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13 | https://zh.wikipedia.org/wiki?curid=13 | 數學 | 數學，是研究數量、結構以及空間等概念及其變化的一門學科，屬於形式科學的一種。數學利用抽象化和邏輯推理，從計數、計算、量度、對物體形狀及運動的觀察發展而成。數學家們拓展這些概念，以公式化新的猜想，以及從選定的公理及定義出發，嚴謹地推匯出一些定理。基礎數學的知識與運用是生活中不可或缺的一環。對數學基本概念的完善，早在古埃及、美索不達米亞及古印度歷史上的古代數學文字便可觀見，而在古希臘那裡有更為嚴謹的處理。從那時開始，數學的發展便持續不斷地小幅進展，至16世紀的文藝復興時期，因為新的科學發現和數學革新兩者的互動，致使數學的加速發展，直至今日。數學併成為許多國家及地區的教育中的一部分。數學在許多領域都有應用，包括科學、工程、醫學、經濟學和金融學等。數學對這些領域的應用通常被稱為應用數學，有時亦會激起新的數學發現，並導致全新學科的發展，例如物理學的實質性發展中建立的某些理論激發數學家對於某些問題的不同角度的思考。數學家也研究純粹數學，就是數學本身的實質性內容，而不以任何實際應用為目標。許多研究雖然以純粹數學開始，但其過程中也發現許多可用之處。詞源.西方語言中“數學”一詞源自於古希臘語的--），其有“學習”、“學問”、“科學”，還有個較狹義且技術性的意思－「數學研究」，即使在其語源內。其形容詞--），意思為「和學習有關的」或「用功的」，亦會被用來指「數學的」。其在英語中表面上的複數形式，及在法語中的表面複數形式"--"，可溯至拉丁文的中性複數"--"，由西塞羅譯自希臘文複數--漢字表示的「數學」一詞大約產生於中國宋元時期。多指象數之學，但有時也含有今天上的數學意義，例如，秦九韶的《數學九章》。直到1939年，經過中國數學名詞審查委員會研究“算學”與“數學”兩詞的使用狀況後，確認以“數學”表示今天意義上的數學含義。歷史.數學有著久遠的歷史。它被認為起源於人類早期的生產活動：中國古代的六藝之一就有「數」，數學一詞在西方有希臘語詞源--，意思是“學問的基礎”，源於--。史前的人類就已嘗試用自然的法則來衡量物質的多少、時間的長短等抽象的數量關係，比如時間單位有日、季節和年等。算術也自然而然地產生了。古代的石碑及泥版亦證實了當時已有幾何的知識。更進一步則需要寫作或其他可記錄數字的系統，如符木或於印加帝國內用來儲存資料的奇普。歷史上曾有過許多不同的記數系統。在最初有歷史記錄的時候，數學內的主要原理是為了做稅務和貿易等相關計算，為瞭解數字間的關係，為了測量土地，以及為了預測天文事件而形成的。這些可以簡單地被概括為數學對數量、結構、空間及時間方面的研究。到了16世紀，算術、初等代數以及三角學等初等數學已大體完備。17世紀變數概念的產生使人們開始研究變化中的量與量的互相關係和圖形間的互相變換，微積分的概念也在此時形成。隨著數學轉向形式化，為研究數學基礎而產生的集合論和數理邏輯等也開始發展。數學的重心從求解實際問題轉變到對一般形式上的思考。從古至今，數學便一直不斷地延展，且與科學有豐富的相互作用，兩者的發展都受惠於彼此。在歷史上有著許多數學發現，並且直至今日都不斷地有新的發現。據米哈伊爾·B·塞甫留克現已超過了一百九十萬份，而且每年還增加超過七萬五千份。此一學海的絕大部份為新的數學定理及其證明。」形成、純數學與應用數學及美學.每當有涉及數量、結構、空間及變化等方面的問題時，通常就需要用到數學去解決問題，而這往往也拓展了數學的研究範疇。一開始，數學的運用可見於貿易、土地測量及之後的天文學。今日，所有的科學都存在著值得數學家研究的問題，且數學本身亦給出了許多的問題。牛頓和萊布尼茲是微積分的發明者，費曼發明瞭費曼路徑積分，這是推理及物理洞察二者的產物，而今日的弦理論亦引申出新的數學。一些數學只和生成它的領域有關，且用來解答此領域的更多問題。但一般被一領域生成的數學在其他許多領域內也十分有用，且可以成為一般的數學概念。即使是「最純的」數學通常亦有實際的用途，此一非比尋常的事實，被1963年諾貝爾物理獎得主維格納稱為「數學在自然科學中不可想像的有效性」。如同大多數的研究領域，科學知識的爆發導致了數學的專業化。