Zeno's paradoxes

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Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. It is usually assumed, based on Plato's Parmenides (128a–d), that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides' view. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one." Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point. Some of Zeno's nine surviving paradoxes (preserved in Aristotle's Physics and Simplicius's commentary thereon) are essentially equivalent to one another. Aristotle offered a refutation of some of them. Three of the strongest and most famous—that of Achilles and the tortoise, the Dichotomy argument, and that of an arrow in flight—are presented in detail below.

Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum, also known as proof by contradiction. They are also credited as a source of the dialectic method used by Socrates. Some mathematicians and historians, such as Carl Boyer, hold that Zeno's paradoxes are simply mathematical problems, for which modern calculus provides a mathematical solution. Some philosophers, however, say that Zeno's paradoxes and their variations (see Thomson's lamp) remain relevant metaphysical problems. The origins of the paradoxes are somewhat unclear. Diogenes Laërtius, a fourth source for information about Zeno and his teachings, citing Favorinus, says that Zeno's teacher Parmenides was the first to introduce the paradox of Achilles and the tortoise. But in a later passage, Laërtius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.

Paradoxes of motion

Dichotomy paradox

That which is in locomotion must arrive at the half-way stage before it arrives at the goal.

— as recounted by Aristotle, Physics VI:9, 239b10

Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on.

The dichotomy

The resulting sequence can be represented as:

${\displaystyle \left\{\cdots ,{\frac {1}{16}},{\frac {1}{8}},{\frac {1}{4}},{\frac {1}{2}},1\right\}}$

This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.

This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can be neither completed nor begun, and so all motion must be an illusion.

This argument is called the "Dichotomy" because it involves repeatedly splitting a distance into two parts. An example with the original sense can be found in an asymptote. It is also known as the Race Course paradox.

Achilles and the tortoise

"Achilles and the Tortoise" redirects here. For other uses, see Achilles and the Tortoise (disambiguation).

Achilles and the tortoise

In a race, the quickest runner can never over­take the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

— as recounted by Aristotle, Physics VI:9, 239b15

In the paradox of Achilles and the tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. Suppose that each racer starts running at some constant speed, one faster than the other. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say 2 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise. As Aristotle noted, this argument is similar to the Dichotomy. It lacks, however, the apparent conclusion of motionlessness.

Arrow paradox

The arrow

Not to be confused with other paradoxes of the same name.

If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion.

— as recounted by Aristotle, Physics VI:9, 239b5

In the arrow paradox, Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one (duration-less) instant of time, the arrow is neither moving to where it is, nor to where it is not. It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.

Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points.

Three other paradoxes as given by Aristotle

Paradox of place

From Aristotle:

If everything that exists has a place, place too will have a place, and so on ad infinitum.

Paradox of the grain of millet

Description of the paradox from the Routledge Dictionary of Philosophy:

The argument is that a single grain of millet makes no sound upon falling, but a thousand grains make a sound. Hence a thousand nothings become something, an absurd conclusion.

Aristotle's refutation:

Zeno is wrong in saying that there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially.

Description from Nick Huggett:

This is a Parmenidean argument that one cannot trust one's sense of hearing. Aristotle's response seems to be that even inaudible sounds can add to an audible sound.

The moving rows (or stadium)

The moving rows

From Aristotle:

... concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This...involves the conclusion that half a given time is equal to double that time.

For an expanded account of Zeno's arguments as presented by Aristotle, see Simplicius's commentary On Aristotle's Physics.

Proposed solutions

Diogenes the Cynic

According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions (see solvitur ambulando). To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes.

Aristotle

Aristotle (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities"). Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles."

Archimedes

Before 212 BC, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. (See: Geometric series, 1/4 + 1/16 + 1/64 + 1/256 + · · ·, The Quadrature of the Parabola.) His argument, applying the method of exhaustion to prove that the infinite sum in question is equal to the area of a particular square, is largely geometric but quite rigorous. Today's analysis achieves the same result, using limits (see convergent series). These methods allow the construction of solutions based on the conditions stipulated by Zeno, i.e. the amount of time taken at each step is geometrically decreasing.

Thomas Aquinas

Thomas Aquinas, commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."

Bertrand Russell

Bertrand Russell offered what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is just change in position over time.

Hermann Weyl

Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy), which is that between any two different points in space (or time), there is always another point. Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. According to Hermann Weyl, the assumption that space is made of finite and discrete units is subject to a further problem, given by the "tile argument" or "distance function problem". According to this, the length of the hypotenuse of a right angled triangle in discretized space is always equal to the length of one of the two sides, in contradiction to geometry. Jean Paul Van Bendegem has argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox.

Henri Bergson

An alternative conclusion, proposed by Henri Bergson in his 1896 book Matter and Memory, is that, while the path is divisible, the motion is not. In this argument, instants in time and instantaneous magnitudes do not physically exist. An object in relative motion cannot have an instantaneous or determined relative position, and so cannot have its motion fractionally dissected.

Peter Lynds

In 2003, Peter Lynds put forth a very similar argument: all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist.  Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes. For more about the inability to know both speed and location, see Heisenberg uncertainty principle.

Nick Huggett

Nick Huggett argues that Zeno is assuming the conclusion when he says that objects that occupy the same space as they do at rest must be at rest.

Paradoxes in modern times

Infinite processes remained theoretically troublesome in mathematics until the late 19th century. With the epsilon-delta definition of limit, Weierstrass and Cauchy developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.

While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Kevin Brown and Francis Moorcroft claim that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes raise.

