The psychology of invention in the mathematical field. Transl. from the English by L. A. Santaló Sors. Reprint of the 1947 Spanish translation. (Psicología de la invención en el campo matemático.) (Spanish) Zbl 1247.01056

Historia y Filosofía de la Ciencia. Buenos Aires: Espasa-Calpe. 234 p. (2011).
This book is a small wonderful treasure in the mathematical field. The initiative by the Real Sociedad Matemática Española in reprinting the 1947 Spanish translation enables us to rethink Hadamard’s ideas on mathematical invention and deserves our congratulation. It is worth remarking the personality of the translator, the Catalan mathematician Lluis A. Santaló Sors (1911–2001). Santaló was a great mathematician who contributed to integral geometry, and his results were used subsequently in medicine in the interpretation of axial tomography. Santaló was a fine professor who transmitted to his pupils his enthusiasm for the mathematics, which in his own words is at once art, science and technique.
As Hadamard explains at the beginning of the book, the idea of this investigation into the psychology of invention in the mathematical field proceeds from Poincaré’s lecture on mathematical invention at the “Société de Psychologie” of Paris, in 1908. The book, a 234-page text, consists of IX chapters and two annexes (a questionnaire on the method of mathematical work and Einstein’s answers) discussing the process of invention in mathematics and analyzing Poincaré’s explanations.
After the introduction, in which Hadamard distinguishes between invention and discovery, the author reproduces the contents of the particular discovery explained by Poincaré in his lecture. In the Chapters II and III he discusses the relation between the unconscious and discovery in mathematics. The first step in the process of discovery is the combination of ideas, while in the second step the discovery is produced when one chooses the most fruitful combinations. Hadamard then arrives at a twofold conclusion: to invent is to choose, and the invention has to be guided by a sense of scientific beauty. He goes on to discuss how invention occurs, either between the moment of illumination or after a spell of hard work or a period of rest. In Chapter IV entitled: “The Preparation Stage. Logic and Chance” and in Chapter V entitled: “The Later Conscious Work”, on the basis of the analysis of Poincaré’s lecture, the author describes the four stages in the process of invention: preparation, incubation, illumination and later work. In this regard, the author makes remarks on the importance of verifying the results, specifying the results and considering these results as provisional for continuing the research. In addition, both these chapters are full of examples of inventions by several authors like Wallis, Newton, Gauss, etc. In Chapter VI Hadamard relates thought with the use of signs like words, figures and mental images throughout the different stages of the process of invention. The author again gives several examples from books, from scientists or from his own inventions. Chapter VII, in which Hadamard attempts to differentiate between different types of mathematical intelligence based on ideas on the use of signs, is also very interesting. The author distinguishes roughly between logical and intuitive thought, and in Chapter VIII analyzes four examples of intuitive thought: Fermat, Riemann, Galois and Poincaré. In the last chapter, Hadamard focuses on the direction that research may take as well as how to choose a subject. He presents several examples of inventions made without any aim of applicability, but which subsequently contributed to development in other fields. In his final remarks, Hadamard refers to subjects he has not considered in his analysis, such as social and historical influences or the relation of invention with neurology.
I believe that no mathematician or historian of mathematics will remain indifferent to a reading of this book, which poses several questions and provides much food for thought. In addition, although the book was first published many years ago, the subject is still fascinating and relevant to the present day.


01A75 Collected or selected works; reprintings or translations of classics
00A30 Philosophy of mathematics
03A05 Philosophical and critical aspects of logic and foundations