主要的分歧為純數學和應用數學。在應用數學內，又被分成兩大領域，並且變成了它們自身的學科——統計學和電腦科學。許多數學家談論數學的"優美"，其內在的美學及美。「簡單」和「一般化」即為美的一種。另外亦包括巧妙的證明，如歐幾裡得對存在無限多質數的證明；又或者是加快計算的數值方法，如快速傅立葉變換。高德菲·哈羅德·哈代在《一個數學家的自白》一書中表明他相信單單是美學上的意義，就已經足夠成為數學研究的正當理由。符號、語言與精確性.我們現今所使用的大部分數學符號在16世紀後才被發明出來的。在此之前，數學以文字的形式書寫出來，這種形式會限制了數學的發展。現今的符號使得數學對於專家而言更容易掌握，但初學者卻常對此望而卻步。它被極度的壓縮：少量的符號包含著大量的訊息。如同音樂符號一般，現今的數學符號有明確的語法，並且有效地對訊息作編碼，這是其他書寫方式難以做到的。符號化和形式化使得數學迅速發展，並幫助各個科學領域建立基礎支撐理論。數學語言亦對初學者而言感到困難。如“或”和“只”這些字有著比日常用語更精確的意思。亦困惱著初學者的，如“開放”和“域”等字在數學裡有著特別的意思。數學術語亦包括如“同胚”及“可積性”等專有名詞。但使用這些特別符號和專有術語是有其原因的：數學需要比日常用語更多的精確性。數學家將此對語言及邏輯精確性的要求稱為「嚴謹」。但在現實應用中，捨棄一些嚴謹性往往會得到更好的結果。嚴謹是數學證明中很重要且基本的一部份。數學家希望他們的定理以系統化的推理依著公理被推論下去。這是為了避免依著不可靠的直觀而推出錯誤的「定理」，而這情形在歷史上曾出現過許多的例子。在數學中被期許的嚴謹程度因著時間而不同：希臘人期許著仔細的論證，但在牛頓的時代，所使用的方法則較不嚴謹。牛頓為瞭解決問題所做的定義，到了十九世紀才重新以小心的分析及正式的證明來處理。今日，數學家們則持續地在爭論電腦協助證明的嚴謹度。當大量的計算難以被驗證時，其證明亦很難說是足夠地嚴謹。公理在傳統的思想中是「不證自明的真理」，但這種想法是有問題的。在形式上，公理只是一串符號，其只對可以由公理系統導出的公式之內容有意義。希爾伯特計劃即是想將所有的數學放在堅固的公理基礎上，但依據哥德爾不完備定理，每一相容且能蘊涵皮亞諾公理的公理系統必含有一不可決定的公式；因而所有數學的最終公理化是不可能的。儘管如此，數學常常被想像成只是某種公理化的集合論，在此意義下，所有數學敘述或證明都可以寫成集合論的公式。數學作為科學.卡爾·弗里德里希·高斯稱數學為「科學的皇后」。在拉丁原文"--"，以及其德語"--"中，對應於「科學」的單字的意思皆為知識。而實際上，science一詞在英語內本來就是這個意思，且無疑問地數學在此意義下確實是一門「科學」。將科學限定在自然科學則是在此之後的事。若認為科學是隻指物理的世界時，則數學，或至少是純數學，不會是一門科學。愛因斯坦曾如此描述：「數學定律越和現實有關，它們越不確定；若它們越是確定的話，它們和現實越不會有關。」許多哲學家相信數學在經驗上不具可否證性，且因此不是卡爾·波普爾所定義的科學。但在1930年代時，在數理邏輯上的重大進展顯示數學不能歸併至邏輯內，且波普爾推斷「大部份的數學定律，如物理及生物學一樣，是假設演繹的：純數學因此變得更接近其假設為猜測的自然科學，比它現在看起來更接近。」然而，其他的思想家，如較著名的拉卡託斯，便提供了一個關於數學本身的可否證性版本。另一觀點則為某些科學領域即認為科學是一種公眾知識，因此亦包含著數學。在任何的情況下，數學和物理科學的許多領域都有著很多相同的地方，尤其是從假設所得的邏輯推論之探索。直覺和實驗在數學和科學的猜想建構上皆扮演著重要的角色。實驗數學在數學中的重要性正持續地在增加，且計算和模擬在科學及數學中所扮演的角色也越來越加重，減輕了數學不使用科學方法的缺點。在史蒂芬·沃爾夫勒姆2002年的著作《一種新科學》中他提出，計算數學應被視為其自身的一科學領域來探索。數學家對此的態度並不一致。一些研究應用數學的數學家覺得他們是科學家，而那些研究純數學的數學家則時常覺得他們是在一門較接近邏輯的領域內工作，且因此基本上是個哲學家。許多數學家認為稱他們的工作是一種科學，是低估了其美學方面的重要性，以及其做為七大博雅教育之一的歷史；另外亦有人認為若忽略其與科學之間的關聯，是假裝沒看到數學和其在科學與工程之間的互動影響，進而促進了數學在許多科學上的發展此一事實。