Popular literature often misrepresents Zeno's arguments. For example, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite. However, none of the original ancient sources has Zeno discussing the sum of any infinite series. Simplicius has Zeno saying "it is impossible to traverse an infinite number of things in a finite time". This presents Zeno's problem not with finding the sum, but rather with finishing a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"?

A humorous take is offered by Tom Stoppard in his 1972 play Jumpers, in which the principal protagonist, the philosophy professor George Moore, suggests that according to Zeno's paradox, Saint Sebastian, a 3rd Century Christian saint martyred by being shot with arrows, died of fright.

Debate continues on the question of whether or not Zeno's paradoxes have been resolved. In The History of Mathematics: An Introduction (2010) Burton writes, "Although Zeno's argument confounded his contemporaries, a satisfactory explanation incorporates a now-familiar idea, the notion of a 'convergent infinite series.'"

Bertrand Russell offered a "solution" to the paradoxes based on the work of Georg Cantor, but Brown concludes "Given the history of 'final resolutions', from Aristotle onwards, it's probably foolhardy to think we've reached the end. It may be that Zeno's arguments on motion, because of their simplicity and universality, will always serve as a kind of 'Rorschach image' onto which people can project their most fundamental phenomenological concerns (if they have any)."

A similar ancient Chinese philosophic consideration

Ancient Chinese philosophers from the Mohist School of Names during the Warring States period of China (479-221 BC) developed equivalents to some of Zeno's paradoxes. The scientist and historian Sir Joseph Needham, in his Science and Civilisation in China, describes an ancient Chinese paradox from the surviving Mohist School of Names book of logic which states, in the archaic ancient Chinese script, "a one-foot stick, every day take away half of it, in a myriad ages it will not be exhausted." Several other paradoxes from this philosophical school (more precisely, movement) are known, but their modern interpretation is more speculative.

Quantum Zeno effect

Main article: Quantum Zeno effect

In 1977, physicists E. C. George Sudarshan and B. Misra discovered that the dynamical evolution (motion) of a quantum system can be hindered (or even inhibited) through observation of the system. This effect is usually called the "quantum Zeno effect" as it is strongly reminiscent of Zeno's arrow paradox. This effect was first theorized in 1958.

Zeno behaviour

In the field of verification and design of timed and hybrid systems, the system behaviour is called Zeno if it includes an infinite number of discrete steps in a finite amount of time. Some formal verification techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour. In systems design these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.

Lewis Carroll and Douglas Hofstadter

What the Tortoise Said to Achilles, written in 1895 by Lewis Carroll, was an attempt to reveal an analogous paradox in the realm of pure logic. If Carroll's argument is valid, the implication is that Zeno's paradoxes of motion are not essentially problems of space and time, but go right to the heart of reasoning itself. Douglas Hofstadter made Carroll's article a centrepiece of his book Gödel, Escher, Bach: An Eternal Golden Braid, writing many more dialogues between Achilles and the Tortoise to elucidate his arguments. Hofstadter connects Zeno's paradoxes to Gödel's incompleteness theorem in an attempt to demonstrate that the problems raised by Zeno are pervasive and manifest in formal systems theory, computing and the philosophy of mind.

 

Philosophy of space and time

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https://en.wikipedia.org/wiki/Philosophy_of_space_and_time 

Philosophy of space and time is the branch of philosophy concerned with the issues surrounding the ontology, epistemology, and character of space and time. While such ideas have been central to philosophy from its inception, the philosophy of space and time was both an inspiration for and a central aspect of early analytic philosophy. The subject focuses on a number of basic issues, including whether time and space exist independently of the mind, whether they exist independently of one another, what accounts for time's apparently unidirectional flow, whether times other than the present moment exist, and questions about the nature of identity (particularly the nature of identity over time).

Ancient and medieval views

The earliest recorded philosophy of time was expounded by the ancient Egyptian thinker Ptahhotep (c. 2650–2600 BC) who said:

Follow your desire as long as you live, and do not perform more than is ordered, do not lessen the time of the following desire, for the wasting of time is an abomination to the spirit...

— 11th maxim of Ptahhotep 

The Vedas, the earliest texts on Indian philosophy and Hindu philosophy, dating back to the late 2nd millennium BC, describe ancient Hindu cosmology, in which the universe goes through repeated cycles of creation, destruction, and rebirth, with each cycle lasting 4,320,000 years. Ancient Greek philosophers, including Parmenides and Heraclitus, wrote essays on the nature of time.

Incas regarded space and time as a single concept, named pacha (Quechua: pacha, Aymara: pacha).

Plato, in the Timaeus, identified time with the period of motion of the heavenly bodies, and space as that in which things come to be. Aristotle, in Book IV of his Physics, defined time as the number of changes with respect to before and after, and the place of an object as the innermost motionless boundary of that which surrounds it.

In Book 11 of St. Augustine's Confessions, he ruminates on the nature of time, asking, "What then is time? If no one asks me, I know: if I wish to explain it to one that asketh, I know not." He goes on to comment on the difficulty of thinking about time, pointing out the inaccuracy of common speech: "For but few things are there of which we speak properly; of most things we speak improperly, still, the things intended are understood." But Augustine presented the first philosophical argument for the reality of Creation (against Aristotle) in the context of his discussion of time, saying that knowledge of time depends on the knowledge of the movement of things, and therefore time cannot be where there are no creatures to measure its passing (Confessions Book XI ¶30; City of God Book XI ch.6).

