Colocvial, un fractal
este "o figură geometrică fragmentată sau frântă care poate fi divizată în părţi, astfel încât fiecare dintre acestea să fie (cel puţin aproximativ) o copie miniaturală a întregului". Termenul a fost introdus de Benoît Mandelbrot în 1975 şi este derivat din latinescul fractus, însemnând "spart" sau "fracturat".
Fractalul, ca obiect geometric, are în general următoarele caracteristici:
* Are o structură fină la scări arbitrar de mici.
* Este prea neregulat pentru a fi descris în limbaj geometric euclidian tradiţional.
* Este autosimilar (măcar aproximativ sau stochastic).
* Are dimensiunea Hausdorff mai mare decât dimensiunea topologică (deşi această cerinţă nu este îndeplinită de curbele Hilbert).
* Are o definiţie simplă şi recursivă.
Deoarece par identici la orice nivel de magnificare, fractalii sunt de obicei consideraţi ca fiind infinit complecşi (în termeni informali). Printre obiectele naturale care aproximează fractalii până la un anumit nivel se numără norii, lanţurile montane, arcele de fulger, liniile de coastă şi fulgii de zăpadă. Totuşi, nu toate obiectele autosimilare sunt fractali—de exemplu, linia reală (o linie dreaptă Euclidiană) este autosimilară, dar nu îndeplineşte celelalte caracteristici.
Matematica din spatele fractalilor a apărut în secolul 17, când filosoful Gottfried Leibniz a considerat autosimilaritatea recursivă (deşi greşise gândindu-se că numai liniile drepte sunt autosimilare în acest sens).
Abia în 1872 a apărut o funcţie al cărei grafic este considerat azi fractal, când Karl Weierstrass a dat un exemplu de funcţie cu proprietatea că este continuă, dar nediferenţiabilă. În 1904, Helge von Koch, nesatisfăcut de definiţia abstractă şi analitică a lui Weierstrass, a dat o definiţie geometrică a unei funcţii similare, care se numeşte astăzi fulgul lui Koch. În 1915, Waclaw Sierpinski a construit triunghiul şi, un an mai târziu, covorul lui Sierpinski. La origine, aceşti fractali geometrici au fost descrişi drept curbe în loc de forme bidimensionale, aşa cum sunt cunoscute astăzi. Ideea de curbe autosimilare a fost preluată de Paul Pierre Lévy, care, în lucrarea sa Curbe şi suprafeţe în plan sau spaţiu formate din parţi similare întregului din 1938, a descris o nouă curbă fractal, curba C a lui Lévy.
a dat, de asemenea, exemple de submulţimi ale axei reale cu proprietăţi neobişnuite — aceste mulţimi Cantor sunt numite astăzi fractali.
Funcţiile iterate în planul complex au fost investigate la sfârşitul secolului 19 şi începutul secolului 20 de Henri Poincaré, Felix Klein, Pierre Fatou şi Gaston Julia. Totuşi, fără ajutorul graficii pe calculator moderne, ei nu puteau vizualiza frumuseţea numeroaselor obiecte pe care le descoepriseră.
În anii 1960, Benoît Mandelbrot a început să cerceteze autosimilaritatea în lucrări precum Cât de lungă este coasta Marii Britanii? Autosimilaritate statistică şi dimensiune fracţională. În sfârşit, în 1975, Mandelbrot
a inventat termenul "fractal" pentru a denumi un obiect al cărei dimensiune Hausdorff-Besicovitch este mai mare decât dimensiunea topologică a sa. A ilustrat această definiţie matematică cu imagini construite pe calculator.
O clasă de exemple simple este dată de mulţimile Cantor, triunghiul şi covorul lui Sierpinski, buretele lui Menger, curba dragon, curba lui Peano şi curba Koch. Alte exemple de fractali sunt fractalul lui Lyapunov şi mulţimile limită ale grupurilor Kleiniene. Fractalii pot fi determinişti (toţi cei anteriori) sau stocastici (adică nedeterminişti). De exemplu, traiectoriile mişcării browniene în plan au dimensiunea Hausdorff 2.
Sistemele haotice dinamice sunt uneori asociate cu fractalii. Obiectele din spaţiul fazelor dintr-un sistem dinamic pot fi fractali (vezi atractor). Obiectele din spaţiul parametrilor al unei familii de sisteme pot fi de asemenea fractali. Un exemplu interesant este mulţimea lui Mandelbrot. Această mulţime conţine discuri întregi, deci are dimensiunea Hausdorff egală cu dimensiunea topologică (adică 2) — dar ceea ce este surprinzător este că graniţa mulţimii lui Mandelbrot are de asemenea dimensiunea Hausdorff 2 (în timp ce dimensiunea topologică este 1), un rezultat demonstrat de Mitsuhiro Shishikura în 1991. Un fractal foarte înrudit este mulţimea Julia.
Chiar şi la curbele simple se poate observa proprietatea de autosimilaritate. De exemplu, distribuţia Pareto produce forme similare la diferite niveluri de grosisment.
