Mathematics Today Twelve Informal Essays
336 Books an evolutionary framework colored by 19thcentury optimism also would eliminate from art elements such as the ugly, which 1find are necessary to a larger harmony. I especially appreciated the second chapter, Psychophysics in Perspective, as it shows this child of 19thcentury scientificthoughtin relation to thinking sincethe Renaissance. There is a tendency in the book at times to make of Henry a somewhat larger than life hero. This might be somewhat accounted for by the scarcity of biographical material. Also, some of the footnotes might have been incorporated in the text, for they would have enriched the narrative. Aside from this nitpicking, it is a worthwhile book. TheVoice of the Symbol. Martin Grotjahn. Mara Books, LOS Angeles, Calif., 1971. 224 pp., illus. $8.00. Symbolism and Art Nouveau. Alistair Mackintosh. Thames & Hudson, London, 1975. 64 pp., illus. Paper, f0.95. Reviewed by Peter Fingaten* Symbol interpreting, although often accompanied by much knowledge, is an act similar to that of a musician who interprets a sonata. Sigmund Freud was in the unique position of having been the originator of psychoanalysisas well as the first skilled practitioner of an approach forged by himself. In Freud a cybernetic process was operative as he changed, modified and improved his basic insights and interpretive methods. In his latest book, Grotjahn brings a broad intellectual culture to bear upon the symbols that he analyzes. The 10 chapters are not connected and do not represent a systematic treatment of the subject but reflect a wide selection of his interests. One of the key remarks of the book is: ‘To listen to the silent voices of the symbol is a difficult undertaking. The more one tries to make the symbol clear in writing, the farther one moves away from it. The more one listens, the less one understands. The closer you look, the less you see it’ (p. vi). His approach is descriptive or diagnostic, without that fruitful interplay between symbol and analyst that is called cybernetic. Nevertheless, he offers many valid insights and a felicitous rephrasing of classic Freudian concepts. Of special interest to artists is the author’s attempt to decipher the symbolic message of Hieronymus Bosch’s painting, ‘The Millennium’ (Prado). He acts as if he alone knows what Bosch really meant with his allegories. ‘The middle part of the triptych, the socalled “Garden of Delight ”, is a cryptogram. It is a secret message given in secret pictorial language’ (p. 116). It is extremely risky for a psychoanalyst to read secret messages from equally secret pictorial symbols! A psychoanalyst attempts to make order out of disorder, which, incidentally, is also the classic definition of art. The author’s remarks on relationships between artists and their symbols are sensitive and insightful, even if he himself is more of a craftsman working in the shadow of a genius rather than an original interpreter. The small book by Alistair Mackintosh has a large scope, correlating Art Nouveau with the ‘public eroticism’ of Ingres and the aesthetics of Delacroix. The author traces the several strands from the occultism of Eliphas Levy and H. P. Blavatsky to the esoteric fantasies of Peladan, who had such an enormous influenceover the symbolists in Paris. His miniature essays on Gauguin, Puvis de Chavannes, Moreau, Redon, the PreRaphaelites, Beardsley and Picasso are as sensitive as they are insightful. Fine color plates round out this surprisingly informative and entirely satisfying book. Style in Art History: An Introduction to Theoriesof Style and Sequence. Margaret Finch. ScarecrowPress, Metuchen, N.J., 1974. 170pp., illus. $6.00. Reviewed by Alan C. Birnholz** Margaret Finch’s little book is precisely what she says it is: an introduction to various theories of style and stylistic *Art and Music Dept., Pace College, 41 Park Row, New York, NY 10003, U.S.A. **Dept. of Art History, State University of New York, Buffalo, NY 14214, U.S.A. change. As such, this book‘s usefulnessis limited to students and teachers who want a comprehensive, decent survey of what has been said on the subject of style. Style in Art Hisfory,then, gives us little that is new with few exceptions, such as Finch’s effort ‘to combine especially...
Add your own favourite book of extension material for teachers and lecturers of mathematics students aged 16 to 19, with a brief description. Follow the style of the entries below.
Inhaltsverzeichnis

Felix Klein, Elementary Mathematics from an Advanced Standpoint (Dover 2004), 2 Vols
These are the first two volumes of the threevolume German edition, Elementarmathematik vom höheren Standpunkte aus (J. Springer, Berlin, 19241928).
