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  「萊布尼茨」重新導向至此。關於其他用法，請見「萊布尼茨 (消歧義)」。

戈特弗里德·威廉·萊布尼茨
萊布尼茨
出生	1646年7月1日 德國萊比錫
逝世	1716年11月14日(70歲) 德國漢諾威
時代	17世紀哲學家
地區	西方哲學家
學派	理性主義
主要領域	數學、認識論、形而上學、科學、神義論
著名思想	單子論
受影響於 柏拉圖、笛卡爾、亞里士多德、經院哲學、盧爾（Ramon Llull）、惠更斯
影響於 康德、沃爾夫、羅素、德勒茲、亞伯拉罕·魯濱遜（Abraham Robinson）以及其他眾多後代數學家
簽名

戈特弗里德·威廉·萊布尼茨（Gottfried Wilhelm Leibniz，1646年—1716年），德國哲學家、數學家。他的著作主要用拉丁語和法語寫成。萊布尼茨是歷史上少見的通才，被譽為十七世紀的亞里士多德。他本人是一名律師，經常往返於各大城鎮，他許多的公式都是在顛簸的馬車上完成的，他也自稱具有男爵的貴族身份。

萊布尼茨在數學史和哲學史上都佔有重要地位。在數學上，他和牛頓先後獨立發明了微積分。有人認為，萊布尼茨最大的貢獻不是發明微積分，而是發明了微積分中使用的數學符號，因為牛頓使用的符號被普遍認為比萊布尼茨的差。萊布尼茨還對二進制的發展做出了貢獻。

在哲學上，萊布尼茨的樂觀主義最為著名，例如他認為，「我們的宇宙，在某種意義上是上帝所創造的最好的一個。」他和笛卡爾、巴魯赫·斯賓諾莎被認為是十七世紀三位最偉大的理性主義哲學家。萊布尼茨在哲學方面的工作在預見了現代邏輯學和分析哲學誕生的同時，也顯然深受經院哲學傳統的影響，更多地應用第一性原理或先驗定義，而不是實驗證據來推導以得到結論。

萊布尼茨對物理學和技術的發展也做出了重大貢獻，並且提出了一些後來涉及廣泛——包括生物學、醫學、地質學、概率論、心理學、語言學和信息科學——的概念。萊布尼茨在政治學、法學、倫理學、神學、哲學、歷史學、語言學諸多方向都留下了著作。

萊布尼茨對如此繁多的學科方向的貢獻分散在各種學術期刊、成千上萬封信件、和未發表的手稿中，截止至2010年，萊布尼茨的所有作品還沒有收集完全。戈特弗里德·威廉·萊布尼茨圖書館的萊布尼茨手稿藏品——Niedersächische Landesbibliothek 2007年被收入聯合國教科文組織編寫的世界記憶項目。

由於萊布尼茨曾在德國漢諾威生活和工作了近四十年，並且在漢諾威去世，為了紀念他和他的學術成就，2006年7月1日，也就是萊布尼茨360周年誕辰之際，漢諾威大學正式改名為漢諾威萊布尼茨大學。

生平

早年

萊布尼茨於1646年7月1日出生於神聖羅馬帝國薩克森州的萊比錫，父親是弗雷德里希·萊布尼茨，母親是嘉芙蓮娜·舒馬克。萊布尼茨的索布血統的父親在萊布尼茨6歲時去世，在那之後小萊布尼茨由他的母親撫養長大。Her teachings influenced Leibniz's philosophical thoughts in his later life.

萊布尼茨的父親，弗雷德里希·萊布尼茨，是萊比錫大學的一名倫理學教授，萊布尼茨繼承了他父親的個人圖書館，在七歲的時候就被允許自由進入閱覽書籍。While Leibniz's schoolwork focused on a small canon of authorities, his father's library enabled him to study a wide variety of advanced 哲學和神學作品 - ones that he would not have otherwise been able to read until his college years.他父親圖書館內的藏書相當一部分是以拉丁文寫就，also lead to his proficiency of the Latin language. 萊布尼茨在12歲時就已精通拉丁文，and he composed three hundred hexameters of Latin verse in a single morning for a special event at school at the age of 13.