Fractalii în natură
Fractali aproximativi sunt uşor de observat în natură. Aceste obiecte afişează o structură auto-similară la o scară mare, dar finită. Exemplele includ norii, fulgii de zăpadă, cristalele, lanţurile montane, fulgerele, reţelele de râuri, conopida sau broccoli şi sistemul de vase sanguine şi vase pulmonare.
Arborii şi ferigile sunt fractali naturali şi pot fi modelaţi pe calculator folosind un algoritm recursiv. Natura recursivă este evidentă în aceste exemple — o ramură a unui arbore sau o frunză a unei ferigi este o copie în miniatură a întregului: nu identice, dar similare.
În 1999, s-a demonstrat despre anumite forme de fractali auto-similari că au o proprietate de "frequency invariance" — aceleaşi proprietăţi electromagnetice indiferent de frecvenţă — din Ecuaţiile lui Maxwell. 
Fractalii în artă
Tipare de fractali au fost descoperite în picturile artistului american Jackson Pollock
. Deşi picturile lui Pollock's par a fi doar stropi haotici, analiza computerizată a descoperit tipare de fractali în opera sa.
Fractalii sunt de asemenea predominanţi în arta şi arhitectura africană. Casele circulare apar în cercuri de cercuri, casele dreptunghiulare în dreptunghiuri de dreptunghiuri şi aşa mai departe. Astfel de tipare se găsesc şi în textile şi sculpturile africane, precum şi în părul împletit în codiţe.
1. ^ Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W. H. Freeman and Company. ISBN 0-7167-1186-9.
2. ^ Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications, xxv, John Wiley & Sons, Ltd. ISBN 0-470-84862-6.
3. ^ Hohlfeld,R., and Cohen,N.,"SELF-SIMILARITY AND THE GEOMETRIC REQUIREMENTS FOR FREQUENCY INDEPENDENCE IN ANTENNAE ", Fractals, Vol. 7, No. 1 (1999) 79-84
4. ^ Richard Taylor, Adam P. Micolich and David Jonas. Fractal Expressionism : Can Science Be Used To Further Our Understanding Of Art?
5. ^ Ron Eglash. African Fractals: Modern Computing and Indigenous Design
. New Brunswick: Rutgers University Press 1999.
6. ^ Peng, Gongwen, Decheng Tian (21 July 1990). "The fractal nature of a fracture surface
". Journal of Physics A (14): 3257-3261. doi:10.1088/0305-4470/23/14/022. Retrieved on 2007-06-02.
As described above
, random fractals can be used to describe many highly irregular real-world objects. Other applications of fractals include:
* Classification of histopathology slides in medicine
* Fractal landscape or Coastline complexity
* Enzyme/enzymology (Michaelis-Menten kinetics)
* Generation of new music
* Generation of various art forms
* Signal and image compression
* Fractal in Soil Mechanics
* Computer and video game design, especially computer graphics for organic environments and as part of procedural generation
* Fractography and fracture mechanics
* Fractal antennas — Small size antennas using fractal shapes
* Small angle scattering theory of fractally rough systems
* Neo-hippies' t-shirts and other fashion
* Generation of patterns for camouflage, such as MARPAT
* Digital sundial
* Technical analysis of price series (see Elliott wave principle)
* Bifurcation theory
* Butterfly effect
* Chaos theory
* Constructal theory
* Contraction mapping theorem
* Diamond-square algorithm
* Droste effect
* Feigenbaum function
* Fractal art
* Fractal compression
* Fractal flame
* Fractal landscape
* List of fractals by Hausdorff dimension
* Publications in fractal geometry
* Newton fractal
Recursion in language
The use of recursion in linguistics, and the use of recursion in general, dates back to the ancient Indian linguist Pāṇini in the 5th century BC
, who made use of recursion in his grammar rules of Sanskrit
Linguist Noam Chomsky
theorizes that unlimited extension of a language such as English is possible only by the recursive device of embedding sentences in sentences. Thus, a chatty person may say, "Dorothy, who met the wicked Witch of the West in Munchkin Land where her wicked Witch sister was killed, liquidated her with a pail of water." Clearly, two simple sentences — "Dorothy met the Wicked Witch of the West in Munchkin Land" and "Her sister was killed in Munchkin Land" — can be embedded in a third sentence, "Dorothy liquidated her with a pail of water," to obtain a very verbose sentence.
However, if "Dorothy met the Wicked Witch" can be analyzed as a simple sentence, then the recursive sentence "He lived in the house Jack built" could be analyzed that way too, if "Jack built" is analyzed as an adjective, "Jack-built", that applies to the house in the same way "Wicked" applies to the Witch. "He lived in the Jack-built house" is unusual, perhaps poetic sounding, but it is not clearly wrong.
