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class="row"> <div class="col"> <div class="text-muted"> An Entity of Type: <a href="http://www.w3.org/2002/07/owl#Thing">Thing</a>, from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, within Data Space: <a href="http://dbpedia.org">dbpedia.org</a> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is </div> <div class="col-xs-3 col-sm-2"> <a href="#" class="thumbnail"> <img src="http://commons.wikimedia.org/wiki/Special:FilePath/P._Oxy._I_29.jpg?width=300" alt="thumbnail" class="img-fluid" /> </a> </div> </div> </div> </section>   <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract">dbo:abstract</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">En matemàtiques, una demostració, també dita prova, és un raonament lògic que estableix la veritat d'una proposició matemàtica. A partir d'axiomes (suposats vertaders per definició) o teoremes (altres proposicions matemàtiques demostrades a partir dels axiomes) i en aplicar regles d'inferència vàlides, hom mostra que una altra proposició també és vertadera. Com que les matemàtiques són una ciència formal, no empírica, la demostració ha de ser un argument que no apel·li a cap fet empíric. Les proves s'obtenen de raonaments deductius, en lloc de raonaments inductius o arguments empírics. Això vol dir que una prova ha de demostrar que una proposició és certa en tots els casos sense una sola excepció. El rigor de la metodologia matemàtica exigeix que cap proposició no s'accepti com a vàlida fins que hom no en conegui una demostració correcta. Els enunciats matemàtics demostrats que es consideren d'una especial rellevància s'anomenen teoremes i els d'importància menor lemes o corol·laris. Una proposició matemàtica que no ha estat provada ni refutada i per la qual hi ha algunes intuïcions que fan pensar que és vertadera, és una conjectura o una hipòtesi. Les demostracions matemàtiques normalment s'escriuen i s'expliquen en llenguatge natural (amb un cert nivell d'artifici formal propi del quefer matemàtic), i això pot deixar lloc a una certa ambigüitat. És per això que la teoria de la demostració ha desenvolupat mètodes per formalitzar completament el raonament matemàtic i poder així determinar amb tota precisió la noció de demostració matemàtica vàlida. Aquestes investigacions han tingut sens dubte un gran valor per a la fonamentació i la filosofia de la matemàtica. Entre els mètodes de demostració habituals en matemàtiques podem enumerar la demostració directa, la demostració per inducció en els naturals, la demostració per contradicció o reducció a l'absurd, la demostració per separació de casos, la demostració constructiva, etcètera. (ca)</li> <li style="display:none;">في الرياضيات، البرهان أو الإثبات هي حجة استدلالية لتحديد صحة عبارة رياضية تستند على مُسلَّمات (بالإنجليزية: Axiom)‏ ومبرهنات (بالإنجليزية: Theorem)‏. يقال لعبارة رياضية أو علاقة رياضية بأنها صحيحة منطقيا إذا تم التوصل إليها في ظل هذه المجموعة من البدهيات باستخدام الاستنتاج وطرق الاستنباط المنطقية. البرهان الرياضي إذا عبارة عن حجة argument أو تعليل منطقي، ليس تجريبيا. ضمن هذا التعريف فإن مقولة أو عبارة رياضية يجب أن تبرهن على صحتها في جميع الظروف والحالات قبل أن يتم اعتبارها مبرهنة theorem رياضية. أما المقولة غير المبرهنة التي تلقى نوعا من الدعم التجريبي فتعرف بالحدسية conjecture. افتراضيا في جميع فروع الرياضيات، تكون البدهيات المفترضة هي بدهيات أي Zermelo–Fraenkel set theory (و هي نظرية مجموعات زيرميلو-فرينكل مع بدهيات الاختيار) ما لم يشار إلى بدهيات مختلفة. نظرية مجموعة زيرميلو-فرينكل تقوم بمشاكلة formalize (أي تجعله شكليا formal) الحدس الرياضي حول نظرية المجموعات، وفي نفس الوقت تقوم نظرية المجموعات بوصف الجبر والتحليل الرياضي. عندما يراد إثبات قضية رياضية يستحسن، في حال الإمكان، وضعها في صيغة اقتضاء ق ¬ ك، إن ذلك يتيح صياغة عكس هذه القضية بسهولة. يسمى العنصر الأيمن (المقدم) «ق» في الاقتضاء فرضاً، ويسمى العنصر الأيسر (التالي) «ك» طلباً. وعلى سبيل المثال تكتب المبرهنة: في كل متوازي أضلاع: ينصف كل من القطرين القطر الآخر، في صيغة اقتضاء كما يأتي: إذا كان الرباعي متوازي أضلاع، فإن قطريه ينصِّف كل منهما الآخر. فالفرض هو أن الرباعي متوازي الأضلاع، والطلب هو