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Mathematics and Logic (Dover Books on Mathematics) Paperback – May 1, 1992

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Mark Kac (Author)
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Mark Kac (Author), Stanislaw M. Ulam (Author)
4.9 out of 5 stars 7 customer reviews
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Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 192 pages
  • Publisher: Dover Publications; Reprint edition (May 1, 1992)
  • Language: English
  • ISBN-10: 0486670856
  • ISBN-13: 978-0486670850
  • Product Dimensions: 0.5 x 5.8 x 8.5 inches
  • Shipping Weight: 12.6 ounces (View shipping rates and policies)
  • Average Customer Review: 4.9 out of 5 stars  See all reviews (7 customer reviews)
  • Amazon Best Sellers Rank: #236,801 in Books (See Top 100 in Books)

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27 of 27 people found the following review helpful
Important and Useful but not Engaging
By Neal Jameson on March 11, 2001
Format: Paperback
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.
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17 of 17 people found the following review helpful
Excellent overview by two outstanding mathematicians
By Marvin J. Greenberg on June 27, 2004
Format: Paperback
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!
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11 of 12 people found the following review helpful
A Classic of Popular Mathematics
By Roger Bagula on July 28, 2001
Format: Paperback
With new mathematics growing wild in the trees in the new century, one should review those things that have given modern 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 dam of mathematics...
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5 of 5 people found the following review helpful
a pleasant journey of mathematical synthesis
By muddy glass on November 1, 2010
Format: Paperback Verified Purchase
when learning mathematics or any subject in general, it is important to draw some connections between what you're currently studying with what's out there. these bridges between different areas give you a sense of the greater beauty at work as you start to see the whole picture. kac and ulam have done this masterfully with their gem of a book, "mathematics and logic." in so doing, the authors convey to the reader some idea of why mathematicians think the way they do.

imagine yourself in a room with two seasoned mathematicians. a conversation is struck and the two mathematicians start talking about some topic that they find interesting. this topic has some connection to another topic, so they begin talking about that. now, a third topic shows up on the scene, so the mathematicians start expounding a bit on this newly arrived animal, only to get to yet another animal on the horizon. and so forth. that's how things go for the first hundred pages or so as the zoo grows! in general, such a style could quickly become an incoherent rambling mess, but this is not the case here. the transitions are not too abrupt and the reader does get a sense of why things pop up when they do. the book even closes with some chapters explicitly laying out the common threads that have woven these selected topics together. very nice.

the topics covered include the usual suspects that often show up in popular expository math books, subjects like elementary number theory, combinatorics, basic group theory, probability, gödel's incompleteness theorems, turing machines, special relativity, and so on. however, the authors also throw in some topics from left field such as braid theory, information theory, and homology.Read more ›
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