Infinity

:For the automobile brand, see Infiniti. For the radio company, see Infinity Broadcasting.

History

Ancient view of infinity

The Yajurveda (c 1800 BC - 800 BC) states that if you remove a part from infinity or add a part to infinity still what remains is infinity. The Jaina mathematical text Surya Prajinapti (c 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. This theory was not realised in Europe until the late 19th century work of George Cantor.

Related Topics:
Yajurveda - 1800 BC - 800 BC - Jain - 400 BC - George Cantor

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In Europe, the traditional view derives from Aristotle:

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:"... it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number."

Related Topics:
Magnitude - Bisected

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This is often called potential infinity; however there are two ideas mixed up with this. One is that it is always possible to find a number of things that surpasses any given number, even if there are not actually such things. The other is that we may quantify over infinite sets without restriction. For example, ?n?Z(?m?Z), which reads, "for any integer n, there exists an integer m > n such that P(m)". The second view is found in a clearer form by medieval writers such as William of Ockham:

Related Topics:
Integer - William of Ockham

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:"Sed omne continuum est actualiter existens. Igitur quaelibet pars sua est vere existens in rerum natura. Sed partes continui sunt infinitae quia non tot quin plures, igitur partes infinitae sunt actualiter existentes." (But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent.)

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The parts are actually there, in some sense. However, on this view, no infinite magnitude can have a number, for whatever number we can imagine, there is always a larger one: "There are not so many (in number) that there are no more". Aquinas also argued against the idea that infinity could be in any sense complete, or a totality.

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Views from the Renaissance to modern times

Galileo (during his long house arrest in Siena after his condemnation by the Inquisition) was the first to notice that we can place an infinite set into one-to-one correspondence with one of its proper subsets (any part of the set, that is not the whole). For example, we can match up the "set" of even numbers {2, 4, 6, 8 ...} with the natural numbers {1, 2, 3, 4 ...} as follows:

Related Topics:
Galileo - Siena - Inquisition - One-to-one correspondence - Proper subset

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:1, 2, 3, 4, ...

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:2, 4, 6, 8, ...

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It appeared, by this reasoning, as though a set which is naturally smaller than the set of which it is a part (since it does not contain all the members of that set) is in some sense the same size. He thought this was one of the difficulties which arise when we try, "with our finite minds", to comprehend the infinite.

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:"So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal", "greater", and "less", are not applicable to infinite, but only to finite, quantities."

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The idea that size can be measured by one-to-one correspondence is today known as Hume's principle, although Hume, like Galileo, believed the principle could not be applied to infinite sets.

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Locke, in common with most of the empiricist philosophers, also believed that we can have no proper idea of the infinite. They believed all our ideas were derived from sense data or "impressions", and since all sensory impressions are inherently finite, so too are our thoughts and ideas. Our idea of infinity is merely negative or privative.

Related Topics:
Locke - Empiricist - Sense data

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:"Whatever positive ideas we have in our minds of any space, duration, or number, let them be never so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever completing the idea, there we have our idea of infinity ... yet when we would frame in our minds the idea of an infinite space or duration, that idea is very obscure and confused, because it is made up of two parts very different, if not inconsistent. For let a man frame in his mind an idea of any space or number, as great as he will, it is plain the mind rests and terminates in that idea; which is contrary to the idea of infinity, which consists in a supposed endless progression." (Essay, II. xvii. 7., author's emphasis)

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Famously, the ultra-empiricist Hobbes tried to defend the idea of a potential infinity in the light of the discovery by Evangelista Torricelli, of a figure (Gabriel's horn) whose surface area is infinite, but whose volume is finite. Not reported, this motivation of Hobbes came too late as curves having infinite length yet bounding finite areas were known much before. Such seeming paradoxes are resolved by taking any finite figure and stretching its content infinitely in one direction; the magnitude of its content is unchanged as its divisions drop off geometrically but the magnitude of its bounds increases to infinity by necessity. Potentiality lies in the definitions of this operation, as well-defined and interconsistent mathematical axioms. A potential infinity is allowed by letting an infinitely-large quantity be cancelled out by an infinitely-small quantity.

Related Topics:
Hobbes - Evangelista Torricelli - Gabriel's horn - Surface area - Volume - Curve - Well-defined

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Modern philosophical views

Modern discussion of the infinite is now regarded as part of set theory and mathematics, and generally avoided by philosophers. An exception was Wittgenstein, who made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period". (see also Logic of antinomies)

Related Topics:
Wittgenstein - Axiomatic set theory

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:"Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes ... In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of ambiguous grammar." (Philosophical Remarks § 141, cf Philosophical Grammar p. 465)

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Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience.

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:"... I can see in space the possibility of any finite experience ... we recognise essential infinity of space in its smallest part." " is infinite in the same sense as the three-dimensional space of sight and movement is infinite, even if in fact I can only see as far as the walls of my room."

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:"... what is infinite about endlessness is only the endlessness itself."

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Infinity symbol

The precise origins of the infinity symbol infty are unclear. One possibility is suggested by the name it is sometimes called — the lemniscate, from the Latin lemniscus, meaning "ribbon". One can imagine walking forever along a simple loop formed from a ribbon.

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A popular explanation is that the infinity symbol is derived from the shape of a Möbius strip. Again, one can imagine walking along its surface forever. This possible explanation is probably incorrect, however, since the symbol had been in use to represent infinity for over two hundred years before August Ferdinand Möbius and Johann Benedict Listing discovered the Möbius strip in 1858.

Related Topics:
Möbius strip - August Ferdinand Möbius - Johann Benedict Listing

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John Wallis is usually credited with introducing infty as a symbol for infinity in 1655 in

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his De sectionibus conicus. One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the Etruscan numeral for 1000, which looked somewhat like CIƆ and was sometimes used to mean "many". Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet.

Related Topics:
Roman numeral - Etruscan numeral - Omega - Greek alphabet

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