أن ينصف كل من قطريه القطر الآخر. للبرهان الرياضي عدة طرق : البرهان المباشر، العكسي، البرهان بالتناقض، البرهان بالاختيار، البرهان بالاستقراء... إلخ. مثلا البرهان المباشر وتعتمد هذه الطريقة على الاقتناع بأن متعدية ونعني بذلك أنه إذا كان : * أ تقتضي ب، ب تقتضي جـ فإن أ تقتضي جـ مثال: * أثبت أنه إذا كان س = 3 فإن 2(4 س + 5) – 1 = 33 البرهان س = 3 تقتضي 4 س = 12 تقتضي 4س + 5 = 17 تقتضي 2 (4س + 5) = 34 تقتضي 2 (4س + 5) – 1 = 33 (ar)</li> <li style="display:none;">V matematice je důkaz demonstrace nutné pravdivosti nějakého tvrzení za určitých předpokladů (axiomů). Matematický důkaz musí být založen výhradně na nezpochybnitelných pravidlech rozumu (ta jsou vyjádřena v matematické logice ve formě logických axiomů), nepřipouští žádný postup založený na názoru, experimentu, intuici či zkušenosti. Tato skutečnost dělá z matematického důkazu nejjistější známý způsob ověření pravdivosti nějakého tvrzení. Tvrzení, ke kterému je znám matematický důkaz, se nazývá matematická věta. (cs)</li> <li style="display:none;">Στα μαθηματικά, απόδειξη είναι μια μη κενή διαδικασία (δηλαδή, είναι μία διαδικασία η οποία περιλαμβάνει ένα τουλάχιστον λογικό βήμα) η οποία επικυρώνει ότι κάποια είναι ορθή, μέσα στα αποδεκτά πλαίσια του πεδίου των μαθηματικών. Η απόδειξη παράγεται αναγωγικά και όχι . Δηλαδή, η απόδειξη πρέπει να δείχνει ότι μια πρόταση είναι αληθής για όλες τις περιπτώσεις που εφαρμόζεται, χωρίς καμία εξαίρεση. Μια πρόταση χωρίς απόδειξη για την οποία πιστεύεται ή υπάρχουν ισχυρές υποψίες ότι ισχύει, λέγεται εικασία. Οι αποδείξεις χρησιμοποιούν τη λογική αλλά συνήθως περιέχουν σε κάποιο βαθμό φυσική γλώσσα, που συνήθως επιτρέπει κάποια ορισμένη αμφισημία. Όντως, η συντριπτική πλειονότητα των αποδείξεων στα γραπτά μαθηματικά μπορούν να θεωρηθούν εφαρμογές της . Αμιγώς μελετώνται από τη . Η διάκριση μεταξύ άτυπης και τυπικής απόδειξης έχει οδηγήσει σε επανεξέταση της τρέχουσας και ιστορικής , ημι-εμπειρικά μαθηματικά και τα λεγόμενα . Η φιλοσοφία των μαθηματικών ασχολείται με το ρόλο της γλώσσας και της λογικής στις αποδείξεις, καθώς και των μαθηματικών ως γλώσσα. Άσχετα από το βαθμό της τυπικότητας που ακολουθείται, το αποτέλεσμα που αποδεικνύεται λέγεται θεώρημα. Σε μια εντελώς τυπική απόδειξη αυτό είναι η τελευταία γραμμή, και η απόδειξη δείχνει πως αυτό ακολουθεί από τα αξιώματα μόνο, με εφαρμογή των κανόνων συναγωγής. Όταν ένα θεώρημα έχει αποδειχθεί, μπορεί να χρησιμοποιηθεί ως βάση για την απόδειξη άλλων προτάσεων. Ένα θεώρημα μπορεί να λέγεται και αν χρησιμοποιείται ως βήμα στην απόδειξη ενός θεωρήματος. Τα αξιώματα είναι οι προτάσεις αυτές που δεν γίνεται, ή δεν χρειάζεται, να αποδεικτούν. Αυτά ήταν στο παρελθόν η βασική μελέτη των φιλόσοφων των μαθηματικών, ενώ πρόσφατα εστιάζουν περισσότερο στη , δηλαδή τι αποτελεί αποδεκτή τακτική. (el)</li> <li style="display:none;">Ein Beweis ist in der Mathematik die als fehlerfrei anerkannte Herleitung der Richtigkeit bzw. der Unrichtigkeit einer Aussage aus einer Menge von Axiomen, die als wahr vorausgesetzt werden, und anderen Aussagen, die bereits bewiesen sind. Man spricht daher auch von axiomatischen Beweisen. Umfangreichere Beweise von mathematischen Sätzen werden in der Regel in mehrere kleine Teilbeweise aufgeteilt, siehe dazu Satz und Hilfssatz. In der Beweistheorie, einem Teilgebiet der mathematischen Logik, werden Beweise formal als Ableitungen aufgefasst und selbst als mathematische Objekte betrachtet, um etwa die Beweisbarkeit oder Unbeweisbarkeit von Sätzen aus gegebenen Axiomen selbst zu beweisen. (de)</li> <li style="display:none;">En matematiko pruvo estas demonstro de deviga vereco de iu aserto surbaze de certaj supozoj (aksiomoj). La matematika pruvo devas esti fondita eksplicite sur nedubeblaj reguloj de prudento (tiuj estas esprimataj en matematika logiko en formo de ), ĝi allasas nenian procedon fonditan en opinio, eksperimento, intuicio aŭ sperto. Tiu ĉi fakto farigas el la matematika pruvo la plej certan konatan manieron de verkontrolo de la vereco de iu aserto. Sed la samaj ecoj faras la matematikan pruvon tute ne eluzeblan en aliaj terenoj ol en la matematiko mem. La aserto, al