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Countably

Countably

In mathematics the term countable is used to describe the size of a set, i.e. the number of elements it contains. The notion of an infinite set is not elementary; it requires a strong sense of abstraction and precision. A set is called countable if the number of elements is finite or if it has the same number of elements as the natural numbers. (Cantor defined a countable set as a set which can be put into one-to-one correspondence with a subset of the natural numbers). The term countable stems from the fact that the natural numbers are often called counting numbers. A set with more elements is called uncountable; not all uncountable sets have the same size. The different sizes of infinite sets are investigated in the theory of cardinal numbers.

Definition

A set S is called countable if there exists an injective function :f\colon S \to \mathbb If f is also bijective then S is called countably infinite or denumerable. The terminology is not universal: some authors define denumerable to mean what we have called "countable"; some define countable to mean what we have called "countably infinite". The next result offers an alternative definition of a countable set S in terms of a surjective function: THEOREM: Let S be a nonempty set. The following statements are equivalent: (1) S is countable (2) There exists an injective function f\colon S \to \mathbb (3) There exists a surjective function g\colon \mathbb \to S

Gentle introduction

The elements of a finite set can be listed, say . However, insofar as a set is a logical description of the properties of its members, it need not be finite. To understand this, imagine that I ask you: how many words can you make out of Scrabble pieces if you are allowed to ask me for more pieces no matter how many you used up? The answer? As many as you like; you can go forever. But that doesn't mean they won't each of them be a word made out of scrabble blocks, rather than apple pies or racecars. Thus an infinite set is still a set, insofar as it is a tool for separating out things with different properties. Now what is a countably infinite set? Technically, a countably infinite set is any set which, in spite of its boundlessness, can be shown equivalent to the natural numbers - nothing more, nothing less. This makes it possible to set apart elements of a countably infinite set using natural numbers as indices, and in turn puts the logic associated with them in very close proximity to the logic associated with the natural numbers themselves; and this makes such sets easily logically tractable.

