Space (mathematics)
In mathematics, a space is a set endowed with a structure defining the relationships among the elements of the set.
A subspace is a subset of the parent space which retains the same structure.
While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.
A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can represent numbers, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space. More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships between the points of a three-dimensional Euclidean space are uniquely determined by Euclid's axioms, and all three-dimensional Euclidean spaces are considered identical.
Topological notions such as continuity have natural definitions for every Euclidean space. However, topology does not distinguish straight lines from curved lines, and the relation between Euclidean and topological spaces is thus "forgetful". Relations of this kind are treated in more detail in the [|"Types of spaces"] section.
It is not always clear whether a given mathematical object should be considered as a geometric "space", or an algebraic "structure". A general definition of "structure", proposed by Bourbaki, embraces all common types of spaces, provides a general definition of isomorphism, and justifies the transfer of properties between isomorphic structures.
History
| Classic | Modern |
| 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 |
| space is three-dimensional | different concepts of dimension apply to different kind of spaces |
| space is the universe of geometry | spaces are just mathematical structures, they occur in various branches of mathematics |
Before the golden age of geometry
In ancient Greek mathematics, "space" was a geometric abstraction of the three-dimensional reality observed in everyday life. About 300 BC, Euclid gave axioms for the properties of space. Euclid built all of mathematics on these geometric foundations, going so far as to define numbers by comparing the lengths of line segments to the length of a chosen reference segment.The method of coordinates was adopted by René Descartes in 1637. At that time, geometric theorems were treated as absolute objective truths knowable through intuition and reason, similar 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 Gaspard Monge in 1795, occurs in projective geometry: 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 cannot appear in theorems of projective geometry, since these notions are neither mentioned in the axioms of projective geometry nor defined from the notions mentioned there. The question "what is the sum of the three angles of a triangle" is meaningful in Euclidean geometry but meaningless in projective geometry.
A different situation appeared in the 19th century: in some geometries the sum of the three angles of a triangle is well-defined but different from the classical value. Non-Euclidean hyperbolic geometry, introduced by Nikolai Lobachevsky in 1829 and János Bolyai in 1832 stated that the sum depends on the triangle and is always less than 180 degrees. Eugenio Beltrami in 1868 and Felix Klein in 1871 obtained Euclidean "models" of the non-Euclidean hyperbolic geometry, and thereby completely justified this theory as a logical possibility.
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 no longer has anything 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 choice of some objects existing in Euclidean space and some relations between these objects that satisfy all axioms of the non-Euclidean geometry. These Euclidean objects and relations "play" the non-Euclidean geometry like contemporary actors playing an ancient performance. Actors can imitate a situation that never occurred in reality. Relations between the actors on the stage imitate relations between the characters in the play. Likewise, the chosen relations between the chosen objects of the Euclidean model imitate 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 of geometry and afterwards
The word "geometry" initially meant a practical way of processing lengths, regions and volumes in the space in which we live, but was then extended widely.According to Bourbaki, the period between 1795 and 1872 can be called "the golden age of geometry". The original space investigated by Euclid is now called three-dimensional Euclidean space. Its axiomatization, started by Euclid 23 centuries ago, was reformed with Hilbert's axioms, Tarski's axioms and Birkhoff's axioms. These axiom systems describe the space via primitive notions constrained by a number of axioms.
Analytic geometry made 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 have been of more interest to amateurs than to professional mathematicians. However, the heritage of classical geometry was not lost. 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".
Simultaneously, numbers began to displace geometry as the foundation of mathematics. For instance, in Richard Dedekind's 1872 essay Stetigkeit und irrationale Zahlen, he asserts that points on a line ought to have the properties of Dedekind cuts, and that therefore a line was the same thing as the set of real numbers. Dedekind is careful to note that this is an assumption that is incapable of being proven. In modern treatments, Dedekind's assertion is often taken to be the definition of a line, thereby reducing geometry to arithmetic. Three-dimensional Euclidean space is defined to be an affine space whose associated vector space of differences of its elements is equipped with an inner product. A definition "from scratch", as in Euclid, is now not often used, since it does not reveal the relation of this space to other spaces. Also, a three-dimensional projective space is now defined as the space of all one-dimensional subspaces of a four-dimensional vector space. This shift in foundations requires a new set of axioms, and if these axioms are adopted, the classical axioms of geometry become theorems.
A space now consists of selected mathematical objects treated as points, and selected relationships between these points. Therefore, spaces are just mathematical structures of convenience. One may expect that the structures called "spaces" are perceived more geometrically than other mathematical objects, but this is not always true.
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. Contemporary mathematicians follow this idea routinely and find it extremely suggestive to use the terminology of classical geometry nearly everywhere.
Functions are important mathematical objects. Usually they form infinite-dimensional function spaces, as noted already by Riemann and elaborated in the 20th century by functional analysis.
Taxonomy of spaces
Three taxonomic ranks
While each type of space has its own definition, the general idea of "space" evades formalization. Some structures are called spaces, other are not, without a formal criterion. Moreover, there is no consensus on the general idea of "structure".According to Pudlák, "Mathematics cannot be explained completely by a single concept such as the mathematical structure. Nevertheless, Bourbaki's structuralist approach is the best that we have."
We will return to Bourbaki's structuralist approach in the last section "Spaces and structures", while we now outline a possible classification of spaces in the spirit of Bourbaki.
We classify spaces on three levels. Given that each mathematical theory describes its objects by some of their properties, the first question to ask is: which properties? This leads to the first classification level. On the second level, one takes into account answers to especially important questions. On the third level of classification, one takes into account answers to all possible questions.
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. 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.
The second-level classification distinguishes, for example, between Euclidean and non-Euclidean spaces; between finite-dimensional and infinite-dimensional spaces; between compact and non-compact spaces, etc.
In Bourbaki's terms, the second-level classification is the classification by "species". Unlike biological taxonomy, a space may belong to several species.
The third-level classification distinguishes, for example, 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 stipulated according to the first level. Mutually isomorphic spaces are thought of as copies of a single space. If one of them belongs to a given species then they all do.
The notion of isomorphism sheds light on the upper-level classification. Given a one-to-one correspondence between two spaces of the same upper-level class, one may ask whether it is an isomorphism or not. This question makes no sense for two spaces of different classes.
An isomorphism to itself is called an automorphism. Automorphisms of a Euclidean space are shifts, rotations, reflections and compositions of these. Euclidean space is homogeneous in the sense that every point can be transformed into every other point by some automorphism.
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. In Bourbaki's terms, the corresponding theory is univalent. In contrast, topological spaces are generally non-isomorphic; their theory is multivalent. A similar idea occurs in mathematical logic: a theory is called categorical if all its models of the same cardinality are mutually isomorphic. According to Bourbaki, the study of multivalent theories is the most striking feature which distinguishes modern mathematics from classical mathematics.