萊布尼茨於14歲進入了他父親曾任教的萊比錫大學學習，並於1662年12月獲得了哲學學士學位。 He defended his Disputatio Metaphysica de Principio Individui, which addressed the Principle of individuation, on June 9, 1663. 1664年2月7日萊布尼茨獲得了哲學碩士學位。He published and defended a dissertation Specimen Quaestionum Philosophicarum ex Jure collectarum, arguing for both a theoretical and a pedagogical relationship between philosophy and Law, in December 1664.1665年9月，在一年的法學學習後，他獲得了法學的學士學位。

1666年，20歲的萊布尼茨出版了他的第一本書，書名為《論組合術》（de arte combinatoria）。the first part of which was also his habilitation thesis in philosophy. His next goal was to earn his license and doctorate in Law, which normally required three years of study then. Older students in the law school blocked his early graduation plans, prompting Leibniz to 於1666年9月離開了萊比錫 in disgust。

然後萊布尼茨入讀 阿爾特多夫大學，and almost immediately he submitted a thesis, which he had probably been working on earlier in Leipzig. 他的論文標題是 Disputatio de Casibus perplexis in Jure. 1666年11月，萊布尼茨獲得了法學的博士學位和執照。他在拒絕了阿爾特多夫大學的教職邀請後，and he spent the rest of his life in the paid service of two main German families of nobleman.

1666-1674

Leibniz's first position was as a salaried alchemist in 紐倫堡, even though he knew nothing about the subject. He soon met Johann Christian von Boineburg (1622–1672), the dismissed chief minister of the Elector of 美因茨, Johann Philipp von Schönborn. Von Boineburg hired Leibniz as an assistant, and shortly thereafter reconciled with the Elector and introduced Leibniz to him. Leibniz then dedicated an essay on law to the Elector in the hope of obtaining employment. The stratagem worked; the Elector asked Leibniz to assist with the redrafting of the legal code for his Electorate. In 1669, Leibniz was appointed Assessor in the Court of Appeal. Although von Boineburg died late in 1672, Leibniz remained under the employment of his widow until she dismissed him in 1674.

Von Boineburg did much to promote Leibniz's reputation, and the latter's memoranda and letters began to attract favorable notice. Leibniz's service to the Elector soon followed a diplomatic role. He published an essay, under the pseudonym of a fictitious 波蘭 nobleman, arguing (unsuccessfully) for the German candidate for the Polish crown. The main European geopolitical reality during Leibniz's adult life was the ambition of 路易十四, backed by French military and economic might. Meanwhile, the Thirty Years' War had left German-speaking Europe exhausted, fragmented, and economically backward. Leibniz proposed to protect German-speaking Europe by distracting Louis as follows. France would be invited to take 埃及 as a stepping stone towards an eventual conquest of the Dutch East Indies. In return, France would agree to leave Germany and the Netherlands undisturbed. This plan obtained the Elector's cautious support. In 1672, the French government invited Leibniz to 巴黎 for discussion, but the plan was soon overtaken by the outbreak of the Franco-Dutch War and became irrelevant. Napoleon's failed invasion of Egypt in 1798 can be seen as an unwitting implementation of Leibniz's plan.

Thus Leibniz began several years in Paris. Soon after arriving, he met 荷蘭 physicist and mathematician 克里斯蒂安·惠更斯 and realised that his own knowledge of mathematics and physics was spotty. With Huygens as mentor, he began a program of self-study that soon pushed him to making major contributions to both subjects, including inventing his version of the differential and integral calculus. He met Nicolas Malebranche and Antoine Arnauld, the leading French philosophers of the day, and studied the writings of Descartes and 帕斯卡, unpublished as well as published. He befriended a German mathematician, Ehrenfried Walther von Tschirnhaus; they corresponded for the rest of their lives.

When it became clear that France would not implement its part of Leibniz's Egyptian plan, the Elector sent his nephew, escorted by Leibniz, on a related mission to the English government in 倫敦, early in 1673. There Leibniz came into acquaintance of Henry Oldenburg and John Collins. After demonstrating a calculating machine he had been designing and building since 1670 to the Royal Society, the first such machine that could execute all four basic arithmetical operations, the Society made him an external member. The mission ended abruptly when news reached it of the Elector's death, whereupon Leibniz promptly returned to Paris and not, as had been planned, to Mainz.