這兩種觀點之間的差異在哲學上產生了數學是「被創造」的爭議。大學院系劃分中常見「科學和數學系」，這指出了這兩個領域被看作有緊密聯絡而非一樣。實際上，數學家通常會在大體上與科學家合作，但在細節上卻會分開。此爭議亦是數學哲學眾多議題的其中一個。數學的各領域.如上所述，數學主要的學科最先產生於商業上計算的需要、瞭解數字間的關係、測量土地及預測天文事件。這四種需要大致地與數量、結構、空間及變化、及較近代的至不確定性的嚴格研究。基礎與哲學.為了闡明數學基礎，數學邏輯和集合論等領域被發展了出來。數學邏輯專注於將數學置在一堅固的公理架構上，並研究此一架構的結果。就數學邏輯本身而言，其為哥德爾第二不完備定理所屬的領域，而這或許是邏輯學中最廣為流傳的成果：總是存在不能被證明的真命題。現代邏輯被分成遞迴論、模型論和證明論，且和理論電腦科學有著密切的關連性，千禧年大獎難題中的P/NP問題就是理論電腦科學中的著名問題。純粹數學.數量.數量的研究起於數，一開始為熟悉的自然數及整數與被描述在算術內的自然數及整數的算術運算。整數更深的性質於數論中有詳細的研究，此一理論包括瞭如費馬最後定理等著名的結果。數論還包括兩個被廣為探討的未解問題：孿生質數猜想及哥德巴赫猜想。當數系更進一步發展時，整數被視為有理數的子集，而有理數則包含於實數中，連續的量即是以實數來表示的。實數則可以被進一步廣義化成複數。數的進一步廣義化可以持續至包含四元數及八元數。從自然數亦可以推廣到超限數，它形式化了計數至無限的這一概念。另一個研究的領域為大小，這個導致了基數和之後對無限的另外一種概念：阿列夫數，它允許無限集合之間的大小可以做有意義的比較。結構.許多如數及函式的集合等數學物件都有著內含的結構。這些物件的結構性質被探討於群、環、-{zh-cn:域;zh-tw:體}-等抽象系統中，該些物件事實上也就是這樣的系統。此為代數的領域。在此有一個很重要的概念，即廣義化至向量空間的向量，它於線性代數中被研究。向量的研究結合了數學的三個基本領域：數量、結構及空間。向量分析則將其擴充套件至第四個基本的領域內，即變化。創立於二十世紀三十年代的法國的布林巴基學派認為：純粹數學，是研究抽象結構的理論。結構，就是以初始概念和公理出發的演繹系統。布林巴基學派認為，有三種基本的抽象結構：代數結構。空間.空間的研究源自於幾何－尤其是歐幾里得幾何。三角學則結合了空間及數，且包含有著名的勾股定理。現今對空間的研究更推廣到了更高維的幾何、非歐幾裡得幾何及拓撲學。數和空間在解析幾何、微分幾何和代數幾何中都有著很重要的角色。在微分幾何中有著纖維叢及流形上的微積分等概念。在代數幾何中有著如多項式方程的解集等幾何物件的描述，結合了數和空間的概念；亦有著拓撲群的研究，結合了結構與空間。李群被用來研究空間、結構及變化。在其許多分支中，拓撲學可能是二十世紀數學中有著最大進展的領域，並包含有存在已久的龐加萊猜想，以及有爭議的四色定理。龐加萊猜想已在2006年確認由俄羅斯數學家格里戈裡·佩雷爾曼證明，而四色定理已在1976年由凱尼斯·阿佩爾和沃夫岡·哈肯用電腦證明，而從來沒有由人力來驗證過。變化.瞭解及描述變化在自然科學裡是一普遍的議題，而微積分更為研究變化的有利工具。函式誕生於此，作為描述一變化的量的核心概念。對於實數及實變函式的嚴格研究為實分析，而複分析則為複數的等價領域。黎曼猜想——數學最基本的未決問題之一——便是以複分析來描述的。泛函分析注重在函式的空間上。泛函分析的眾多應用之一為量子力學。許多的問題很自然地會匯出一個量與其變化率之間的關係，而這在微分方程中被研究。在自然界中的許多現象可以被動力系統所描述；混沌理論則是對系統的既不可預測而又是決定的行為作明確的描述。離散數學.離散數學是指對理論電腦科學最有用處的數學領域之總稱，這包含有可計算理論、計算複雜性理論及資訊理論。可計算理論檢驗電腦的不同理論模型之極限，這包含現知最有力的模型——圖靈機。複雜性理論研究可以由電腦做為較易處理的程度；有些問題即使理論是可以以電腦解出來，但卻因為會花費太多的時間或空間而使得其解答仍然不為實際上可行的，儘管電腦硬體的快速進步。最後，資訊理論專注在可以儲存在特定媒介內的資料總量，且因此有壓縮及熵等概念。