A more formal introduction

It might then seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view is not tenable, however, under the natural definition of size. To elaborate this we need the concept of a bijection. Do the sets and have the same size? :"Obviously, yes." :"How do you know?" :"Well it's obvious. Look, they've both got 3 elements". :"What's a 3?" This may seem a strange situation but, although a "bijection" seems a more advanced concept than a "number", the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets. This is where the concept of a bijection comes in: define the correspondence :a ↔ 1, b ↔ 2, c ↔ 3 Since every element of is paired with precisely one element of (and vice versa) this defines a bijection. We now generalise this situation and define two sets to be of the same size precisely when there is a bijection between them. For all finite sets this gives us the usual definition of "the same size". What does it tell us about the size of infinite sets? Consider the sets A = , the set of positive integers and B = , the set of even positive integers. We claim that, under our definition, these sets have the same size, and that therefore B is countably infinite. Recall that to prove this we need to exhibit a bijection between them. But this is easy: 1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ... As in the earlier example, every element of A has been paired off with precisely one element of B, and vice versa. Hence they have the same size. This gives an example of a set which is of the same size as one of its proper subsets, a situation which is impossible for finite sets. Likewise, the set of all ordered pairs of natural numbers is countably infinite, as can be seen by following a path like this one: :\begin (0,0) & \rightarrow & (0,1) & & (0,2) & \rightarrow & (0,3) & \\ & \swarrow & & \nearrow & & \swarrow & & \\ (1,0) & & (1,1) & & (1,2) & & \ddots & \\ \downarrow & \nearrow & & \swarrow & & & & \\ (2,0) & & (2,1) & & \ddots & & & \\ & \swarrow & & & & & & \\ (3,0) & & \ddots & & & & & \\ \downarrow & & & & & & & \\ \vdots & & & & & & & \end The resulting mapping is like this: 0 ↔ (0,0), 1 ↔ (0,1), 2 ↔ (1,0), 3 ↔ (2,0), 4 ↔ (1,1), 5 ↔ (0,2), … It is evident that this mapping will cover all such ordered pairs. Interestingly: if you treat each pair as being the numerator and denominator of a vulgar fraction, then for every possible fraction, we can come up with a distinct number corresponding to it. Since every natural number is also a fraction N/1, we can conclude that there are the same number of fractions as there are of whole numbers. THEOREM: The Cartesian product of finitely many countable sets is countable. This form of triangular mapping recursively generalizes to vectors of finitely many natural numbers by repeatedly mapping the first two elements to a natural number. For example, (2,0,3) maps to (5,3) which maps to 41. Sometimes more than one mapping is useful. This is where you map the set which you want to show countably infinite, onto another set; and then map this other set to the natural numbers. For example, the positive rational numbers can easily be mapped to (a subset of) the pairs of natural numbers because p/q maps to (p, q). What about infinite subsets of countably infinite sets? Do these have less elements than N? THEOREM: Every subset of a countable set is countable. In particular, every infinite subset of a countably infinite set is countably infinite. For example, the set of prime numbers is countable, by mapping the nth prime number to n:
- 2 maps to 1
- 3 maps to 2
- 5 maps to 3
- 7 maps to 4
- 11 maps to 5
- 13 maps to 6
- 17 maps to 7
- 19 maps to 8
- 23 maps to 9
- etc. What about sets being "larger than" N? An obvious place to look would be Q, the set of all rational numbers, which is "clearly" much bigger than N. But looks can be deceiving, for we assert THEOREM: Q (the set of all rational numbers) is countable. Q can be defined as the set of all fractions a/b where a and b are integers and b > 0. This can be mapped onto the subset of ordered triples of natural numbers (a, b, c) such that b > 0, a and b are coprime, and c ∈ , and if a = 0 then c = 0.
- 0 maps to (0,1,0)
- 1 maps to (1,1,0)
- −1 maps to (1,1,1)
- 1/2 maps to (1,2,0)
- −1/2 maps to (1,2,1)
- 2 maps to (2,1,0)
- −2 maps to (2,1,1)
- 1/3 maps to (1,3,0)
- −1/3 maps to (1,3,1)
- 3 maps to (3,1,0)
- −3 maps to (3,1,1)
- 1/4 maps to (1,4,0)
- −1/4 maps to (1,4,1)
- 2/3 maps to (2,3,0)
- −2/3 maps to (2,3,1)
- 3/2 maps to (3,2,0)
- −3/2 maps to (3,2,1)
- 4 maps to (4,1,0)
- −4 maps to (4,1,1)
- ... By a similar development, the set of algebraic numbers is countable, and so is the set of definable numbers. THEOREM: (Assuming the axiom of choice) The union of countably many countable sets is countable. For example, given countable sets a, b, c ... Using a variant of the triangular enumeration we saw above:
- a0 maps to 0
- a1 maps to 1
- b0 maps to 2
- a2 maps to 3
- b1 maps to 4
- c0 maps to 5
- a3 maps to 6
- b2 maps to 7
- c1 maps to 8
- d0 maps to 9
- a4 maps to 10
- ... Note that this only works if the sets a, b, c,... are disjoint. If not, then the union is even smaller and is therefore also countable by a previous theorem. THEOREM: The set of all finite-length sequences of natural numbers is countable. This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which is a countable set (finite Cartesian product). So we are talking about a countable union of countable sets, which is countable by the previous theorem. THEOREM: The set of all finite subsets of the natural numbers is countable. If you have a finite subset, you can order the elements into a finite sequence. There are only countably many finite sequences, so also there are only countably many finite subsets.

Further theorems about uncountable sets


- The set of real numbers is uncountable, and so is the set of all sequences of natural numbers and the set of all subsets of N (see Cantor's diagonal argument). Remember our example of the scrabble words. Although we can keep asking for more letters from the bag, each word we form is finitely long. The number of possible words is the same as the number of natural numbers. If we permit infinitely long words, then the number of possible "words" is greater than this. In fact, with infinitely long words, the number of words is the same as the number of real numbers. We noted earlier that there are no more fractions than there are natural numbers. The decimal expansion if a fraction is always a finitely long decimal number followed by a repeating decimal.
- 0.33333333333 ...
- 12.648986986986986986 ...
- 1.75 Let's say we use our decimal point to also indicate the start of the repeater:
- ..3
- 12.648.986
- 1.75. Then we can express any fraction using a finitely long decimal expansion with repeating bit. It's clear that this is the same situation as with our finitely long scrabble words, and so once again the number of possible fractions is not greater than the number of natural numbers.