The idea that recursion is the essential property that enables language is challenged by linguist Daniel Everett in his work Cultural Constraints on Grammar and Cognition in Pirahã: Another Look at the Design Features of Human Language in which he hypothesizes that cultural factors made recursion unnecessary in the development of the Pirahã language. This concept challenges Chomsky's idea and accepted linguistic doctrine that recursion is the only trait which differentiates human and animal communication and is currently under intense debate.
Recursion in linguistics enables 'discrete infinity
' by embedding phrases within phrases of the same type in a hierarchical structure. Without recursion, language does not have 'discrete infinity' and cannot embed sentences into infinity (with a 'Russian doll' effect). Everett contests that language must have discrete infinity, and that the Piraha language - which he claims lacks recursion - is in fact finite. He likens it to the finite game of Chess, which has a finite number of moves but is nevertheless very productive, with novel moves being discovered throughout history.
Recursion in plain English
Recursion is the process a procedure goes through when one of the steps of the procedure involves rerunning the procedure. A procedure that goes through recursion is said to be recursive. Something is also said to be recursive when it is the result of a recursive procedure.
E.g., shampoo directions:
To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps that are to be taken based on a set of rules. The running of a procedure involves actually following the rules and performing the steps. An analogy might be that a procedure is like a menu in that it is the possible steps, while running a procedure is actually choosing the courses for the meal from the menu.
A procedure is recursive if one of the steps that makes up the procedure calls for a new running of the procedure. Therefore a recursive four course meal would be a meal in which one of the choices of appetizer, salad, entrée, or dessert was an entire meal unto itself. So a recursive meal might be potato skins, baby greens salad, chicken Parmesan, and for dessert, a four course meal, consisting of crab cakes, Caesar salad, for an entrée, a four course meal, and chocolate cake for dessert, so on until each of the meals within the meals is completed.
A recursive procedure must complete every one of its steps. Even if a new running is called in one of its steps, each running must run through the remaining steps. What this means is that even if the salad is an entire four course meal unto itself, you still have to eat your entrée and dessert.
A common geeky joke (for example recursion in the Jargon File) is the following "definition" of recursion.
Another example occurs in Kernighan and Ritchie's "The C Programming Language." The following index entry is found on page 269:
recursion 86, 139, 141, 182, 202, 269
This is a parody on references in dictionaries, which in some careless cases may lead to circular definitions. Jokes often have an element of wisdom, and also an element of misunderstanding. This one is also the second-shortest possible example of an erroneous recursive definition of an object, the error being the absence of the termination condition (or lack of the initial state, if looked at from an opposite point of view). Newcomers to recursion are often bewildered by its apparent circularity, until they learn to appreciate that a termination condition is key. A variation is:
If you still don't get it, See: "Recursion".
which actually does terminate, as soon as the reader "gets it".
Other examples are recursive acronyms, such as GNU, PHP or HURD.
GNU: GNU's Not Unix
PHP: PHP: Hypertext Preprocessor
HURD: HIRD of Unix-Replacing Daemons (where HIRD stands for HURD of Interfaces Representing Depth. It is also a play on the words herd of gnus)
In the philosophy of Subhash Kak, recursionism refers to the idea that replicated forms and self-similar forms are common in the physical world, and that this has some mystical significance. Kak describes recursionism as follows:
Patterns repeat across space, time, scale and fields. Recursion is an expression of the fundamental laws of nature, and it is to be seen both at the physical and the abstract levels as also across relational entities. Recursionism provides a way of knowing since it helps us to find meaning by a shift in perspective and by abstraction.
The idea of recursionism also occurs in Hindu Vedanta philosophy, where it is seen most prominently in the Upanishads
. There are recursionist strands in the works of Fichte, Schopenhauer, Nietzsche.
Subhash Kak. Recursionism and Reality
* Sacred geometry
related to the concepts of breaking the fourth wall
and meta-reference, which often involve self-reference.
- Gödel's theorem: "This sentence is false"
+ Repeat after me. We are all individuals. - (Graham Chapman) "Life of Brian"
+ I'm sorry, am I repeating myself? Am I being redundant? Am I saying things over and over?
My brilliant and beautiful wife without whom I would be nothing. She always comforts and consoles, never complains or interferes, asks nothing and endures all, and writes my dedications.
+ With a curvaceous figure that Venus would have envied, a tanned, unblemished oval face framed with lustrous think brown hair, deep azure-blue eyes fringed with long black lashes, perfect teeth that vied for competition, and a small straight nose, Marilee had a beauty that defied description
* Strange loop
- Escher. Drawing Hands
- The myriad time-travel paradoxes of classic Science Fiction can be perceived as creative versions of Strange Loops that fail by self-cancelling feedback. The two base forms of this are the Predestination paradox and the Ontological paradox
* Barnsley, Michael F., and Hawley Rising. Fractals Everywhere. Boston: Academic Press Professional, 1993. ISBN 0-12-079061-0
* Falconer, Kenneth. Techniques in Fractal Geometry. John Willey and Sons, 1997. ISBN 0-471-92287-0
* Jürgens, Hartmut, Heins-Otto Peitgen, and Dietmar Saupe. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992. ISBN 0-387-97903-4
* Benoît B. Mandelbrot The Fractal Geometry of Nature. New York: W. H. Freeman and Co., 1982. ISBN 0-7167-1186-9
* Peitgen, Heinz-Otto, and Dietmar Saupe, eds. The Science of Fractal Images. New York: Springer-Verlag, 1988. ISBN 0-387-96608-0
* Clifford A. Pickover, ed. Chaos and Fractals: A Computer Graphical Journey - A 10 Year Compilation of Advanced Research. Elsevier, 1998. ISBN 0-444-50002-2
* Jesse Jones, Fractals for the Macintosh, Waite Group Press, Corte Madera, CA, 1993. ISBN 1-878739-46-8.
