Introduction to the special issue commemorating Liu Hui

Liu Hui, active in the 3rd century CE, was the main founder of classical Chinese mathematics. He was not only the most important mathematician in Chinese history but also held an extremely important position in the history of world mathematics. In November 2023, the 42nd session of the UNESCO General Conference reviewed and approved a series of commemorative activities marking the anniversary of Liu Hui's birth, recommended by the China Association for Science and Technology (CAST). This was the first series of commemorative activities for a Chinese scientist approved by UNESCO. From 23 to 27 September 2024, the special exhibition ‘Liu Hui, a Leading Founder of Classical Chinese Mathematical Theories’ was held at the Réfectoire des Cordeliers in Paris, France. On the morning of 24 September 2024, the launch ceremony of the International Series of Commemorative Activities for Liu Hui was held at UNESCO Headquarters in Paris. On the afternoon of the same day, the ‘International Symposium on Liu Hui's Academic Thoughts’ took place at the Réfectoire des Cordeliers. Subsequently, commemorative activities for Liu Hui, including exhibitions, academic conferences, academic lectures and popular science lectures, were held in several cities of China, Seoul of South Korea, Baku of Azerbaijan and Rome of Italy, highlighting Liu Hui's contributions to the history of world mathematics and the international recognition he has garnered.

Since September 2023, the team led by Researcher Professor Zou Dahai, the leader of the major project ‘Study on China’s Scientific and Technological Traditions and Their Practical Significance’ (中国科技传统及其现实意义研究) under the 14th Five-Year Plan of the Institute for the History of Natural Sciences, Chinese Academy of Sciences (IHNS, CAS), has actively collaborated with the National Communication Center for Science and Technology (NCCST) of CAST to provide academic support for the International Series of Commemorative Activities for Liu Hui.

In ancient China, although mathematics was regarded as an indispensable basic discipline, its social status was consistently low. There were very few professional mathematicians, and the social status of mathematicians often had almost no correlation with their mathematical achievements, but was associated with their influence in politics, economy and culture, especially their official positions. Although Liu Hui made extraordinary mathematical achievements and wielded far-reaching influence, his social status was relatively low. There is no biography of him in official dynastic histories, and only scattered historical accounts in various documents mention his life and deeds. Fortunately, his commentaries on the Nine Chapters on Mathematical Procedures (hereafter the Nine Chapters) and his work Chongcha (later known as Sea Island Mathematical Classic since the Tang Dynasty) became textbooks in the national mathematical education system from the Sui and Tang dynasties onwards and have been preserved to the present, serving as credible evidence for modern scholars studying Liu Hui. It is precisely the comprehensive and multifaceted research on Liu Hui conducted by the international academic community, especially the upsurge of research on the Nine Chapters and Liu Hui in the 1980s and 1990s, that has enabled him to gain international recognition and grounded the series of commemorative activities dedicated to him on a solid academic foundation. Naturally, research on Liu Hui is far from exhaustive; summarizing existing research results, exploring the unknown, and proposing and solving new problems remain inherent responsibilities of academia. The seven articles we solicited and included in this special issue are new achievements in this regard.

Professor Guo Shuchun from IHNS of CAS is the most important representative in the field of Liu Hui studies. His article ‘Liu Hui, the founder of the theoretical system of classical mathematics in China’ combines his own research experience to introduce the upsurge of Liu Hui studies in the 1980s and 1990s. He argues that the Nine Chapters compiled in the 1st century BCE only established the framework of classical Chinese mathematics, while Liu Hui in the 3rd century, influenced by the ethos of academic debate during the Wei and Jin dynasties, defined some mathematical concepts, comprehensively demonstrated the correctness of the algorithms in the Nine Chapters mainly through deductive reasoning, and thus established the theoretical system of classical mathematics in China. He also reviews how,he, on the basis of Yan Dunjie's research, reached the conclusion that Liu Hui was from Zouping, Shandong Province. These have provided important foundations for today's commemorative activities for Liu Hui.