See also


- Infinity
- Aleph number Category:Set theory ko:가산집합 ja:可算無限集合

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. : :Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure. : :Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics. : :Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

See also list of mathematics history topics :History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

See also


- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle

Bibliography


- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links


- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Infinity

Infinity is a term with very distinct, separate meanings 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. The word infinity comes from Latin : "Infinito", unending. 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 limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals, Russell's paradox, hyperreal numbers, projective geometry, extended real numbers 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 cardinals. For a discussion about infinity and the physical universe, see Universe.

History

Ancient view of infinity

The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800 BC - 800 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 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. Jaina mathematicians were the first to conceive of different orders of infinity, including one they called unenumerable (innumerable).[http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_jaina.htm] [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html] The concept of different orders of infinity would remain unknown in Europe until the late 19th century. 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." [Physics 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, ∀n∈Z(∃m∈Z[m>n∧P(m)]), 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 subsets (any part of the set, that is not the whole). For example, we can match up the "set" of even numbers with the natural numbers 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." [On 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 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.

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 :"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 [the] essential infinity of space in its smallest part." "[Time] 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." :"... what is infinite about endlessness is only the endlessness itself."

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. 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. John Wallis is usually credited with introducing \infty 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 ∞ (∞).

Mathematical infinity

Infinity in real analysis

In real analysis, the symbol \infty, called "infinity", denotes an unbounded limit. x \rightarrow \infty means that x grows beyond any assigned value, and x \rightarrow -\infty means x is eventually less than any assigned value. Points labeled \infty and -\infty 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 numbers. We can also treat \infty and -\infty 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
- \int_^ \, f(t) dt \ = \infty means that f(t) does not bound a finite area from 0 to 1
- \int_^ \, f(t) dt \ = \infty means that the area under f(t) is not finite
- \int_^ \, f(t) dt \ = 1 means that the area under f(t) approaches 1

Infinity in complex analysis

As in real analysis, in complex analysis the symbol \infty, called "infinity", denotes an unbounded limit. x \rightarrow \infty means that the magnitude |x| of x grows beyond any assigned value. A point labeled \infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is still a one-dimensional complex manifold and called the extended complex plane or the Riemann sphere. In this context is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of \infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations.

Arithmetic properties of infinity

Infinity is not a real number but may be considered part of the extended real number line, in which arithmetic operations involving infinity may be performed.

Infinity with itself

# \infty + \infty = \infty \cdot \infty = (-\infty) \cdot (-\infty) = \infty # (-\infty) + (-\infty) = \infty \cdot (-\infty) = (-\infty) \cdot \infty = (-\infty)

Operations involving infinity and real numbers

# -\infty < x < \infty # x + \infty = \infty and x + (-\infty) = (-\infty) # x - \infty = -\infty # x - (-\infty) = \infty # = 0 and = 0 # If 0 then x \cdot \infty = \infty and x \cdot (-\infty) = (-\infty). # If -\infty then x \cdot \infty = -\infty and x \cdot (-\infty) = \infty.

Undefined operations

# 0 \cdot \infty and 0 \cdot (-\infty) # \infty + (-\infty) and (-\infty) + \infty # # ^0 # 1^ Notice that [ = 0] \not\equiv [0 \cdot \infty = x]. This is because zero times infinity is undefined.

Infinity in set theory

A different type of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (\aleph_0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its "proper" parts; this notion of infinity is called Dedekind infinite. Cantor defined two kinds of infinite numbers, the ordinal numbers and the cardinal numbers. Ordinal numbers may be identified with well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one to one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes. Our intuition gained from finite sets breaks down when dealing with infinite sets. One example of this is Hilbert's paradox of the Grand Hotel.

Mathematics without infinity

Leopold Kronecker rejected the notion of infinity and began a school of thought in the philosophy of mathematics called finitism, which led to the philosophical and mathematical school of mathematical constructivism.

Use of infinity in common speech

In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forever to get tickets." In video games, "infinite lives" and "infinite ammo" usually mean a never-ending supply of lives and ammunition. An infinite loop in computer programming is a conditional loop construction whose condition always evaluates to true. As long as there is no external interaction (such as switching the computer off, or the heat death of the universe), the loop will continue to run for all time. In practice however, most programming loops considered as infinite will halt by exceeding the (finite) number range of one of its variables. See halting problem. These terms describe things that are only theoretically infinite; it is impossible to play a video game for an infinite period of time or keep a computer running for an infinite period of time. The number Infinity plus 1 is also used sometimes in common speech.