* Hans Lauwerier, Fractals: Endlessly Repeated Geometrical Figures, Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991. ISBN 0-691-08551-X, cloth. ISBN 0-691-02445-6 paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.
* Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850839-5 and ISBN 978-0-19-850839-7.
* Bernt Wahl, Peter Van Roy, Michael Larsen, and Eric Kampman Exploring Fractals on the Macintosh
, Addison Wesley, 1995. ISBN 0-201-62630-6
* Nigel Lesmoir-Gordon. "The Colours of Infinity: The Beauty, The Power and the Sense of Fractals." ISBN 1-904555-05-5 (The book comes with a related DVD of the Arthur C. Clarke documentary introduction to the fractal concept and the Mandelbrot set.
* Gouyet, Jean-François. Physics and Fractal Structures (Foreword by B. Mandelbrot); Masson, 1996. ISBN 2-225-85130-1, and New York: Springer-Verlag, 1996. ISBN 0-387-94153-1. Out-of-print. Available in PDF version at .
at the Open Directory Project
The Sierpiński triangle has Hausdorff dimension log(3)/log(2) ≈ 1.585, which follows from the fact that it is a union of three copies of itself, each scaled by a factor of 1/2.
If one takes Pascal's triangle with 2n rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpiński triangle. More precisely, the limit as n approaches infinity of this parity-colored 2n-row Pascal triangle is the Sierpiński triangle.
The area of a Sierpiński triangle is zero (in Lebesgue measure). This can be seen from the infinite iteration, where we remove 25% of the area left at the previous iteration. Therefore the proportion of even numbers in Pascal's triangle must tend to 1 as the number of rows of the triangle tends to infinity.
* Eric W. Weisstein, Sierpinski Sieve at MathWorld.
* Sierpinski Triangle C++ code
* Paul W. K. Rothemund, Nick Papadakis, and Erik Winfree, Algorithmic Self-Assembly of DNA Sierpinski Triangles, PLoS Biology, volume 2, issue 12, 2004.
* Sierpinski Gasket by Trema Removal at cut-the-knot
* Sierpinski Gasket and Tower of Hanoi at cut-the-knot
* Article explaining Sierpinski's Triangle created with a bitwise XOR (example program in Macromedia Flash ActionScript)
* Article explaining Sierpinski's Triangle created with the Chaos Game (example program in Macromedia Flash ActionScript)
* Sierpinski Triangle and Machu Picchu. Fractal illustration with animation and sound.
* VisualBots - Freeware multi-agent simulator in Microsoft Excel. Sample programs include Sierpinski Triangle.
* IFS Fractal fern and Sierpinski triangle - JAVA applet
* Contains a section where the Sierpinski triangle can be seen step by step -- Shockwave
* The artist Richard Marquis has created murrine Sierpinski triangles which can be viewed on his website here, here, and elsewhere on his recent work page.
* Another reference, removed triangles are open sets
* Online Sierpinski Triangle Generator
Calitatea informaţiilor sau a exprimării din acest articol trebuie îmbunătăţită.
Acest articol a fost etichetat în octombrie 2007
George Ferdinand Ludwig Philipp Cantor (3 martie 1854 – 6 ianuarie 1918) a fost un matematician german. El a devenit faimos datorită teoriei sale despre mulţimi, ce a devenit o teorie fundamentală în matematică. Cantor a stabilit importanţa corespondenţei unu la unu între mulţimi, definind mulţimile infinite şi cele bine-ordonate şi dovedind totodată că numerele reale sunt “mult mai numeroase” decât numerele naturale. De fapt, teoremele lui Cantor implică existenţa unei “infinităţi de infinităţi”. El a definit numerele cardinale şi cele ordinale şi aritmetica lor. Opera lui Cantor era de un mare interes filosofic, lucru de care el era foarte conştient.[necesită citare]
Teoria lui Cantor privind numerele transfinite a fost de la început privită ca aşa de intuitiv contrară – şi chiar şocantă – încât a întâmpinat rezistenţă din partea matematicienilor contemporani, precum Leopold Kronecker, Henri Poincaré, iar mai târziu Hermann Weyl şi Luitzen Egbertus Jan Brouwer, în timp ce Ludwig Wittgenstein a ridicat obiecţii filosofice. Câţiva teologi creştini (mai exct neo-scolasticii) au văzut opera lui Cantor ca o provocare la unicitatea infinităţii absolute a naturii lui Dumnezeu, etichetând cu prima ocazie teoria numerelor transfinite cu panteismul.
