Editing Space (section)

== Philosophy of space ==

Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like the ''[[Timaeus (dialogue)|Timaeus]]'' of [[Plato]], or [[Socrates]] in his reflections on what the Greeks called ''[[khôra]]'' (i.e. "space"), or in the ''[[Physics (Aristotle)|Physics]]'' of [[Aristotle]] (Book IV, Delta) in the definition of ''topos'' (i.e. place), or in the later "geometrical conception of place" as "space ''qua'' extension" in the ''Discourse on Place'' (''Qawl fi al-Makan'') of the 11th-century Arab [[polymath]] [[Alhazen]].<ref>Refer to Plato's ''Timaeus'' in the Loeb Classical Library, [[Harvard University]], and to his reflections on ''khora''. See also Aristotle's ''Physics'', Book IV, Chapter 5, on the definition of ''topos''. Concerning Ibn al-Haytham's 11th century conception of "geometrical place" as "spatial extension", which is akin to [[Descartes]]' and Leibniz's 17th century notions of ''extensio'' and ''analysis situs'', and his own mathematical refutation of Aristotle's definition of ''topos'' in natural philosophy, refer to: [[Nader El-Bizri]], "In Defence of the Sovereignty of Philosophy: al-Baghdadi's Critique of Ibn al-Haytham's Geometrisation of Place", ''Arabic Sciences and Philosophy'' ([[Cambridge University Press]]), Vol. 17 (2007), pp. 57–80.</ref> 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]].

[[Isaac Newton]] viewed space as absolute, existing permanently and independently of whether there was any matter in it.<ref>French, A.J.; Ebison, M.G. (1986). ''Introduction to Classical Mechanics''. Dordrecht: Springer, p. 1.</ref> In contrast, other [[natural philosopher]]s, notably [[Gottfried Leibniz]], thought that space was in fact a collection of relations between objects, given by their [[distance]] and [[direction (geometry)|direction]] from one another. In the 18th century, the philosopher and theologian [[George Berkeley]] attempted to refute the "visibility of spatial depth" in his ''Essay Towards a New Theory of Vision''. Later, the [[metaphysic]]ian [[Immanuel Kant]] said that the concepts of space and time are not empirical ones derived from experiences of the outside world—they are elements of an already given systematic framework that humans possess and use to structure all experiences. Kant referred to the experience of "space" in his ''[[Critique of Pure Reason]]'' as being a subjective "pure ''[[a priori and a posteriori|a priori]]'' form of intuition".

=== Galileo ===
[[Galileo Galilei|Galilean]] and [[René Descartes|Cartesian]] theories about space, matter, and motion are at the foundation of the [[Scientific Revolution]], which is understood to have culminated with the publication of [[Isaac Newton|Newton]]'s ''[[Philosophiæ Naturalis Principia Mathematica|Principia Mathematica]]'' in 1687.<ref name="Huggett-1999">{{Cite book|title=Space from Zeno to Einstein: classic readings with a contemporary commentary|date=1999|publisher=MIT Press|editor=Huggett, Nick |isbn=978-0-585-05570-1|location=Cambridge, MA |oclc=42855123|bibcode = 1999sze..book.....H}}</ref> Newton's theories about space and time helped him explain the movement of objects. While his theory of space is considered the most influential in physics, it emerged from his predecessors' ideas about the same.<ref>{{Cite journal|last=Janiak|first=Andrew|year=2015|title=Space and Motion in Nature and Scripture: Galileo, Descartes, Newton|journal=Studies in History and Philosophy of Science|volume=51|pages=89–99|pmid=26227236|doi=10.1016/j.shpsa.2015.02.004|bibcode=2015SHPSA..51...89J}}</ref>