Physical infinity

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value, for instance by taking an infinite value in an extended real number system (see also: hyperreal number), or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. There exists the concept of infinite entities (such as an infinite plane wave) but there are no means to generate such things. Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system. This point of view does not mean that infinity cannot be used in physics. For convenience sake, calculations, equations, theories and approximations often use infinite series, unbounded functions, etc., and may involve infinite quantities. Physicists however require that the end result be physically meaningful. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called renormalization.

Infinity in cosmology

An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Does space "go on forever"? This is an important open question of cosmology. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By walking/sailing/driving straight long enough, you'll return to the exact spot you started from. The universe, at least in principle, might have a similar topology; if you fly your space ship straight ahead long enough, perhaps you would eventually revisit your starting point. If the universe is indeed ever expanding as science suggests then you could never get back to your starting point even on an infinite time scale.

Three types of infinities

Besides the mathematical infinity and the physical infinity, there could also be a philosophical infinity. There are scientists who hold that all three really exist and there are scientists who hold that none of the three exist. And in between there are the various possibilities. Rudy Rucker, in his book Infinity and the Mind -- the science and philosophy of the mind (1982), has worked out a model list of representatives of each of the eight possible standpoints. The footnote on p.335 of his book suggests the consideration of the following names: Abraham Robinson, Plato, Thomas Aquinas, L.E.J. Brouwer, David Hilbert, Bertrand Russell, Kurt Gödel and Georg Cantor.

Infinity in science fiction

The Hitchhiker's Guide to the Galaxy contains the following definition of infinity: :"Bigger than the biggest thing ever and then some, much bigger than that, in fact really amazingly immense, a totally stunning size, real 'Wow, that's big!' time. Infinity is just so big that by comparison, bigness itself looks really titchy. Gigantic multiplied by colossal multiplied by staggeringly huge is the sort of concept we are trying to get across here." Another quote from The Hitchhiker's Guide to the Galaxy states: "Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity -- distance is incomprehensible and therefore meaningless." Rudy Rucker's novel White Light describes a mathematician who leaves his body and travels to a kind of afterworld that includes a mountain whose Absolute Infinite height matches that of the class of all ordinals. Georg Cantor makes an appearance as a character, and the hero finds a physical correlate for Cantor's Continuum Problem.

See also


- Infinitesimal
- Axiom of infinity

References


-
- #

External links


- [http://www.earlham.edu/~peters/writing/infapp.htm A Crash Course in the Mathematics of Infinite Sets], by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1-59. The stand-alone appendix to Infinite Reflections, below. A concise introduction to Cantor's mathematics of infinite sets.
- [http://www.earlham.edu/~peters/writing/infinity.htm Infinite Reflections], by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1-59.
- [http://pespmc1.vub.ac.be/INFINITY.html Infinity, Principia Cybernetica]
- [http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html Hotel Infinity]
- [http://samvak.tripod.com/infinite.html The concepts of finiteness and infinity in philosophy]
- [http://uk.geocities.com/frege@btinternet.com/cantor/Phil-Infinity.htm Source page on medieval and modern writing on Infinity]

Note

Large cardinals are quantitative infinities defining the number of things in a collection, which are so large that they cannot be proven to exist in the ordinary mathematics of Zermelo-Fraenkel plus Choice (ZFC). Category:Mathematics Category:Philosophical terminology Category:Philosophy of mathematics Category:Theology ko:무한 ja:無限 simple:Infinity

Abstraction

An abstraction is an idea, concept, or word which defines the phenomena that make up its referents (those concrete events or things to which the abstraction refers).

Thought process

In philosophical terminology abstraction is the thought process wherein ideas are distanced from objects. object Abstraction uses a strategy of simplification of detail, wherein formerly concrete details are left ambiguous, vague, or undefined; thus speaking of things in the abstract demands that the listener have an intuitive or common experience with the speaker, if the speaker expects to be understood (as in picture 1, to the right). For example, many different things have the property of redness: lots of things are red. red And we find the relation sitting-on everywhere: many things sit on other things. The property of redness and the relation sitting-on are therefore abstract (as represented by the notation of graph 1, to the right). Specifically, the conceptual diagram graph 1 identifies only 3 boxes, 2 ellipses, and 4 arrows (and their 9 labels), whereas the picture 1 shows much more pictorial detail, with the scores of implied relationships as implicit in the picture rather than with the 9 explicit details in the graph.

