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Fascinating study considers the origins and nature of mathematics, its development and role in the history of scientific thinking, impact of high-speed computers, 20th-century changes in the foundations of mathematics and mathematical logic, mathematization of science and technology, much more. Compelling reading for mathematicians, physicists, students—anyone interested in the evolution of mathematical thought. 34 illustrations. 1968 edition.
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Customer Reviews (5)

Excellent overview by two outstanding mathematicians
The audience for ths book is people with background in mathematics. They teach many branches briefly with their examples in Chapter 1, but the presentation is at a fairly sophisticated level.

I found the historical and philosophical remarks very valuable, coming from authorities like these two men. Examples (slightly edited by me):

The great analysts of the 18th and 19th centuries (e.g., Newton, Leibniz, Bernoulli, Euler, Lagrange ...) had an almost unerring instinct for presenting valid results and plausible proofs without a firm basis in formal systems and without strict adherence to standards of logical rigor... Mathematical intuition in the hands of people of genius has such a clarity and unity that it anticipates special formalisms.

Mathematics is a science; it is also an art. The criteria of judgment in mathematics are always aesthetic, at least in part... One looks for `usefulness,' for `interest,' and also for `beauty.' Beauty is subjective, yet it is surprising that there is usually considerable agreement among mathematicians concerning aesthetic values.

It is a distinctive feature of mathematics that it can operate effectively and efficiently without defining its objects. Points, straight lines and planes are not defined... One need not know what things are so long as one knows what statements about them one is allowed to make [the axioms]... Other statements involving these undefined words can then be deduced by logic alone. This permits geometry to be taught to a blind man and even to a computer!

A Classic of Popular Mathematics
With new mathematics growing wild in the trees in the new century, one should review those things that have givenmodern mathematics direction and flavor. This classic little book sits along side of Sawyer's " Prelude to Mathematics" as the go to books for basic understanding. To these books I have added an unlikely candidate " Elliptical Curves" by McKean and Moll.If one has these three books, he will have a crack in the damof mathematics...

Important and Useful but not Engaging
The book shows what the power of mathematics is, how it changes, and how it expands to new areas. Unlike books that aim to popularize math, the book does not pontificate a mystic view of the meaning of mathematics; rather, it gives a sober perspective of what has happened in mathematics and what can be expected of the field.

The book requires somewhat serious mathematical thinking.

A great strength of the book is the diverse mathematical concepts that it presents: homology groups, group theory, Turing machines, undecidability, Monte Carlo method.In a compact book, you learn a little about some important ideas in advanced mathematics.

Important!! This book is not written to popularize its subject matter, so it is different and definitely less entertaining than most popular science books.

why math is interesting, written for laypeople
Kac and Ulam were both strong mathematicians; this encyclopedia articlewhich they coauthored provides a nice counterweightto the answers-without-questions view of mathwhich can result from a little indifferentcoursework.

Easy and lively reading, written for everyone.

A bargain indeed
Just read the editorial review (Book News).It's all true (except for the publishing date,which is 1968, not 1986...) ... Read more

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Integral geometry originated with problems on geometrical probability and convex bodies. Its later developments have proved to be useful in several fields ranging from pure mathematics (measure theory, continuous groups) to technical and applied disciplines (pattern recognition, stereology). The book is a systematic exposition of the theory and a compilation of the main results in the field. The volume can be used to complement courses on differential geometry, Lie groups, or probability or differential geometry. It is ideal both as a reference and for those wishing to enter the field. ... Read more

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This is a volume of essays and reviews that delightfuly explore mathematics in all its moods - from the light and the witty, and humorous to serious, rational, and cerebral. Topics include: logic, combinatorics, statistics, economics, artifical intellligence, computer science, and applications of mathematics broadly. You will also find history and philosophy covered, including discussion of the work of Ulam, Kant, and Heidegger among others. ... Read more

Customer Reviews (1)

An enlightening book on mathematics
This is an unusual book on the discipline of mathematics. Don't look here for the usual fluffy collages of fractals, chaos theory and hagiographies.

What this book offers is a collection of essays on the state and nature of mathematics by three top mathematicians. They're honestly worried about the schism between math and science, interested in the effect the computer could have on math, critical about the mathematical establishment. They're completely honest, and their criticisms are incisive and well reasoned. This book is food for thought, and not only about math.

The authors differ in their writing styles. While all three have good essays, Rota is a fantastic writer, and his mixture of serious thought, pith and gossip is completely refreshing. You should also read (before this book) Rota's "Indiscrete Thoughts". ... Read more

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A Classic in Statistical Physics
The book contains too much on the topic of irreversibility from the Boltzmann standpoint to be of much interest today, excepting one chapter: the one by Kac on functional integration and partial differential equations. Here, Kac makes Feynman's connection rigorous for the case of a Wiener process (Brownian motion) and presents the 'Feynman-Kac formula' that so many financial math texts now rave about without pushing the boundaries further mathematically to do something useful for empirical modelling in option pricing. For more in this direction, see Durrett's "Brownian Motion and Martingales in Analysis" and the final chapter in Steele's "Financial Calculus". See also papers by me now in preparation. ... Read more

Customer Reviews (1)

It should not surprise anyone. . .
Enignas of Chance is distinctly and clearly written by a very interesting person, and one could hope that there are works which go further in explaining his contributions to Mathematics and Physics. ... Read more