Criticile dure au fost acoperite însă de consideraţiile internaţionale. În 1904 Societatea regală din Londra i-a acordat lui Cantor Medalia Silvestru[necesită citare], cea mai mare onoare ce o putea oferi. Astăzi, majoritatea matematicienilor, care nu sunt nici constructivişti, nici finişti acceptă opera lui Cantor privind mulţimile transfinite şi aritmetica, recunoscându-le ca o schimbare de paradigmă.[necesită citare]
Cantor s-a născut în 1845 în vestul coloniei Sankt Petersburg din Rusia. George, cel mai mare din cei şase fraţi, era un violonist eminent, moştenind de la părinţii săi considerabile talente muzicale şi artistice. Tatăl lui Cantor a fost membrul Bursei de valori din Sankt Petersburg; când s-a îmbolnăvit familia s-a mutat în Germania, în 1856, mai întîi la Wiesbad, apoi la Frankfurt, căutând ierni mai blînde decât cele din Sankt Petersburg. În 1860, Cantor a absolvit cu distincţii şcoala reală din Darmstadt; calificările sale excepţionale în matematică, cu precădere în trigonometrie, au fost remarcate. În 1862 Cantor a intrat la Institutul politehnic federal din Zurich. După ce a primit o moştenire substanţială după moartea tatălui din 1863, Cantor s-a mutata cu studiile la Universitatea din Berlin, luând parte la cursurile lui Kronecker, Karl Weierstrass şi Ernst Kummer. A petrecut vara anului 1866 la Universitatea din Gottingen, pe atunci un foarte important centru de cercetare matematică. În 1867 Berlin l-a numit doctor în folosofie pentru teza în teoria numerelor, De aequationibus secundi gradus indeterminati.[necesită citare]
Cantor a primit un post la Universitatea din Halle, unde si-a desfăşurat întreaga sa carieră profesională. Cantor a fost promovat ca profesor extraordinar în 1872 şi a fost făcut profesor pe deplin în 1879. Pentru a deţine un astfel de grad la vârsta de numai 34 de ani era o realizare nemaipomenită, dar Cantor visa la scaunul unei universităţi mult mai prestigioase, şi anume cea din Berlin. Totuşi opera sa a întâmpinat prea multă opoziţie pentru ca acest lucru să fie posibil. Cantor a suferit o depresie destul de gravă în 1884, criza sa emoţională îndreptându-l înspre filosofie. De asemenea a început un studiu intens privind literatura elisabetană, urmărind să arate că Francis Bacon a scris piesele de teatru atribuite lui Shakespeare.
În 1890 Cantor a contribuit la fondarea Deutsche Mathematiker-Vereinigung, prima întrunire având loc la Halle în 1891. În 1911, Cantor a fost unul din distinşii oameni de ştiinţă străini invitaţi să participe la a 500-a aniversare de la fondarea Universităţii St.Andrews din Scoţia. Anul următor această universitate i-a acordat lui Cantor titlul de Doctor Honoris Causa, dar acesta a fost prea bolnav pentru a primi personal această numire. A murit pe 6 ianuarie 1918 în sanatoriul unde şi-a petrecut ultimul an de viaţă.
Acest articol legat de matematică este deocamdată un ciot. Poţi ajuta Wikipedia prin completarea lui!
Number theory and function theory
Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this difficult problem in 1869. Between 1870 and 1872, Cantor published more papers on trigonometric series, including one defining irrational numbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts.
An illustration of Cantor's diagonal argument for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the infinite list of sequences above.
The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper, "Über eine Eigenschaft des Imbegriffes aller reellen algebraischen Zahlen" ("On a Characteristic Property of All Real Algebraic Numbers"). The paper, published in Crelle's Journal thanks to Dedekind's support (and despite Kronecker's opposition), was the first to formulate a mathematically rigorous proof that there was more than one kind of infinity
. This demonstration is a centerpiece of his legacy as a mathematician, helping lay the groundwork for both calculus and the analysis of the continuum of real numbers. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements). He then proved that the real numbers were not countable, albeit employing a proof more complex than the remarkably elegant and justly celebrated diagonal argument he set out in 1891. Prior to this, he had already proven that the set of rational numbers is countable.
Joseph Liouville had established the existence of transcendental
numbers in 1851, and Cantor's paper established that the set of transcendental numbers is uncountable. The logic is as follows: Cantor had shown that the union of two countable sets must be countable. The set of all real numbers is equal to the union of the set of algebraic numbers with the set of transcendental numbers (that is, every real number must be either algebraic or transcendental). The 1874 paper showed that the algebraic numbers (that is, the roots of polynomial equations with integer coefficients), were countable. In contrast, Cantor had also just shown that the real numbers were not countable
. If transcendental numbers were countable, then the result of their union with algebraic numbers would also be countable. Since their union (which equals the set of all real numbers) is uncountable, it logically follows that the transcendentals must be uncountable. The transcendentals have the same "power" (see below) as the reals, and "almost all" real numbers must be transcendental. Cantor remarked that he had effectively reproved a theorem, due to Liouville, to the effect that there are infinitely many transcendental numbers in each interval.
Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Kronecker, who admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole. Cantor also discovered the Cantor set during this period.
The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre" ("Foundations of a General Theory of Aggregates"), published in 1883, was the most important of the six and was also published as a separate monograph. It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.
In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel's first incompleteness theorem.
In 1895 and 1897, Cantor published a two-part paper in Mathematische Annalen under Felix Klein's editorship; these were his last significant papers on set theory. The first paper begins by defining set, subset, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B are equivalent. Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schroeder theorem.
Main article: Bijection
Cantor's 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Dedekind, Cantor proved a far stronger result: for any positive integer n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n-dimensional space. About this discovery Cantor famously wrote to Dedekind: "Je le vois, mais je ne le crois pas!
" ("I see it, but I don't believe it!") The result that he found so astonishing has implications for geometry and the notion of dimension.
In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence, and introduced the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined countable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space
Rn has the same power as the real numbers R, as does a countably infinite product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.
This paper, like the 1874 paper, displeased Kronecker, and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Weierstrass also supported its publication. Nevertheless, Cantor never again submitted anything to Crelle.
Main article: Continuum hypothesis
Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.
The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can neither be proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC").
Paradoxes of set theory
Discussions of set-theoretic paradoxes began to appear around the end of the nineteenth century. Some of these implied fundamental problems with Cantor's set theory program. In an 1897 paper on an unrelated topic, Cesare Burali-Forti set out the first such paradox, the Burali-Forti paradox: the ordinal number of the set of all ordinals must be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an 1896 letter to Hilbert. Criticism mounted to the point where Cantor launched counter-arguments in 1903, intended to defend the basic tenets of his set theory.
In 1899, Cantor discovered his eponymous paradox: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any set A, the cardinal number of the power set of A is strictly larger than the cardinal number of A (this fact is now known as Cantor's theorem). This paradox, together with Burali-Forti's, led Cantor to formulate a concept called limitation of size, according to which the collection of all ordinals, or of all sets, was an "inconsistent multiplicity" that was "too large" to be a set. Such collections later became known as proper classes.
One common view among mathematicians is that these paradoxes, together with Russell's paradox
, demonstrate that it is not possible to take a "naive", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for Zermelo and others to produce axiomatizations of set theory. Others note, however, that the paradoxes
do not obtain in an informal view motivated by the iterative hierarchy, which can be seen as explaining the idea of limitation of size. Some also question whether the Fregean formulation of naive set theory (which was the system directly refuted by the Russell paradox) is really a faithful interpretation of the Cantorian conception.
Philosophy, religion and Cantor's mathematics
The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's. He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one. To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications
—he identified the Absolute Infinite with God, and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.
Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence. Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind. Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set. Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all." Finally, Wittgenstein
's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set.
Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. In particular, Neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity". Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:
“ "…the transfinite species are just as much at the disposal of the intentions of the Creator and His absolute boundless will as are the finite numbers." ”
Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism—and was shocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophical beliefs.
In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim, as well as theologians such as Cardinal Johannes Franzelin, who once replied by equating the theory of transfinite numbers with pantheism. Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.
Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this metaphysical system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his famous assertion that "the essence of mathematics is its freedom
." These ideas parallel those of Edmund Husserl.
Cantor's 1883 paper reveals that he was well aware of the opposition his ideas were encountering:
“ "…I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers." ”
Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of contradiction and defined in terms of previously accepted concepts. He also cites Aristotle, Descartes, Berkeley, Leibniz, and Bolzano on infinity.
Cantor's paternal grandparents were from Copenhagen, and fled to Russia from the disruption of the Napoleonic Wars. In his letters, Cantor referred to them as "Israelites". However, there is no direct evidence on whether his grandparents practiced Judaism; there is very little direct information on them of any kind. Jakob Cantor, Cantor's grandfather, gave his children Christian saints' names. Further, several of his grandmother's relatives were in the Czarist civil service, which would not welcome Jews, unless they, or their ancestors, converted to Orthodox Christianity. Cantor's father, Georg Woldemar Cantor, was educated in the Lutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. His mother, Maria Anna Böhm, was an Austrian born in Saint Petersburg and baptized Roman Catholic; she converted to Protestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, saying
“ "Even if we were descended from Jews ten times over, and even though I may be, in principle, completely in favour of equal rights for Hebrews, in social life I prefer Christians..." ”
which could imply that she was of Jewish ancestry.
Thus Cantor was not himself Jewish
by faith, but has nevertheless been called variously German, Jewish, Russian, and Danish.