Conceptual schemes for abstraction

Problems begin to arise, however, when we try to define specific rules by which we can determine which things are abstract, and which concrete. This might be illustrated by the difference between graph 1 and picture 1 above, in which the description sitting-on (graph 1) might be considered more abstract than the graphic instance of an image (picture 1).

Referents

Abstractions sometimes have ambiguous referents; for example, "happiness" (when used as an abstraction) can refer to many things as there are people and events or states of being which make them happy. A further example; suppose one attempts to define the term architecture and what it refers to. Architecture is more than simply designing safe functional buildings, it also involves elements of creation and innovation which aim at elegant solutions to problems of construction, the use of space, and at its best, to evoke an emotional response in the builders, owners, viewers and users of the building.

Instantiation

Something is often considered abstract if it does not exist at any particular place and time; instead instances, or members, of it might exist in many different places and/or times (we say that what is abstract can be multiply instantiated, in the sense of picture 1, picture 2, etc., shown above). If however we just say that what is abstract is what can be instantiated, and that abstraction is simply the movement in the opposite direction to instantiation, we haven't explained everything. That would make 'cat' and 'telephone' abstract ideas; but note that even small children can recognise an instance of a cat or a telephone, despite their varying appearances in particular cases. You could say that these concepts are abstractions but are not found to be as abstract as in the sense of the objects listed in graph 1, shown above. We might look at other graphs, in a progression from cat to mammal to animal, and see that animal is more abstract than mammal; but on the other hand mammal is a harder idea to express, certainly in relation to marsupial.

Physicality

Things are often said to be concrete, that is, not abstract, when they have physical existence or when they occupy space. In general, a concept is considered concrete if it is not abstract: it must be both particular and an individual, and hence occupy both space and time. To say that a physical object is concrete is to say, approximately, that it is a particular individual that is located at a particular place and time.[All these features relate to the known world on a level of both "consciousness and in regarding our sight" -that which we share on the physical plane- yet there is substantial arrangements also on the quantum level that also has effect (and affect) on our physical 'appearable' level, to which physicality is expanded both from us and beyond us, so to speak.]

Realness

Abstract things are sometimes defined as those things that do not exist in reality or exist only as sensory experience, like red. The problem begins to arise here when we try to decide which things are, in fact, real. Is God real, or abstract? Even if real, could God also be abstract? Is the number 3 real? Is goodness real, or only its effects, or is it just an abstract idea created by humans? [-All things must be assumed to be 'real' in respects to consciousness....] One approach to these questions is to consider the use of predicates, as a general term for whether things are variously real, abstract, concrete, good, etc. In this sense, the questions are then propositions about predicates, which remain to be evaluated by the investigator. In the graph 1 above, the predicates might be denoted by graphical relationships between objects, as in the arrows joining boxes and ellipses. The different levels of abstraction might be denoted by a progression of arrows joining boxes or ellipses in multiple rows, where the arrows point from one row to another, in a series of other graphs, say graph 2, etc.

Precise semantic meaning

Graph 1 details some explicit relationships between the objects of the diagram. For example the arrow between the agent and CAT:Elsie depicts an example of an is-a relationship, as does the arrow between the location and the MAT. The arrows between the SITTING gerund and the nouns agent and location express the diagram's basic relationship; "agent is SITTING on location"; Elsie is an instance of CAT.

Abstraction used in philosophy

Abstraction in philosophy is the (oft-alleged) process, in concept-formation, of recognizing some common feature among a number of individuals, and on that basis forming the concept of that feature. The notion of abstraction is important to understanding some philosophical controversies surrounding empiricism and the problem of universals. It has also recently become popular in formal logic under predicate abstraction.

Ontological status

If we say that properties of abstract concepts / relations are, or have being, clearly we mean they have a different sort of being from that which physical objects, like rocks and trees, have, in much the sense that picture 1 differs from graph 1. That accounts for the usefulness of the word abstract. We apply it to properties and relations to mark the fact that if they exist, they do not exist in space or time, but that instances of them can exist in many different places. On the other hand the apple and an individual human being are said to be concrete, and particulars, and individuals. Confusingly, philosophers sometimes refer to tropes, or property-instances (e.g., the particular redness of this particular apple), as abstract particulars.