As one of the pioneers of [[modern science]], Galileo revised the established [[Aristotelianism|Aristotelian]] and [[Ptolemy|Ptolemaic]] ideas about a [[Geocentric model|geocentric]] cosmos. He backed the [[Nicolaus Copernicus|Copernican]] theory that the universe was [[Heliocentrism|heliocentric]], with a stationary Sun at the center and the planets—including the Earth—revolving around the Sun. If the Earth moved, the Aristotelian belief that its natural tendency was to remain at rest was in question. Galileo wanted to prove instead that the Sun moved around its axis, that motion was as natural to an object as the state of rest. In other words, for Galileo, celestial bodies, including the Earth, were naturally inclined to move in circles. This view displaced another Aristotelian idea—that all objects gravitated towards their designated natural place-of-belonging.<ref>{{Cite book|title=Time and space|last=Dainton |first=Barry |date=2001|publisher=McGill-Queen's University Press|isbn=978-0-7735-2302-9|location=Montreal|oclc=47691120}}</ref>

=== René Descartes ===
[[René Descartes|Descartes]] set out to replace the Aristotelian worldview with a theory about space and motion as determined by [[natural law]]s. In other words, he sought a [[Metaphysics|metaphysical]] foundation or a [[Mechanics|mechanical]] explanation for his theories about matter and motion. [[Cartesian space]] was [[Euclidean space|Euclidean]] in structure—infinite, uniform and flat.<ref>{{Cite book|title=Time and Space|last=Dainton|first=Barry|publisher=McGill-Queen's University Press|year=2014|pages=164}}</ref> It was defined as that which contained matter; conversely, matter by definition had a spatial extension so that there was no such thing as empty space.<ref name="Huggett-1999" />

The Cartesian notion of space is closely linked to his theories about the nature of the body, mind and matter. He is famously known for his "cogito ergo sum" (I think therefore I am), or the idea that we can only be certain of the fact that we can doubt, and therefore think and therefore exist. His theories belong to the [[Rationalism|rationalist]] tradition, which attributes knowledge about the world to our ability to think rather than to our experiences, as the [[Empiricism|empiricists]] believe.<ref>{{Cite book|title=Descartes: a very short introduction|last=Tom.|first=Sorell|date=2000|publisher=Oxford University Press|isbn=978-0-19-154036-3|location=Oxford|oclc=428970574}}</ref> He posited a clear distinction between the body and mind, which is referred to as the [[Mind–body dualism|Cartesian dualism]].

=== Leibniz and Newton ===
[[File:Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg|[[Gottfried Leibniz]]|upright|thumb]]
Following Galileo and Descartes, during the seventeenth century the [[philosophy of space and time]] revolved around the ideas of [[Gottfried Leibniz]], a German philosopher–mathematician, and [[Isaac Newton]], who set out two opposing theories of what space is. Rather than being an entity that independently exists over and above other matter, Leibniz held that space is no more than the collection of spatial relations between objects in the world: "space is that which results from places taken together".<ref>Leibniz, Fifth letter to Samuel Clarke. By H.G. Alexander (1956). ''The Leibniz-Clarke Correspondence''. Manchester: Manchester University Press, pp. 55–96.</ref> Unoccupied regions are those that ''could'' have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an idealised [[abstraction]] from the relations between individual entities or their possible locations and therefore could not be [[continuous probability distribution|continuous]] but must be [[discrete probability distribution|discrete]].<ref>Vailati, E. (1997). ''Leibniz & Clarke: A Study of Their Correspondence''. New York: Oxford University Press, p. 115.</ref>
Space could be thought of in a similar way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people.<ref>Sklar, L. (1992). ''Philosophy of Physics''. Boulder: Westview Press, p. 20.</ref>
Leibniz argued that space could not exist independently of objects in the world because that implies a difference between two universes exactly alike except for the location of the material world in each universe. But since there would be no observational way of telling these universes apart then, according to the [[identity of indiscernibles]], there would be no real difference between them. According to the [[principle of sufficient reason]], any theory of space that implied that there could be these two possible universes must therefore be wrong.<ref>Sklar, L. ''Philosophy of Physics''. p. 21.</ref>

