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Foundations and Fundamental Concepts of Mathematics (Dover Books on Mathematics) Kindle Edition
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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 ›
Most Helpful Customer Reviews on Amazon.com (beta) (May include reviews from Early Reviewer Rewards Program)
Chapter 1 is somewhat speculative because we don't really know much about mathematics before Euclid. Such history is mostly derived second, third or fourth hand from writers who lived many centuries later. Chapter 2 looks at the logical, axiomatic aspects of Euclid and Archimedes, but I think that modern criticism of the soundness of logic at that time is unfair. The logical achievements of the 13 books of Euclid were a stunning revolution in human thought.
Chapter 3 is on non-Euclidean geometry, which really means Euclid without the 5th postulate on parallel lines. This issue is important only because it provided a focus for mathematical logic for so long, eventually leading to a general mobilisation of effort to put all of mathematics on a safe axiomatic footing. Chapter 4 is about the investigations by Hilbert and others into the axiomatic basis of synthetic geometry.
Chapter 5 on algebraic structure is concerned with the axiomatic approach to algebra, which nowadays we take for granted, but it was really only relatively recently that algebra received the axiomatic treatment, eventually leading to extreme abstraction away from the original motivation to solve practical equations. Thus the axiomatic approach which was seen to be so successful in Euclid was applied to algebra. Chapter 6 then informally discusses axiomatization in general.
Chapter 7 discusses how the very serious issues regarding the real numbers at the end of the 19th century were resolved. It must be remembered that until that time, real numbers were still considered to be more a branch of geometry than arithmetic. The modern concept of a complete ordered field which is in one-to-one correspondence with "the continuum" was developed quite late. Eves presents the axiomatic aspects of the real numbers. Even today, the real number system is not fully understood. If you don't have the axiom of choice, for example, or you weaken AC in various ways, many of the intuitively "obvious" properties of the real numbers are lost, although Eves does not mention such deep questions here.
Chapter 8 gives a brief discussion of boolean algebra, infinite sets, and a bit of topology. Not much detail there. Chapter 9 gives a set of 4 axioms for propositional logic based on the implication and negation operators, but denoted in terms of the disjunction operator. Some samples of theorems are given for this, followed by some very brief discussion of the history of axiomatic approaches to logic.
All in all, this is a very relaxing overview of many of the issues which arise in the foundations of mathematics. I think it is more suitable as background to a mathematics course, to give a broader context to the subject. It is not a serious head-banging, cranium-crushing technical introduction to modern mathematical logic. But I think the stress of a serious technical book could be greatly reduced by reading this book by Eves first.
James L. McHard
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