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Counterexamples in Analysis (Dover Books on Mathematics) by John M. H. Olmsted
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- Counterexamples in Analysis (Dover Books on Mathematics) by John M. H. Olmsted Amazon, The Most Trusted These counterexamples, arranged according to difficulty or sophistication, deal mostly with the part of analysis known as real variables, starting at the level of calculus. The first half of the book concerns functions of a real variable; topics include the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, uniform convergence, and sets and measure on the real axis. The second half, encompassing higher dimensions, examines functions of two variables, plane sets, area, metric and topological spaces, and function spaces. This volume contains much that will prove suitable for students who have not yet completed a first course in calculus, and ample material of interest to more advanced students of analysis as well as graduate students. 1962 edition. 12 figures. Bibliography. Index. Errata.
- Personal Review: Counterexamples in Analysis (Dover Books on Mathematics) by John M. H. Olmsted I have owned this book for years and have quite enjoyed reading in it. I must admit that I have not read it through; it is tough going. Although it is surely a great book, to my taste it is too thoroughly focused on details of pure analysis and not sufficiently attentive to pedagogy and logic. In fact, it is so focused on the internals of pure analysis that it does not even bother to tell the reader what this field is. It does not note that analytic geometry is the application of pure analysis to geometry or that analytic number theory is the application of pure analysis to the theory of numbers. For more on the nature of pure analysis see page 540 of the 1999 CAMBRIDGE DICTIONARY OF PHILOSOPHY, which included an article "Mathematical Analysis" because of the importance of the subject to philosophy of mathematics. John Corcoran might have had this book in mind when he wrote the following in his abstract "Counterexamples and Proexamples", page 460 in the 2005 BULLETIN OF SYMBOLIC LOGIC. I quote the whole abstract because I think that readers will enjoy the book even more if they have Corcoran's ideas in mind--not to imply that I agree with everything Corcoran says there. "Abstract: Every perfect number that is not even is a counterexample for the universal proposition that every perfect number is even. Conversely, every counterexample for the proposition "every perfect number is even" is a perfect number that is not even. Every perfect number that is odd is a proexample for the existential proposition that some perfect number is odd. Conversely, every proexample for the proposition "some perfect number is odd" is a perfect number that is odd. As trivial these remarks may seem, they can not be taken for granted, even in mathematical and logical texts designed to introduce their respective subjects. One well-reviewed book on counterexamples in analysis says that in order to demonstrate that a universal proposition is false it is necessary and sufficient to construct a counterexample. It is easy to see that it is not necessary to construct a counterexample to demonstrate that the proposition "every true proposition is known to be true" is false - necessity fails. Moreover the mere construction of an object that happens to be a counterexample does not by itself demonstrate that it is a counterexample - sufficiency fails. In order to demonstrate that a universal proposition is false it is neither necessary nor sufficient to construct a counterexample. Likewise, of course, in order to demonstrate that an existential proposition is true it is neither necessary nor sufficient to construct a proexample. This article defines the above relational concepts of counterexample and of proexample, it discusses their surprising history and philosophy, it gives many examples of uses of these and related concepts in the literature and it discusses some of the many errors that have been made as a result of overlooking the challenging subtlety of the proper use of these two basic and indispensable concepts." For More 5 Star Customer Reviews and Lowest Price:
- Counterexamples in Analysis (Dover Books on Mathematics) by John M. H. Olmsted 5 Star Customer Reviews and Lowest Price!
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