Space Through Time: A Constantly Evolving Concept

History of the Concept of Space

From Origins to Antiquity: Space as Extension

The notion of space has dozens of definitions depending on context, but none fully captures its essence. Its evolution, tied to the history of mathematics and physics, reflects a journey from primitive intuition to extreme complexity. How has this concept transformed our understanding of the world?

For primitive societies, studied by ethnologists, space does not exist as an abstract concept: the center of the world coincides with the center of the village. In ancient Greece, particularly during the time of Pericles (around 495 BCE), space was conceived as a limited extension occupied by bodies. With Plato (428–348 BCE), philosophy and science emerged: space became the void receptacle in which everything exists (as described in the Timaeus). Aristotle (384–322 BCE) defined it as the sum of places.

From Euclidean Geometry to the Renaissance: "Geometrized" Space

The first classical formalization of space appeared with Euclidean geometry (around 300 BCE). Space was idealized: points, lines, polyhedra, and conic sections deployed according to precise mathematical rules (length, area, volume). This model dominated during the Islamic Golden Age (≈750 to ≈1250), a period marked by the translation of Greek, Indian, and Persian scientific works into Arabic.

During the Renaissance, artists like Piero della Francesca revolutionized the representation of space through perspective, foreshadowing projective geometry. The 17th century marked a turning point with the Newtonian revolution: Newton laid the foundations for an absolute physical space and universal time, immutable and independent of the phenomena they contain.

Newtonian Space: An Absolute Framework

Newton established that:

• Space is a rigid, infinite container where all phenomena occur.
• Time flows uniformly and irreversibly.

This framework, dominant until the 20th century, was challenged by the theory of relativity. Today, in black holes, space and time merge, illustrating the limits of the Newtonian model.

The Revolution of Non-Euclidean Geometries

Space Becomes Curved: Breaking with Euclid

In the 19th century, mathematics radically redefined space: it was no longer flat but curved, with variable curvature (zero, positive, or negative). For example:

• A line or plane has zero curvature.
• A sphere has constant curvature equal to \( \frac{1}{R^2} \) (greater as its radius \( R \) decreases).

Curvature is described by a tensor, a complex mathematical object that varies across regions of space. "Bumpy" surfaces, like that of an ocean, are concrete examples: each point has a distinct curvature.

N.B.:
A tensor is a mathematical object generalizing vectors and matrices, capable of describing geometric or physical properties independent of the coordinate system. In general relativity, the Riemann curvature tensor (denoted \( R^\rho_{\sigma\mu\nu} \)) measures how spacetime curves under the influence of matter, while the metric tensor \( g_{\mu\nu} \) defines distances and angles in this space. For example, the curvature of a sphere (always positive) or that of a saddle (negative) is expressed via these tensors.

Einstein and Spacetime: The Fusion of Time and Space

In the 20th century, Einstein integrated time into space: spacetime, a concept introduced by Hermann Minkowski in 1908, is a 4-dimensional variety (3 spatial + 1 temporal) with variable curvature. This curvature is gravity (General Theory of Relativity, 1915). All objects, from planets to photons, follow this dynamic geometry. "Between the Earth and the Moon, there is geometry." The differential geometry of Bernhard Riemann (1826–1866) enabled this advance by allowing non-Euclidean spaces, where the angles of a triangle do not necessarily sum to 180°.

N.B.:
Differential geometry is a branch of mathematics that studies the geometric properties of curves, surfaces, and manifolds using differential calculus. It describes curved spaces (like the Earth's surface or spacetime) using tools such as derivatives, tensors, and differential equations. For example, it explains why the sum of the angles of a triangle drawn on a sphere exceeds 180° (unlike Euclidean geometry), or how the curvature of spacetime influences the motion of planets in general relativity.

Toward Increasingly Abstract Spaces

Non-Commutative Geometry and Quantum Mechanics

In 1985, Alain Connes developed non-commutative geometry, where:

• The notion of a point disappears: locating a particle becomes impossible.
• Coordinates (such as position) no longer commute (e.g., \( x \times y \neq y \times x \)).

This approach reflects the principles of quantum mechanics: Heisenberg's uncertainty principle forbids simultaneously knowing a particle's position and velocity with precision. In 1924, Louis de Broglie (1892–1987) postulated that all matter has an associated wave, whose wavelength depends on temperature (de Broglie wavelength).

N.B.:
Non-commutative geometry is a mathematical theory developed by Alain Connes in the 1980s, where the coordinates of space no longer commute (i.e., \(x \cdot y \neq y \cdot x\)). This revolutionary approach describes spaces where the classical notion of a point disappears, as in the quantum world. For example, in quantum mechanics, the position and velocity of a particle cannot be measured simultaneously with precision (Heisenberg's uncertainty principle), reflecting this fundamental non-commutativity of space at the microscopic scale.

Fiber Spaces and Modern Physics

Fiber spaces introduce a complex structure:

• A base space (e.g., a surface).
• Fibers attached to each point of the base (e.g., the Möbius strip).
• Connections that link these elements via mathematical "charts."

Applications:

• In quantum field theory, to describe fundamental interactions.
• In general relativity, to model the curvature of spacetime.

Toward a Unified Theory?

Over 2500 years, the concept of space has evolved: from Euclid to non-Euclidean varieties, from non-commutative geometry to fiber spaces. Yet, no theory successfully unifies spacetime at subatomic and macroscopic scales. Physicists like Carlo Rovelli (loop theory) suggest that space and time might emerge from quantum processes, rather than existing a priori. What if these notions were merely perceptual illusions, convenient tools for interpreting reality?

Table of the Evolution of the Concept of Space Through Time