This is an exemplary presentation of historical scholarship. It is well written and researched, focusing on the evolution of Wu Wen-Tsun's mathematical and political ideas. The author also discusses basic questions about the social role of science and mathematics. Wu was born in Shanghai in 1919 and attained international recognition in three overlapping fields: algebraic topology; the history of mathematics, particularly ancient Chinese mathematics; and the theory of mathematics mechanization. This book is not only a biography, or a report of Wu's mathematical achievements; it is much more: an innovative analysis of the complexity of social and political history, and of its repercussions on the development of modern science and mathematics, focusing on China. The author analyses how Chinese philosophers, scientists, artists and politicians struggled to reconcile the efforts to modernize science and technology in China, incorporating Western advances, with the valorization of the historically formed Chinese identity.

Wu Wen-Tsun graduated from the National Chiao Tung University in 1940 and taught in Chinese universities. In 1947 he went to France, where he studied in Strasbourg and in 1949 received his doctorate with a thesis ‘Sur les classes caractéristiques des structures fibrées sphériques’ under Charles Ehresmann. He continued this research in collaboration with René Thom. In 1951 Wu returned to China for a position in the University of Peking. In 1955 he introduced the Wu classes of manifolds, internationally recognized as a major contribution to algebraic topology, and in 1957 was elected an academician of the Chinese Academy of Sciences.

According to Wu, the beginning of the Cultural Revolution was in 1965 and he says that his research in algebraic topology ended then. Although he did some sporadic research during the Cultural Revolution, he stopped completely in 1976, the end of the Cultural Revolution. He refers to a second stage of his mathematical research during the Cultural Revolution, when he joined computer manufacturing companies, learning and working with labourers, and at the same time devoted himself to studying the history of Chinese mathematics in ancient times.

Jiri Hudecek brings for reflection some basic questions about the social role of mathematics. How can mathematical research be directed to the benefit of the people? Is research in pure mathematics of lesser priority? In an interview given to the author, Wu Wen-Tsun told him that by reading Mao Zedong's Selected Works from cover to cover, he started to side with Mao's strategy: ‘You fight your way, I fight my way, the enemy advances, I retreat, the enemy retreats, I pursue. I think it is the same with mathematics. The enemy in mathematics is nature’ (p. 3). This led Wu to recognize the complex relationship between science and Maoist ideology and he described his scientific work in terms of a permanent struggle with situations and problems.

Wu devoted himself to the history of mathematics in ancient China and recognized that its line of development, methods of thinking and style of presentation were quite different from Western mathematics. He claims that in the ancient Greek tradition shapes and numbers were separated, while in the ancient Chinese tradition spatial forms and quantitative relations were blended. Wu became recognized as a most innovative historian of mathematics, claiming that Chinese mathematics development reveals a drive towards mechanization and the development of algorithms, rather than axiomatization and abstraction. There were no deductive systems in the Euclidean style, based on definition–axiom–theorem–proof. Ancient Chinese formulated a few general plausible principles and discovered geometrical results which were proved in a deductive manner.

Wu claimed that while working in computer manufacturing companies he was struck by the power of computers and began to consider applying computers to the study of mathematics. The result was a method of proving geometric theorems by means of computers, which became known as mathematics mechanization, and has an immense variety of applications both in mathematics and in science and technology. Wu became internationally acclaimed for having inaugurated this new research area.

This book is very well organized into seven chapters. The introduction sketches the structure and methodology of the author's research project. Chapter 2 describes Wu's career, which led him to international recognition in algebraic topology. Chapters 3 and 4 (which together make up about one-third of the book) cover about twenty-five years of the political situation in China and events in the Institute of Mathematics of the Chinese Academy of Sciences since Wu's return in 1950. These two chapters are devoted mainly to the analysis of the difficult relations of scientists and of the institute with the orientations of the Party and the political movements and policy changes in China, and how much this influenced Wu's new thinking about mathematics. Chapter 5 shows how Wu developed an original approach to the history and philosophy of mathematics. Chapter 6 is devoted to explaining the basics of Wu's method of the mechanization of mathematics, followed by his rise to public recognition in China and to international acclaim for this work. Chapter 7 has interesting quantitative data on Wu's production, clustered in what we might consider three careers.

The last sentence of the book claims that although Wu ‘stands apart from most other Chinese mathematicians by looking to the past, he shares the same goal with them of opening the future to a revived Chinese mathematics, a mathematics serving the symbolic and practical needs of the society that nurtures it’ (p.169). The history of world civilizations shows us that the human species struggles in a variety of natural conditions and cultural traditions, to survive and to transcend survival, which encompasses environmental conditions and mythological, linguistic, religious, economic and several socio-political aspects. The strategies to survive and to transcend constitute the historically formed cultural identity of a civilization. The main contribution of Wu Wen-Tsun was a new reading of the history of ancient Chinese mathematics and a reconciliation of the efforts to modernize science and technology in China, proposing a theory of mechanical theorem proving which incorporates Western advances and historically formed Chinese cultural identities.