convergence, open sets, closed sets etc.) are defined naturally in every Euclidean space. In other words, every Euclidean space is also a topological space. Every isomorphism between two Euclidean spaces is also an isomorphism between the corresponding topological spaces (called "homeomorphism"), but the converse is wrong: a homeomorphism may distort distances. In terms of Bourbaki, "topological space" is an underlying structure of the "Euclidean space" structure. Similar ideas occur in category theory: the category of Euclidean spaces is a concrete category over the category of topological spaces; the forgetful (or "stripping") functor maps the former category to the latter category.

A three-dimensional Euclidean space is a special case of a Euclidean space. In terms of Bourbaki, the species of three-dimensional Euclidean space is richer than the species of Euclidean space. Likewise, the species of compact topological space is richer than the species of topological space.

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).

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.