# Absolute space and time

Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame.

## Before Newton

A version of the concept of absolute space (in the sense of a preferred frame) can be seen in Aristotelian physics.[1] Robert S. Westman writes that a "whiff" of absolute space can be observed in Copernicus's classic work De revolutionibus orbium coelestium, where Copernicus uses the concept of an immobile sphere of stars.[2]

## Newton

Originally introduced by Sir Isaac Newton in Philosophiæ Naturalis Principia Mathematica, the concepts of absolute time and space provided a theoretical foundation that facilitated Newtonian mechanics.[3] According to Newton, absolute time and space respectively are independent aspects of objective reality:[4]

Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ...

According to Newton, absolute time exists independently of any perceiver and progresses at a consistent pace throughout the universe. Unlike relative time, Newton believed absolute time was imperceptible and could only be understood mathematically. According to Newton, humans are only capable of perceiving relative time, which is a measurement of perceivable objects in motion (like the Moon or Sun). From these movements, we infer the passage of time.

Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies: and which is vulgarly taken for immovable space ... Absolute motion is the translation of a body from one absolute place into another: and relative motion, the translation from one relative place into another ...

— Isaac Newton

These notions imply that absolute space and time do not depend upon physical events, but are a backdrop or stage setting within which physical phenomena occur. Thus, every object has an absolute state of motion relative to absolute space, so that an object must be either in a state of absolute rest, or moving at some absolute speed.[5] To support his views, Newton provided some empirical examples: according to Newton, a solitary rotating sphere can be inferred to rotate about its axis relative to absolute space by observing the bulging of its equator, and a solitary pair of spheres tied by a rope can be inferred to be in absolute rotation about their center of gravity (barycenter) by observing the tension in the rope.

Absolute time and space continue to be used in classical mechanics, but modern formulations by authors such as Walter Noll and Clifford Truesdell go beyond the linear algebra of elastic moduli to use topology and functional analysis for non-linear field theories.[6]

## Differing views

Two spheres orbiting around an axis. The spheres are distant enough for their effects on each other to be ignored, and they are held together by a rope. The rope is under tension if the bodies are rotating relative to absolute space according to Newton, or because they rotate relative to the universe itself according to Mach, or because they rotate relative to local geodesics according to general relativity.

Historically, there have been differing views on the concept of absolute space and time. Gottfried Leibniz was of the opinion that space made no sense except as the relative location of bodies, and time made no sense except as the relative movement of bodies.[7] George Berkeley suggested that, lacking any point of reference, a sphere in an otherwise empty universe could not be conceived to rotate, and a pair of spheres could be conceived to rotate relative to one another, but not to rotate about their center of gravity,[8] an example later raised by Albert Einstein in his development of general relativity.

A more recent form of these objections was made by Ernst Mach. Mach's principle proposes that mechanics is entirely about relative motion of bodies and, in particular, mass is an expression of such relative motion. So, for example, a single particle in a universe with no other bodies would have zero mass. According to Mach, Newton's examples simply illustrate relative rotation of spheres and the bulk of the universe.[9]

When, accordingly, we say that a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe.
—Ernst Mach; as quoted by Ciufolini and Wheeler: Gravitation and Inertia, p. 387

These views opposing absolute space and time may be seen from a modern stance as an attempt to introduce operational definitions for space and time, a perspective made explicit in the special theory of relativity.

Even within the context of Newtonian mechanics, the modern view is that absolute space is unnecessary. Instead, the notion of inertial frame of reference has taken precedence, that is, a preferred set of frames of reference that move uniformly with respect to one another. The laws of physics transform from one inertial frame to another according to Galilean relativity, leading to the following objections to absolute space, as outlined by Milutin Blagojević:[10]

• The existence of absolute space contradicts the internal logic of classical mechanics since, according to Galilean principle of relativity, none of the inertial frames can be singled out.
• Absolute space does not explain inertial forces since they are related to acceleration with respect to any one of the inertial frames.
• Absolute space acts on physical objects by inducing their resistance to acceleration but it cannot be acted upon.