In contrast to ancient Greek philosophers who believed that the universe had an infinite past with no beginning, medieval philosophers and theologians developed the concept of the universe having a finite past with a beginning, now known as Temporal finitism. The Christian philosopher John Philoponus presented early arguments, adopted by later Christian philosophers and theologians of the form "argument from the impossibility of the existence of an actual infinite", which states:

"An actual infinite cannot exist."

"An infinite temporal regress of events is an actual infinite."

"∴ An infinite temporal regress of events cannot exist."

In the early 11th century, the Muslim physicist Ibn al-Haytham (Alhacen or Alhazen) discussed space perception and its epistemological implications in his Book of Optics (1021). He also rejected Aristotle's definition of topos (Physics IV) by way of geometric demonstrations and defined place as a mathematical spatial extension. His experimental proof of the intro-mission model of vision led to changes in the understanding of the visual perception of space, contrary to the previous emission theory of vision supported by Euclid and Ptolemy. In "tying the visual perception of space to prior bodily experience, Alhacen unequivocally rejected the intuitiveness of spatial perception and, therefore, the autonomy of vision. Without tangible notions of distance and size for correlation, sight can tell us next to nothing about such things."

Realism and anti-realism

A traditional realist position in ontology is that time and space have existence apart from the human mind. Idealists, by contrast, deny or doubt the existence of objects independent of the mind. Some anti-realists, whose ontological position is that objects outside the mind do exist, nevertheless doubt the independent existence of time and space.

In 1781, Immanuel Kant published the Critique of Pure Reason, one of the most influential works in the history of the philosophy of space and time. He describes time as an a priori notion that, together with other a priori notions such as space, allows us to comprehend sense experience. Kant holds that neither space nor time are substance, entities in themselves, or learned by experience; he holds, rather, that both are elements of a systematic framework we use to structure our experience. Spatial measurements are used to quantify how far apart objects are, and temporal measurements are used to quantitatively compare the interval between (or duration of) events. Although space and time are held to be transcendentally ideal in this sense, they are also empirically real—that is, not mere illusions.

Some idealist writers, such as J. M. E. McTaggart in The Unreality of Time, have argued that time is an illusion (see also The flow of time, below).

The writers discussed here are for the most part realists in this regard; for instance, Gottfried Leibniz held that his monads existed, at least independently of the mind of the observer.

Absolutism and relationalism

Leibniz and Newton

The great debate between defining notions of space and time as real objects themselves (absolute), or mere orderings upon actual objects (relational), began between physicists Isaac Newton (via his spokesman, Samuel Clarke) and Gottfried Leibniz in the papers of the Leibniz–Clarke correspondence.

Arguing against the absolutist position, Leibniz offers a number of thought experiments with the purpose of showing that there is contradiction in assuming the existence of facts such as absolute location and velocity. These arguments trade heavily on two principles central to his philosophy: the principle of sufficient reason and the identity of indiscernibles. The principle of sufficient reason holds that for every fact, there is a reason that is sufficient to explain what and why it is the way it is and not otherwise. The identity of indiscernibles states that if there is no way of telling two entities apart, then they are one and the same thing.

The example Leibniz uses involves two proposed universes situated in absolute space. The only discernible difference between them is that the latter is positioned five feet to the left of the first. The example is only possible if such a thing as absolute space exists. Such a situation, however, is not possible, according to Leibniz, for if it were, a universe's position in absolute space would have no sufficient reason, as it might very well have been anywhere else. Therefore, it contradicts the principle of sufficient reason, and there could exist two distinct universes that were in all ways indiscernible, thus contradicting the identity of indiscernibles.

Standing out in Clarke's (and Newton's) response to Leibniz's arguments is the bucket argument: Water in a bucket, hung from a rope and set to spin, will start with a flat surface. As the water begins to spin in the bucket, the surface of the water will become concave. If the bucket is stopped, the water will continue to spin, and while the spin continues, the surface will remain concave. The concave surface is apparently not the result of the interaction of the bucket and the water, since the surface is flat when the bucket first starts to spin, it becomes concave as the water starts to spin, and it remains concave as the bucket stops.

In this response, Clarke argues for the necessity of the existence of absolute space to account for phenomena like rotation and acceleration that cannot be accounted for on a purely relationalist account. Clarke argues that since the curvature of the water occurs in the rotating bucket as well as in the stationary bucket containing spinning water, it can only be explained by stating that the water is rotating in relation to the presence of some third thing—absolute space.

Leibniz describes a space that exists only as a relation between objects, and which has no existence apart from the existence of those objects. Motion exists only as a relation between those objects. Newtonian space provided the absolute frame of reference within which objects can have motion. In Newton's system, the frame of reference exists independently of the objects contained within it. These objects can be described as moving in relation to space itself. For almost two centuries, the evidence of a concave water surface held authority.

Mach

Another important figure in this debate is 19th-century physicist Ernst Mach. While he did not deny the existence of phenomena like that seen in the bucket argument, he still denied the absolutist conclusion by offering a different answer as to what the bucket was rotating in relation to: the fixed stars.

Mach suggested that thought experiments like the bucket argument are problematic. If we were to imagine a universe that only contains a bucket, on Newton's account, this bucket could be set to spin relative to absolute space, and the water it contained would form the characteristic concave surface. But in the absence of anything else in the universe, it would be difficult to confirm that the bucket was indeed spinning. It seems equally possible that the surface of the water in the bucket would remain flat.