The sudden deaths of Leibniz's two patrons in the same winter meant that Leibniz had to find a new basis for his career. In this regard, a 1669 invitation from the Duke of Brunswick to visit Hanover proved fateful. Leibniz declined the invitation, but began corresponding with the Duke in 1671. In 1673, the Duke offered him the post of Counsellor which Leibniz very reluctantly accepted two years later, only after it became clear that no employment in Paris, whose intellectual stimulation he relished, or with the Habsburg imperial court was forthcoming.

House of Hanover, 1676–1716

Leibniz managed to delay his arrival in Hanover until the end of 1676, after making one more short journey to London, where he possibly was shown some of Newton's unpublished work on the calculus. This fact was deemed evidence supporting the accusation, made decades later, that he had stolen the calculus from Newton. On the journey from London to Hanover, Leibniz stopped in 海牙 where he met 列文虎克,the discoverer of microorganisms. He also spent several days in intense discussion with 斯賓諾莎, who had just completed his masterwork, the Ethics. Leibniz respected Spinoza's powerful intellect, but was dismayed by his conclusions that contradicted both Christian and Jewish orthodoxy.

In 1677, he was promoted, at his request, to Privy Counselor of Justice, a post he held for the rest of his life. Leibniz served three consecutive rulers of the House of Brunswick as historian, political adviser, and most consequentially, as librarian of the ducal library. He thenceforth employed his pen on all the various political, historical, and theological matters involving the House of Brunswick; the resulting documents form a valuable part of the historical record for the period.

Among the few people in north Germany to accept Leibniz were the Electress Sophia of Hanover (1630–1714), her daughter Sophia Charlotte of Hanover (1668–1705), the Queen of Prussia and his avowed disciple, and Caroline of Ansbach, the consort of her grandson, the future George II. To each of these women he was correspondent, adviser, and friend. In turn, they all approved of Leibniz more than did their spouses and the future king George I of Great Britain.

The population of Hanover was only about 10,000, and its provinciality eventually grated on Leibniz. Nevertheless, to be a major courtier to the House of Brunswick was quite an honor, especially in light of the meteoric rise in the prestige of that House during Leibniz's association with it. In 1692, the Duke of Brunswick became a hereditary Elector of the 神聖羅馬帝國. The British Act of Settlement 1701 designated the Electress Sophia and her descent as the royal family of the United Kingdom, once both King William III and his sister-in-law and successor, Queen Anne, were dead. Leibniz played a role in the initiatives and negotiations leading up to that Act, but not always an effective one. For example, something he published anonymously in England, thinking to promote the Brunswick cause, was formally censured by the British Parliament.

The Brunswicks tolerated the enormous effort Leibniz devoted to intellectual pursuits unrelated to his duties as a courtier, pursuits such as perfecting the calculus, writing about other mathematics, logic, physics, and philosophy, and keeping up a vast correspondence. He began working on the calculus in 1674; the earliest evidence of its use in his surviving notebooks is 1675. By 1677 he had a coherent system in hand, but did not publish it until 1684. Leibniz's most important mathematical papers were published between 1682 and 1692, usually in a journal which he and Otto Mencke founded in 1682, the Acta Eruditorum. That journal played a key role in advancing his mathematical and scientific reputation, which in turn enhanced his eminence in diplomacy, history, theology, and philosophy.

The Elector Ernst August commissioned Leibniz to write a history of the House of Brunswick, going back to the time of Charlemagne or earlier, hoping that the resulting book would advance his dynastic ambitions. From 1687 to 1690, Leibniz traveled extensively in Germany, Austria, and Italy, seeking and finding archival materials bearing on this project. Decades went by but no history appeared; the next Elector became quite annoyed at Leibniz's apparent dilatoriness. Leibniz never finished the project, in part because of his huge output on many other fronts, but also because he insisted on writing a meticulously researched and erudite book based on archival sources, when his patrons would have been quite happy with a short popular book, one perhaps little more than a genealogy with commentary, to be completed in three years or less. They never knew that he had in fact carried out a fair part of his assigned task: when the material Leibniz had written and collected for his history of the House of Brunswick was finally published in the 19th century, it filled three volumes.