Newton himself recognized the role of inertial frames.[11]

The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line.

As a practical matter, inertial frames often are taken as frames moving uniformly with respect to the fixed stars.[12] See Inertial frame of reference for more discussion on this.

## Special relativity

The concepts of space and time were separate in physical theory prior to the advent of special relativity theory, which connected the two and showed both to be dependent upon the reference frame's motion. In Einstein's theories, the ideas of absolute time and space were superseded by the notion of spacetime in special relativity, and curved spacetime in general relativity.

Absolute simultaneity refers to the concurrence of events in time at different locations in space in a manner agreed upon in all frames of reference. The theory of relativity does not have a concept of absolute time because there is a relativity of simultaneity. An event that is simultaneous with another event in one frame of reference may be in the past or future of that event in a different frame of reference,[7]:59 which negates absolute simultaneity.

## Einstein

Quoted below from his later papers, Einstein identified the term aether with "properties of space", a terminology that is not widely used. Einstein stated that in general relativity the "aether" is not absolute anymore, as the geodesic and therefore the structure of spacetime depends on the presence of matter.[13]

To deny the ether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view. For the mechanical behaviour of a corporeal system hovering freely in empty space depends not only on relative positions (distances) and relative velocities, but also on its state of rotation, which physically may be taken as a characteristic not appertaining to the system in itself. In order to be able to look upon the rotation of the system, at least formally, as something real, Newton objectivises space. Since he classes his absolute space together with real things, for him rotation relative to an absolute space is also something real. Newton might no less well have called his absolute space “Ether”; what is essential is merely that besides observable objects, another thing, which is not perceptible, must be looked upon as real, to enable acceleration or rotation to be looked upon as something real.

— Albert Einstein, Ether and the Theory of Relativity (1920)[14]

Because it was no longer possible to speak, in any absolute sense, of simultaneous states at different locations in the aether, the aether became, as it were, four-dimensional, since there was no objective way of ordering its states by time alone. According to special relativity too, the aether was absolute, since its influence on inertia and the propagation of light was thought of as being itself independent of physical influence....The theory of relativity resolved this problem by establishing the behaviour of the electrically neutral point-mass by the law of the geodetic line, according to which inertial and gravitational effects are no longer considered as separate. In doing so, it attached characteristics to the aether which vary from point to point, determining the metric and the dynamic behaviour of material points, and determined, in their turn, by physical factors, namely the distribution of mass/energy. Thus the aether of general relativity differs from those of classical mechanics and special relativity in that it is not ‘absolute’ but determined, in its locally variable characteristics, by ponderable matter.

— Albert Einstein, Über den Äther (1924)[15]

## General relativity

Special relativity eliminates absolute time (although Gödel and others suspect absolute time may be valid for some forms of general relativity)[16] and general relativity further reduces the physical scope of absolute space and time through the concept of geodesics.[7]:207–223 There appears to be absolute space in relation to the distant stars because the local geodesics eventually channel information from these stars, but it is not necessary to invoke absolute space with respect to any system's physics, as its local geodesics are sufficient to describe its spacetime.[17]