Mach argued that, in effect, the water experiment in an otherwise empty universe would remain flat. But if another object were introduced into this universe, perhaps a distant star, there would now be something relative to which the bucket could be seen as rotating. The water inside the bucket could possibly have a slight curve. To account for the curve that we observe, an increase in the number of objects in the universe also increases the curvature in the water. Mach argued that the momentum of an object, whether angular or linear, exists as a result of the sum of the effects of other objects in the universe (Mach's Principle).

Einstein

Albert Einstein proposed that the laws of physics should be based on the principle of relativity. This principle holds that the rules of physics must be the same for all observers, regardless of the frame of reference that is used, and that light propagates at the same speed in all reference frames. This theory was motivated by Maxwell's equations, which show that electromagnetic waves propagate in a vacuum at the speed of light. However, Maxwell's equations give no indication of what this speed is relative to. Prior to Einstein, it was thought that this speed was relative to a fixed medium, called the luminiferous ether. In contrast, the theory of special relativity postulates that light propagates at the speed of light in all inertial frames, and examines the implications of this postulate.

All attempts to measure any speed relative to this ether failed, which can be seen as a confirmation of Einstein's postulate that light propagates at the same speed in all reference frames. Special relativity is a formalization of the principle of relativity that does not contain a privileged inertial frame of reference, such as the luminiferous ether or absolute space, from which Einstein inferred that no such frame exists.

Einstein generalized relativity to frames of reference that were non-inertial. He achieved this by positing the Equivalence Principle, which states that the force felt by an observer in a given gravitational field and that felt by an observer in an accelerating frame of reference are indistinguishable. This led to the conclusion that the mass of an object warps the geometry of the space-time surrounding it, as described in Einstein's field equations.

In classical physics, an inertial reference frame is one in which an object that experiences no forces does not accelerate. In general relativity, an inertial frame of reference is one that is following a geodesic of space-time. An object that moves against a geodesic experiences a force. An object in free fall does not experience a force, because it is following a geodesic. An object standing on the earth, however, will experience a force, as it is being held against the geodesic by the surface of the planet.

Einstein partially advocates Mach's principle in that distant stars explain inertia because they provide the gravitational field against which acceleration and inertia occur. But contrary to Leibniz's account, this warped space-time is as integral a part of an object as are its other defining characteristics, such as volume and mass. If one holds, contrary to idealist beliefs, that objects exist independently of the mind, it seems that relativistics commits them to also hold that space and temporality have exactly the same type of independent existence.

Conventionalism

The position of conventionalism states that there is no fact of the matter as to the geometry of space and time, but that it is decided by convention. The first proponent of such a view, Henri Poincaré, reacting to the creation of the new non-Euclidean geometry, argued that which geometry applied to a space was decided by convention, since different geometries will describe a set of objects equally well, based on considerations from his sphere-world.

This view was developed and updated to include considerations from relativistic physics by Hans Reichenbach. Reichenbach's conventionalism, applying to space and time, focuses around the idea of coordinative definition.

Coordinative definition has two major features. The first has to do with coordinating units of length with certain physical objects. This is motivated by the fact that we can never directly apprehend length. Instead we must choose some physical object, say the Standard Metre at the Bureau International des Poids et Mesures (International Bureau of Weights and Measures), or the wavelength of cadmium to stand in as our unit of length. The second feature deals with separated objects. Although we can, presumably, directly test the equality of length of two measuring rods when they are next to one another, we can not find out as much for two rods distant from one another. Even supposing that two rods, whenever brought near to one another are seen to be equal in length, we are not justified in stating that they are always equal in length. This impossibility undermines our ability to decide the equality of length of two distant objects. Sameness of length, to the contrary, must be set by definition.

Such a use of coordinative definition is in effect, on Reichenbach's conventionalism, in the General Theory of Relativity where light is assumed, i.e. not discovered, to mark out equal distances in equal times. After this setting of coordinative definition, however, the geometry of spacetime is set.

As in the absolutism/relationalism debate, contemporary philosophy is still in disagreement as to the correctness of the conventionalist doctrine.

Structure of space-time

Building from a mix of insights from the historical debates of absolutism and conventionalism as well as reflecting on the import of the technical apparatus of the General Theory of Relativity, details as to the structure of space-time have made up a large proportion of discussion within the philosophy of space and time, as well as the philosophy of physics. The following is a short list of topics.

Relativity of simultaneity

According to special relativity each point in the universe can have a different set of events that compose its present instant. This has been used in the Rietdijk–Putnam argument to demonstrate that relativity predicts a block universe in which events are fixed in four dimensions.

Invariance vs. covariance

Bringing to bear the lessons of the absolutism/relationalism debate with the powerful mathematical tools invented in the 19th and 20th century, Michael Friedman draws a distinction between invariance upon mathematical transformation and covariance upon transformation.

Invariance, or symmetry, applies to objects, i.e. the symmetry group of a space-time theory designates what features of objects are invariant, or absolute, and which are dynamical, or variable.

Covariance applies to formulations of theories, i.e. the covariance group designates in which range of coordinate systems the laws of physics hold.

This distinction can be illustrated by revisiting Leibniz's thought experiment, in which the universe is shifted over five feet. In this example the position of an object is seen not to be a property of that object, i.e. location is not invariant. Similarly, the covariance group for classical mechanics will be any coordinate systems that are obtained from one another by shifts in position as well as other translations allowed by a Galilean transformation.

