Lixue 理學, literally "teachings of the principle", was the most influential one of the many schools of Neo-Confucianism, and this term is therefore often used as the Chinese term for the expression "Neo-Confucianism" per se. The word xin ruxue 新儒學 "new Confucianism" was coined by Feng Youlan 馮友蘭 (1895-1990) and Chen Yinke 陳寅恪 (1890-1969), and found acceptance in Western studies on Chinese philosophy, yet with a very comprehensive and generalised meaning. This wide meaning of the term lixue returned to China and refers not just to the teachings of a particular school, but in a narrower sense either to all Song-period teachings, or in a wider sense even to all philosophers from the 11th to the 18th centuries.
The original term lixue is used for the teachings of the most important philosophers of the Song period 宋 (960-1279), namely Zhou Dunyi 周敦頤 (1017-1073), Zhang Zai 張載 (1020-1077), the brothers Cheng Hao 程顥 (1031-1085) and Cheng Yi 程頤 (1033-1107), as well as Zhu Xi 朱熹 (1130-1200). Their teachings are therefore subsumed under the term Cheng-Zhu lixue 程朱理學. Their Lixue School of the "universal principle" (li 理) is often perceived as the counterpart of the Xinxue 心學, the School of the "human mind" (xin 心), the most important representative of which were Lu Jiuyuan 陸九淵 (1139-1192) and Wang Shouren 王守仁 (Wang Yangming 王陽明, 1472-1529), therefore called Lu-Wang xinxue 陸王心學.
The Lixue emerged in the teachings of the Northern Song period 北宋 (960-1126) and gained predominance from around 1200. It thus prevailed during the Song and Ming 明 (1368-1644) periods, and is therefore also known as Song-Ming lixue 宋明理學 or Songxue 宋學 "Teachings of the Song period". Philosophers of the Qing period 清 (1644-1911) went over to different methods and topics of research. The so-called Hanxue 漢學 "Han-[methods] School" or philological school (see Qian-Jia xuepai 乾嘉學派) ended the scholarly dominance of the Lixue School, even if its influence on society and policy remained.
Subbranches of the Lixue School were the "School of Matter" (qixue 氣學) with representatives like Zhang Zai, Luo Qinshun 羅欽順 (1465-1547) and Wang Fuzhi 王夫之 (1619-1692), the school of symbols and numbers (xiangshuxue 象數學) of the numerologist Shao Yong 邵雍 (1011-1077), the Wuzhou School (Wuxue 婺學) or Lü School (Lüxue 呂學) or Jinhua Branch (Jinhua xuepai 金華學派) of Lü Zuqian 呂祖謙 (1137-1181), the Hunan School (Huxiangxue 湖湘學) of Zhang Shi 張栻 (1133-1180), the Lian Creek School (Lianxue 濂學, Zhou Dunyi), the Guanzhong School (Guanxue 關學) of Shen Yan 申顏, Hou Ke 侯可 and Zhang Zai, the River Luo School (Luoxue 洛學) of the Cheng brothers, the New School (Xinxue 新學) or Jinggong xinxue 荆公新学) of Wang Anshi 王安石 (1021-1086) or the Sichuan School (Shuxue 蜀學) founded by Su Xun 蘇洵 (1009-1066) and maintained by the brothers Su Shi 蘇軾 (1037-1101) and Su Zhe 蘇轍 (1039-1112), Huang Tingjian 黃庭堅 (1045-1105), Zhang Lei 張耒 (1054-1114), Qin Guan 秦觀 (1049-1100), Du Zheng 度正 (b. 1167) and Wei Liaoweng 魏了翁 (1178-1237).
The "universal principle" (li 理) is also called dao 道, for which reason the school is also known as daoxue 道學 "teachings of the Way". The latter has nothing to do with Daoism, barring the adaption of the concept of the Way (dao) as a fundamental factor in cosmology.
The term lixue appeared during the Southern Song period 南宋 (1127-1279) and was used for two different meanings, namely the philosophical teachings of the relation between the universal principles or order (li) and the human character (xing 性), or the scholarly method of discussing metaphysical dependencies and meanings (yi 義) as opposed to the older method of interpreting the Classical writings by textual methods of paragraph-and-sentence exegesis (zhangju xungu 章句訓詁), with a focus on semantics and phonology.
Han Yu 韓愈 (768-824) and Li Ao 李翱 (774-836) can be seen as the forerunners of Neo-Confucianism. They contributed to the revival of Confucian studies after centuries of Daoist and Buddhist predominance. In his book Huangshi richao 黃氏日鈔, the late Southern Song-period scholar Huang Zhen 黃震 (1213-1281) explained that the shift of methods and intellectual contents emerged under the guidance of Hu Yuan 胡瑗, Sun Fu 孫復 (992-1057) and Shi Jie 石介 (1005-1045), and was intensified by Fan Zhongyan 范仲淹 (989-1052) and Ouyang Xiu 歐陽修 (1007-1072). The new trend culminated in the teachings of the masters of Yi-Luo 伊洛, i.e. scholars who resided Luoyang 洛陽, a city located at the banks of the rivers Luo and Yi. These were Shao Yong, Zhou Dunyi, Zhang Zai, and the brothers Cheng Hao and Cheng Yi. These "five great masters of the Northern Song" (Beisong wu zi 北宋五子) introduced the study of the relation between the universal principles and the human character. Shao Yong and Zhou Dunyi approached this issue from a numerological perspective.
