Encyclopedia of Distances

Lee distance
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The tower is taken to be of a determinate size and shape, but the visual appearance continually changes. How can that be? Berkeley claims that visual ideas are merely signs of tactile ideas. There is no resemblance between visual and tactile ideas. Their relationship is like that between words and their meanings. If one hears a noun, one thinks of an object it denotes. Similarly, if one sees an object, one thinks of a corresponding idea of touch, which Berkeley deems the secondary mediate object of sight. In both cases, there are no necessary connections between the ideas. His discussion of magnitude is analogous to his discussion of distance.

Berkeley explores the relationships between the objects of sight and touch by introducing the notions of minimum visibles and tangibles, the smallest points one actually can perceive by sight and touch, points which must be taken to be indivisible. The apparent size of the visual object, its confusion or distinctness, and its faintness or vigor play roles in judging the size of the tangible object. All things being equal, if it appears large, it is taken to be large. If it be distinct and clear, I judge it greater.

As in the case of distance, there are no necessary connections between the sensory elements of the visual and tangible object. Berkeley argues that the objects of sight and touch - indeed, the objects of each sensible modalities — are distinct and incommensurable. The tower that visually appears to be small and round from a distance is perceived to be large and square by touch. So, one complex tactual object corresponds to the indefinitely large number of visual objects.

Since there are no necessary connections between the objects of sight and touch, the objects must be distinct.

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Distances in Applied Social Sciences. If the instructor is to do the job it cannot be a relationship between equals. Before the development of advanced computer and printing techniques, maps were drawn by hand. But the z -spin disposition of the R-particle changes immediately after an earlier z -spin measurement on the L-particle: The R-particles see Fig. Supporters of the Everett interpretation disagree. The paper version is clearly a welcome asset in a mathematics library.

Before turning to the discussions of Berkeley's idealism and immaterialism, there are several points we should notice. First, there are various points in the New Theory of Vision where Berkeley writes as if ideas of touch are or are of external objects cf. Since the Berkeley of the Principles and Dialogues contends that all ideas are mind-dependent and all physical objects are composed of ideas, some have questioned whether the position in the New Theory of Vision is consistent with the work that immediately follows.

Some scholars suggest that either that the works on vision are scientific works which, as such, make no metaphysical commitments or that allusions to "external objects" are cases of speaking with the vulgar. Secondly, insofar as in his later works Berkeley claims that ordinary objects are composed of ideas, his discussion of the correlation of ideas of sight and touch tends to anticipate his later view by explaining how one "collects" the ideas of distinct senses to form one thing. In the Introduction to the Principles of Human Knowledge , Berkeley laments the doubt and uncertainty found in philosophical discussions Intro.

He finds the source of skepticism in the theory of abstract ideas, which he criticizes. It is agreed on all hands, that the qualities or modes of things do never really exist each of them apart by it self, and separated from all others, but are mixed, as it were, and blended together, several in the same object.

But we are told, the mind being able to consider each quality singly, or abstracted from those other qualities with which it is united, does by that means frame to it self abstract ideas. Although theories of abstraction date back at least to Aristotle Metaphysics , Book K, Chapter 3, ab4 , were prevalent among the medievals cf.

First, Locke's work was recent and familiar. Second, Berkeley seems to have considered Locke's account the best available. According to Locke, the doctrine of abstract ideas explains how knowledge can be communicated and how it can be increased. It explains how general terms obtain meaning Locke, 3. A general term, such as 'cat' refers to an abstract general idea, which contains all and only those properties that one deems common to all cats, or, more properly, the ways in which all cats resemble each other.

The connection between a general term and an abstract idea is arbitrary and conventional, and the relation between an abstract idea and the individual objects falling under it is a natural relation resemblance. If Locke's theory is sound, it provides a means by which one can account for the meaning of general terms without invoking general objects universals. Berkeley's attack on the doctrine of abstract ideas follows three tracks.

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On the face of it, his argument is weak. At most it shows that insofar as he cannot form the idea, and assuming that all humans have similar psychological abilities, there is some evidence that no humans can form abstract ideas of the sort Locke described.

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But there is a remark made in passing that suggests there is a much stronger argument implicit in the section. Berkeley writes:. To be plain, I own my self able to abstract in one sense, as when I consider some particular parts or qualities separated from others, with which though they are united in some object, yet, it is possible they may really exist without them.

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But I deny that I can abstract one from another, or conceive separately, those qualities which it is impossible should exist so separated; or that I can frame a general notion by abstracting from particulars in the manner aforesaid. Which two last are the proper acceptations of abstraction. This three-fold distinction among types of abstraction is found in Arnauld and Nicole's Logic or the Art of Thinking. The first type of abstraction concerns integral parts. The head, arms, torso, and legs are integral parts of a body: each can exist in separation from the body of which it is a part Arnauld and Nicole, p.