In the classical case, the invariance, or symmetry, group and the covariance group coincide, but they part ways in relativistic physics. The symmetry group of the general theory of relativity includes all differentiable transformations, i.e., all properties of an object are dynamical, in other words there are no absolute objects. The formulations of the general theory of relativity, unlike those of classical mechanics, do not share a standard, i.e., there is no single formulation paired with transformations. As such the covariance group of the general theory of relativity is just the covariance group of every theory.

Historical frameworks

A further application of the modern mathematical methods, in league with the idea of invariance and covariance groups, is to try to interpret historical views of space and time in modern, mathematical language.

In these translations, a theory of space and time is seen as a manifold paired with vector spaces, the more vector spaces the more facts there are about objects in that theory. The historical development of spacetime theories is generally seen to start from a position where many facts about objects are incorporated in that theory, and as history progresses, more and more structure is removed.

For example, Aristotelian space and time has both absolute position and special places, such as the center of the cosmos, and the circumference. Newtonian space and time has absolute position and is Galilean invariant, but does not have special positions.

Holes

With the general theory of relativity, the traditional debate between absolutism and relationalism has been shifted to whether spacetime is a substance, since the general theory of relativity largely rules out the existence of, e.g., absolute positions. One powerful argument against spacetime substantivalism, offered by John Earman is known as the "hole argument".

This is a technical mathematical argument but can be paraphrased as follows:

Define a function d as the identity function over all elements over the manifold M, excepting a small neighbourhood H belonging to M. Over H d comes to differ from identity by a smooth function.

With use of this function d we can construct two mathematical models, where the second is generated by applying d to proper elements of the first, such that the two models are identical prior to the time t=0, where t is a time function created by a foliation of spacetime, but differ after t=0.

These considerations show that, since substantivalism allows the construction of holes, that the universe must, on that view, be indeterministic. Which, Earman argues, is a case against substantivalism, as the case between determinism or indeterminism should be a question of physics, not of our commitment to substantivalism.

Direction of time

The problem of the direction of time arises directly from two contradictory facts. Firstly, the fundamental physical laws are time-reversal invariant; if a cinematographic film were taken of any process describable by means of the aforementioned laws and then played backwards, it would still portray a physically possible process. Secondly, our experience of time, at the macroscopic level, is not time-reversal invariant. Glasses can fall and break, but shards of glass cannot reassemble and fly up onto tables. We have memories of the past, and none of the future. We feel we can't change the past but can influence the future.

Causation solution

One solution to this problem takes a metaphysical view, in which the direction of time follows from an asymmetry of causation. We know more about the past because the elements of the past are causes for the effect that is our perception. We feel we can't affect the past and can affect the future because we can't affect the past and can affect the future.

There are two main objections to this view. First is the problem of distinguishing the cause from the effect in a non-arbitrary way. The use of causation in constructing a temporal ordering could easily become circular. The second problem with this view is its explanatory power. While the causation account, if successful, may account for some time-asymmetric phenomena like perception and action, it does not account for many others.

However, asymmetry of causation can be observed in a non-arbitrary way which is not metaphysical in the case of a human hand dropping a cup of water which smashes into fragments on a hard floor, spilling the liquid. In this order, the causes of the resultant pattern of cup fragments and water spill is easily attributable in terms of the trajectory of the cup, irregularities in its structure, angle of its impact on the floor, etc. However, applying the same event in reverse, it is difficult to explain why the various pieces of the cup should fly up into the human hand and reassemble precisely into the shape of a cup, or why the water should position itself entirely within the cup. The causes of the resultant structure and shape of the cup and the encapsulation of the water by the hand within the cup are not easily attributable, as neither hand nor floor can achieve such formations of the cup or water. This asymmetry is perceivable on account of two features: i) the relationship between the agent capacities of the human hand (i.e., what it is and is not capable of and what it is for) and non-animal agency (i.e., what floors are and are not capable of and what they are for) and ii) that the pieces of cup came to possess exactly the nature and number of those of a cup before assembling. In short, such asymmetry is attributable to the relationship between i) temporal direction and ii) the implications of form and functional capacity.

The application of these ideas of form and functional capacity only dictates temporal direction in relation to complex scenarios involving specific, non-metaphysical agency which is not merely dependent on human perception of time. However, this last observation in itself is not sufficient to invalidate the implications of the example for the progressive nature of time in general.

Thermodynamics solution

The second major family of solutions to this problem, and by far the one that has generated the most literature, finds the existence of the direction of time as relating to the nature of thermodynamics.

The answer from classical thermodynamics states that while our basic physical theory is, in fact, time-reversal symmetric, thermodynamics is not. In particular, the second law of thermodynamics states that the net entropy of a closed system never decreases, and this explains why we often see glass breaking, but not coming back together.

But in statistical mechanics things become more complicated. On one hand, statistical mechanics is far superior to classical thermodynamics, in that thermodynamic behavior, such as glass breaking, can be explained by the fundamental laws of physics paired with a statistical postulate. But statistical mechanics, unlike classical thermodynamics, is time-reversal symmetric. The second law of thermodynamics, as it arises in statistical mechanics, merely states that it is overwhelmingly likely that net entropy will increase, but it is not an absolute law.

Current thermodynamic solutions to the problem of the direction of time aim to find some further fact, or feature of the laws of nature to account for this discrepancy.