[[Image:GodfreyKneller-IsaacNewton-1689.jpg|[[Isaac Newton]]|upright|thumb]]
Newton took space to be more than relations between material objects and based his position on [[observation]] and experimentation. For a [[Relational theory|relationist]] there can be no real difference between [[inertial frame of reference|inertial motion]], in which the object travels with constant [[velocity]], and [[non-inertial reference frame|non-inertial motion]], in which the velocity changes with time, since all spatial measurements are relative to other objects and their motions. But Newton argued that since non-inertial motion generates [[force]]s, it must be absolute.<ref>Sklar, L. ''Philosophy of Physics''. p. 22.</ref> He used the example of [[Bucket argument|water in a spinning bucket]] to demonstrate his argument. Water in a [[bucket]] is hung from a rope and set to spin, starts with a flat surface. After a while, as the bucket continues to spin, the surface of the water becomes concave. If the bucket's spinning is stopped then the surface of the water remains concave as it continues to spin. The concave surface is therefore apparently not the result of relative motion between the bucket and the water.<ref>{{cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Newton_bucket.html|title=Newton's bucket|work=st-and.ac.uk|access-date=20 July 2008|archive-url=https://web.archive.org/web/20080317062957/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Newton_bucket.html|archive-date=17 March 2008|url-status=live}}</ref> Instead, Newton argued, it must be a result of non-inertial motion relative to space itself. For several centuries the bucket argument was considered decisive in showing that space must exist independently of matter.

=== Kant ===
[[Image:Immanuel Kant portrait c1790.jpg|upright|thumb|[[Immanuel Kant]]]]
In the eighteenth century the German philosopher [[Immanuel Kant]] published his theory of space as "a property of our mind" by which "we represent to ourselves objects as outside us, and all as in space" in the [[Critique of Pure Reason]]<ref>{{cite book |last=Allison |first=Henry E. |author-link=Henry E. Allison |title=Kant's Transcendental Idealism: An Interpretation and Defense; Revised and Enlarged Edition | page=97-132 |isbn=978-0300102666 |year=2004 |publisher=Yale University Press}}</ref> On his view the nature of spatial predicates are "relations that only attach to the form of intuition alone, and thus to the subjective constitution of our mind, without which these predicates could not be attached to anything at all."<ref>{{cite book |last=Kant |first=Immanuel |author-link=Immanuel Kant |title=Critique of Pure Reason (The Cambridge Edition of the Works of Immanuel Kant) | page=A3/B37-38 |isbn=978-0-5216-5729-7 |year=1999 |publisher=Cambridge University Press}}</ref> This develops his theory of [[knowledge]] in which knowledge about space itself can be both ''a priori'' and ''[[analytic-synthetic distinction|synthetic]]''.<ref>Carnap, R. ''An Introduction to the Philosophy of Science''. pp. 177–178.</ref>
According to Kant, knowledge about space is ''synthetic'' because any proposition about space cannot be true ''merely'' in virtue of the meaning of the terms contained in the proposition. In the counter-example, the proposition "all unmarried men are bachelors" ''is'' true by virtue of each term's meaning. Further, space is ''a priori'' because it is the form of our receptive abilities to receive information about the external world. For example, someone without sight can still perceive spatial attributes via touch, hearing, and smell. Knowledge of space itself is ''a priori'' because it belongs to the subjective constitution of our mind as the form or manner of our intuition of external objects.

