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Foundations and Fundamental Concepts of Mathematics (Dover Books on Mathematics) Paperback – May 20, 1997
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My few semesters of calculus, differential equations, and other applied math failed to formally introduce me to abstract algebras, non-Euclidian geometries, projective geometry, symbolic logic, and mathematical philosophy. I generally considered algebra and geometry to be singular nouns. Howard Eves corrected my grammar.
"Foundations and Fundamental Concepts" is not a traditional history of mathematics, but an investigation of the philosophical context in which new developments emerged. Eves paints a clear picture of the critical ideas and turning points in mathematics and he does so without requiring substantial mathematics by the reader. Calculus is not required.
The first two chapters, titled "Mathematics Before Euclid" and "Euclid's Elements", consider the origin of mathematics and the remarkable development of the Greek axiomatic method that dominated mathematics for nearly 2000 years.
In chapter three Eves introduces non-Euclidian geometry. Mathematics is transformed from an empirical method focused on describing our real, three-dimensional world to a creative endeavor that manufactures new, abstract geometries.
This discussion of geometries, as opposed to geometry, continues in chapter four.Read more ›
The book goes on with chapters on Hilbert's Grundlagen, Algebraic Structure etc, always showing not only the substance of these periods but also the shift in the way of thinking and the development towards rigor. The last chapter is titled Logic and Philosophy. Eves divides "contemporary" philosophies of mathematics into three schools: logistic (Russel/Whitehead), intuitionist (Brouwer) and the formalist (Hilbert).
The book ends with some interesting appendices on specific problems like the first propositions of Euclid, nonstandard analysis and even Gödel's incompleteness theorem. Bibliography, solutions to selected problems and an index are carefully prepared to round up an excellent book.
Should you buy this book ? Yes.Read more ›
In the first chapter Eves gives a brief but good historical overview of mathematics in ancient civilizations. He deals with the early Egyptians, Babylonians, and of course the Greeks. This approach naturally segues into an emphasis upon Euclid and his monumental Elements. Eves pays particular attention to Euclid's methodology, the material axiomatic, discussing its origin and ensuing problems.
Other texts that I have read on the subject of mathematical logic tend to give quite a bit of time to Euclid's fifth (or parallel) postulate. Not until reading Eves' book have I understood why though. Euclid's fifth postulate has the appearance of being quite different from the first four; any non-mathematician can perceive this fact from a mere browsing of the first several postulates. Euclid needed this fifth statement for his geometry; and since he could never prove it as a theorem, he made it a postulate in his system. Eves notes that a good deal of mathematical history is devoted to this same exact project that Euclid failed to accomplish.Read more ›
Most Recent Customer Reviews
Important book in the sense that mathematics as a field works off of assumptions that may or may not be true or cannot be proved internally, and this is almost always taken for... Read morePublished 4 months ago by Luke
Absolutely the best Math book ever.Unfortunately the Kindle version has lots of errors in the formulas.Published 13 months ago by C. Portwood
It appears that all of mathematical knowledge may be available to brute force at the speed of light:
Nice blend of historical development of mathematics and going through the concepts themselves. Easy to read. Read morePublished 23 months ago by Carl Beard
This book provides a good, clear introduction to a number of topics for which it is hard to find good, clear introductions. Read morePublished on January 5, 2012 by Steve L
The reviewer who said see Page 2 is really hitting below the belt. Page 2 says nothing can be said about indian and chinese maths as they wrote on perishable items. Read morePublished on May 17, 2007 by Carol Meade
Back in the days when I thought that mathematics could be understood by a through understanding of fundamental logic, I thought this book would help me on that.. Read morePublished on June 17, 2006 by Humberto Mejia
The author reviews mathematical history but mentions no India nor China. He presented a biased view of mathematical history.
The books is misleading in that regard.