Mathematics Today: Twelve Informal Essays ed. by Lynn Arthur Steen (review)

158 Books Quite a few of the numerous illustrations in the book (there is a central section with 24 full-page color plates) are of artworks. Concerning some of these and in connection with one's ability to recognize blurred pictures, Frisby speculates that 'one strand in modern art (Cubism in particular) has been, perhaps unwittingly, to stretch to the limit the visual system's capability for building up a satisfactorily structural description of what a painting contains.' But, I think this is not so, because for many painters subject matter is mostly a pretext, their essential aim being to construct an autonomous system of formal relationships between pictorial elements. To concentrate on the recognition of subject matter in a cubist painting is no more relevant than looking for depicted objects in one of the random-dot stereograms produced by Bela Julesz and by Frisby himself. The large selection of these stereograms (to be viewed through the red/green spectacles provided with the book) provides a remarkable feast for the eyes, as well as a source for new important conclusions concerning depth perception. The book is a rich and stimulating collection of information and of ideas for those artists seriously concerned with expanding their 'field of vision'. Mathematics Today: Twelve Informal Essays. Lynn Arthur Steen, ed. Springer-Verlag, New York, 1978. 367 pp., iIIus. $14.90. Reviewed by Magnus J. Wenninger' The objective of this book is clearly stated in the opening words of the Preface, 'to convey to the intelligent nonmathematician something of the nature, development, and use of mathematical concepts, particularly those that have found application in current scientific research.' As a collection of essays it strikes me as admirably suited for this purpose. The author of each essay gives some historical background, then enters into some detail in a style as nontechnical as possible, introducing general readers to the present status of each topic but also maintaining an open-endedness that admits the possibility of future developments in each field. The authors present mathematics from the aspect of biographical sketches, introducing chosen topics in terms of the mathematicians who have worked on them and made significant contributions to their development. Portraits enhance the presentation. The book is not what might be called easy reading. The choice of mathematical topics is such that general readers will immediately have the impression of being in the realm of so-called higher mathematics. The editor's introductory essay takes note of this by saying: 'Those who complete a strong high school mathematics program reach roughly the middle of the seventeenth century, while a first-year of college calculus carries some students only as far as eighteenth century mathematics.' But any reader with at least that much mathematical knowledge will find these essays very enlightening. They are written by mathematicians who teach at leading universities in the U.S.A. and whose aim evidently is to give general readers the 'flavour', if not a complete grasp, of mathematics as it is today. Hence the title of the book: Mathematics Today. The topics range from number theory, groups and symmetry , assorted geometries, combinatorial and statistical theory to econometrics. The essay on 'the four color problem' was particularly interesting to me, because it discusses the place of digital computers in mathematical proof. The essayists in the book frankly grapple with the problem of both the advantages and the limitations of 'mathematicizing' any topic, be it either weather prediction, population growth or economic development . The higher mathematics of these essays becomes so abstract that general readers are mostly left with no foothold in the reality of everyday experience, yet the impression is just as clearly left that this mathematics is still a useful tool in the hands o~ those who ply it. Keenly interested readers may even find an incentive here to pursue the topics further by making 'St. Augustine's College, Nassau, Bahamas. use of the references listed with each essay. In conclusion it may be worthwhile to note that the book also shows that mathematics need not always be directed toward useful applications. The opening essay alludes to this fact; mathematics has a beauty and an elegance that mathematicians experience in some way...