Reification

Reification, also called hypostatization, might be considered a logical fallacy whenever an abstract concept, such as "society" or "technology" might be treated as if it were a concrete thing which can be photographed in a picture rather than sketched in a graph. It is important to note that reification necessarily occurs linguistically in the English language and many other languages wherein abstract objects are referred to using the same sorts of nouns that signify concrete objects. This can further confuse us about which things are abstract and which are concrete, as our loose use of language would tend to influence us toward examples of reification:
- England expects that every man will do his duty -- Horatio Nelson

Compression

An abstraction can be seen as a process of mapping multiple different pieces of constituent data to a single piece of abstract data based on similarities in the constituent data, for example many different physical cats map to the abstraction "CAT". This conceptual scheme emphasizes the inherent equality of both constituent and abstract data, thus avoiding problems arising from the distinction between "abstract" and "concrete". In this sense the process of abstraction entails the identification of similarities between objects and the process of associating these objects with an abstraction (which is itself an object).
- For example, picture 1 above illustrates the concrete relationship "Cat sits on Mat". Chains of abstractions can therefore be constructed moving from neural impulses arising from sensory perception to basic abstractions such as color or shape to experiential abstractions such as a specific cat to semantic abstractions such as the "idea" of a CAT to classes of objects such as "mammals" and even categories such as "object" as opposed to "action".
- For example, graph 1 above expresses the abstraction "agent sits on location". This conceptual scheme entails no specific hierarchical taxonomy (such as the one mentioned involving cats and mammals), only a progressive compression of detail.

The neurology of abstraction

Some research into the human brain suggests that the left and right hemispheres differ in their handling of abstraction. One side handles collections of examples (eg: examples of a tree) whereas the other handles the concept itself.

Abstraction in Art

Most typically abstraction is used in the arts as a synonym of Abstract art in general. It can, however, refer to any object or image which has been distilled from the real world, or indeed another work. Artist Robert Stark wrote,"Ten years after abandoning formal landscape painting to explore the more direct act of freely applying paint to a surface without a representational motive, I have developed a new vocabulary; light and dark, warm and cool, making marks, brush-strokes like heart-rhythms. Every day is a test of each painting's ability to stand on its own. Each painting is subject to being changed, to being reworked or scraped and repainted as long as it remains in the studio. Where I often used to spend weeks on a painting, attempting to 'make a picture,' now my concerns are more about the energy of light, the mass of space, the emotions of shadows. I want the painting to meet the viewer somewhere in the middle, where the viewer brings his own experiences to bear in understanding and feeling what he is seeing. I want my paintings to achieve the complexity and density of poetry or of a symphony, to build suggestive layers, implicit felt meaning, not merely to be entertaining bit of color to seduce the eye. I want my paintings to be accessible to children as well as adults, and to be so simply and directly painted that it shows the act of painting for the joy and excitement of it."

See also


- Abstract art
- Abstraction (computer science)
- Abstraction (mathematics)
- Abstract structure
- Abstract (summary), model (abstract)
- Abstract interpretation
- Gottlob Frege
- Ontology

External links


- [http://www.utm.edu/research/iep/f/frege.htm Internet Encyclopedia of Philosophy: Gottlob Frege]
- [http://plato.stanford.edu/entries/abstract-objects/ Stanford Encyclopedia of Philosophy: Abstract Objects]
- [http://originresearch.com/sd/sd1.cfm Discussion at The Well concerning Abstraction hierarchy]
- [http://www.cs.hmc.edu/claremont/keller/webBook/ch01/sec01.html The Purpose of Abstraction (a must read)]

References


- Eugene Raskin, Architecturally Speaking, 2nd edition, a Delta book, Dell (1966), trade paperback, 129 pages
- The American Heritage Dictionary of the English Language, 3rd edition, Houghton Mifflin (1992), hardcover, 2140 pages, ISBN 0395448956 Category:Philosophical terminology Category:Thought th:นามธรรม