During the Southern Song period, the various teachings of the Lixue were amalgamated by the famous scholar Zhu Xi, who became the Saint of Neo-Confucianism and entered the pantheon of Confucianism (see Confucius temples). Yet he also met resistance among contemporary scholars, mainly Lu Jiuyuan, who had a different approach as to the ability of the human mind (xin) to perceive what was correct in the sense of the universal principle and how to transform this knowledge into practice. Lu founded the so-called school of the mind (xinxue 心學), which had a late representative with Wang Shouren during the high Ming period.
In the very late decades of the Southern Song, Zhu Xi's teachings were adopted by the government as the official, state-backed version of Confucianism. The writings of the brothers Cheng and Zhu Xi were canonized and were to be studied for the preparation for the state examinations. Even if the Cheng-Zhu teaching remained dominant in practice, for instance, in the use of Zhu Xi's Sishu zhangju jizhu 四書章句集注 as textbook for the preparation for the state examinations, philosophers of the Qing period left the realm of metaphysics and returned to the interpretation of the Confucian Classics by means of philological studies.
The emergence of the "School of the universal principle" is related to the challenge by Buddhist metaphysics, which offered a deeper understanding of the underlying principles of cosmology than the teachings of Confucius (Kongzi 孔子, 551-479 BCE) and his disciples and successors were ever able to present. The new trend of Confucian scholars to interpret the world was also influenced by Daoist teachings which included the concept of the "Way" (dao). This concept was adopted by the Confucians and transformed into a model by which the propositions of a Confucian society could be explained out of the existence of general principles inherent not just in humans, but also in animals, objects and the world of physics or cosmogony. In this way, Confucianism left the world of society and politics and became a concept of explaining the whole cosmos with all its (physical and metaphysical) mechanisms.
The Dao or universal principle li were a constant and never changing principle inherent in all objects that had come into the world. This principle had been there from the beginning of the cosmos, but was in the real world bound to matter, just as all matter (inanimate as well as animate) inherently contained the rules determined of the principle in completeness. The proposition of the "completeness of the universal principle in all different objects" (li yi fen shu 理一分殊) is derived from a similar thesis in Buddhism. These principles were not just the rules of physics, but moral principles (in the Confucian sense) which were implanted into the character (xing 性, i.e. the Heavenly mind or mind of the Dao, daoxin 道心) of every person. It was therefore theoretically possible that every man might become perfectly good. Yet this potential was in fact often obstructed by contaminations in matter (qi 氣) and environment, leading to a tainted human mind (renxin 人心). While this thesis is described in the teachings of Meng Ke 孟軻 (385-304 or 372-289 BCE, Mengzi 孟子) and Xun Kuang 荀況 (trad. 313-238, Xunzi 荀子), the Neo-Confucians of the Lixue School added a metaphysical explanation. The only way to "purify" the human mind was self-cultivation. The success of it would lead to order at home and good governance.
Among humans, only the perfect man (shengren 聖人) was able to thoroughly perceive all aspects of the universal principle (li), and to transform them rightly into the conduct in the three (or five) types of social relations (sangang 三綱, wulun 五倫) and the Confucian virtues (wuchang 五常) of kindness (ren 仁), propriety (yi 義), rites (li 禮), wisdom (zhi 智), and trust (xin 信). The perfect man knew how to "preserve the principles of Heaven" (cun tianli 存天理) and to "discard [improper and selfish] human desires" (qu renyu 去人欲). All other persons had to carry out studies and self-cultivation in order to achieve a higher stage of consciousness related to Confucian principles.
The canon of the Lixue philosophers was the numerological Classic Yijing 易經 "Book of Changes", and the Four Books (sishu 四書) Daxue 大學 "Great Learning", Zhongyong 中庸 "Doctrine of the Mean", Lunyu 論語 "Confucian Analects", and Mengzi "Master Meng". Out of these texts, the Lixue School found their arguments for discussing the relation between the universal principle and matter or substance (li qi 理氣), the human heart and human character (xin xing 心性), the investigation of things (ge wu 格物) to gain perfect knowledge (zhi zhi 致知), the focus on respect (zhu jing 主敬) and on quiescence (zhu jing 主靜), self-restraint and self-cultivation and preservation of knowledge about the universal principle (hanyang 涵養), the relation between knowledge and practice (zhi xing 知行), the stirring up and reservation (yi fa 已發, wei fa 未發) of emotions (qing 情), the [pure and good] Heaven-endowed character of man and things (tianming zhi xing 天命之性), the [tainted] substance-related character of man and things (qizhi zhi xing 氣質之性), the natural mind and the human mind (daoxin 道心, renxin 人心), the Heavenly principle and human desires (tianli 天理, renyu 人欲), and the way of virtuous royal rulership (wang 王) by righteousness (yi 義) and benevolence (shu 恕), and the rule of selfish hegemons (ba 霸), by striving for interests and profit (li 利).
The school of the universal principle remained the most important Neo-Confucian branch for several centuries to come and was able to adapt to various challenges by slightly adapting to the necessities of the time. Yet the general propositions withstood the onslaught of time and contributed to the cementation of the Confucian ideology not just in China, but also in the "Confucian" countries Korea, Japan and Vietnam.