## References and notes

1. ^ Absolute and Relational Theories of Space and Motion
2. ^ Robert S. Westman, The Copernican Achievement, University of California Press, 1975, p. 45.
3. ^ Knudsen, Jens M.; Hjorth, Poul (2012). Elements of Newtonian Mechanics (illustrated ed.). Springer Science & Business Media. p. 30. ISBN 978-3-642-97599-8. Extract of page 30
4. ^ In Philosophiae Naturalis Principia Mathematica See the Principia on line at Andrew Motte Translation
5. ^ Space and Time: Inertial Frames (Stanford Encyclopedia of Philosophy)
6. ^ C. Truesdell (1977) A First Course in Rational Continuum Mechanics, Academic Press ISBN 0-12-701301-6
7. ^ a b c Ferraro, Rafael (2007), Einstein's Space-Time: An Introduction to Special and General Relativity, Springer Science & Business Media, Bibcode:2007esti.book.....F, ISBN 9780387699462
8. ^ Paul Davies; John Gribbin (2007). The Matter Myth: Dramatic Discoveries that Challenge Our Understanding of Physical Reality. Simon & Schuster. p. 70. ISBN 978-0-7432-9091-3.
9. ^ Ernst Mach; as quoted by Ignazio Ciufolini; John Archibald Wheeler (1995). Gravitation and Inertia. Princeton University Press. pp. 386–387. ISBN 978-0-691-03323-5.
10. ^ Milutin Blagojević (2002). Gravitation and Gauge Symmetries. CRC Press. p. 5. ISBN 978-0-7503-0767-3.
11. ^ Isaac Newton: Principia, Corollary V, p. 88 in Andrew Motte translation. See the Principia on line at Andrew Motte Translation
12. ^ C Møller (1976). The Theory of Relativity (Second ed.). Oxford UK: Oxford University Press. p. 1. ISBN 978-0-19-560539-6. OCLC 220221617.
13. ^ Kostro, L. (2001), "Albert Einstein's New Ether and his General Relativity" (PDF), Proceedings of the Conference of Applied Differential Geometry: 78–86, archived from the original (PDF) on 2010-08-02.
14. ^ Einstein, Albert: "Ether and the Theory of Relativity" (1920), Sidelights on Relativity (Methuen, London, 1922)
15. ^ A. Einstein (1924), "Über den Äther", Verhandlungen der Schweizerischen Naturforschenden Gesellschaft, 105 (2): 85–93. English translation: Concerning the Aether Archived 2010-11-04 at the Wayback Machine
16. ^ Savitt, Steven F. (September 2000), "There's No Time Like the Present (in Minkowski Spacetime)", Philosophy of Science, 67 (S1): S563–S574, CiteSeerX 10.1.1.14.6140, doi:10.1086/392846
17. ^ Gilson, James G. (September 1, 2004), Mach's Principle II, arXiv:physics/0409010, Bibcode:2004physics...9010G
Absolute theory

In philosophy, absolute theory (or absolutism) usually refers to a theory based on concepts (such as the concept of space) that exist independently of other concepts and objects. The absolute point of view was advocated in physics by Isaac Newton.An absolute theory is the opposite of a relational theory.

Aether theories

In physics, aether theories (also known as ether theories) propose the existence of a medium, a space-filling substance or field, thought to be necessary as a transmission medium for the propagation of electromagnetic or gravitational forces. Since the development of special relativity, theories using a substantial aether fell out of use in modern physics, and are now joined by more abstract models.This early modern aether has little in common with the aether of classical elements from which the name was borrowed. The assorted theories embody the various conceptions of this medium and substance.

Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.

In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to describe superstring theory, eleven dimensions can describe supergravity and M-theory, and the state-space of quantum mechanics is an infinite-dimensional function space.

The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.

Dual number

In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 (ε is nilpotent). The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε where a and b are uniquely determined real numbers. The dual numbers can also be thought of as the exterior algebra of a one-dimensional vector space; the general case of n dimensions leads to the Grassmann numbers.

The algebra of dual numbers is a ring that is a local ring since the principal ideal generated by ε is its only maximal ideal.

Dual numbers form the coefficients of dual quaternions.

Like the complex numbers and split-complex numbers, the dual numbers form an algebra that is 2-dimensional over the field of real numbers.

Ernst Mach

Ernst Waldfried Josef Wenzel Mach (; German: [ˈɛɐ̯nst max]; 18 February 1838 – 19 February 1916) was an Austrian physicist and philosopher, noted for his contributions to physics such as study of shock waves. The ratio of one's speed to that of sound is named the Mach number in his honor. As a philosopher of science, he was a major influence on logical positivism and American pragmatism. Through his criticism of Newton's theories of space and time, he foreshadowed Einstein's theory of relativity.

Fluent (mathematics)

A fluent is a time-varying quantity or variable. The term was used by Isaac Newton in his early calculus to describe his form of a function. The concept was introduced by Newton in 1665 and detailed in his mathematical treatise, Method of Fluxions. Newton described any variable that changed its value as a fluent – for example, the velocity of a ball thrown in the air. The derivative of a fluent is known as a fluxion, the main focus of Newton's calculus. A fluent can be found from its corresponding fluxion through integration.