Until the 1970s, the chief academic publications on Cantor were two short monographs by Schönflies (1927)—largely the correspondence with Mittag-Leffler—and Fraenkel (1930). Both were at second and third hand; neither had much on his personal life. The gap was largely filled by Eric Temple Bell's Men of Mathematics (1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the history of mathematics"; and as "one of the worst". Bell presents Cantor's relationship with his father as Oedipal, Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics, and fills the picture with stereotypes. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell—including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents.
* Cantor's back-and-forth method
* Cantor function
* Heine–Cantor theorem
* Cantor medal—award by the Deutsche Mathematiker-Vereinigung in honor of Georg Cantor.
1. ^ In the Gregorian calendar (Grattan-Guinness 2000, p. 351). Some modern Russian sources give February 19, 1845, the equivalent date according to the Julian calendar, which was in use in Saint Petersburg at the time.
2. ^ The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert and Ilgauds 1985 are useful additional sources.
3. ^ Dauben 2004, p. 1.
4. ^ a b Dauben, 1977, p. 86; Dauben, 1979, pp. 120 & 143.
5. ^ a b Dauben, 1977, p. 102.
6. ^ a b c Dauben 1979, p. 266.
7. ^ Dauben 2004, p. 1. See also Dauben 1977, p. 89 15n.
8. ^ a b Rodych 2007
9. ^ a b c Dauben 1979, p. 280:"…the tradition made popular by [Arthur Moritz Schönflies] blamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression.
10. ^ a b Dauben 2004, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor's examining physicians at Halle Nervenklinik, referring to Cantor's mental illness as "cyclic manic-depression".
11. ^ a b c Dauben 1979, p. 248.
12. ^ a b Dauben 2004, pp. 8, 11 & 12-13.
13. ^ a b Reid 1996, p. 177.
14. ^ Dauben 1979, p. 163.
15. ^ Dauben 1979, p. 34.
16. ^ Dauben 1977, p. 89 15n.
17. ^ Dauben 1979, pp. 2–3; Grattan-Guinness 1971, pp. 354–355.
18. ^ Dauben 1979, p. 138.
19. ^ Dauben 1979, p. 139.
20. ^ a b Dauben 1979, p. 282.
21. ^ Dauben 1979, p. 136; Grattan-Guinness 1971, pp. 376–377. Letter dated June 21, 1884.
22. ^ Dauben 1979, pp. 281–283.
23. ^ Dauben 1979, p. 283.
24. ^ For a discussion of König's paper see Dauben 1979, 248–250. For Cantor's reaction, see Dauben 1979, p. 248; 283.
25. ^ Dauben 1979, p. 283–284.
26. ^ Dauben 1979, p. 284.
27. ^ a b Johnson 1972, p. 55.
28. ^ This paragraph is a highly abbreviated summary of the impact of Cantor's lifetime of work. More details and references can be found later.
29. ^ A countable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this terminology is not universally followed, and sometimes "denumerable" is used as a synonym for "countable".
30. ^ This follows closely the first part of Cantor's 1891 paper.
31. ^ Moore 1995, pp. 112 & 114; Dauben 2004, p. 1.
32. ^ For example, geometric problems posed by Galileo and John Duns Scotus suggested that all infinite sets were equinumerous — see Moore 1995, p. 114.
33. ^ For this, and more information on the mathematical importance of Cartan's work on set theory, see e.g., Suppes 1972.
34. ^ Dauben 1977, p. 89.
35. ^ The English translation is Cantor 1955.
36. ^ Wallace 2003, p. 259.
37. ^ Dauben 1979, p. 69; 324 63n. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it.
38. ^ Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidence for or against CH itself; among the most prominent of these is W. Hugh Woodin. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.
39. ^ Dauben 1979, pp. 240–270; see especially pp. 241 & 259.
40. ^ Hallett 1986.
41. ^ Weir 1998, p. 766: "…it may well be seriously mistaken to think of Cantor's Mengenlehre [set theory] as naive…"
42. ^ Dauben 1979, p. 295.
43. ^ Dauben, 1979, p. 120.
44. ^ Hallett 1986, p. 13. Compare to the writings of Thomas Aquinas.
45. ^ Dauben 1979, p. 225
46. ^ Snapper 1979, p. 3
47. ^ Davenport 1997, p.3
48. ^ a b Dauben, 1977, p. 85.
49. ^ Cantor 1932, p. 404. Translation in Dauben 1977, p. 95.
50. ^ Dauben 1979, p. 296.
51. ^ Dauben, 1979, p. 144.
52. ^ Dauben 1977 pp. 91–93.
53. ^ On Cantor, Husserl, and Gottlob Frege, see Hill and Rosado Haddock (2000).
54. ^ Dauben 1979, p. 96.
55. ^ E.g., Grattan-Guinness's only evidence on the grandfather's date of death is that he signed papers at his son's engagement.
56. ^ Purkert and Ilgauds 1987, p. 15.