In 1711, John Keill, writing in the journal of the Royal Society and with Newton's presumed blessing, accused Leibniz of having plagiarized Newton's calculus. Thus began the calculus priority dispute which darkened the remainder of Leibniz's life. A formal investigation by the Royal Society (in which Newton was an unacknowledged participant), undertaken in response to Leibniz's demand for a retraction, upheld Keill's charge. Historians of mathematics writing since 1900 or so have tended to acquit Leibniz, pointing to important differences between Leibniz's and Newton's versions of the calculus.

In 1711, while traveling in northern Europe, the Russian 沙皇 Peter the Great stopped in Hanover and met Leibniz, who then took some interest in Russian matters for the rest of his life. In 1712, Leibniz began a two year residence in 維也納, where he was appointed Imperial Court Councillor to the Habsburgs. On the death of Queen Anne in 1714, Elector Georg Ludwig became King George I of Great Britain, under the terms of the 1701 Act of Settlement. Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of the Princess of Wales, Caroline of Ansbach, George I forbade Leibniz to join him in London until he completed at least one volume of the history of the Brunswick family his father had commissioned nearly 30 years earlier. Moreover, for George I to include Leibniz in his London court would have been deemed insulting to Newton, who was seen as having won the calculus priority dispute and whose standing in British official circles could not have been higher. Finally, his dear friend and defender, the dowager Electress Sophia, died in 1714.

Leibniz died in 漢諾威 in 1716: at the time, he was so out of favor that neither George I (who happened to be near Hanover at the time) nor any fellow courtier other than his personal secretary attended the funeral. Even though Leibniz was a life member of the Royal Society and the Berlin Academy of Sciences, neither organization saw fit to honor his passing. His grave went unmarked for more than 50 years. Leibniz was eulogized by Fontenelle, before the Academie des Sciences in Paris, which had admitted him as a foreign member in 1700. The eulogy was composed at the behest of the Duchess of Orleans, a niece of the Electress Sophia.

Leibniz never married. He complained on occasion about money, but the fair sum he left to his sole heir, his sister's stepson, proved that the Brunswicks had, by and large, paid him well. In his diplomatic endeavors, he at times verged on the unscrupulous, as was all too often the case with professional diplomats of his day. On several occasions, Leibniz backdated and altered personal manuscripts, actions which put him in a bad light during the calculus controversy. On the other hand, he was charming, well-mannered, and not without humor and imagination; He had many friends and admirers all over Europe.

原條目

1646年7月1日萊布尼茨出生於神聖羅馬帝國的萊比錫，祖父三代人均曾在薩克森政府供職，父親是Friedrich Leibnütz，母親是Catherina Schmuck。長大後，萊布尼茨名字的拼法才改成「Leibniz」，但是一般人習慣寫成「Leibnitz」。晚年時期，他的簽名通常寫成「von Leibniz」，以示貴族身份。萊布尼茨死後，他的作品才公諸於世，作者名稱通常是「Freiherr [Baron] G. W. von Leibniz.」，但沒有人確定他是否確實有男爵的貴族頭銜。

萊布尼茨的父親是萊比錫大學的倫理學教授，在萊布尼茨6歲時去世，留下了一個私人的圖書館。12歲時自學拉丁文，並着手學習希臘文。14歲時進入萊比錫大學唸書，20歲時完成學業，專攻法律和一般大學課程。1666年他出版第一部有關於哲學方面的書籍，書名為《論組合術》（de arte combinatoria）。

1666年萊布尼茨於阿爾特多夫拿到博士學位後，拒絕了教職的邀請，並經由當時政治家博尼伯格男爵的介紹，任職服務於美茵茨選帝侯大主教約翰·菲利普·馮·舍伯恩的高等法庭。

1671年發表兩篇論文《抽象運動的理論》（Theoria motus abstracti）及《新物理學假說》（Hypothesis physica nova），分別題獻給巴黎的科學院和倫敦的皇家學會，在當時歐洲學術界增加了知名度。

