Information AboutInfinity |
| CATEGORIES ABOUT INFINITY | |
| infinity | |
| philosophical terminology | |
| philosophy of mathematics | |
| theologyinfinity | |
| philosophical terminology | |
| philosophy of mathematics | |
| theology | |
| mathematics | |
|
The word infinity comes from the Latin ''infinitas'' or "unboundedness". It refers to several distinct concepts which arise in Theology , Philosophy , Mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. In Theology , for example in the work of Theologians such as Duns Scotus , the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In Philosophy , infinity can be attributed to space and time, as for instance in Kant 's first Antinomy . In both theology and philosophy, infinity is explored in articles such as The Ultimate , The Absolute , God , and Zeno's Paradoxes . In mathematics, infinity is relevant to, or the subject matter of, articles such as mathematical Limit s, Aleph Number s, Class es in Set Theory , Dedekind-infinite Set s, Large Cardinal s, Russell's Paradox , Hyperreal Number s, Projective Geometry , Extended Real Number s and the Absolute Infinite . By some, infinity is considered to be not a number but a concept of increase beyond bounds. In Popular Culture , we have Buzz Lightyear 's rallying cry, "To infinity — and beyond!", which may also be viewed as the rallying cry of Set Theorists considering Large Cardinal s. For a discussion about infinity and the physical universe, see Universe . HISTORY Early Indian views of infinity The earliest known documented knowledge of infinity was presented in Ancient India in the '' Yajur Veda '' (c. 1200 – 900 BC ) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jain a Mathematical text ''Surya Prajinapti'' (c. 400 BC ) classifies all numbers into three sets: enumerable, innumerable and infinite. Each of these was further subdivided into three orders:
The ), infinite in area (two dimensions), infinite everywhere (three dimensions), and infinite perpetually (infinite number of dimensions). According to Singh (1987), Joseph (2000) and Agrawal (2000), the highest enumerable number ''N'' of the Jains corresponds to the modern concept of Aleph-null (the Cardinal Number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of infinite cardinal numbers, of which the highest enumerable number ''N'' is the smallest. In the Jaina work on the Theory Of Sets , two basic types of infinite numbers are distinguished. On both physical and Ontological grounds, a distinction was made between ''asmkhyata'' and ''ananata'', between rigidly bounded and loosely bounded infinities. Early European views of infinity In Europe, the traditional view derives from Aristotle : :"... 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." 207b8 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, , 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 : 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. 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. 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 Subset s (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: :1, 2, 3, 4, ... :2, 4, 6, 8, ... 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. :"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." two New Sciences'', 1638 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. 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. :"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) 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 Curve s 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. 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". :"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) Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience. :"... I can see in space the possibility of any finite experience ... we recognise essential infinity of space in its smallest part." "[Time is infinite in the same sense as the three- Dimension al space of sight and movement is infinite, even if in fact I can only see as far as the walls of my room." :"... what is infinite about endlessness is only the endlessness itself." Infinity symbol The precise origins of the infinity symbol ∞ 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. 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. However, this explanation is improbable, 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 . It is also possible that it is inspired by older Religious / Alchemical Symbolism . For instance, it has been found in Tibetan Rock Carvings , and the Ouroboros , or infinity snake, is often depicted in this shape. In the Tarot , the lemniscate represents the Balance of Forces and is often associated with the Magician Card . John Wallis is usually credited with introducing ∞ as a symbol for infinity in 1655 in 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 . The infinity symbol is represented in Unicode by the character ∞ (U+221E). MATHEMATICAL INFINITY Infinity is the state of being greater than any finite (real) number however large. Infinity in real analysis In Real Analysis , the symbol , called "infinity", denotes an unbounded Limit . means that ''x'' grows beyond any assigned value, and means x is eventually less than any assigned value. Points labeled and can be added to the real numbers as a Topological Space , producing the two-point Compactification of the real numbers. Adding algebraic properties to this gives us the Extended Real Number s. We can also treat and as the same, leading to the '''one-point compactification''' of the real numbers, which is the Real Projective Line . Projective Geometry also introduces a Line At Infinity in Plane Geometry , and so forth for higher dimensions. Infinity is often used not only to define a limit but as if it were a value in the extended real numbers in Real Analysis ; if ''f''(''t'') ≥ 0 then
Infinity in complex analysis As in real analysis, in Complex Analysis the symbol , called "infinity", denotes an unbounded Limit . means that the magnitude |
|
|