Laws solution

A third type of solution to the problem of the direction of time, although much less represented, argues that the laws are not time-reversal symmetric. For example, certain processes in quantum mechanics, relating to the weak nuclear force, are not time-reversible, keeping in mind that when dealing with quantum mechanics time-reversibility comprises a more complex definition. But this type of solution is insufficient because 1) the time-asymmetric phenomena in quantum mechanics are too few to account for the uniformity of macroscopic time-asymmetry and 2) it relies on the assumption that quantum mechanics is the final or correct description of physical processes.

One recent proponent of the laws solution is Tim Maudlin who argues that the fundamental laws of physics are laws of temporal evolution (see Maudlin [2007]). However, elsewhere Maudlin argues: "[the] passage of time is an intrinsic asymmetry in the temporal structure of the world... It is the asymmetry that grounds the distinction between sequences that runs from past to future and sequences which run from future to past" [ibid, 2010 edition, p. 108]. Thus it is arguably difficult to assess whether Maudlin is suggesting that the direction of time is a consequence of the laws or is itself primitive.

Flow of time

The problem of the flow of time, as it has been treated in analytic philosophy, owes its beginning to a paper written by J. M. E. McTaggart, in which he proposes two "temporal series". The first series, which means to account for our intuitions about temporal becoming, or the moving Now, is called the A-series. The A-series orders events according to their being in the past, present or future, simpliciter and in comparison to each other. The B-series eliminates all reference to the present, and the associated temporal modalities of past and future, and orders all events by the temporal relations earlier than and later than. In many ways, the debate between proponents of these two views can be seen as a continuation of the early modern debate between the view that there is absolute time (defended by Isaac Newton) and the view that there is only merely relative time (defended by Gottfried Leibniz).

McTaggart, in his paper "The Unreality of Time", argues that time is unreal since a) the A-series is inconsistent and b) the B-series alone cannot account for the nature of time as the A-series describes an essential feature of it.

Building from this framework, two camps of solution have been offered. The first, the A-theorist solution, takes becoming as the central feature of time, and tries to construct the B-series from the A-series by offering an account of how B-facts come to be out of A-facts. The second camp, the B-theorist solution, takes as decisive McTaggart's arguments against the A-series and tries to construct the A-series out of the B-series, for example, by temporal indexicals.

Dualities

Quantum field theory models have shown that it is possible for theories in two different space-time backgrounds, like AdS/CFT or T-duality, to be equivalent.

Presentism and eternalism

Main articles: Presentism (philosophy of time) and Eternalism (philosophy of time)

According to Presentism, time is an ordering of various realities. At a certain time some things exist and others do not. This is the only reality we can deal with and we cannot for example say that Homer exists because at the present time he does not. An Eternalist, on the other hand, holds that time is a dimension of reality on a par with the three spatial dimensions, and hence that all things—past, present and future—can be said to be just as real as things in the present. According to this theory, then, Homer really does exist, though we must still use special language when talking about somebody who exists at a distant time—just as we would use special language when talking about something far away (the very words near, far, above, below, and such are directly comparable to phrases such as in the past, a minute ago, and so on).

Endurantism and perdurantism

Main articles: Endurantism and Perdurantism

The positions on the persistence of objects are somewhat similar. An endurantist holds that for an object to persist through time is for it to exist completely at different times (each instance of existence we can regard as somehow separate from previous and future instances, though still numerically identical with them). A perdurantist on the other hand holds that for a thing to exist through time is for it to exist as a continuous reality, and that when we consider the thing as a whole we must consider an aggregate of all its "temporal parts" or instances of existing. Endurantism is seen as the conventional view and flows out of our pre-philosophical ideas (when I talk to somebody I think I am talking to that person as a complete object, and not just a part of a cross-temporal being), but perdurantists such as David Lewis have attacked this position. They argue that perdurantism is the superior view for its ability to take account of change in objects.

On the whole, Presentists are also endurantists and Eternalists are also perdurantists (and vice versa), but this is not a necessary relation and it is possible to claim, for instance, that time's passage indicates a series of ordered realities, but that objects within these realities somehow exist outside of the reality as a whole, even though the realities as wholes are not related. However, such positions are rarely adopted.

Community

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Community 

A community of interest gathers at Stonehenge, England, for the summer solstice.

A community is a social unit (a group of living things) with commonality such as norms, religion, values, customs, or identity. Communities may share a sense of place situated in a given geographical area (e.g. a country, village, town, or neighbourhood) or in virtual space through communication platforms. Durable relations that extend beyond immediate genealogical ties also define a sense of community, important to their identity, practice, and roles in social institutions such as family, home, work, government, society, or humanity at large. Although communities are usually small relative to personal social ties, "community" may also refer to large group affiliations such as national communities, international communities, and virtual communities.

The English-language word "community" derives from the Old French comuneté (currently "Communauté"), which comes from the Latin communitas "community", "public spirit" (from Latin communis, "common").

Human communities may have intent, belief, resources, preferences, needs, and risks in common, affecting the identity of the participants and their degree of cohesiveness.

Perspectives of various disciplines

Archaeology

Archaeological studies of social communities use the term "community" in two ways, paralleling usage in other areas. The first is an informal definition of community as a place where people used to live. In this sense it is synonymous with the concept of an ancient settlement - whether a hamlet, village, town, or city. The second meaning resembles the usage of the term in other social sciences: a community is a group of people living near one another who interact socially. Social interaction on a small scale can be difficult to identify with archaeological data. Most reconstructions of social communities by archaeologists rely on the principle that social interaction in the past was conditioned by physical distance. Therefore, a small village settlement likely constituted a social community and spatial subdivisions of cities and other large settlements may have formed communities. Archaeologists typically use similarities in material culture—from house types to styles of pottery—to reconstruct communities in the past. This classification method relies on the assumption that people or households will share more similarities in the types and styles of their material goods with other members of a social community than they will with outsiders.