1672年萊布尼茨被約翰·菲利普派至巴黎，以動搖路易十四對入侵荷蘭及其它西歐日爾曼鄰國的興趣，並轉投注精力於埃及。這項政治計劃並沒有成功，但萊布尼茨卻進入了巴黎的知識圈，結識了馬勒伯朗士和數學家惠更斯等人。這一時期的萊布尼茨專注於數學研究，並發明了微積分。

1672及1673年博尼伯格和約翰·菲利普卻相繼過世，迫使萊布尼茨最後於1676年離開巴黎而轉任職服務於漢諾威的約翰·弗雷德里希公爵。於上任時，於海牙拜訪斯賓諾莎，與其討論了數天哲學。之後萊布尼茨來到漢諾威管理圖書館，並擔任公爵的法律顧問。

1680至1685年間，擔任哈茨山銀礦礦採工程師。在這期間，萊布尼茨致力於風車設計，以抽取礦坑中的地下水。然而受限於技術問題和礦工傳統觀念的阻力，計劃沒有成功。

1685年起，再受繼任的公爵Ernst August所託，轉而開始做其Braunschweig-Lüneburg貴族族譜研究。這項計劃一直到萊布尼茨去世前都沒有完成。

1686年完成《形而上學論》(Discours de métaphysique)。

1689年為完成Braunschweig-Lüneburg族譜研究，遊歷於意大利。其時結識耶穌會派遣於中國的傳教士，而開始對中國事物有更強烈的興趣。

1695年於期刊發表《新系統》，進而使萊布尼茨哲學中，關於實體間與心物間之「預定和諧」理論，被廣泛認識。

1700年萊布尼茨說服勃蘭登堡選帝侯腓特烈三世於柏林成立科學院，並擔任首任院長。

1704年完成《人類理智新論》。本文針對洛克的《人類理智論》，用對話的體裁，逐章節提出批評。然因洛克的突然過世，萊布尼茨不願被落入欺負死者的口實，所以本書在萊布尼茨生前一直都沒有出版。

1710年，出於對1705年過世的普魯士王后索菲·夏洛特的感念，出版《神義論》(Essais de Théodicée)。

1714年於維也納著寫《單子論》(Monadologie;標題為後人所加)及《建立於理性上之自然與恩惠的原理》。同年，漢諾瓦公爵Georg Ludwig繼任為英國國王喬治一世，卻拒絕將萊布尼茨帶至倫敦，而將他疏遠於漢諾威。

1716年11月14日萊布尼茨於漢諾威孤獨地過世，除了他自己的秘書外，即使Georg Ludwig本人正巧在漢諾威，宮廷無其他人參加他的喪禮。直到去世前幾個月，才寫完一份關於中國人宗教思想的手稿：《論中國人的自然神學》。

哲學家

單子論

除了是一位出眾的天才數學家之外，萊布尼茨亦是歐陸理性主義哲學的高峰。承斷了西方哲學傳統的思想，他認為世界，因其確定（換句話說，有關世界的知識是客觀普遍和必然的）之故，必然是由自足的實體所構成。所謂的自足，是不依他物存在和不依他物而被認知。萊布尼茨的前輩斯賓諾莎以為實體只有一個，就是神／自然。萊布尼茨對此不敢苟同，原因之一是斯氏的泛神觀和聖經的神學有明顯衝突，其次，是因為斯氏的理論沒有能夠解決由笛卡兒以降的二元論，令世界出現了斷層（他雖然強調世界為一，但沒有說明這一個看來是二元對立的世界的一統是如何可能）。

萊布尼茨以為實體是多的，是無限多的。跟隨亞里士多德的實體觀，他以為實體是一命題的主語。在一個命題S是P中，S就是實體。因為實體是自足的，則它要包含所有可能的謂語，即是「...是P」。由此，我們可以推出，實體有四個特徵：不可分割性、封閉性、統有性和道德性。

不可分割性是指，任何有廣延的東西，即有長度的東西，都可以被分割。被分割了的東西分別包含了自己的全部可能性，並且自足，則有廣延的東西的內容，即可能性要依附於他的部份的可能性。如此類推，則只要有廣延性，就不自足，而要依他物而被知（對萊布尼茨來說，真正的知識就是要窮一物的可能性），就不是實體。故實體不可分割，是一沒有廣延的東西，在萊布尼茨的晚年著作中（Monadology），他稱之為單子（Monad），單子的性質就是思（thought）。這廣延的世界就是由無限多的單子構成。

