Space (mathematics)

In mathematics, a space is a set of objects that possess a specific structure. This structure defines how the elements within the set relate to one another. While we often think of space as an empty area, a mathematical space is actually a collection of points. These points can represent many different things. They might be simple numbers, complex functions, or even smaller subspaces.

The identity of a space is determined by these relationships rather than the nature of the points themselves. If two different spaces share the same internal relationships, they are considered identical through a one-to-one correspondence called an isomorphism.

To understand how a space functions, one must look at its defining rules, known as axioms. A subspace is a smaller part of a larger parent space that maintains that same structure. Different types of spaces exist depending on which relationships are prioritized. For example, a Euclidean space uses specific rules to determine distances and angles. In contrast, a topological space focuses on different properties, such as continuity. Topology is often described as a "forgetful" relation to Euclidean geometry because it does not distinguish between straight and curved lines.

There are several ways to classify these mathematical structures. One method involves looking at which properties are being described. This creates a hierarchy of classification. At the highest level, we can distinguish between Euclidean and projective spaces. In a Euclidean space, the distance between two points is a fundamental measurement. However, in a projective space, the concept of distance is not defined.

This means a question about the sum of a triangle's angles is meaningful in Euclidean geometry, but it is meaningless in projective geometry.

Historically, the concept of space has undergone a massive transformation. In ancient Greece, space was seen as a geometric abstraction of our three-dimensional reality. Around 300 BC, Euclid provided axioms to describe these properties. For centuries, mathematicians believed these geometric truths were absolute and objective. They viewed axioms as obvious implications of definitions. During this era, mathematicians used relations like congruence and similarity to compare figures.

For instance, a homothety can scale a figure to create a similar shape, such as turning one circle into another.

In 1637, René Descartes introduced analytic geometry by using the method of coordinates. This allowed for more precise computations. However, the 19th century brought a revolution that challenged the absolute nature of Euclidean geometry. In 1829, Nikolai Lobachevsky introduced non-Euclidean hyperbolic geometry. Around the same time, János Bolyai and Carl Friedrich Gauss also explored these ideas. They discovered that in certain spaces, the sum of the angles in a triangle is always less than 180 degrees.

This proved that axioms were not absolute truths, but rather hypotheses that could change the entire nature of a space.

This discovery was further validated by Eugenio Beltrami in 1868 and Felix Klein in 1871. They created Euclidean models of non-Euclidean geometry. These models showed that non-Euclidean rules could exist logically within a Euclidean framework. This shifted the view of mathematics away from experimental reality and toward logical consistency. It proved that the nature of the objects is less important than the relations between them.

Mathematicians realized that even if a geometry does not match our physical world, its theorems remain mathematically true.

Modern mathematics has moved toward a structuralist approach. The mathematician Nicolas Bourbaki proposed a general definition of structure that encompasses many types of spaces. Today, many mathematicians view spaces as convenient structures used across various branches of math. For example, Richard Dedekind suggested in 1872 that a line could be defined as the set of real numbers. This effectively reduced geometry to arithmetic.

Now, we see spaces as diverse tools, ranging from finite-dimensional models to infinite-dimensional function spaces used in functional analysis.

Ultimately, the study of space allows mathematicians to organize vast amounts of information. Whether studying Hilbert spaces, metric spaces, or probability spaces, the goal remains the same. We use these structures to define how different mathematical objects interact. By choosing different axioms, we can build entirely different mathematical universes. This flexibility is what makes the concept of a mathematical space one of the most powerful ideas in science.