Ecology

Main article: Community (ecology)

In ecology, a community is an assemblage of populations - potentially of different species - interacting with one another. Community ecology is the branch of ecology that studies interactions between and among species. It considers how such interactions, along with interactions between species and the abiotic environment, affect social structure and species richness, diversity and patterns of abundance. Species interact in three ways: competition, predation and mutualism:

• Competition typically results in a double negative—that is both species lose in the interaction.
• Predation involves a win/lose situation, with one species winning.
• Mutualism sees both species co-operating in some way, with both winning.

The two main types of ecological communities are major communities, which are self-sustaining and self-regulating (such as a forest or a lake), and minor communities, which rely on other communities (like fungi decomposing a log) and are the building blocks of major communities.

A simplified example of a community. A community includes many populations and how they interact with each other. This example shows interaction between the zebra and the bush, and between the lion and the zebra, as well as between the bird and the organisms by the water, like the worms.

Semantics

The concept of "community" often has a positive semantic connotation, exploited rhetorically by populist politicians and by advertisers to promote feelings and associations of mutual well-being, happiness and togetherness - veering towards an almost-achievable utopian community, in fact.

In contrast, the epidemiological term "community transmission" can have negative implications; and instead of a "criminal community" one often speaks of a "criminal underworld" or of the "criminal fraternity".

Key concepts

Gemeinschaft and Gesellschaft

Main article: Gemeinschaft and Gesellschaft

In Gemeinschaft und Gesellschaft (1887), German sociologist Ferdinand Tönnies described two types of human association: Gemeinschaft (usually translated as "community") and Gesellschaft ("society" or "association"). Tönnies proposed the Gemeinschaft–Gesellschaft dichotomy as a way to think about social ties. No group is exclusively one or the other. Gemeinschaft stress personal social interactions, and the roles, values, and beliefs based on such interactions. Gesellschaft stress indirect interactions, impersonal roles, formal values, and beliefs based on such interactions.

Sense of community

Main article: Sense of community

In a seminal 1986 study, McMillan and Chavis identify four elements of "sense of community":

1. membership: feeling of belonging or of sharing a sense of personal relatedness,
2. influence: mattering, making a difference to a group and of the group mattering to its members
3. reinforcement: integration and fulfillment of needs,
4. shared emotional connection.

To what extent do participants in joint activities experience a sense of community?

A "sense of community index (SCI) was developed by Chavis and colleagues, and revised and adapted by others. Although originally designed to assess sense of community in neighborhoods, the index has been adapted for use in schools, the workplace, and a variety of types of communities.

Studies conducted by the APPA indicate that young adults who feel a sense of belonging in a community, particularly small communities, develop fewer psychiatric and depressive disorders than those who do not have the feeling of love and belonging.

Socialization

Main article: Socialization

Lewes Bonfire Night procession commemorating 17 Protestant martyrs burnt at the stake from 1555 to 1557

The process of learning to adopt the behavior patterns of the community is called socialization. The most fertile time of socialization is usually the early stages of life, during which individuals develop the skills and knowledge and learn the roles necessary to function within their culture and social environment. For some psychologists, especially those in the psychodynamic tradition, the most important period of socialization is between the ages of one and ten. But socialization also includes adults moving into a significantly different environment where they must learn a new set of behaviors.

Socialization is influenced primarily by the family, through which children first learn community norms. Other important influences include schools, peer groups, people, mass media, the workplace, and government. The degree to which the norms of a particular society or community are adopted determines one's willingness to engage with others. The norms of tolerance, reciprocity, and trust are important "habits of the heart," as de Tocqueville put it, in an individual's involvement in community.

Community development

Main article: Community development

Community development is often linked with community work or community planning, and may involve stakeholders, foundations, governments, or contracted entities including non-government organisations (NGOs), universities or government agencies to progress the social well-being of local, regional and, sometimes, national communities. More grassroots efforts, called community building or community organizing, seek to empower individuals and groups of people by providing them with the skills they need to effect change in their own communities. These skills often assist in building political power through the formation of large social groups working for a common agenda. Community development practitioners must understand both how to work with individuals and how to affect communities' positions within the context of larger social institutions. Public administrators, in contrast, need to understand community development in the context of rural and urban development, housing and economic development, and community, organizational and business development.

Formal accredited programs conducted by universities, as part of degree granting institutions, are often used to build a knowledge base to drive curricula in public administration, sociology and community studies. The General Social Survey from the National Opinion Research Center at the University of Chicago and the Saguaro Seminar at the John F. Kennedy School of Government at Harvard University are examples of national community development in the United States. The Maxwell School of Citizenship and Public Affairs at Syracuse University in New York State offers core courses in community and economic development, and in areas ranging from non-profit development to US budgeting (federal to local, community funds). In the United Kingdom, the University of Oxford has led in providing extensive research in the field through its Community Development Journal, used worldwide by sociologists and community development practitioners.

At the intersection between community development and community building are a number of programs and organizations with community development tools. One example of this is the program of the Asset Based Community Development Institute of Northwestern University. The institute makes available downloadable tools to assess community assets and make connections between non-profit groups and other organizations that can help in community building. The Institute focuses on helping communities develop by "mobilizing neighborhood assets" – building from the inside out rather than the outside in. In the disability field, community building was prevalent in the 1980s and 1990s with roots in John McKnight's approaches.