Fluxion

The fluxion of a "fluent" (a time-varying quantity, or function) is its instantaneous rate of change, or gradient, at a given point. Fluxions were introduced by Isaac Newton to describe his form of a time derivative (a derivative with respect to time). Newton introduced the concept in 1665 and detailed them in his mathematical treatise, Method of Fluxions. Fluxions and fluents made up Newton's early calculus.

Fundamental interaction

In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and electromagnetic interactions, which produce significant long-range forces whose effects can be seen directly in everyday life, and the strong and weak interactions, which produce forces at minuscule, subatomic distances and govern nuclear interactions. Some scientists speculate that a fifth force might exist, but this is neither widely accepted nor proven.

Each of the known fundamental interactions can be described mathematically as a field. The gravitational force is attributed to the curvature of spacetime, described by Einstein's general theory of relativity. The other three are discrete quantum fields, and their interactions are mediated by elementary particles described by the Standard Model of particle physics.

Within the Standard Model, the strong interaction is carried by a particle called the gluon, and is responsible for quarks binding together to form hadrons, such as protons and neutrons. As a residual effect, it creates the nuclear force that binds the latter particles to form atomic nuclei. The weak interaction is carried by particles called W and Z bosons, and also acts on the nucleus of atoms, mediating radioactive decay. The electromagnetic force, carried by the photon, creates electric and magnetic fields, which are responsible for the attraction between orbital electrons and atomic nuclei which holds atoms together, as well as chemical bonding and electromagnetic waves, including visible light, and forms the basis for electrical technology. Although the electromagnetic force is far stronger than gravity, it tends to cancel itself out within large objects, so over large distances (on the scale of planets and galaxies), gravity tends to be the dominant force.

All four fundamental forces are believed to be related, and to unite into a single force at high energies on a minuscule scale, the Planck scale, but particle accelerators cannot produce the enormous energies required to experimentally probe this. Efforts to devise a common theoretical framework that would explain the relation between the forces are perhaps the greatest goal of today's theoretical physicists. The weak and electromagnetic forces have already been unified with the electroweak theory of Sheldon Glashow, Abdus Salam, and Steven Weinberg for which they received the 1979 Nobel Prize in physics. Progress is currently being made in uniting the electroweak and strong fields within a Grand Unified Theory (GUT). A bigger challenge is to find a way to quantize the gravitational field, resulting in a theory of quantum gravity (QG) which would unite gravity in a common theoretical framework with the other three forces. Some theories, notably string theory, seek both QG and GUT within one framework, unifying all four fundamental interactions along with mass generation within a theory of everything (ToE).

History of special relativity

The history of special relativity consists of many theoretical results and empirical findings obtained by Albert A. Michelson, Hendrik Lorentz, Henri Poincaré and others. It culminated in the theory of special relativity proposed by Albert Einstein and subsequent work of Max Planck, Hermann Minkowski and others.

Hourglass

An hourglass (or sandglass, sand timer, sand clock or egg timer) is a device used to measure the passage of time. It comprises two glass bulbs connected vertically by a narrow neck that allows a regulated trickle of material (historically sand) from the upper bulb to the lower one. Factors affecting the time it measured include sand quantity, sand coarseness, bulb size, and neck width. Hourglasses may be reused indefinitely by inverting the bulbs once the upper bulb is empty. Depictions of hourglasses in art survive in large numbers from antiquity to the present day, as a symbol for the passage of time. These were especially common sculpted as epitaphs on tombstones or other monuments, also in the form of the winged hourglass, a literal depiction of the well-known Latin epitaph tempus fugit ("time flies").

Isaac Newton Group of Telescopes

The Isaac Newton Group of Telescopes or ING consists of three optical telescopes: the William Herschel Telescope, the Isaac Newton Telescope, and the Jacobus Kapteyn Telescope, operated by a collaboration between the UK Science and Technology Facilities Council, the Dutch NWO and the Spanish IAC. The telescopes are located at Roque de los Muchachos Observatory on La Palma in the Canary Islands.