The second kind of abstraction "arises when we consider a mode without paying attention to its substance, or two modes which are joined together in the same substance, taking each one separately" Arnauld and Nicole, p. The third concerns distinctions of reason, for example, conceiving of a triangle as equilateral without conceiving of it as equiangular Arnauld and Nicole, p. Berkeley grants that he can abstract in the first sense - "I can consider the hand, the eye, the nose, each by it self abstracted or separated from the rest of the body" Intro. The latter two cases represent impossible states of affairs.

Many abstractionists also accepted a conceivability criterion of possibility: If one can clearly and distinctly conceive of a state of affairs, then it is possible for that state of affairs to exist as conceived cf. Descartes, This principle entails that impossible states of affairs are inconceivable. So, granting it is impossible for a mode to exist apart from a substance Intro.

And if the second falls, the third falls as well, since the third requires that alternative descriptions of an object pick out no differences in reality. So, a traditional theory of modes and substances, the conceivability criterion of possibility, and abstraction are an inconsistent triad. The inconsistency can be resolved by dropping the doctrine of abstract ideas.

Berkeley made this point explicitly in the first draft of the Introduction:. It is, I think, a receiv'd axiom that an impossibility cannot be conceiv'd. For what created intelligence will pretend to conceive, that which God cannot cause to be?

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Now it is on all hands agreed, that nothing abstract or general can be made really to exist, whence it should seem to follow, that it cannot have so much as an ideal existence in the understanding. Works One of the marks of the modern period is an adherence to the principle of parsimony Ockham's Razor. The principle holds that the theoretically simpler of two explanations is more probably true. In the seventeenth and eighteen centuries, this was sometimes expressed as "God does nothing in vain" cf. DHP2 So, if it is possible to construct a theory of meaning that does not introduce abstract ideas as a distinct kind of idea, that theory would be simpler and deemed more probably true.

Granting Locke that all existents are particulars Locke 3. Ideas remain particular, although a particular idea can function as a general idea. For example, when a geometer draws a line on a blackboard, it is taken to represent all lines, even though the line itself is particular and has determinate qualities. Similarly, a particular idea can represent all similar ideas. So, whether one takes Berkeley to mean that words apply immediately to objects or that meaning is mediated by paradigmatic ideas, the theory is simpler than the abstractionists' insofar as all ideas are particular and determinate.

In effect, it is something imperfect that cannot exist, an idea wherein some parts of several different and inconsistent ideas are put together" Locke 4.

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Upon quoting the passage, Berkeley merely asks his reader whether he or she can form the idea, but his point seems to be much stronger. The described idea is inconsistent , and therefore represents an impossible state of affairs, and it is therefore inconceivable , since whatever is impossible is inconceivable. This is explicit in a parallel passage in the New Theory of Vision. After quoting the triangle passage, Berkeley remarks, "But had he called to mind what he says in another place, to wit, 'That ideas of mixed modes wherein any inconsistent ideas are put together cannot so much as exist in the mind, i.

If abstract ideas are not needed for communication - Berkeley takes the fact that infants and poorly educated people communicate, while the formation of abstract ideas is said to be difficult, as a basis for doubting the difficulty thesis Intro. The abstractionists maintain that abstract ideas are needed for geometrical proofs. Berkeley argues that only properties concerning, for example, a triangle as such are germane to a geometric proof.

So, even if one's idea of a triangle is wholly determinate consider a diagram on a blackboard , none of the differentiating properties prevent one from constructing a proof, since a proof is not concerned solely with the idea or drawing with which one begins. He maintains that it is consistent with his theory of meaning to selectively attend to a single aspect of a complex, determinate idea Intro.

Berkeley concludes his discussion of abstraction by noting that not all general words are used to denote objects or kinds of objects. His discussion of the nondenotative uses of language is often taken to anticipate Ludwig Wittgenstein's interest in meaning-as-use.

Berkeley's famous principle is esse is percipi , to be is to be perceived. Berkeley was an idealist. He held that ordinary objects are only collections of ideas, which are mind-dependent. Berkeley was an immaterialist. He held that there are no material substances. There are only finite mental substances and an infinite mental substance, namely, God. On these points there is general agreement. There is less agreement on Berkeley's argumentative approach to idealism and immaterialism and on the role of some of his specific arguments.

His central arguments are often deemed weak. The account developed here is based primarily on the opening thirty-three sections of the Principles of Human Knowledge. It assumes, contrary to some commentators, that Berkeley's metaphysics rests on epistemological foundations. This approach is prima facie plausible insofar as it explains the appeal to knowledge in the title of the Principles cf.

There will be occasional digressions concerning the problems perceived by those who claim that Berkeley's approach was more straightforwardly metaphysical. It is evident to any one who takes a survey of the objects of human knowledge, that they are either ideas actually imprinted on the senses, or else such as are perceived by attending to the passions and operations of the mind, or lastly ideas formed by help of memory and imagination, either compounding, dividing, or barely representing those originally perceived in the aforesaid ways.