封閉性是說每一單子必然是自足的，不依他而存在，而又包含了自己的全部可能性。則一單子不可能和另一單子有交互作用（interaction）。若一單子作用於另一單子，則後一單子有一可能性沒有包括在該單子之內，即該單子沒能自足的包含自己的全部內容，而要依附於他物。因為實體的定義，這是不可能的。故萊布尼茨說：「單子之間沒有窗戶。」

統有性是指每一單子都必然以某種角度（perspective）包括了全世界。因為世界是緊密的由因果所構成，故A作用於B，其實不單單是作用於B，而是全世界。如果說一單子的內容包括自身的全部可能，則每一單子均以該單子自身為中心指向全世界。而這個世界是一的，不等於說所有單子都是一樣的，因為同一世界可以不同的角度來認知，而不失為一一統的世界。

最後，單子的道德性則較複雜。這個特性的提出是基於兩個理由，一、是世界的一統性（unity），二、是世界的確定性。對於前者，所有的單子都包含全世界，但各以自己的角度，世界的一統性是不是假的呢？如果我們要說一統，可以如何說起呢？對於後者，世界是由單子構成，單子只是其可能性的集合，世界亦只是一可能。那我們是不是不可能有一種不僅僅是可能，而是必然的知識呢？我們可以在甚麼意義下說有關世界的知識是真的、確定的呢？萊布尼茨將之歸功於一神，世界的創造者。從一個方面說，神在創造之前，沒有已成的材料，故沒有既成的有限處境，則創造是一純意志的創造，神是單憑其至善而創造這一個世界的。故此，如萊布尼茨的名言，這一個確切成就了的世界是「眾多可能的世界之中最好的一個。」這乎合了萊布尼茨的信仰要求。另一方面，要確定的了解一事物，則要了解其原因。要理解這一個原因，又要追索該原因的原因。如此類推，則世界的確定性知識不可能是一世界之內的動因（efficient cause），而是一超越的形上因（metaphysical cause）。萊布尼茨稱這個理論上必要設置的形上因為神。故，這一個世界之所以是如此，就是因為這是最好的，是至善的可能世界。人，要完全理解這神的至善意志，是不可能的，但可朝這一個方向邁進，因為人的心靈作一特殊的單子，是有記憶的，可以基於過去，疇劃自己的未來，這是人類分享的神性，即道德的可能性。人可以透過開放可能性，了解這個神創造的世界，而了解如何成為一個道德的人。

這一種世界的道德觀，可以被視為康德的先驅，分別在於萊布尼茨獨斷的提出了神為道德的完滿，把可能性說成了是在神的目光之下的實在，而沒有真正的將世界的可能性看作為可能性。而且萊布尼茨對天賦觀念（innate idea）的批評，正是黑格爾對康德的批評，在這個意義上說，康德一方面是被休謨（Hume）從萊布尼茨的獨斷夢中喚醒，可是同時亦到由洛克（Locke）起的哲學病變--對理性界限的審查--所污染。在這一方面，萊布尼茨卻比康德走前了一步。

符號思維

萊布尼茨有個顯著的信仰，大量的人類推理可以被歸約為某類運算，而這種運算可以解決看法上的差異:

“ 精煉我們的推理的唯一方式是使它們同數學一樣切實，這樣我們能一眼就找出我們的錯誤，並且在人們有爭議的時候，我們可以簡單的說，讓我們計算「calculemus」，而無須進一步的忙亂，就能看出誰是正確的。 ” ——《發現的藝術》1685, W 51

萊布尼茨的演算推論器，很能讓人想起符號邏輯，可以被看作使這種計算成為可行的一種方式。萊布尼茨寫的備忘錄（帕金森於1966年翻譯了它們）可以被看作是對符號邏輯的探索--所以他的演算--上路了。但是Gerhard和Couturat沒有出版這些著作，直到現代形式邏輯在1880年代於弗雷格的《概念文字》和查爾斯·皮爾士及其學生的著作中形成，所以就更在喬治·布爾和德·摩根在1847年開創這種邏輯之後了。

