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Intuitionism. (L’intuitionisme.) (French) Zbl 0980.03001

Que Sais-Je? Paris: Presses Universitaires de France. 125 p. (1992).
Foundations of mathematics became a major field of research after the introduction of the notion of a set in the late 19th century and the paradoxes which were soon to be formulated. The controversies, which arouse among mathematicians, philosophers, historians, psychologists, anthropologists, among others, can be synthesized in one question: how are mathematical objects generated? Three main strands were proposed in the early part of the 20th century to answer this basic question: logicism, whose main proponent was B. Russell, intuitionism, of L. E. J. Brouwer, and formalism, of D. Hilbert. Since then, mathematicians have passionately adhered to one of these schools or proceed creating mathematics ignoring the feud. This is clear when one looks in the requirements for graduate and postgraduate degrees in mathematics all over the world. But a question is always latent even among those that do not side with any of these schools: what is the nature of mathematics?
The small book by Jean Largeault, following the tradition of the excellent collection Que sais-je?, offers an introduction to intuitionism, perhaps the most controversial of these three strands. It is written with competence, clearness and breadth. The author succeeds in dealing with a subject that has been regarded by many as arid in a pleasant and attractive way, without compromising scholarship. Surely, it can be profited by philosophers and other scholars and by mathematicians as well. The presentation is accessible, and useful, to a broad range of scholars. Particularly mathematicians will find the treatment accessible, since the usually strange nomenclature and symbolism characteristic of intuitionism is reduced to a minimum. Although this does seem to have been the intention of the author, the book serves as an excellent invitation for mathematicians to look into the foundations of mathematics.
It is amazing how in such a small book the author touches so many points. The book may be seen as organized in three parts. In the first part the author begins with a chapter on intuitionism and constructivism, offering an overview of the problems associated with foundations, with appropriate historical remarks. This chapter should be particularly attractive and accessible to mathematics educators. In the second chapter, Brouwer’s intuitionism is presented. Particularly interesting is the way Largeault presents and treats the law of the excluded middle, probably the most controversial and key issue in Brouwer’s proposal.
Those not initiated in the theme, which are the typical readers of this collection, will surely want to know how does this theory relate to what they have been doing in mathematics. Chapter 3 deals with classical analysis. Some of the basic results are presented in the intuitionistic framework. Equally important is Chapter 4, which deals with the continuum. It starts with a discussion of the difficulties encountered by the Greeks in dealing with infinity and the consequent issues about the continuum. The author succeeds in giving an enriching discussion of the mathematical, philosophical and psychological issues related to infinity and the continuum, and makes a clear presentation of the way Brouwer and Hermann Weyl dealt with the question.
Chapter 5 and the Conclusion are more explicitly philosophical. And a short essential bibliography is provided. Although philosophical considerations permeate the entire book, in the last pages the author ventures into questions rarely raised among mathematicians, such as an ethics intrinsic to mathematics. It opens up a new direction of discussions which, probably due to the limitations imposed to books of this collection, is not pursued further.

MSC:

03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03A05 Philosophical and critical aspects of logic and foundations
03F55 Intuitionistic mathematics
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
00A30 Philosophy of mathematics
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