Professor Joseph W Dauben, at the City University of New York, presents his paper ‘Commemorating Liu Hui: His significance as a UNESCO World Heritage mathematician’, which summarizes academic research on Liu Hui and develops his own discussions on this basis. In addition to the work of Chinese scholars, he pays special attention to absorbing the research achievements of scholars from other countries on Liu Hui. From the perspective of comparison and exchange, he points out the characteristics of Liu Hui's achievements as well as his contributions and influence in the history of world mathematics, affirming that Liu Hui ‘was a pioneering mathematician of outstanding significance’ and a ‘World Heritage mathematician’. His discussions highlight Liu Hui's status in the history of world mathematics and the significance of Liu Hui studies as an international academic research field.

Professor Donald B Wagner from the University of Copenhagen contributes his paper ‘The development of the classical Chinese algebra of polynomials: From the Nine Chapters and Liu Hui through Wang Xiaotong, Li Ye and Zhu Shijie’, which discusses the development of classical Chinese polynomial algebra. He summarizes a series of breakthroughs in this process reflected in existing documents and argues that the square-root extraction and cube-root extraction methods in the Nine Chapters, along with Liu Hui's geometric interpretations, marked the beginning of this journey. This study places Liu Hui's work on root extraction within a diachronic developmental context, thereby providing a specific case study of Liu Hui's influence on the development of mathematics in subsequent generations.

Professor Andrea Bréard from the Friedrich-Alexander-Universität Erlangen-Nürnberg presents her paper ‘Incognito: The secret lives of Liu Hui in 19th-century China’, which explores an interesting question: In the late Qing Dynasty, Liu Hui's name rarely appeared in mathematical documents of the time—did this mean that Liu Hui had no influence on mathematical practice in the 19th century and beyond? By examining the language, argumentative structures and research objects used by mathematicians, she demonstrates that Liu Hui's ideas actually exerted a lasting influence. She advocates tracing the context of mathematical inheritance and the trajectory of conceptual development rooted in Liu Hui's commentaries on the Nine Chapters in the 3rd century, and exploring the potential ‘soft power’ of ancient Chinese mathematical knowledge.

Professor Zhu Yiwen from the Department of Philosophy of Sun Yat-sen University offers his paper ‘A new perspective on Liu Hui's mathematical achievements’, which focuses on three issues that have been discussed by previous scholars but remain insufficiently explored. First, Liu Hui's discussions on the early development of Chinese mathematics, which profoundly shaped the foundation of Chinese mathematical culture. Second, Liu Hui's contributions to the so-called ‘geometric algebra’ that continued to develop in China until the 13th century. Third, Liu Hui's innovative ideas and practices regarding the use of counting rods, which were early precursors to the textualization and symbolization of Chinese mathematics in the 13th century. He also argues that Liu Hui's achievements should be understood in the context of his interactions with the political power and philosophical thought of his time.

Professor Zhou Xiaohan from IHNS of CAS presents his paper ‘Translations and interpretations of mathematical works in ancient China: A case study of Liu Hui's commentaries on The Nine Chapters’, which reviews the translation and introduction of Chinese mathematics by European missionaries and sinologists in China since the mid-19th century. He argues that the Nine Chapters and Liu Hui's commentary on it have been a long-term focus of attention, and that faithfully conveying the profound ideas of ancient scholars is a challenge faced by researchers. He emphasizes that the consistency of terminology and the faithful preservation of original mathematical structures are crucial for conveying the mathematical ideas of the original works, and explains how translators should strike a balance between historical accuracy and modern readability.

Professor Zou Dahai from IHNS of CAS contributes his paper ‘The motivation, methods and ideological origins behind the formulation and demonstration of the Liu Hui Principle’, which argues that the proposal of Liu Hui's Principle is closely related to the strong awareness of proof formed by Liu Hui in his era. He notes that the demonstration of this principle is the result of the innovative application and organic integration of China's algorithmic tradition and philosophical tradition. The achievement of this integration is related to the synergy of various factors, but the most important among them is Liu Hui's personal abilities and qualities.

The above seven articles include both macro-level discussions and in-depth explorations of specific themes, covering various aspects such as Liu Hui's life, academic achievements, ideological origins, historical influence and innovative thinking. It is believed that they will inject new vitality into future Liu Hui studies and provide an academic foundation and driving force for the dissemination and promotion of Liu Hui's profound ideas and scientific legacy in the new era.