Product Description
To hear the shape of a drum is to infer information aboutthe shape of the drumhead from the sound it makes, i.e.,from the list of basic harmonics, via the use ofmathematical theory. "Can One Hear the Shape of a Drum?" wasthe witty title of an article by Mark Kac in the AmericanMathematical Monthly 1966 (see the references below), butthese questions can be traced back all the way to HermannWeyl.The frequencies at which a drumhead can vibrate dependon its shape. Known mathematical formulas tell us thefrequencies if we know the shape. A central question is: canthey tell us the shape if we know the frequencies? No othershape than a square vibrates at the same frequencies as asquare. Is it possible for two different shapes to yield thesame set of frequencies? Kac did not know the answer to thatquestion ... Read more

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High Quality Content by WIKIPEDIA articles! Mark Kac was a Polish mathematician. His main interest was probability theory. His question, "Can you hear the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry. Kac completed his Ph.D. in mathematics at the Polish Jan Kazimierz University of Lwów in 1937 under the direction of Hugo Steinhaus. While there, he was a member of the Lwów School of Mathematics. After receiving his degree he began to look for a position abroad, and in 1938 was granted a scholarship from the Parnas Foundation which enabled him to go work in the United States. He arrived in New York City in November, 1938. From 1939 until 1961 he was at Cornell University, first as an instructor, then from 1943 as assistant professor and from 1947 as full professor. While there, he became a naturalized US citizen in 1943. In 1961 he left Cornell and went to Rockefeller University in New York City. After twenty years there, he moved to the University of Southern California where he spent the rest of his career. ... Read more

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Der Erwerb des Buches enthält gleichzeitig die kostenlose Mitgliedschaft im Buchklub des Verlags zum Ausprobieren - dort können Sie von über einer Million Bücher ohne weitere Kosten auswählen. Das Buch besteht aus Wikipedia-Artikeln: Don Zagier, Cornelius Lanczos, Shiing-Shen Chern, Barry Mazur, Mark Kac, Günter M. Ziegler, Andrew Granville, Heinz Bauer, Martin Davis, Shreeram Abhyankar, Joan Birman, Norman Levinson, Leon Henkin, Gilbert Ames Bliss, John Stillwell, Thomas W. Hawkins, Philip Davis, Neil Sloane, W. B. R. Lickorish, Jonathan Borwein, François Treves, Carl Pomerance, Guido Weiss, Michael Rosen, Gordon Thomas Whyburn, Dunham Jackson, Reuben Hersh, David Harold Bailey, Richard Bruck,. Online finden Sie die kostenlose Aktualisierung der Bücher. Nicht dargestellt. Auszug: Don Bernhard Zagier (* 29. Juni 1951 in Heidelberg) ist ein amerikanischer Mathematiker. Derzeit ist er einer der Direktoren des Max-Planck-Instituts für Mathematik in Bonn und Professor am französischen Collège de France in Paris. Seine Hauptarbeitsgebiete sind Zahlentheorie, Theorie der Modulformen und Verbindungen zur Topologie. Zagier ist 1951 in Heidelberg als Sohn amerikanischer Eltern geboren und in den USA aufgewachsen. Er bestand im Alter von 13 Jahren sein Abitur. Er studierte am MIT Mathematik und Physik und wurde 1967 - im Alter von 16 Jahren - Putnam Fellow (im Jahr zuvor gewann er den ersten Preis in der Mathematik-Olympiade). 1968 erhielt er den B.A., ging dann an die Oxford University und an die Universität Bonn, wo er bei Friedrich Hirzebruch mit 19 promovierte (offiziell in Oxford). Nach zweijährigem Aufenthalt an der ETH Zürich und am IHES in Bures-sur-Yvette bei Paris kam er 1974 nach Bonn, habilitierte sich 1975 und wurde 1976 Deutschlands jüngster Professor. 1984 wurde er als Wissenschaftliches Mitglied der Max-Planck-Gesellschaft an das Max-Planck-Institut für Mathematik in Bonn berufen, wo er 1995 zum Direktor ernannt wurde. ...http://booksllc.net/?l=de&id=1263139 ... Read more

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This digital document, covering the life and work of Mark Kac, is an entry from Contemporary Authors, a reference volume published by Thomson Gale. The length of the entry is 351 words. The page length listed above is based on a typical 300-word page. Although the exact content of each entry from this volume can vary, typical entries include the following information:

• Place and date of birth and death (if deceased)
• Family members
• Education
• Professional associations and honors
• Employment
• Writings, including books and periodicals
• A description of the author's work
• References to further readings about the author

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"‘Mathematicians, like Proust and everyone else, are at their best when writing about their first love’ … They are among the very best we have; and their best is very good indeed. … One approaches this book with high hopes. Happily, one is not disappointed.""…In paperback it might well have become a best seller. …read it."From The Mathematical Intelligencer"Mathematics is shaped by the consistent concerns and styles of powerful minds —three of which are represented here. Kac’s work is marked by deep commitment and breadth of inquiry."Rota is the easiest of these authors to read…a delight: witty and urbane, with a clear and interesting agenda and an astonishing intellectual range. To read him is to be a part of a pleasant and rewarding conversation."Jacob T. Schwartz attacks problems in…computer science, mathematical economics, computer and instructional … No matter how slippery the problem, Schwartz manages to capture a substantial chunk of it in his mathematical net."—Mathematics MagazineThis is a volume of essays and reviews that delightfully explore mathematics in all its moods — from the light and the witty, and humorous to serious, rational, and cerebral. Topics include: logic, combinatorics, statistics, economics, artificial intelligence, computer science, and applications of mathematics broadly. You will also find history and philosophy covered, including discussion of the work of Ulam, Kant, Heidegger among others. ... Read more