kiu estas konata matematika pruvo, nomiĝas teoremo. (eo)</li> <li style="display:none;">Matematikan, bat egiaztatzeko erabiltzen den argudio deduktibo bat da froga edo frogapen bat. Argumentazioan aurretik ezarritako baieztapenak erabil daitezke, hala nola teoremak eta hasierako baieztapenak edo axiomak. Froga, printzipioz, funtsean frogarik behar ez duten baieztapenak lortu arte garatu daiteke, axioma izenekoak. Frogantzak dedukziozko arrazoibideen adibideak dira, eta indukzio bidezkoak eta enpirikoak bereizten dira. Frogapen batek baieztapen bat beti egiazkoa dela frogatu behar du (batzuetan, kasu posible guztiak zerrendatu eta horietako bakoitzean baliozkoa dela ikusita), eta ez kasu askotan onartzen dela. Egiazkotzat jotzen den frogatu gabeko baieztapen bati aieru deritzo. Frogapenetan, logikaz gain, normalean, hizkuntza naturala ere erabiltzen da, anbiguotasunen bat izan dezakeena. Egia esan, frogantza matematiko gehienak zorrotzaren aplikaziotzat jo daitezke. , (ez hizkuntza naturalean) idatziak, hartzen dira kontuan. bereizketak logika matematiko historikoa eta gaur egunekoa, eta ikertzea eragin du. Matematikaren filosofia hizkuntzaren eginkizunaz eta frogapenen logikaz arduratzen da. Teorema baten frogapena ez ezagutzeak ez du esan nahi egiazkoa ez denik; faltsua dela esango bada, ezeztapena frogatu behar da. (eu)</li> <li style="display:none;">En matemáticas, una demostración o bien una prueba es un argumento deductivo para asegurar la verdad de una proposición matemática. En la argumentación se pueden usar otras afirmaciones previamente establecidas, tales como teoremas o bien las afirmaciones iniciales o axiomas. En principio una demostración se puede rastrear hasta afirmaciones generalmente aceptadas, conocidas como axiomas. Las demostraciones son ejemplos de razonamiento deductivo y se distinguen de argumentos inductivos o empíricos; una demostración debe demostrar que una afirmación es siempre verdadera (ocasionalmente al listar todos los casos posibles y mostrar que es válida en cada uno), más que enumerar muchos casos confirmatorios. Una afirmación no probada que se cree verdadera se conoce como conjetura. Las locas demostraciones emplean lógica pero normalmente incluyen una buena parte de lenguaje natural, el cual usualmente admite alguna ambigüedad. De hecho, la gran mayoría de las demostraciones en las matemáticas escritas puede ser considerada como aplicaciones de lógica informal rigurosa. Las , escritas en lenguaje simbólico en lugar de lenguaje natural, se consideran en teoría de la demostración. La distinción entre demostraciones formales e informales ha llevado a examinar la lógica matemática histórica y actual, el cuasi-empirismo matemático y el formalismo matemático. La filosofía de las matemáticas concierne al rol del lenguaje y la lógica en las demostraciones, y en las matemáticas como lenguaje. El hecho de no conocer ninguna demostración de un teorema no implica su no veracidad; solo la demostración de la negación de este resultado implica que es falso. (es)</li> <li>A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language. (en)</li> <li style="display:none;">En mathématiques et en logique, une démonstration est un ensemble structuré d'étapes correctes de raisonnement. Dans une démonstration, chaque étape est soit un axiome (un fait acquis), soit l'application d'une règle qui permet d'affirmer qu'une proposition, la conclusion, est une conséquence logique d'une ou plusieurs autres propositions, les prémisses de la règle. Les prémisses sont soit des axiomes, soit des propositions déjà obtenues comme conclusions de l'application d'autres règles. Une proposition qui est la conclusion de l'étape ultime d'une démonstration est un théorème. Le terme « preuve » est parfois employé comme un synonyme de « démonstration » par attraction de l'anglais proof[réf. souhaitée]. La démonstration est foncièrement différente de l'argumentation, qui est une autre forme de raisonnement, employant des arguments qualitatifs, en faisant référence éventuellement à des données chiffrées, dans le but de pousser quelqu'un