=== Non-Euclidean geometry ===
{{main|Non-Euclidean geometry}}

[[Image:Sphere closed path.svg|thumb|150px|left|[[Spherical geometry]] is similar to [[elliptical geometry]]. On a [[sphere]] (the [[Surface (topology)|surface]] of a [[ball (mathematics)|ball]]) there are no [[parallel line]]s.]]Euclid's ''Elements'' contained five postulates that form the basis for Euclidean geometry. One of these, the [[parallel postulate]], has been the subject of debate among mathematicians for many centuries. It states that on any [[Plane (mathematics)|plane]] on which there is a straight line ''L<sub>1</sub>'' and a point ''P'' not on ''L<sub>1</sub>'', there is exactly one straight line ''L<sub>2</sub>'' on the plane that passes through the point ''P'' and is parallel to the straight line ''L<sub>1</sub>''. Until the 19th century, few doubted the truth of the postulate; instead debate centered over whether it was necessary as an axiom, or whether it was a theory that could be derived from the other axioms.<ref>Carnap, R. ''An Introduction to the Philosophy of Science''. p. 126.</ref> Around 1830 though, the Hungarian [[János Bolyai]] and the Russian [[Nikolai Ivanovich Lobachevsky]] separately published treatises on a type of geometry that does not include the parallel postulate, called [[hyperbolic geometry]]. In this geometry, an [[Infinity|infinite]] number of parallel lines pass through the point ''P''. Consequently, the sum of angles in a triangle is less than 180° and the ratio of a [[circle]]'s [[circumference]] to its [[diameter]] is greater than [[pi]]. In the 1850s, [[Bernhard Riemann]] developed an equivalent theory of [[elliptical geometry]], in which no parallel lines pass through ''P''. In this geometry, triangles have more than 180° and circles have a ratio of circumference-to-diameter that is less than [[pi]].

{| class="wikitable" style="margin: 1em auto 1em auto" style="text-align:center"
! style="width:100px" | Type of geometry || style="width:90px" | Number of parallels || style="width:100px" | Sum of angles in a triangle || style="width:120px" | Ratio of circumference to diameter of circle || style="width:90px" | Measure of curvature
|-
! Hyperbolic
| Infinite || < 180° || > π || < 0
|-
! Euclidean
| 1 || 180° || π || 0
|-
! Elliptical
| 0 || > 180° || < π || > 0
|}

=== Gauss and Poincaré ===
[[Image:Carl Friedrich Gauss.jpg|upright|thumb|[[Carl Friedrich Gauss]]]]
[[Image:Young Poincare.jpg|left|upright|thumb|[[Henri Poincaré]]]]

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 [[triangulation|triangulating]] mountain tops in Germany.<ref>Carnap, R. ''An Introduction to the Philosophy of Science''. pp. 134–136.</ref>

[[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.<ref>Jammer, Max (1954). ''Concepts of Space. The History of Theories of Space in Physics''. Cambridge: Harvard University Press, p. 165.</ref> 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.<ref>A medium with a variable [[index of refraction]] could also be used to bend the path of light and again deceive the scientists if they attempt to use light to map out their geometry.</ref> 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 [[Conventionalism|convention]].<ref>Carnap, R. ''An Introduction to the Philosophy of Science''. p. 148.</ref> 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.<ref>Sklar, L. ''Philosophy of Physics''. p. 57.</ref>

=== Einstein ===
[[File:Albert Einstein 1947.jpg|right|upright|thumb|[[Albert Einstein]]]]
In 1905, [[Albert Einstein]] published his [[special theory of relativity]], which led to the concept that space and time can be viewed as a single construct known as ''[[spacetime]]''. In this theory, the [[speed of light]] in [[vacuum]] is the same for all observers—which has [[relativity of simultaneity|the result]] that two events that appear simultaneous to one particular observer will not be simultaneous to another observer if the observers are moving with respect to one another. Moreover, an observer will measure a moving clock to [[time dilation|tick more slowly]] than one that is stationary with respect to them; and objects are measured [[length contraction|to be shortened]] in the direction that they are moving with respect to the observer.

Subsequently, Einstein worked on a [[general theory of relativity]], which is a theory of how [[gravity]] interacts with spacetime. Instead of viewing gravity as a [[Force field (physics)|force field]] acting in spacetime, Einstein suggested that it modifies the geometric structure of spacetime itself.<ref>Sklar, L. ''Philosophy of Physics''. p. 43.</ref> According to the general theory, time [[gravitational time dilation|goes more slowly]] at places with lower gravitational potentials and rays of light bend in the presence of a gravitational field. Scientists have studied the behaviour of [[binary pulsar]]s, confirming the predictions of Einstein's theories.{{Citation needed|date=December 2024}} Non-Euclidean geometry is usually used to describe spacetime.{{Citation needed|date=December 2024}}