作為一相對較新的領域，離散數學有許多基本的未解問題。其中最有名的為P/NP問題——千禧年大獎難題之一。一般相信此問題的解答是否定的。應用數學.應用數學思考將抽象的數學工具運用在解答科學、工商業及其他領域上之現實問題。應用數學中的一重要領域為統計學，它利用機率論為其工具並允許對含有機會成分的現象進行描述、分析與預測。大部份的實驗、調查及觀察研究需要統計對其資料的分析。數值分析研究有什麼計算方法，可以有效地解決那些人力所限而算不出的數學問題；它亦包含了對計算中捨入誤差或其他誤差之研究數學獎項.數學獎通常和其他科學的獎項分開。數學上最有名的獎為菲爾茲獎，創立於1936年，每四年頒獎一次。它通常被認為是數學領域的諾貝爾獎。另一個國際上主要的獎項為阿貝爾獎，創立於2003年。兩者都頒獎於特定的工作主題，包括數學新領域的創新或已成熟領域中未解決問題的解答。著名的23個問題，稱為希爾伯特的23個問題，於1900年由德國數學家大衛·希爾伯特所提出。這一連串的問題在數學家之間有著極高的名望，且至少有九個問題已經被解答了出來。另一新的七個重要問題，稱為千禧年大獎難題，發表於2000年。對其每一個問題的解答都有著一百萬美元的獎金，而當中只有一個問題和希爾伯特的問題重複。參考書目. 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18 | https://zh.wikipedia.org/wiki?curid=18 | 哲學 | "哲學是研究普遍的、基本問題的學科，包括存在、知識、價值、理智、心靈(...TRUNCATED) | [[0,2],[0,6,190865,2],[0,6,354,2],[0,6,4418,2],[0,6,49789,2],[0,6,43,2],[0,6,37,2],[0,6,10396,2],[0,(...TRUNCATED) | ["▁","zhe2 xue2","shi4","yan2 jiu1","pu3 bian4","de5","、","ji1 ben3","wen4 ti2","de5","xue2","ke(...TRUNCATED) |
21 | https://zh.wikipedia.org/wiki?curid=21 | 文學 | "文學，在狹義上，是一種語言藝術，亦即使用語言文字為手段，形象化地反(...TRUNCATED) | [[0,2],[0,6,104344,2],[0,6,4,2],[0,6376,2],[0,6,243487,2],[0,6,24492,2],[0,6,575,2],[0,6,4,2],[0,6,8(...TRUNCATED) | ["▁","wen2 xue2",",","zai4","xia2","yi4","shang4",",","shi4 yi1 zhong3","yu3 yan2","yi4 shu4",",",(...TRUNCATED) |
22 | https://zh.wikipedia.org/wiki?curid=22 | 歷史 | "歷史，指人類社會過去的事件和行動，以及對這些事件行為有系統的記錄、(...TRUNCATED) | [[0,2],[0,6,34744,2],[0,6,4,2],[0,6,8171,2],[0,6,52067,2],[0,6,12727,2],[0,6,18976,2],[0,6,43,2],[0,(...TRUNCATED) | ["▁","li4 shi3",",","zhi3","ren2 lei4","she4 hui4","guo4 qu4","de5","shi4 jian4","he2","xing2 dong(...TRUNCATED) |
25 | https://zh.wikipedia.org/wiki?curid=25 | 計算機科學 | "計算機科學設計、部署計算機和計算機系統通常被認為是非計算機科學學科(...TRUNCATED) | [[0,2],[0,6,43431,2],[0,6,3752,2],[0,6,41425,2],[0,6,6156,2],[0,6,37,2],[0,6,49053,2],[0,6,43431,2],(...TRUNCATED) | ["▁","ji4 suan4","ji1","ke1 xue2","she4 ji4","、","bu4 shu3","ji4 suan4","ji1","he2","ji4 suan4",(...TRUNCATED) |