形式邏輯

萊布尼茨是在亞里士多德和1847年喬治·布爾和德·摩根分別出版開創現代形式邏輯的著作之間最重要的邏輯學家。萊布尼茨闡明了我們現在叫做合取、析取、否定、同一、集合包含和空集的首要性質。萊布尼茨的邏輯原理和他的整個哲學可被歸約為兩點:

1. 所有的我們的觀念（概念）都是由非常小數目的簡單觀念複合而成，它們形成了人類思維的字母。
2. 複雜的觀念來自這些簡單的觀念，通過模擬算術運算的統一的和對稱的組合。

數學家

目前微積分領域使用的符號仍是萊布尼茨所提出的。在高等數學和數學分析領域，萊布尼茨判別法是用來判別交錯級數的收斂性的。

微積分

萊布尼茨與牛頓誰先發明微積分的爭論是數學界至今最大的公案。萊布尼茨於1684年發表第一篇微分論文，定義了微分概念，採用了微分符號dx，dy。1686年他又發表了積分論文，討論了微分與積分，使用了積分符號∫。依據萊布尼茨的筆記本，1675年11月11日他便已完成一套完整的微分學。

然而1695年英國學者宣稱：微積分的發明權屬於牛頓；1699年又說：牛頓是微積分的「第一發明人」。1712年英國皇家學會成立了一個委員會調查此案，1713年初發佈公告：「確認牛頓是微積分的第一發明人。」萊布尼茨直至去世後的幾年都受到了冷遇。由於對牛頓的盲目崇拜，英國學者長期固守於牛頓的流數術，只用牛頓的流數符號，不屑採用萊布尼茨更優越的符號，以致英國的數學脫離了數學發展的時代潮流。

不過萊布尼茨對牛頓的評價非常的高，在1701年柏林宮廷的一次宴會上，普魯士國王腓特烈詢問萊布尼茨對牛頓的看法，萊布尼茨說道：

“ 在從世界開始到牛頓生活的時代的全部數學中，牛頓的工作超過了一半 ”

牛頓在1687年出版的《自然哲學的數學原理》的第一版和第二版也寫道：「十年前在我和最傑出的幾何學家萊布尼茨的通信中，我表明我已經知道確定極大值和極小值的方法、作切線的方法以及類似的方法，但我在交換的信件中隱瞞了這方法，……這位最卓越的科學家在回信中寫道，他也發現了一種同樣的方法。他並訴述了他的方法，它與我的方法幾乎沒有什麼不同，除了他的措詞和符號而外」（但在第三版及以後再版時，這段話被刪掉了）。

因此，後來人們公認牛頓和萊布尼茨是各自獨立地創建微積分的。

牛頓從物理學出發，運用集合方法研究微積分，其應用上更多地結合了運動學，造詣高於萊布尼茨。萊布尼茨則從幾何問題出發，運用分析學方法引進微積分概念、得出運算法則，其數學的嚴密性與系統性是牛頓所不及的。

萊布尼茨認識到好的數學符號能節省思維勞動，運用符號的技巧是數學成功的關鍵之一。因此，他所創設的微積分符號遠遠優於牛頓的符號，這對微積分的發展有極大影響。1714至1716年間，萊布尼茨在去逝前，起草了《微積分的歷史和起源》一文（本文直到1846年才被發表），總結了自己創立微積分學的思路，說明了自己成就的獨立性。

拓撲學

拓撲學最早稱之「位相分析學」（analysis situs），是萊布尼茨1679年提出的，這是一門研究地形、地貌相類似的學科，當時主要研究的是出於數學分析的需要而產生的一些幾何問題。關於萊布尼茨對拓撲學的貢獻，尚存爭論。Mates引用Jacob Freudenthal1954年一篇論文裏的話說：

儘管萊布尼茨認為一列點在空間中的位置是由其間距離唯一決定的——若且唯若距離發生變化時點的位置發生相應的改變——他的仰慕者歐拉，在他著名的一篇論文（1736年發表，解決了柯尼斯堡七橋問題及其推廣）中，卻是在「拓撲變形時點的位置不發生變化」的意義下使用「幾何位置」這個名詞的。他誤信了萊布尼茨是這個概念的創始者。……人們常常意識不到萊布尼茨是在完全不同的意義下使用這個名詞的，因此被尊為數學的這個分支領域的奠基人並不恰當。