These telescopes were formerly under the control of the Royal Greenwich Observatory before UK government cutbacks in 1998.

Isaac Newton Telescope

The Isaac Newton Telescope or INT is a 2.54 m (100 in.) optical telescope run by the Isaac Newton Group of Telescopes at Roque de los Muchachos Observatory on La Palma in the Canary Islands since 1984.

Originally the INT was situated at Herstmonceux Castle in Sussex, England, which was the site of the Royal Greenwich Observatory after it moved away from Greenwich due to light pollution. It was inaugurated in 1967 by Queen Elizabeth II.

Herstmonceux suffered from poor weather, and the advent of mass air travel made it plausible for UK astronomers to run an overseas observatory. In 1979, the INT was shipped to La Palma, where it has remained ever since. It saw its second first light in 1984, with a video camera.Today, it is used mostly with the Wide Field Camera (WFC), a four CCD instrument with a field of view of 0.56x0.56 square degrees which was commissioned in 1997. The other main instrument available at the INT is the Intermediate Dispersion Spectrograph (IDS), recently re-introduced having been unavailable for a period of several years.

Lorentz transformation

In physics, the Lorentz transformations are a one-parameter family of linear transformations from a coordinate frame in space time to another frame that moves at a constant velocity, the parameter, within the former. The transformations are named after the Dutch physicist Hendrik Lorentz. The respective inverse transformation is then parametrized by the negative of this velocity.

The most common form of the transformation, parametrized by the real constant ${\displaystyle v,}$ representing a velocity confined to the x-direction, is expressed as

{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}}

where (t, x, y, z) and (t′, x′, y′, z′) are the coordinates of an event in two frames, where the primed frame is seen from the unprimed frame as moving with speed v along the x-axis, c is the speed of light, and ${\displaystyle \gamma =\textstyle \left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{-1}}$ is the Lorentz factor.

Expressing the speed as ${\displaystyle \beta ={\frac {v}{c}},}$ an equivalent form of the transformation is

{\displaystyle {\begin{aligned}ct'&=\gamma \left(ct-\beta x\right)\\x'&=\gamma \left(x-\beta ct\right)\\y'&=y\\z'&=z.\end{aligned}}}

Frames of reference can be divided into two groups: inertial (relative motion with constant velocity) and non-inertial (accelerating, moving in curved paths, rotational motion with constant angular velocity, etc.). The term "Lorentz transformations" only refers to transformations between inertial frames, usually in the context of special relativity.

In each reference frame, an observer can use a local coordinate system (most exclusively Cartesian coordinates in this context) to measure lengths, and a clock to measure time intervals. An observer is a real or imaginary entity that can take measurements, say humans, or any other living organism—or even robots and computers. An event is something that happens at a point in space at an instant of time, or more formally a point in spacetime. The transformations connect the space and time coordinates of an event as measured by an observer in each frame.

They supersede the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). The Galilean transformation is a good approximation only at relative speeds much smaller than the speed of light. Lorentz transformations have a number of unintuitive features that do not appear in Galilean transformations. For example, they reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events, but always such that the speed of light is the same in all inertial reference frames. The invariance of light speed is one of the postulates of special relativity.

Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. The Lorentz transformation is in accordance with special relativity, but was derived before special relativity.

The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space, the mathematical model of spacetime in special relativity, the Lorentz transformations preserve the spacetime interval between any two events. This property is the defining property of a Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a hyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.

Ludwig Lange (physicist)

Ludwig Lange (born June 21, 1863 in Gießen; died July 12, 1936 in Weinsberg) was a German physicist.

Solstice

A solstice is an event occurring when the Sun appears to reach its most northerly or southerly excursion relative to the celestial equator on the celestial sphere. Two solstices occur annually, around June 21 and December 21. The seasons of the year are determined by reference to both the solstices and the equinoxes.