Precision

Precision has the following meanings: #In engineering, science, industry and statistics, precision characterises the degree of mutual agreement among a series of individual measurements, values, or results - see accuracy and precision. #In computing, one of: ##the precision (number of digits) with which a value is expressed (e.g., the number of significant decimal digits or bits – a calculation which rounds to three digits is said to have a working precision or rounding precision of 3) ##the units of the least significant digit of a measurement; for example, if a measurement is 17.130 meters then its precision is millimeters (one unit in the last place, or ulp, is 1mm) ##(in some programming languages and databases) the number of decimal places after the decimal point in a fixed-point number; to avoid confusion, this usage is best avoided. #With respect to a set of independent devices of the same design, precision is the ability of these devices to produce the same value or result, given the same input conditions and operating in the same environment. (As defined by Federal Standard 1037C and MIL-STD-188.) #With respect to a single device, put into operation repeatedly without adjustments, precision is the ability to produce the same value or result, given the same input conditions and operating in the same environment. (As defined by Federal Standard 1037C and MIL-STD-188.) #In information retrieval, a measure of the number of relevant documents in the set of all documents returned by a search. It forms a natural pair with recall. #The Precision Club bidding system in the game of contract bridge. ----

Uncountable set

In mathematics, an uncountable or nondenumerable set is a set which is not countable. Here, "countable" means countably infinite or finite, so by definition, all uncountable sets are infinite. Explicitly, a set X is uncountable if and only if there does not exist a surjective function from the natural numbers N to X. Not all uncountable sets have the same size; the sizes of infinite sets are analyzed with the theory of cardinal numbers. Formally, an uncountable set is defined as one whose cardinality is strictly greater than \aleph_0 (aleph-null), the cardinality of the natural numbers). The best known example of an uncountable set is the set R of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable as well, for instance the set of all infinite sequences of natural numbers (and even the set of all infinite sequences consisting only of zeros and ones) and the set of all subsets of natural numbers. The cardinality of R is often called the cardinality of the continuum and denoted by c or \beth_1 (beth-one). The Cantor set is an uncountable subset of R. The Cantor set is a fractal and has Hausdorff dimension greater than zero but less than one (R has dimension one). This is an example of the following fact: any subset of R of Hausdorff dimension strictly greater than zero must be uncountable. Another example of a uncountable set is the set of all functions from R to R. It is not hard to believe that this set is even "more uncountable" than R. The cardinality of this set is \beth_2 (beth-two) and is, indeed, larger than \beth_1. A much more abstract example of an uncountable set is the set of all countably infinite ordinal numbers, denoted Ω. The cardinality of Ω is denoted \aleph_1 (aleph-one). It can be shown that \aleph_1 is the smallest uncountable cardinal number. One might naturally wonder whether \beth_1, the cardinality of the reals, is equal to \aleph_1 or if it is strictly larger. The statement that \aleph_1 = \beth_1 is called the continuum hypothesis. This hypothesis is now known to be independent from the ordinary axioms of set theory (cf. Zermelo-Frankel axioms). Which is to say, that one can either assume the continuum hypothesis is true, or assume that is false without running into any contradictions.

See also


- cardinal number
- aleph number
- beth number Category:Set theory

Injective

In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.
- A function f: \; A \to B is injective (one-to-one) if f(x)=f(y) \; \to \; x=y or, equivalently, if x \ne y \; \to \; f(x) \ne f(y). One could also say that every element of the codomain (sometimes called range) is mapped to by at most one element (argument) of the domain; not every element of the codomain, however, need have an argument mapped to it. An injective function is an injection.
- A function is surjective (onto) if every element of the codomain is mapped to by some element (argument) of the domain; some images may be mapped to by more than one argument. (Equivalently, a function where the range is equal to the codomain.) A surjective function is a surjection.
- A function is bijective (one-to-one and onto) if and only if (iff) it is both injective and surjective. (Equivalently, every element of the codomain is mapped to by exactly one element of the domain.) A bijective function is a bijection (one-to-one correspondence). (Note: a one-to-one function is injective, but may fail to be surjective, while a one-to-one correspondence is both injective and surjective.) An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). The four possible combinations of injective and surjective features are illustrated in the following diagrams.