à agir. (fr)</li> <li style="display:none;">Pembuktian Matematika adalah sebuah demonstrasi yang meyakinkan atas , teorema itu benar, dengan bantuan logika dan matematika. Pembuatan bukti telah lama mendapatkan perhatian besar dalam . (in)</li> <li style="display:none;">Una dimostrazione matematica è un processo di deduzione che, partendo da premesse assunte come valide (ipotesi) o da proposizioni dimostrate in virtù di queste premesse, determina la necessaria validità di una nuova proposizione in virtù della (sola) correttezza formale del ragionamento. (it)</li> <li style="display:none;">数学における証明 (しょうめい、英語: Mathematical proof) とは、ある命題が正しいことを主張するための一連の演繹のこと。証明の各段階においては、前提（公理、定理等の認められた事実）や仮定から推論規則によって新たな命題を導くという形態をとる。ある証明の中で導入された仮定は、証明の別の部分で証明されるか、その証明の中で否定されなければならない（背理法）。命題 P を証明したいとき、P を直に証明することを直接証明と言う。それに対して P が真であることを直接証明する代わりに、P と同値な別の命題が真であることを証明する方法を間接証明と言う（これらはあくまで直観的な分類に過ぎず、数学的な定義があるわけではない）。 (ja)</li> <li style="display:none;">수학에서 증명(證明, 영어: proof)은 특정한 공리들을 가정하고, 그 가정 하에서 어떤 명제가 참이라는 것을 보여주는 것을 가리킨다. (특정한 공리는 별다른 언급이 없으면 선택 공리를 추가한 체르멜로-프렝켈 집합론으로 가정한다.) 증명은 논리를 통해 이루어져야 하지만 일반적으로 자연어를 포함하게 되며, 따라서 애매한 부분을 포함하기도 한다. 실제로 수학에서 대부분의 증명은 비형식 논리를 통해 이루어진다. 순수 형식논리를 통한 증명은 증명이론에서 다룬다. 형식적인 증명과 비형식적인 증명 사이의 구별은 형식주의에 철저히 입각해 있건 그렇지 않건, 참이라는 것이 밝혀진 명제는 정리라고 한다. 정리는 그 정의로서 참이라고 밝혀졌기에 다른 명제를 증명하는 데 사용할 수 있다. 증명 기법에는 다음과 같은 것들이 있다. * 직접 증명: 공리와 정의, 그리고 이미 증명된 정리를 논리적으로 직접 연결하여 증명한다.(연역적) * 수학적 귀납법: 바탕 명제(base case)가 참일 때, 귀납 규칙(induction rule)을 증명하여 무한히 많은 다른 명제들도 참이라는 것을 보인다. (귀납적) * (proof by construction): 어떤 성질을 만족하는 구체적인 예제를 하나 만들어 그 성질을 만족하는 어떤 것이 실제로 존재함을 증명한다. * 귀류법(reductio ad absurdum): 어떤 명제가 거짓이라고 가정하면 모순이 발생한다는 것을 증명하면, 그 명제가 참이어야 함을 알 수 있다. * 반례 은 확률론적 방법을 동원해 어떤 예제가 존재함을 보이는 것이다. 이것은 어떤 명제가 참이라는 것을 '확률적'으로 보이는 것과 전혀 다르며, 명제가 참일 확률을 계산하는 것과 실제로 그것을 증명하는 것은 거리가 멀다는 것을 콜라츠 추측을 통해 알 수 있다. 은 두 개의 서로 다른 식이 같은 값이라는 것을 보일 때 두 식이 같은 물체의 개수를 다른 방식으로 세는 것임을 증명한다. 참일 것으로 추정하지만 아직 참이라는 것이 밝혀지지 않은 명제는 추측이라고 한다. 때로는 주어진 공리계에서 어떤 명제를 증명하는 것은 불가능하다고 밝혀지는 경우도 있다. 연속체 가설이 그 예이다. 대부분의 공리계에는 증명도 반증도 불가능한 명제가 존재한다. 더 자세한 설명은 불완전성 정리를 참고 (ko)</li> <li style="display:none;">Een wiskundig bewijs is het volgens formele regels aantonen dat, gegeven bepaalde axioma's, een bepaalde bewering waar is. (nl)</li> <li style="display:none;">Dowód – wykazanie, że pewne zdanie jest prawdziwe. Dowód należy odróżnić od empirycznego lub heurystycznego rozumowania. Każdy krok dowodu musi jasno wynikać z poprzednich lub być przyjętym aksjomatem; rozumowanie niespełniające tego warunku nie jest dowodem. Ostatni krok dowodu to udowodnione zdanie, które w ten sposób staje się twierdzeniem danej teorii. Zwyczajowo koniec dowodu oznacza się skrótem q.e.d. (quod erat demonstrandum), c.n.d. (co należało dowieść), c.b.d.o. (co było do okazania) lub podobnym. (pl)</li> <li style="display:none;">Em matemática, uma prova é uma demonstração de que, dados certos axiomas, algum enunciado de interesse é necessariamente verdadeiro. Utiliza como base premissas intrínsecas a um modelo conceitual e um silogismo que, a partir de uma série de operações, chega ao resultado. Costuma-se marcar o final de uma prova com a abreviação c. q. d. (como queríamos demonstrar). (pt)</li> <li style="display:none;">Ett bevis eller mer generellt en härledning, är en följd av slutledningar, vilka från bestämda axiom och givna premisser leder fram till en slutsats. I matematiken kallas ett påstående som formellt kan bevisas, för ett teorem eller en sats. Ett påstående, som är obevisat, kallas för en förmodan. Hjälpsatser, som används vid bevisföringen kallas för lemman. I praktiken är bevisföring en kompromiss mellan stringens och enkelhet. Det har i matematikhistorien vid flera tillfällen hänt, att fel upptäckts i publicerade bevisförsök för satser, som tidigare betraktats som giltiga. Beviset för fyrfärgssatsen var under en period kontroversiellt eftersom det innehöll för tiden nya (datorberoende) kontrollmetoder, men numera accepteras dessa. Ett matematiskt bevis kan inte jämföras med bevis i andra vetenskaper, vars grundsatser kan förändras. (sv)</li> <li style="display:none;">Математическое