The term solstice can also be used in a broader sense, as the day when this occurs. The day of a solstice in either hemisphere has either the most sunlight of the year (summer solstice) or the least sunlight of the year (winter solstice) for any place other than the Equator. Alternative terms, with no ambiguity as to which hemisphere is the context, are "June solstice" and "December solstice", referring to the months in which they take place every year. The word solstice is derived from the Latin sol ("sun") and sistere ("to stand still"), because at the solstices, the Sun's declination appears to "stand still"; that is, the seasonal movement of the Sun's daily path (as seen from Earth) stops at a northern or southern limit before reversing direction.

Space

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 (i.e. "space"), or in the 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. 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. In Isaac Newton's view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and 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 metaphysician 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 form of intuition".

In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space.

Term (time)

A term is a period of duration, time or occurrence, in relation to an event. To differentiate an interval or duration, common phrases are used to distinguish the observance of length are near-term or short-term, medium-term or mid-term and long-term.

It is also used as part of a calendar year, especially one of the three parts of an academic term and working year in the United Kingdom: Michaelmas term, Hilary term / Lent term or Trinity term / Easter term, the equivalent to the American semester. In America there is a midterm election held in the middle of the four-year presidential term, there are also academic midterm exams.

In economics, it is the period required for economic agents to reallocate resources, and generally reestablish equilibrium. The actual length of this period, usually numbered in years or decades, varies widely depending on circumstantial context. During the long term, all factors are variable.

In finance or financial operations of borrowing and investing, what is considered long-term is usually above 3 years, with medium-term usually between 1 and 3 years and short-term usually under 1 year. It is also used in some countries to indicate a fixed term investment such as a term deposit.

In law, the term of a contract is the duration for which it is to remain in effect (not to be confused with the meaning of "term" that denotes any provision of a contract). A fixed-term contract is one concluded for a pre-defined time.

The Principles of Mathematics

The Principles of Mathematics (PoM) is a 1903 book by Bertrand Russell, in which the author presented his famous paradox and argued his thesis that mathematics and logic are identical.The book presents a view of the foundations of mathematics and has become a classic reference. It reported on developments by Giuseppe Peano, Mario Pieri, Richard Dedekind, Georg Cantor, and others.

In 1905 Louis Couturat published a partial French translation that expanded the book's readership. In 1937 Russell prepared a new introduction saying, "Such interest as the book now possesses is historical, and consists in the fact that it represents a certain stage in the development of its subject." Further editions were printed in 1938, 1951, 1996, and 2009.

Watchmaker analogy

The watchmaker analogy or watchmaker argument is a teleological argument which states, by way of an analogy, that a design implies a designer. The analogy has played a prominent role in natural theology and the "argument from design," where it was used to support arguments for the existence of God and for the intelligent design of the universe, in both Christianity and Deism.

Sir Isaac Newton, among other leaders in the scientific revolution, including René Descartes, upheld "that the physical laws he had uncovered revealed the mechanical perfection of the workings of the universe to be akin to a watch, wherein the watchmaker is God."The 1859 publication of Charles Darwin's theory of natural selection put forward an explanation for complexity and adaptation, which reflects scientific consensus on the origins of biological diversity. In the eyes of some, this provided a counter-argument to the watchmaker analogy: for example, the evolutionary biologist Richard Dawkins referred to the analogy in his 1986 book The Blind Watchmaker giving his explanation of evolution. Others, however, consider the watchmaker analogy to be compatible with evolutionary creation, opining that the two concepts are not mutually exclusive. In the 19th century, deists, who championed the watchmaker analogy, held that Darwin's theory fit with "the principle of uniformitarianism—the idea that all processes in the world occur now as they have in the past" and that deistic evolution "provided an explanatory framework for understanding species variation in a mechanical universe."In the United States, starting in the 1960s, creationists revived versions of the argument to dispute the concepts of evolution and natural selection, and there was renewed interest in the watchmaker argument. The most famous statement of this teleological argument using the watchmaker analogy was given by William Paley in his 1802 book Natural Theology or Evidences of the Existence and Attributes of the Deity.

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