This seems to say that ideas are the immediate objects of knowledge in a fundamental sense acquaintance. Following Locke, there are ideas of sense, reflection, and imagination. So, ordinary objects, as known, are collections of ideas marked by a single name. Berkeley's example is an apple. This Berkeley calls this 'mind' or 'spirit'.

Minds as knowers are distinct from ideas as things known. For an idea, to be is to be perceived known. If one construes 'sensible objects' as ideas of sense, and ideas are objects of knowledge, then having a real existence distinct from being perceived would require that an object be known as an idea and unknown as a thing distinct from being perceived , which is inconsistent. Ordinary objects, as known, are nothing but collections of ideas. If, like Descartes, Berkeley holds that claims of existence are justified if and only if the existent can be known, then ordinary objects must be at least collections of ideas.

But notice what has not yet been shown. It has not been shown that ordinary objects are only collections of ideas, nor has it be shown that thinking substances are immaterial. Berkeley's next move is to ask whether there are grounds for claiming ordinary objects are something more than ideas.

The above account is not the only interpretation of the first seven sections of the Principles.

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Many commentators take a more directly metaphysical approach. They assume that ideas are mental images Pitcher, p. Winkler, p. Tipton, p. Works n1. The epistemic interpretation we have been developing seems to avoid these problems. Berkeley holds that ordinary objects are at least collections of ideas. Are they something more? He prefaces his discussion with his likeness principle, the principle that nothing but an idea can resemble an idea. Why is this? A claim that two objects resemble each other can be justified only by a comparison of the objects cf. So, if only ideas are immediately perceived, only ideas can be compared.

So, there can be no justification for a claim that an idea resembles anything but an idea. These distances are particularly crucial, for example, in computational biology, image analysis, speech recognition, and information retrieval. Here, an assessment of the practical questions arising during selection of a "good'' distance function has been left aside in favor of a comprehensive listing of the main available distances, a useful tool for the distance design community.

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This reader-friendly reference offers both independent introductions and definitions, while at the same time making cross-referencing easy through hyperlink-like boldfaced references to original definitions. Instead, one method can be used to measure nearby distances, a second can be used to measure nearby-to-intermediate distances, and so on.

Each rung of the ladder provides information that can be used to determine distances at the next higher rung. At the base of the ladder are fundamental distance measurements, in which distances are determined directly, with no physical assumptions about the nature of the object in question. Beyond the use of parallax, the overlapping chain of distance measurement techniques includes the use of cepheid variables, planetary nebulae, most luminous supergiants, most luminous globular clusters, most luminous HII regions, supernovae, and Hubble constant and red shifts.

In neutral geometry, the minimum distance between two points is the length of the line segment between them. In analytic geometry, one can find the distance between two points of the xy-plane using the distance formula. The distance between x 1 , y 1 and x 2 , y 2 is given by. Similarly, given points x 1 , y 1 , z 1 and x 2 , y 2 , z 2 in three-space , the distance between them is. Which is easily proven by constructing a right triangle with a leg on the hypotenuse of another with the other leg orthogonal to the plane that contains the first triangle and applying the Pythagorean theorem.

In the study of complicated geometries, we call this most common type of distance Euclidean distance, as it is derived from the Pythagorean theorem, which does not hold in Non-Euclidean geometries. This distance formula can also be expanded into the arc-length formula. In the Euclidean space R n , the distance between two points is usually given by the Euclidean distance 2-norm distance. Other distances, based on other norms, are sometimes used instead.

The 2-norm distance is the Euclidean distance, a generalization of the Pythagorean theorem to more than two coordinates. It is what would be obtained if the distance between two points were measured with a ruler: the "intuitive" idea of distance. The 1-norm distance is more colorfully called the taxicab norm or Manhattan distance , because it is the distance a car would drive in a city laid out in square blocks if there are no one-way streets.

The infinity norm distance is also called Chebyshev distance. In 2D it represents the distance kings must travel between two squares on a chessboard. The p -norm is rarely used for values of p other than 1, 2, and infinity, but see super ellipse. In physical space the Euclidean distance is in a way the most natural one, because in this case the length of a rigid body does not change with rotation. Such a distance function is known as a metric. Together with the set, it makes up a metric space. This definition satisfies the three conditions above, and corresponds to the standard topology of the real line.

But distance on a given set is a definitional choice. This also defines a metric, but gives a completely different topology, the "discrete topology"; with this definition numbers cannot be arbitrarily close. Various distance definitions are possible between objects. For example, between celestial bodies one should not confuse the surface-to-surface distance and the center-to-center distance.

If the former is much less than the latter, as for a LEO, the first tends to be quoted altitude , otherwise, e.