László Lovász, Trends in Mathematics: How they could Change Education? (2006)
This paper is especially recommended. IMU President László Lovász' paper delivered at the The Future of Mathematics Education in Europe in Lisbon, 2006. You can download it here: PDF, TEX
H. Behnke, F. Bachmann, K. Fladt (Eds.), Fundamentals of Mathematics (MIT Press, 1974)
Translation from the German of Grundzüge der Mathematik, Vandenhoeck & Ruprecht, Göttingen, 1962. 3 Vols.
The book is the translation of a work commissioned by the ICMI in order to support the scientific foundations of instruction in mathematics, which was one of the topics chosen by the Commission at a meeting in Paris in 1954, in preparation for the International Congress of Mathematicians in Edinburgh in 1958. The English book has three volumes with 42 chapters: the first volume concerns the foundations of Mathematics, the Real Number System and Algebra; the second volume is devoted to geometry and the third to Analysis. There are, in general, two authors for each chapter: one a university researcher, the other a teacher of long experience in the German educational system. And the whole book has been coordinated in repeated conferences, involving altogether about 150 authors and coordinators.
Martin Gardner, Mathematical Puzzles and Diversions (Penguin, 1956)
The first of his series of books derived from his columns in Scientific American from 1956 to 1981 [1]. Categorised as “Recreational Mathematics”, these are accessible to those with secondary school mathematics and a willingness to explore mathematical ideas.
Albert Cuoco, Mathematical Connections: A Companion for Teachers (2005)
This is a resource book for high school teachers. See the review by Steve Maurer.
Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (Basic Books, 1979)
From Wikipedia:
GEB takes the form of an interweaving of various narratives. The main chapters alternate with dialogues between imaginary characters, inspired by Lewis Carroll's "What the Tortoise Said to Achilles" , in which Achilles and the Tortoise discuss a paradox related to modus ponens. Hofstadter bases the other dialogues on this one introducing characters such as a Crab, a Genie, and others. These narratives frequently dip into selfreference and metafiction.
Word play also features prominently in the work. Puns are occasionally used to connect ideas, such as "the Magnificrab, Indeed" with Bach's Magnificat in D; SHRDLU, Toy of Man's Designing" with Bach's Jesu, Joy of Man's Desiring; and "Typographical Number Theory", which inevitably reacts explosively when it attempts to make statements about itself. One Dialogue contains a story about a genie (from the Arabic "Djinn") and various "tonics" (of both the liquid and musical varieties), which is titled "Djinn and Tonic".
One dialogue in the book is written in the form of a crab canon, in which every line before the midpoint corresponds to an identical line past the midpoint. The conversation still makes sense due to uses of common phrases that can be used as either greetings or farewells ("Good day") and the positioning of lines which, upon close inspection, double as an answer to a question in the next line.
Zalman Usiskin, Dick Stanley, Anthony Peressini, Elena Anne Marchisotto, Mathematics for High School Teachers: An Advanced Perspective (Prentice Hall, 2003)
Philip Davis & Reuben Hersh, The Mathematical Experience (Birkhauser, 1981)
From Wikipedia:
[This book] discusses the practice of modern mathematics from a historical and philosophical perspective. It won the 1983 National Book Award in the Science category.
It is frequently cited by mathematicians as a book that was influential in their decision to continue their studies in graduate school and has been hailed as a classic of mathematical literature. The book drew a critical review from Martin Gardner, who disagreed with some of the authors' philosophical opinions, but was wellreceived otherwise.
A study edition and also a study guide for use with the book have been released, both coauthored with Elena A. Marchisotto. The authors wrote a followup book, Descartes' Dream: The World According to Mathematics, and both have separately written other books on related subjects, such as Davis' Mathematics And Common Sense: A Case of Creative Tension and Hersh's What is Mathematics, Really?
Philip Davis & Reuben Hersh, Descartes' Dream: The World According to Mathematics (Houghton Mifflen, 1986)
Martin Aigner & Günter Ziegler, Proofs from THE BOOK (SpringerVerlag, 1998)
Lynn Arthur Steen (Ed), Mathematics Today: Twelve Informal Essays. (SpringerVerlag, 1978)
L.A. Steen has written many books and articles on [http:// mathematics], from this very early one to a followup Mathematics Tomorrow (SpringerVerlag, 1981)
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