page_title: Space
section_title: Gauss and Poincaré
hierarchical_section_title: Space / Philosophy of space / Gauss and Poincaré
caption_attribution_description: Henri Poincaré
context_page_description: Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.
  Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like the Timaeus of Plato, or Socrates in his reflections on what the Greeks called khôra, or in the Physics of Aristotle in the definition of topos, or in the later "geometrical conception of place" as "space qua extension" in the Discourse on Place of the 11th-century Arab polymath Alhazen. Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, particularly during the early development of classical mechanics.
context_section_description: Although there was a prevailing Kantian consensus at the time, once non-Euclidean geometries had been formalised, some began to wonder whether or not physical space is curved. Carl Friedrich Gauss, a German mathematician, was the first to consider an empirical investigation of the geometrical structure of space. He thought of making a test of the sum of the angles of an enormous stellar triangle, and there are reports that he actually carried out a test, on a small scale, by triangulating mountain tops in Germany.
  Henri Poincaré, a French mathematician and physicist of the late 19th century, introduced an important insight in which he attempted to demonstrate the futility of any attempt to discover which geometry applies to space by experiment. He considered the predicament that would face scientists if they were confined to the surface of an imaginary large sphere with particular properties, known as a sphere-world. In this world, the temperature is taken to vary in such a way that all objects expand and contract in similar proportions in different places on the sphere. With a suitable falloff in temperature, if the scientists try to use measuring rods to determine the sum of the angles in a triangle, they can be deceived into thinking that they inhabit a plane, rather than a spherical surface. In fact, the scientists cannot in principle determine whether they inhabit a plane or sphere and, Poincaré argued, the same is true for the debate over whether real space is Euclidean or not. For him, which geometry was used to describe space was a matter of convention. Since Euclidean geometry is simpler than non-Euclidean geometry, he assumed the former would always be used to describe the 'true' geometry of the world.
caption: Henri Poincaré
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