但平野秀秋持有不同看法，他引用曼德勃羅的話說:

在萊布尼茨海量的科學成果中探索是發人深省的體驗。除了微積分以及其他已經完成的研究之外，大量涉及內容廣泛且極富前瞻性的研究對科學發展的推動力勢不可擋。在『填充理論』上即有例子，……在發現萊布尼茨還曾經關注過幾何度量的重要性之後，我對他的狂熱更甚了。在「歐幾里德普羅塔」中……，其使得歐幾里德公理更加嚴格，他陳述道，……『對直線，我有數種不同的定義。直線是曲線的一種，而曲線的任何部分都是和整體相似的，因此直線也具有這種特性；這不僅適用於曲線，而且適用於集合。』這個論斷今天已經可以被證明。

因而分形幾何（由伯努瓦·曼德勃羅發揚光大）理論在萊布尼茨的自相似性思想和連續性原理中尋求支持：大自然沒有跳躍（拉丁語「natura non facit saltus」，英語"nature does not make jumps"）。當萊布尼茨在他的形而上學著作中寫道，「直線是曲線的一種，其任何部分都是和整體類似的」，他實際上提前兩個世紀預言了拓撲學的誕生。至於「填充理論」，萊布尼茨對他的朋友Des Bosses說，「你想像一個圓，然後用三個全等的最大半徑的圓填滿它，後來的三個小圓又可以以同樣的過程被更小的圓填充」。這個過程可以無限地繼續下去，並由此生發出了自相似性的思想。萊布尼茨對於歐氏公理的改進亦包含同樣的概念。

科學家以及工程師

今天人們仍然在討論萊布尼茨的研究成果，不僅是為了其偉大的預見性和可能未被注意到的發現，更是為了探知這種能夠先知的方法。他在物理學上的工作大部分被囊括在格哈特的數學著作中。

物理

參見：物力論(形而上學)

Leibniz contributed a fair amount to the statics and dynamics emerging about him, often disagreeing with Descartes and Newton. He devised a new theory of motion (dynamics) based on kinetic energy and potential energy, which posited space as relative, whereas Newton felt strongly space was absolute. An important example of Leibniz's mature physical thinking is his Specimen Dynamicum of 1695.

Until the discovery of subatomic particles and the quantum mechanics governing them, many of Leibniz's speculative ideas about aspects of nature not reducible to statics and dynamics made little sense. For instance, he anticipated Albert Einstein by arguing, against Newton, that space, time and motion are relative, not absolute. Leibniz's rule is an important, if often overlooked, step in many proofs in diverse fields of physics. The principle of sufficient reason has been invoked in recent cosmology, and his identity of indiscernibles in quantum mechanics, a field some even credit him with having anticipated in some sense. Those who advocate digital philosophy, a recent direction in cosmology, claim Leibniz as a precursor.

The vis viva

Leibniz's vis viva (Latin for living force) is mv², twice the modern kinetic energy. He realized that the total energy would be conserved in certain mechanical systems, so he considered it an innate motive characteristic of matter. Here too his thinking gave rise to another regrettable nationalistic dispute. His vis viva was seen as rivaling the conservation of momentum championed by Newton in England and by Descartes in France; hence academics in those countries tended to neglect Leibniz's idea. Engineers eventually found vis viva useful, so that the two approaches eventually were seen as complementary.

萊布尼茨與中國文化

萊布尼茨是較早接觸中華文化的歐洲人。法國漢學大師若阿基姆·布韋（Joachim Bouvet，漢名白晉，1662－1732年）向萊布尼茨介紹了《周易》和八卦的系統，他們兩人一直是好朋友。在萊布尼茨眼中，「陰」與「陽」基本上就是他的二進制的中國版。他曾斷言言：「二進制乃是具有世界普遍性的、最完美的邏輯語言」。目前在德國圖林根，著名的郭塔王宮圖書館（Schlossbibliothek zu Gotha）內仍保存一份萊氏的手稿，標題寫着「1與0，一切數字的神奇淵源。」