Community building and organizing

The anti-war affinity group "Collateral Damage" protesting the Iraq War

In The Different Drum: Community-Making and Peace (1987) Scott Peck argues that the almost accidental sense of community that exists at times of crisis can be consciously built. Peck believes that conscious community building is a process of deliberate design based on the knowledge and application of certain rules. He states that this process goes through four stages:

1. Pseudocommunity: When people first come together, they try to be "nice" and present what they feel are their most personable and friendly characteristics.
2. Chaos: People move beyond the inauthenticity of pseudo-community and feel safe enough to present their "shadow" selves.
3. Emptiness: Moves beyond the attempts to fix, heal and convert of the chaos stage, when all people become capable of acknowledging their own woundedness and brokenness, common to human beings.
4. True community: Deep respect and true listening for the needs of the other people in this community.

In 1991, Peck remarked that building a sense of community is easy but maintaining this sense of community is difficult in the modern world.

The three basic types of community organizing are grassroots organizing, coalition building, and "institution-based community organizing," (also called "broad-based community organizing," an example of which is faith-based community organizing, or Congregation-based Community Organizing).

Community building can use a wide variety of practices, ranging from simple events (e.g., potlucks, small book clubs) to larger-scale efforts (e.g., mass festivals, construction projects that involve local participants rather than outside contractors).

Community building that is geared toward citizen action is usually termed "community organizing." In these cases, organized community groups seek accountability from elected officials and increased direct representation within decision-making bodies. Where good-faith negotiations fail, these constituency-led organizations seek to pressure the decision-makers through a variety of means, including picketing, boycotting, sit-ins, petitioning, and electoral politics.

Community organizing can focus on more than just resolving specific issues. Organizing often means building a widely accessible power structure, often with the end goal of distributing power equally throughout the community. Community organizers generally seek to build groups that are open and democratic in governance. Such groups facilitate and encourage consensus decision-making with a focus on the general health of the community rather than a specific interest group.

If communities are developed based on something they share in common, whether location or values, then one challenge for developing communities is how to incorporate individuality and differences. Rebekah Nathan suggests in her book, My Freshman Year, we are drawn to developing communities totally based on sameness, despite stated commitments to diversity, such as those found on university websites.

Types of community

Participants in Diana Leafe Christian's "Heart of a Healthy Community" seminar circle during an afternoon session at O.U.R. Ecovillage

A number of ways to categorize types of community have been proposed. One such breakdown is as follows:

1. Location-based Communities: range from the local neighbourhood, suburb, village, town or city, region, nation or even the planet as a whole. These are also called communities of place.
2. Identity-based Communities: range from the local clique, sub-culture, ethnic group, religious, multicultural or pluralistic civilisation, or the global community cultures of today. They may be included as communities of need or identity, such as disabled persons, or frail aged people.
3. Organizationally-based Communities: range from communities organized informally around family or network-based guilds and associations to more formal incorporated associations, political decision making structures, economic enterprises, or professional associations at a small, national or international scale.

The usual categorizations of community relations have a number of problems: (1) they tend to give the impression that a particular community can be defined as just this kind or another; (2) they tend to conflate modern and customary community relations; (3) they tend to take sociological categories such as ethnicity or race as given, forgetting that different ethnically defined persons live in different kinds of communities —grounded, interest-based, diasporic, etc.

In response to these problems, Paul James and his colleagues have developed a taxonomy that maps community relations, and recognizes that actual communities can be characterized by different kinds of relations at the same time:

1. Grounded community relations. This involves enduring attachment to particular places and particular people. It is the dominant form taken by customary and tribal communities. In these kinds of communities, the land is fundamental to identity.
2. Life-style community relations. This involves giving primacy to communities coming together around particular chosen ways of life, such as morally charged or interest-based relations or just living or working in the same location. Hence the following sub-forms:
   1. community-life as morally bounded, a form taken by many traditional faith-based communities.
   2. community-life as interest-based, including sporting, leisure-based and business communities which come together for regular moments of engagement.
   3. community-life as proximately-related, where neighbourhood or commonality of association forms a community of convenience, or a community of place (see below).
3. Projected community relations. This is where a community is self-consciously treated as an entity to be projected and re-created. It can be projected as through thin advertising slogan, for example gated community, or can take the form of ongoing associations of people who seek political integration, communities of practice based on professional projects, associative communities which seek to enhance and support individual creativity, autonomy and mutuality. A nation is one of the largest forms of projected or imagined community.

In these terms, communities can be nested and/or intersecting; one community can contain another—for example a location-based community may contain a number of ethnic communities. Both lists above can used in a cross-cutting matrix in relation to each other.

Internet communities

In general, virtual communities value knowledge and information as currency or social resource. What differentiates virtual communities from their physical counterparts is the extent and impact of "weak ties", which are the relationships acquaintances or strangers form to acquire information through online networks. Relationships among members in a virtual community tend to focus on information exchange about specific topics. A survey conducted by Pew Internet and The American Life Project in 2001 found those involved in entertainment, professional, and sports virtual-groups focused their activities on obtaining information.

An epidemic of bullying and harassment has arisen from the exchange of information between strangers, especially among teenagers, in virtual communities. Despite attempts to implement anti-bullying policies, Sheri Bauman, professor of counselling at the University of Arizona, claims the "most effective strategies to prevent bullying" may cost companies revenue.

Virtual Internet-mediated communities can interact with offline real-life activity, potentially forming strong and tight-knit groups such as QAnon.