39 | https://zh.wikipedia.org/wiki?curid=39 | 民族 | "在中華人民共和國，民族是一個根據文化與血緣傳承而形成的共同體，經中(...TRUNCATED) | [[0,6376,2],[0,6,66772,2],[0,6,13324,2],[0,6,6233,2],[0,6,264,2],[0,6,5444,2],[0,6,4,2],[0,6,27839,2(...TRUNCATED) | ["▁在","zhong1 hua2","ren2 min2","gong4","he2","guo2",",","min2 zu2","shi4 yi1 ge4","gen1 ju4","w(...TRUNCATED) |
45 | https://zh.wikipedia.org/wiki?curid=45 | 戲劇 | "戲劇」和「觀眾」。「演員」是四者當中最重要的元素，他是角色的代言人(...TRUNCATED) | [[0,2],[0,6,178040,2],[0,6,560,2],[0,6,264,2],[0,3691,2],[0,6,75885,2],[0,6,17635,2],[0,3691,2],[0,6(...TRUNCATED) | ["▁","xi4 ju4","」","he2","「","guan1 zhong4","」。","「","yan3 yuan2","」","shi4","si4","zh(...TRUNCATED) |
48 | https://zh.wikipedia.org/wiki?curid=48 | 電影 | "電影，特點是運動／移動的畫面，是一種視覺藝術作品，用來模擬透過使用(...TRUNCATED) | [[0,2],[0,6,17001,2],[0,6,4,2],[0,6,164083,2],[0,6,354,2],[0,6,13463,2],[0,248,2],[0,6,45622,2],[0,6(...TRUNCATED) | ["▁","dian4 ying3",",","te4 dian3","shi4","yun4 dong4","/","yi2 dong4","de5","hua4 mian4",",","shi(...TRUNCATED) |
51 | https://zh.wikipedia.org/wiki?curid=51 | 音樂 | "音樂，廣義而言，就是指任何以聲音組成的藝術。英文Music一詞源於古希臘(...TRUNCATED) | [[0,2],[0,6,18631,2],[0,6,4,2],[0,6,43928,2],[0,6,24492,2],[0,6,27283,2],[0,6,4,2],[0,6,2252,2],[0,6(...TRUNCATED) | ["▁","yin1 le4",",","guang3","yi4","er2 yan2",",","jiu4 shi4","zhi3","ren4 he2","yi3","sheng1 yin1(...TRUNCATED) |
53 | https://zh.wikipedia.org/wiki?curid=53 | 經濟學 | "經濟學，為研究商品和服務關係的社會科學學門，包括其中所有的購買、生(...TRUNCATED) | [[0,2],[0,6,16359,2],[0,6,5020,2],[0,6,4,2],[0,125035,2],[0,6,4418,2],[0,6,6689,2],[0,6,264,2],[0,6,(...TRUNCATED) | ["▁","jing1 ji4","xue2",",","wei4","yan2 jiu1","shang1 pin3","he2","fu2 wu4","guan1 xi4","de5","sh(...TRUNCATED) |
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