[Paper Review] The third approach to the history of mathematics in China

QU Anjing|ArXiv.org|May 1, 2003

History and Theory of Mathematics1 references3 citations

TL;DR

This paper proposes a third research paradigm in the history of Chinese mathematics, shifting focus from 'what' mathematics was done (Li & Qian) and 'how' it was done (Wu Wen-tsun) to 'why' it was done. By integrating the historical, practical, and theoretical traditions, the approach reorients the field from mathematics in history to the broader history of mathematics, offering new avenues for interdisciplinary research and revitalizing the discipline by emphasizing purpose, context, and function in mathematical development.

ABSTRACT

The first approach to the history of mathematics in China led by Li Yan (1892--1963) and Qian Baocong (1892--1974) featured discovering {\it what} mathematics had been done in China's past. From the 1970s on, Wu Wen-tsun and others shifted this research paradigm to one of recovering {\it how} mathematics was done in ancient China. Both approaches, however, focus on the same problem, that is mathematics in history. The theme of the third approach is supposed to be {\it why} mathematics was done. Combining this approach with the former two, the research paradigm will be improved from one of mathematics in history to that of the history of mathematics.

Motivation & Objective

• To address the stagnation in Chinese mathematics history research by proposing a new paradigm beyond 'what' and 'how' mathematics was done.
• To reframe the field from 'mathematics in history' to 'the history of mathematics' by emphasizing the underlying motivations and functions of mathematical practice.
• To stimulate new research by exploring the 'why' behind mathematical developments in ancient China.
• To integrate the practical and theoretical traditions in Chinese mathematics to provide a more comprehensive understanding of its historical development.
• To revitalize the discipline by offering fresh, context-driven research questions that attract younger scholars.

Proposed method

• Proposes a three-part research paradigm: the first approach (discovery of past mathematics), the second (reconstruction of historical methods), and the third (understanding the purpose and function of mathematics).
• Analyzes historical Chinese mathematical texts—such as Liu Hui’s Haidao suanjing and Yi-xing’s Dayan li—to identify mathematical techniques and their functional roles.
• Applies comparative analysis between Chinese and Greek mathematical traditions, contrasting the practical tradition (problem-solving focus) with the theoretical tradition (hypothesis-driven explanation).
• Uses case studies of interpolation and remote measurement to demonstrate that Chinese mathematicians developed sophisticated numerical methods not for theoretical elegance but for practical accuracy in calendrical and astronomical calculations.
• Integrates insights from the history of science and epistemology to frame mathematics as a response to real-world problems, not just abstract inquiry.
• Advocates for a dualistic model of scientific development, where both practical and theoretical traditions co-evolve and inform modern science.

Experimental results

Research questions

• RQ1 Why did ancient Chinese mathematicians develop specific numerical methods like interpolation and remote measurement techniques?
• RQ2 How did the practical needs of calendrical science and astronomy shape the development of mathematical techniques in imperial China?
• RQ3 In what ways did the functional purpose of mathematics in China differ from the theoretical motivations seen in Greek mathematics?
• RQ4 How can the integration of the practical and theoretical traditions provide a more complete picture of the history of mathematics?
• RQ5 What new research directions emerge when historians shift from 'what' and 'how' to 'why' in the study of Chinese mathematics?

Key findings

• The formula in Yi-xing’s Dayan li is a quadratic interpolation function of unequal intervals, equivalent to Gauss’s interpolation, though not originally derived from that framework.
• Liu Zhuo’s earlier formula in Huangji li is a quadratic interpolation of equal intervals, demonstrating advanced numerical technique in 7th-century China.
• The remote measurement formula in Liu Hui’s Haidao suanjing (formula 2) is a practical solution for measuring heights and distances using gnomons, rooted in observational geometry.
• Chinese mathematics developed a strong practical tradition focused on solving concrete problems, with numerical methods valued for their applicability rather than theoretical derivation.
• Theoretical traditions, as seen in Greek mathematics, prioritize root formulas and axiomatic reasoning, whereas Chinese mathematics prioritized numerical approximation and functional utility.
• The synthesis of practical and theoretical traditions reveals that modern science benefits from both approaches, and the history of mathematics must reflect this dualistic evolution.