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Descartes’ dream. The world according to mathematics. (English) Zbl 0643.00001
San Diego - Boston - New York: Harcourt Brace Jovanovich, Publishers (distr. by Academic Press). XVIII, 321 p. $ 19.95 (1986).
This is a follow-up to “The mathematical experience” by the same authors (1980; Zbl 0459.00001). However, the present work has a different goal. After having described rather successfully mathematics and the art of doing it in the first book from the perspective of the professional mathematician, the authors now focus on the impact of mathematics on the world surrounding us.
The investigation proposed in the preface can be summarized by the question: “How does the application of mathematics constrain our lives and change our perception of reality?” Since the computer plays a prominent role in the applications of mathematics, it is natural that a large part of the book is devoted to the “computational experience” and the consequences of the ever-increasing computerization of our world. We proceed to summarize the contents of the individual chapters.
Chapter I, “This mathematical world”, collects evidence from many fields to show the omnipresence of mathematics or at least of numbers, encoding all kinds of information, for easy processing by computers. On the other hand, it is pointed out that the use of computers (and, maybe, even mathematics) meets certain limits when it comes to emotions and music, literature or art. This chapter also takes up such topics as the stochastized world - replacing unknown causes with statistical “evidence” -, the equilibrium machine which illustrates the complexity of systems, and the (hopefully) beautiful world of computer art which is going to be created.
Chapter II contains various facts indicating that numbers influence the reality surrounding us: in social policy, in testing, and computer dating. Also included are remarks on the “rhetorical use” of mathematics and the use of mathematics courses as a filter in the curricula of other fields.
Chapter III is entitled “Cognition and computation”. After shortly recalling the scientific procedure according to Comte the authors describe various aspects of computing and the computer scientist. Managing a Chinese restaurant teaches fundamental ways of thinking, namely algorithmic, modular, and systems thinking, and, above all, metathinking. Metathinking is illustrated again in a separate section interviewing the computer scientist C. M. Strauss (various parts of the interview are scattered through the book). Another part of this dialogue deals with the “Whorfian hypothesis” as applied to computer languages.
Chapter IV deals with the time-dependence of mathematics and computing. It is pointed out that mathematical statements are time dependent since we never have a statement “as such” but only by the help of symbols - these, of course, need interpretation. Another section suggests a certain parallelism in the development of non-Euclidean geometry and ethical relativism. It is plausible that also in mathematics certain ideas can surface only at the ”right” time. Finally, all too obvious time dependent elements in the present computerization of our world are collected.
Chapter V illustrates the relationship between ”Mathematics and ethics”. This relationship being neither obvious nor close the illustrations are somewhat artificial. So it is discussed what results from the description of God given by Philo Judaeus (you may have to consult your encyclopedia) if ”God” is replaced by ”mathematics” throughout. A more down-to-earth version asks: is mathematics eternal, true, or good? It is pointed out that none of these questions has an easy answer. A second subsection interpretes the statement ”The computer thinks” in four possible ways, following a suggestion of Dante Alighieri: literally, allegorically, morally, and anagogically. Though it is hard to check whether Dante’s requirements are fulfilled the discussion of ”natural” and ”automatic” thinking is very useful.
The final Chapter VI is entitled ”Personal meanings”. It is mainly concerned with the role of mathematics in creating and destroying meaning. While meaning is created by imposing structure upon our perception of reality (e.g. geometric structure) meaning is lost, on the other hand, through abstraction and generalization. The book concludes with a discussion of two opposite positions, already quoted at the beginning and (partially) in the title: does mathematics take over the world as our only tool of understanding and managing reality (this is called ”Descartes’ dream”) or will mathematics be just one tool among others all subjected to the rule of ”human institutions” (which is referred to as ”Vico’s position”)? The authors seem to favour the latter position and express their hope that mankind will find a way to prevent ”revolutionary waves of symbols from washing over us”.
This book is certainly useful since it raises many questions that ought to be discussed not only by professional mathematicians and computer scientists. The authors collect many interesting examples and write, in general, clearly and vividly. To mention some shortcomings: most chapters are collections of independently written essays which causes a certain lack of coherence. Also, the discussion does not always live up to the high philosophical level suggested by the title or the opening paragraph of an essay. Finally, I would have liked a better description of the illustrations and their connection with the text surrounding them. Nevertheless, the book can be recommended as interesting and sometimes inspiring reading.
Reviewer: J.Br√ľning

MSC:
00A05 Mathematics in general
00A30 Philosophy of mathematics
00-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general
00-02 Research exposition (monographs, survey articles) pertaining to mathematics in general