- The History of Mathematical Proof in Ancient Traditions ed. by Karine Chemla
The book under review comprises a programmatic introduction by the editor (68 pages) and two collections of essays entitled “Views on the Historiography of Mathematical Proof” (seven essays, 224 pages) and “History of Mathematical Proof in Ancient Traditions: The Other Evidence” (nine essays, 278 pages). The introduction starts with a brief rehearsal of what the author calls the “standard history of mathematical proof in ancient traditions at the present day”—or, briefly, the “standard narrative”: mathematical proof as a well-founded scientific activity was conceived and developed in ancient Greece, culminating in the works of Aristotle (in logic) and Euclid (in geometry); no traces of similar intellectual endeavors can be found in other areas, such as Mesopotamia, Egypt, or India. As alleged protagonists of this view, the reader meets, among others, the French physicist Jean-Baptiste Biot (with a relatively harsh statement from 1841) and the mathematician Morris Kline (with a much more moderate statement of 1972). The purpose of the book, then, is to challenge the standard narrative and design a research program to replace it with a more adequate assessment of the achievements of non-Greek mathematicians in antiquity. The pivotal question is, in what sense and by what methods were mathematical procedures justified in showing that they always produce correct results when applied?
The various essays presented are not unrelated and fit well under the two headings supplied. The first collection examines the difficulties of reconstructing the “original meaning” of a mathematical manuscript through the veils of many editions, in spite of mistakes and alterations; beyond standard philological work, the study of diagrams brings in some new aspects (essays 2 and 3). Other essays discuss undetected justifications in analyzing non-Greek mathematical texts. The second collection elaborates on systematic justifications of old mathematical procedures outside Greece. These essays are interesting, especially essay 8 on various rather unrelated procedures occurring already in classical Greece. But what becomes painfully clear is the lack of information on which to build a coherent and plausible “nonstandard narrative.”
In this respect, a contemporary mathematician might point out a dilemma in mathematical research that is not so different from what emerges from the studies presented. A valid mathematical proof has to be built according to the rules of mathematical logic, although only in principle. Given the complexity of modern mathematics, a formal proof would regularly become too long and too technical to be read or published. What is worse, it might obscure the new insights resulting from the research. Mathematical discoveries are often inspired by analogies between seemingly quite unrelated problems that eventually lead to a new theory or the solution of an old problem, but these insights may not find a [End Page 524] place in the logically correct presentation of the research. “Logic is the hygiene of the mathematician,” as Hermann Weyl said, implying that mathematicians have many other ways to “justify” their results. Hence, the praxis of modern mathematics (and theoretical physics!) offers an interesting field in which to study varieties of justification. Of particular interest for the research program indicated in the book could be recently constructed algorithms that compute the correctness of a mathematical proof. To understand the past, stable patterns still working today might be helpful.
Jochen Brüning, a member of the Berlin-Brandenburg Academy of Sciences and Humanities, is professor of mathematics and director of the Hermann von Helmholtz Center for Cultural Technology at the Humboldt University of Berlin. His research is primarily in the fields of geometric analysis and spectral theory.