Cheng–Zhu school in the context of "Neo-Confucian"

Neo-Confucianism (Chinese: 宋明理學; pinyin: Sòng-Míng lǐxué, often shortened to lǐxué 理學, literally "School of Principle") is the cultural revival of Confucianism as an ethical, social and religious system, which dominated Chinese philosophy from the 13th through the 19th century. Although its origin lie in the Tang dynasty, it was fully developed during the Song dynasty under the formulations of Zhu Xi (1130–1200), the tradition's central figure. Zhu, alongside Cheng Yi and Cheng Hao, comprises the dominant Cheng–Zhu school, in opposition to the later Lu–Wang school led by Wang Yangming and Lu Xiangshan.

Neo-Confucianism could have been an attempt to create a more rationalist and secular form of Confucianism by rejecting mystical elements of Taoism and Buddhism that had influenced Confucianism during and after the Han dynasty. Although the neo-Confucianists were critical of Taoism and Buddhism, the two did have an influence on the philosophy, and the neo-Confucianists borrowed terms and concepts. However, unlike the Buddhists and Taoists, who saw metaphysics as a catalyst for spiritual development, religious enlightenment, and immortality, the neo-Confucianists used metaphysics as a guide for developing a rationalist ethical philosophy. After the Mongol conquest of China in the thirteenth century, Chinese scholars and officials restored and preserved neo-Confucianism as a way to safeguard the cultural heritage of China.

Cheng Yi (1033–1107), also known by various other names and romanizations, was a Chinese classicist, essayist, philosopher, and politician of the Song Dynasty. He worked with his older brother Cheng Hao. Like his brother, he was a student of Zhou Dunyi, a friend of Shao Yong, and a nephew of Zhang Zai. The five of them, along with Sima Guang, are called the Six Great Masters by his follower Zhu Xi. He became a prominent figure in neo-Confucianism, and the philosophy of Cheng Yi, Cheng Hao and Zhu Xi is referred to as the Cheng–Zhu school or the Rationalistic School.

The School of Mind, or the School of Heart (Chinese: 心學; pinyin: xīn xué), or Yangmingism (Chinese: 陽明學; pinyin: yángmíng xué; Japanese: 陽明学, romanized: yōmeigaku), is one of the major philosophical schools of Neo-Confucianism, based on the ideas of the idealist Neo-Confucian philosopher Wang Shouren (whose pseudonym was Yangming Zi and who is thus often referred as Wang Yangming). Throughout the whole Yuan dynasty, as well as in the beginning of the Ming dynasty, the magistral philosophy in China was the Rationalistic School, another school of Neo-Confucianism emphasizing the importance of observational science built by Cheng Yi and especially Zhu Xi. Wang Yangming, on the other hand, developed his philosophy as the main intellectual opposition to the Cheng-Zhu School. Yangmingism is considered to be part of the School of Mind established by Lu Jiuyuan, upon whom Yangming drew inspirations. Yangming argued that one can learn the supreme principle (理, pinyin: Li) from their minds, objecting to Cheng and Zhu's belief that one can only seek the supreme principle in the objective world. Furthermore, Yangmingism posits a oneness of action and knowledge in relation to one's concepts of morality. This idea, "regard the inner knowledge and the exterior action as one" (知行合一) is the main tenet in Yangmingism.

It is sometimes called the Lu-Wang school by people who wish to emphasize the influence of Lu Jiuyuan. The Lu-Wang school (or Xinxue) was seen as a rival to the Cheng–Zhu school (or Lixue) school named after its leading philosophers, Cheng Yi and Zhu Xi.