доказательство — рассуждение с целью обоснования истинности какого-либо утверждения (теоремы), цепочка логических умозаключений, показывающая, что при условии истинности некоторого набора аксиом и правил вывода утверждение верно. В зависимости от контекста, может иметься в виду доказательство в рамках некоторой формальной системы (построенная по специальным правилам последовательность утверждений, записанная на формальном языке) или текст на естественном языке, по которому при необходимости можно восстановить формальное доказательство. Необходимость формального доказательства утверждений — одна из основных характерных черт математики как дедуктивной отрасли знаний, соответственно, понятие доказательства играет центральную роль в предмете математики, а наличие доказательств и их корректность определяют статус любых математических результатов. На протяжении всей истории математики представление о способах и допустимых методах доказательства существенно менялось, в основном, в сторону большей формализации и бо́льших ограничений. Ключевой вехой в вопросе формализации доказательства стало создание математической логики в XIX веке и формализация её средствами основных техник доказательства. В XX веке построена теория доказательств — теория, изучающая доказательство как математический объект. С появлением во второй половине XX века компьютеров особое значение получило применение методов математического доказательства для проверки и синтеза программ, и даже было установлено структурное соответствие между компьютерными программами и математическими доказательствами (соответствие Карри — Ховарда), на основе которого созданы средства автоматического доказательства. Основные приёмы, используемые при построении доказательств: прямое доказательство, математическая индукция и её обобщения, доказательство от противного, контрапозиция, построение, перебор, установление биекции, двойной счёт; в приложениях в качестве математических доказательств привлекаются также методы, не дающие формального доказательства, но обеспечивающие практическую применимость результата — вероятностные, статистические, приближённые. В зависимости от раздела математики, используемого формализма или математической школы не все методы могут приниматься безоговорочно, в частности, конструктивное доказательство предполагает серьёзные ограничения. (ru)</li> <li style="display:none;">在數學上，數學證明（Mathematical proof）是在一個特定的公理系統中，根据一定的规则或标准，由公理和定理推導出某些命題的過程。比起证据，数学证明一般依靠演绎推理，而不是依靠自然归纳和经验性的理据。這樣推導出來的命題也叫做該系統中的定理。數學證明建立在逻辑之上，但通常會包含若干程度的自然語言，因此可能會產生一些含糊的部分。實際上，用文字形式寫成的數學證明，在大多數情況都可以視為非形式邏輯的應用。在證明論的範疇內，則考慮那些用純形式化的语言写出的證明。這個区别导致了对過往到現在的、和的大部分检验。數學哲學就關注語言和邏輯在數學證明中的角色，和。 (zh)</li> <li style="display:none;">Доведення у математиці — процедура, за допомогою якої встановлюють істинність гіпотези чи будь-якого твердження. Принципи доведення вивчаються спеціальною галуззю математики — теорією доказів. 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text-break"><ul> <li><a class="uri" rel="dct:subject" resource="http://dbpedia.org/resource/Category:Mathematical_proofs" href="http://dbpedia.org/resource/Category:Mathematical_proofs">dbc:Mathematical_proofs</a></li> <li><a class="uri" rel="dct:subject" resource="http://dbpedia.org/resource/Category:Sources_of_knowledge" href="http://dbpedia.org/resource/Category:Sources_of_knowledge">dbc:Sources_of_knowledge</a></li> <li><a class="uri" rel="dct:subject" resource="http://dbpedia.org/resource/Category:Mathematical_logic" href="http://dbpedia.org/resource/Category:Mathematical_logic">dbc:Mathematical_logic</a></li> <li><a class="uri" rel="dct:subject" resource="http://dbpedia.org/resource/Category:Mathematical_terminology" href="http://dbpedia.org/resource/Category:Mathematical_terminology">dbc:Mathematical_terminology</a></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://purl.org/linguistics/gold/hypernym">gold:hypernym</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="gold:hypernym" resource="http://dbpedia.org/resource/Argument" prefix="gold: http://purl.org/linguistics/gold/" href="http://dbpedia.org/resource/Argument">dbr:Argument</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/1999/02/22-rdf-syntax-ns#type">rdf:type</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="rdf:type" resource="http://www.w3.org/2002/07/owl#Thing" href="http://www.w3.org/2002/07/owl#Thing">owl:Thing</a></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#comment">rdfs:comment</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">V matematice je důkaz demonstrace nutné pravdivosti nějakého tvrzení za určitých předpokladů (axiomů). Matematický důkaz musí být založen výhradně na nezpochybnitelných pravidlech rozumu (ta jsou vyjádřena v matematické logice ve formě logických axiomů), nepřipouští žádný postup založený na názoru, experimentu, intuici či zkušenosti. Tato skutečnost dělá z matematického důkazu nejjistější známý způsob ověření pravdivosti nějakého tvrzení. Tvrzení, ke kterému je znám matematický důkaz, se nazývá matematická věta. (cs)</li> <li style="display:none;">En matematiko pruvo estas demonstro de deviga vereco de iu aserto surbaze de certaj supozoj (aksiomoj). La matematika pruvo devas esti fondita eksplicite sur nedubeblaj reguloj de prudento (tiuj estas esprimataj en matematika logiko en formo de ), ĝi allasas nenian procedon fonditan en opinio, eksperimento, intuicio aŭ sperto. Tiu ĉi fakto farigas el la matematika pruvo la plej certan konatan manieron de verkontrolo de la vereco de iu aserto. Sed la samaj ecoj faras la matematikan pruvon tute ne eluzeblan en aliaj terenoj ol en la matematiko mem. La aserto, al kiu estas konata matematika pruvo, nomiĝas teoremo. (eo)</li> <li style="display:none;">Pembuktian Matematika adalah sebuah demonstrasi yang meyakinkan atas , teorema itu benar, dengan bantuan logika dan matematika. Pembuatan bukti telah lama mendapatkan perhatian besar dalam . (in)</li> <li style="display:none;">Una dimostrazione matematica è un processo di deduzione che, partendo da premesse assunte come valide (ipotesi) o da proposizioni dimostrate in virtù di queste premesse, determina la necessaria validità di una nuova proposizione in virtù della (sola) correttezza formale del ragionamento. (it)</li> <li style="display:none;">数学における証明 (しょうめい、英語: Mathematical proof) とは、ある命題が正しいことを主張するための一連の演繹のこと。証明の各段階においては、前提（公理、定理等の認められた事実）や仮定から推論規則によって新たな命題を導くという形態をとる。ある証明の中で導入された仮定は、証明の別の部分で証明されるか、その証明の中で否定されなければならない（背理法）。命題 P を証明したいとき、P を直に証明することを直接証明と言う。それに対して P が真であることを直接証明する代わりに、P と同値な別の命題が真であることを証明する方法を間接証明と言う（これらはあくまで直観的な分類に過ぎず、数学的な定義があるわけではない）。 (ja)</li> <li style="display:none;">Een wiskundig bewijs is het volgens formele regels aantonen dat, gegeven bepaalde axioma's, een bepaalde bewering waar is. (nl)</li> <li style="display:none;">Dowód – wykazanie, że pewne zdanie jest prawdziwe. Dowód należy odróżnić od empirycznego lub heurystycznego rozumowania. Każdy krok dowodu musi jasno wynikać z poprzednich lub być przyjętym aksjomatem; rozumowanie niespełniające tego warunku nie jest dowodem. Ostatni krok dowodu to udowodnione zdanie, które w ten sposób staje się twierdzeniem danej teorii. Zwyczajowo koniec dowodu oznacza się skrótem q.e.d. (quod erat demonstrandum), c.n.d. (co należało dowieść), c.b.d.o. (co było do okazania) lub podobnym. (pl)</li> <li style="display:none;">Em matemática, uma prova é uma demonstração de que, dados certos axiomas, algum enunciado de interesse é necessariamente verdadeiro. Utiliza como base premissas intrínsecas a um modelo conceitual e um silogismo que, a partir de uma série de operações, chega ao resultado. Costuma-se marcar o final de uma prova com a abreviação c. q. d. (como queríamos demonstrar). (pt)</li> <li style="display:none;">在數學上，數學證明（Mathematical proof）是在一個特定的公理系統中，根据一定的规则或标准，由公理和定理推導出某些命題的過程。比起证据，数学证明一般依靠演绎推理，而不是依靠自然归纳和经验性的理据。這樣推導出來的命題也叫做該系統中的定理。數學證明建立在逻辑之上，但通常會包含若干程度的自然語言，因此可能會產生一些含糊的部分。實際上，用文字形式寫成的數學證明，在大多數情況都可以視為非形式邏輯的應用。在證明論的範疇內，則考慮那些用純形式化的语言写出的證明。這個区别导致了对過往到現在的、和的大部分检验。數學哲學就關注語言和邏輯在數學證明中的角色，和。 (zh)</li> <li style="display:none;">Доведення у математиці — процедура, за допомогою якої встановлюють істинність гіпотези чи будь-якого твердження. Принципи доведення вивчаються спеціальною галуззю математики — теорією доказів. (uk)</li> <li style="display:none;">في الرياضيات، البرهان أو الإثبات هي حجة استدلالية لتحديد صحة عبارة رياضية تستند على مُسلَّمات (بالإنجليزية: Axiom)‏ ومبرهنات (بالإنجليزية: Theorem)‏. يقال لعبارة رياضية أو علاقة رياضية بأنها صحيحة منطقيا إذا تم التوصل إليها في ظل هذه المجموعة من البدهيات باستخدام الاستنتاج وطرق الاستنباط المنطقية. البرهان الرياضي إذا عبارة عن حجة argument أو تعليل منطقي، ليس تجريبيا. ضمن هذا التعريف فإن مقولة أو عبارة رياضية يجب أن تبرهن على صحتها في جميع الظروف والحالات قبل أن يتم اعتبارها مبرهنة theorem رياضية. أما المقولة غير المبرهنة التي تلقى نوعا من الدعم التجريبي فتعرف بالحدسية conjecture. افتراضيا في جميع فروع الرياضيات، تكون البدهيات المفترضة هي بدهيات أي Zermelo–Fraenkel set theory (و هي نظرية مجموعات زيرميلو-فرينكل مع بدهيات الاختيار) ما لم يشار إلى بدهيات مختلفة. نظرية مجموعة زيرميلو-فرينكل تق (ar)</li> <li style="display:none;">En matemàtiques, una demostració, també dita prova, és un raonament lògic que estableix la veritat d'una proposició matemàtica. A partir d'axiomes (suposats vertaders per definició) o teoremes (altres proposicions matemàtiques demostrades a partir dels axiomes) i en aplicar regles d'inferència vàlides, hom mostra que una altra proposició també és vertadera. (ca)</li> <li style="display:none;">Στα μαθηματικά, απόδειξη είναι μια μη κενή διαδικασία (δηλαδή, είναι μία διαδικασία η οποία περιλαμβάνει ένα τουλάχιστον λογικό βήμα) η οποία επικυρώνει ότι κάποια είναι ορθή, μέσα στα αποδεκτά πλαίσια του πεδίου των μαθηματικών. Η απόδειξη παράγεται αναγωγικά και όχι . Δηλαδή, η απόδειξη πρέπει να δείχνει ότι μια πρόταση είναι αληθής για όλες τις περιπτώσεις που εφαρμόζεται, χωρίς καμία εξαίρεση. Μια πρόταση χωρίς απόδειξη για την οποία πιστεύεται ή υπάρχουν ισχυρές υποψίες ότι ισχύει, λέγεται εικασία. (el)</li> <li style="display:none;">Ein Beweis ist in der Mathematik die als fehlerfrei anerkannte Herleitung der Richtigkeit bzw. der Unrichtigkeit einer Aussage aus einer Menge von Axiomen, die als wahr vorausgesetzt werden, und anderen Aussagen, die bereits bewiesen sind. Man spricht daher auch von axiomatischen Beweisen. Umfangreichere Beweise von mathematischen Sätzen werden in der Regel in mehrere kleine Teilbeweise aufgeteilt, siehe dazu Satz und Hilfssatz. (de)</li> <li style="display:none;">Matematikan, bat egiaztatzeko erabiltzen den argudio deduktibo bat da froga edo frogapen bat. Argumentazioan aurretik ezarritako baieztapenak erabil daitezke, hala nola teoremak eta hasierako baieztapenak edo axiomak. Froga, printzipioz, funtsean frogarik behar ez duten baieztapenak lortu arte garatu daiteke, axioma izenekoak. Frogantzak dedukziozko arrazoibideen adibideak dira, eta indukzio bidezkoak eta enpirikoak bereizten dira. Frogapen batek baieztapen bat beti egiazkoa dela frogatu behar du (batzuetan, kasu posible guztiak zerrendatu eta horietako bakoitzean baliozkoa dela ikusita), eta ez kasu askotan onartzen dela. Egiazkotzat jotzen den frogatu gabeko baieztapen bati aieru deritzo. (eu)</li> <li style="display:none;">En matemáticas, una demostración o bien una prueba es un argumento deductivo para asegurar la verdad de una proposición matemática. En la argumentación se pueden usar otras afirmaciones previamente establecidas, tales como teoremas o bien las afirmaciones iniciales o axiomas. En principio una demostración se puede rastrear hasta afirmaciones generalmente aceptadas, conocidas como axiomas. Las demostraciones son ejemplos de razonamiento deductivo y se distinguen de argumentos inductivos o empíricos; una demostración debe demostrar que una afirmación es siempre verdadera (ocasionalmente al listar todos los casos posibles y mostrar que es válida en cada uno), más que enumerar muchos casos confirmatorios. Una afirmación no probada que se cree verdadera se conoce como conjetura. (es)</li> <li>A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is (en)</li> <li style="display:none;">En mathématiques et en logique, une démonstration est un ensemble structuré d'étapes correctes de raisonnement. Dans une démonstration, chaque étape est soit un axiome (un fait acquis), soit l'application d'une règle qui permet d'affirmer qu'une proposition, la conclusion, est une conséquence logique d'une ou plusieurs autres propositions, les prémisses de la règle. Les prémisses sont soit des axiomes, soit des propositions déjà obtenues comme conclusions de l'application d'autres règles. Une proposition qui est la conclusion de l'étape ultime d'une démonstration est un théorème. (fr)</li> <li style="display:none;">수학에서 증명(證明, 영어: proof)은 특정한 공리들을 가정하고, 그 가정 하에서 어떤 명제가 참이라는 것을 보여주는 것을 가리킨다. (특정한 공리는 별다른 언급이 없으면 선택 공리를 추가한 체르멜로-프렝켈 집합론으로 가정한다.) 