57. ^ For more information, see: Dauben 1979, p. 1 and notes; Grattan-Guinness 1971, pp. 350–352 and notes; Purkert and Ilgauds 1985; the letter is from Aczel 2000, pp. 93–94, from Louis' trip to Chicago in 1863. It is ambiguous in German, as in English, whether the recipient is included.
58. ^ Cantor was frequently described as Jewish in his lifetime. For example, Jewish Encyclopedia, art. "Cantor, Georg"; Jewish Year Book 1896–1897, "List of Jewish Celebrities of the Nineteenth Century", p.119; this list has a star against people with one Jewish parent, but Cantor is not starred.
59. ^ Grattan-Guinness 1971, p. 350.
60. ^ Grattan-Guinness 1971 (quotation from p. 350, note), Dauben 1979, p.1 and notes. (Bell's Jewish stereotypes appear to have been removed from some postwar editions.)
Older sources on Cantor's life should be treated with caution
. See Historiography section above.
Primary literature in English
* Cantor, Georg (1955, 1915). Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover. ISBN 978-0486600451
* Ewald, William B. (ed.) (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. ISBN 978-0198532712
Primary literature in German
* Cantor, Georg (1932). Gesammelte Abhandlungen mathematischen und philosophischen inhalts. (PDF) Almost everything that Cantor wrote.
* Aczel, Amir D. (2000). The mystery of the Aleph
: Mathematics, the Kabbala, and the Human Mind. New York: Four Walls Eight Windows Publishing. ISBN 0760777780. A popular treatment of infinity, in which Cantor is frequently mentioned.
* Dauben, Joseph W. (1977). Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite. Journal of the History of Ideas 38.1.
* Dauben, Joseph W. (1979). Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. The definitive biography to date. ISBN 978-0-691-02447-9
* Dauben, Joseph (1993, 2004). "Georg Cantor and the Battle for Transfinite Set Theory" in Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA) (pp. 1–22). Internet version published in Journal of the ACMS 2004.
* Davenport, Anne A. (1997). The Catholics, the Cathars, and the Concept of Infinity
in the Thirteenth Century. Isis 88.2:263–295.
* Grattan-Guinness, Ivor (1971). Towards a Biography of Georg Cantor. Annals of Science 27:345–391.
* Grattan-Guinness, Ivor (2000). The Search for Mathematical Roots: 1870–1940. Princeton University Press. ISBN 978-0691058580
* Hallett, Michael (1986). Cantorian Set Theory and Limitation of Size. New York: Oxford University Press. ISBN 0-19-853283-0
* Halmos, Paul (1998, 1960). Naive Set Theory. New York & Berlin: Springer. ISBN 3540900926
* Hill, C. O. & Rosado Haddock, G. E. (2000). Husserl or Frege? Meaning, Objectivity, and Mathematics. Chicago: Open Court. ISBN 0812695380 Three chapters and 18 index entries on Cantor.
* Johnson, Phillip E. (1972). The Genesis and Development of Set Theory. The Two-Year College Mathematics Journal 3.1:55–62.
* Moore, A.W. (1995, April). A brief history of infinity
. Scientific American.4:112–116.
* Penrose, Roger (2004). The Road to Reality. Alfred A. Knopf. ISBN 0679776311 Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist.
* Purkert, Walter & Ilgauds, Hans Joachim (1985). Georg Cantor: 1845–1918. Birkhäuser. ISBN 0-8176-1770-1
* Reid, Constance (1996). Hilbert. New York: Springer-Verlag. ISBN 0387049991
* Rucker, Rudy (2005, 1982). Infinity and the Mind. Princeton University Press. ISBN 0553255312 Deals with similar topics to Aczel, but in more depth.
* Rodych, Victor (2007). "Wittgenstein's Philosophy of Mathematics" in Edward N. Zalta (Ed.) The Stanford Encyclopedia of Philosophy.
* Snapper, Ernst (1979). The Three Crises in Mathematics: Logicism, Intuitionism and Formalism. Mathematics Magazine 524:207–216.
* Suppes, Patrick (1972, 1960). Axiomatic Set Theory. New York: Dover. ISBN 0486616304 Although the presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor's results, which demonstrates Cantor's continued importance for the edifice of foundational mathematics.
* Wallace, David Foster (2003). Everything and More: A Compact History of Infinity
. New York: W.W. Norton and Company. ISBN 0393003388
* Weir, Alan (1998). Naive Set Theory is Innocent!. Mind 107.428:763–798.
* O'Connor, John J. & Robertson, Edmund F., “Georg Cantor”, MacTutor History of Mathematics archive
* O'Connor, John J. & Robertson, Edmund F., “A history of set theory”, MacTutor History of Mathematics archive Mainly devoted to Cantor's accomplishment.
* Georg Cantor at the Mathematics Genealogy Project
* Selections from Cantor's philosophical writing.
* Text of Cantor's 1891 diagonal argument.
* Stanford Encyclopedia of Philosophy: Set theory by Thomas Jech.
* Grammar school Georg-Cantor Halle (Saale): Georg-Cantor-Gynmasium Halle
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