Injection

if and only if A function is injective (one-to-one) if every possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is an injection. The formal definition is the following. :The function f: A \to B is injective iff for all a,b \in A, we have f(a) = f(b) \Rarr a = b.
- A function f : A → B is injective if and only if A is empty or f is left-invertible, that is, there is a function g: B → A such that g o f = identity function on A.
- Since every function is surjective when its codomain is restricted to its range, every injection induces a bijection onto its range. More precisely, every injection f : A → B can be factored as a bijection followed by an inclusion as follows. Let fR : A → f(A) be f with codomain restricted to its image, and let i : f(A) → B be the inclusion map from f(A) into B. Then f = i o fR. A dual factorisation is given for surjections below.
- The composition of two injections is again an injection, but if g o f is injective, then it can only be concluded that f is injective. See the figure at right.
- Every embedding is injective.

Surjection

embedding A function is surjective (onto) if every possible image is mapped to by at least one argument. In other words, every element in the codomain has non-empty preimage. Equivalently, a function is surjective if its range is equal to its codomain. A surjective function is a surjection. The formal definition is the following. :The function f: A \to B is surjective iff for all b \in B, there is a \in A such that f(a) = b.
- A function f : A → B is surjective if and only if it is right-invertible, that is, if and only if there is a function g: B → A such that f o g = identity function on B. (This statement is equivalent to the axiom of choice.)
- By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain. More precisely, every surjection f : A → B can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f. Let P(~) : A → A/~ be the projection map which sends each x in A to its equivalence class [x]~, and let fP : A/~ → B be the well-defined function given by fP([x]~) = f(x). Then f = fP o P(~). A dual factorisation is given for injections above.
- The composition of two surjections is again a surjection, but if g o f is surjective, then it can only be concluded that g is surjective. See the figure at right
- .

Bijection

axiom of choice A function is bijective if it is both injective and surjective. A bijective function is a bijection (one-to-one correspondence). A function is bijective if and only if every possible image is mapped to by exactly one argument. This equivalent condition is formally expressed as follows. :The function f: A \to B is bijective iff for all b \in B, there is a unique a \in A such that f(a) = b.
- A function f : A → B is bijective if and only if it is invertible, that is, there is a function g: B → A such that g o f = identity function on A and f o g = identity function on B. This function maps each image to its unique preimage.
- The composition of two bijections is again a bijection, but if g o f is a bijection, then it can only be concluded that f is injective and g is surjective. (See the figure at right and the remarks above regarding injections and surjections.)
- The bijections from a set to itself form a group under composition, called the symmetric group.

Cardinality

Suppose you want to define what it means for two sets to "have the same number of elements". One way to do this is to say that two sets "have the same number of elements" if and only if all the elements of one set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Accordingly, we can define two sets to "have the same number of elements" if there is a bijection between them. We say that the two sets have the same cardinality.

Examples

It is important to specify the domain and codomain of each function since by changing these, functions which we think of as the same may have different jectivity.

Injective and surjective (bijective)


- For every set A the identity function idA and thus specifically \mathbf \to \mathbf : x \mapsto x.
- \mathbf^+ \to \mathbf^+ : x \mapsto x^2 and thus also its inverse \mathbf^+ \to \mathbf^+ : x \mapsto \sqrt.
- The exponential function \exp : \mathbf \to \mathbf^+ : x \mapsto \mathrm^x and thus also its inverse the natural logarithm \ln : \mathbf^+ \to \mathbf : x \mapsto \ln

Injective and non-surjective


- The exponential function \exp : \mathbf \to \mathbf : x \mapsto \mathrm^x

Non-injective and surjective


- \mathbf \to \mathbf : x \mapsto (x-1)x(x+1) = x^3 - x
- The sine function f(x) = sin x

Non-injective and non-surjective


- \mathbf \to \mathbf : x \mapsto x^2

Properties


- For every function f, subset A of the domain and subset B of the codomain we have A ⊂ f −1(fA) and f(f −1B) ⊂ B. If f is injective we have A = f −1(fA) and if f is surjective we have f(f −1B) = B.
- For every function h : A → C we can define a surjection H : A → h(A) : a → h(a) and an injection I : h(A) → C : a → a. It follows that h = I o H. This decomposition is unique up to isomorphism.

Category theory

In the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms, and isomorphisms, respectively.

History

This terminology was originally coined by the Bourbaki group.

See also


- injective module
- permutation
- horizontal line test Category:Set theory ja:全単射

Bijective

In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.
- A function f: \; A \to B is injective (one-to-one) if f(x)=f(y) \; \to \; x=y or, equivalently, if x \ne y \; \to \; f(x) \ne f(y). One could also say that every element of the codomain (sometimes called range)