증명은 논리를 통해 이루어져야 하지만 일반적으로 자연어를 포함하게 되며, 따라서 애매한 부분을 포함하기도 한다. 실제로 수학에서 대부분의 증명은 비형식 논리를 통해 이루어진다. 순수 형식논리를 통한 증명은 증명이론에서 다룬다. 형식적인 증명과 비형식적인 증명 사이의 구별은 형식주의에 철저히 입각해 있건 그렇지 않건, 참이라는 것이 밝혀진 명제는 정리라고 한다. 정리는 그 정의로서 참이라고 밝혀졌기에 다른 명제를 증명하는 데 사용할 수 있다. 증명 기법에는 다음과 같은 것들이 있다. 은 확률론적 방법을 동원해 어떤 예제가 존재함을 보이는 것이다. 이것은 어떤 명제가 참이라는 것을 '확률적'으로 보이는 것과 전혀 다르며, 명제가 참일 확률을 계산하는 것과 실제로 그것을 증명하는 것은 거리가 멀다는 것을 콜라츠 추측을 통해 알 수 있다. 참일 것으로 추정하지만 아직 참이라는 것이 밝혀지지 않은 명제는 추측이라고 한다. (ko)</li> <li style="display:none;">Математическое доказательство — рассуждение с целью обоснования истинности какого-либо утверждения (теоремы), цепочка логических умозаключений, показывающая, что при условии истинности некоторого набора аксиом и правил вывода утверждение верно. В зависимости от контекста, может иметься в виду доказательство в рамках некоторой формальной системы (построенная по специальным правилам последовательность утверждений, записанная на формальном языке) или текст на естественном языке, по которому при необходимости можно восстановить формальное доказательство. Необходимость формального доказательства утверждений — одна из основных характерных черт математики как дедуктивной отрасли знаний, соответственно, понятие доказательства играет центральную роль в предмете математики, а наличие доказательств и (ru)</li> <li style="display:none;">Ett bevis eller mer generellt en härledning, är en följd av slutledningar, vilka från bestämda axiom och givna premisser leder fram till en slutsats. I matematiken kallas ett påstående som formellt kan bevisas, för ett teorem eller en sats. Ett matematiskt bevis kan inte jämföras med bevis i andra vetenskaper, vars grundsatser kan förändras. (sv)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label">rdfs:label</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">برهان رياضي (ar)</li> <li style="display:none;">Demostració (matemàtiques) (ca)</li> <li style="display:none;">Matematický důkaz (cs)</li> <li style="display:none;">Beweis (Mathematik) (de)</li> <li style="display:none;">Μαθηματική απόδειξη (el)</li> <li style="display:none;">Matematika pruvo (eo)</li> <li style="display:none;">Froga matematiko (eu)</li> <li style="display:none;">Demostración en matemática (es)</li> <li style="display:none;">Pembuktian matematika (in)</li> <li style="display:none;">Démonstration (logique et mathématique) (fr)</li> <li style="display:none;">Dimostrazione matematica (it)</li> <li style="display:none;">証明 (数学) (ja)</li> <li>Mathematical proof (en)</li> <li style="display:none;">증명 (수학) (ko)</li> <li style="display:none;">Wiskundig bewijs (nl)</li> <li style="display:none;">Dowód (matematyka) (pl)</li> <li style="display:none;">Prova matemática (pt)</li> <li style="display:none;">Математическое доказательство (ru)</li> <li style="display:none;">Matematiskt bevis (sv)</li> <li style="display:none;">數學證明 (zh)</li> <li style="display:none;">Доведення (uk)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#seeAlso">rdfs:seeAlso</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="rdfs:seeAlso" resource="http://dbpedia.org/resource/History_of_logic" href="http://dbpedia.org/resource/History_of_logic">dbr:History_of_logic</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#sameAs">owl:sameAs</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="owl:sameAs" resource="http://rdf.freebase.com/ns/m.0l46k" href="http://rdf.freebase.com/ns/m.0l46k">freebase:Mathematical proof</a></li> <li><a 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