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Foundations and Fundamental Concepts of Mathematics (Dover Books on Mathematics) by [Eves, Howard]
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Foundations and Fundamental Concepts of Mathematics (Dover Books on Mathematics) Kindle Edition

5.0 out of 5 stars 1 customer review

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Length: 370 pages Enhanced Typesetting: Enabled Page Flip: Enabled

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From the Back Cover

This third edition of a popular, well-received text offers undergraduates an opportunity to obtain an overview of the historical roots and the evolution of several areas of mathematics. The selection of topics conveys not only their role in this historical development of mathematics but also their value as bases for understanding the changing nature of mathematics.

Product details

  • Format: Kindle Edition
  • File Size: 5948 KB
  • Print Length: 370 pages
  • Publisher: Dover Publications; 3rd Revised ed. edition (10 April 2012)
  • Sold by: Amazon Media EU S.à r.l.
  • Language: English
  • Text-to-Speech: Enabled
  • X-Ray:
  • Word Wise: Not Enabled
  • Screen Reader: Supported
  • Enhanced Typesetting: Enabled
  • Average Customer Review: 5.0 out of 5 stars 1 customer review
  • Amazon Bestsellers Rank: #708,335 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
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Format: Paperback
There are several books available on the history of mathematics. Some give an account on the development of a certain area, others focus on a group of persons and some do hardly more than story telling. I was looking for one that tells the story of the development of the main ideas and the understanding of what mathematics and science in general is (or what people thought it is and should be). Howard Eves' book is the first book I bought that gives me the answers I was looking for. Starting with pre-Euclidean fragments, going on with Euclid, Aristotle and the Pythagoreans, straight to non-Euclidean geometry it focuses on the axiomatic method of geometry. What pleased me most here is that the author really takes each epoch for serious. He quotes longer (and well chosen) passages from Euclid, Aristotle and Proclus to demonstrate their approaches. Each chapter ends with a Problems section. I was surprised to see how much these problems reveal of the epoch, its problems and thinking.
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.
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Most Helpful Customer Reviews on Amazon.com (beta) (May include reviews from Early Reviewer Rewards Program)

Amazon.com: 4.3 out of 5 stars 16 reviews
5.0 out of 5 stars A relaxed survey of the underpinnings of mathematics. 15 April 2017
By Alan U. Kennington - Published on Amazon.com
Format: Paperback Verified Purchase
If you ignore the exercises, this 1958 book by Eves is a relaxed survey of a range of fundamental concepts in mathematics. Well, you do need about 2 or 3 years of university mathematics to appreciate it. Otherwise it would not be so relaxing, and then you would definitely learn a lot by doing the exercises, which don't look particularly taxing to me. This is the kind of book that a mathematics student could read to obtain a more rounded understanding of the logical underpinnings of the subject.

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.
2 of 2 people found the following review helpful
5.0 out of 5 stars The best 27 July 2015
By Jeremy M. - Published on Amazon.com
Format: Kindle Edition Verified Purchase
Absolutely the best Math book ever.Unfortunately the Kindle version has lots of errors in the formulas.
4.0 out of 5 stars Kept my attention 23 April 2016
By Luke - Published on Amazon.com
Format: Kindle Edition Verified Purchase
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 granted.
1 of 1 people found the following review helpful
4.0 out of 5 stars Nice blend of historical development of mathematics and going through ... 6 Oct. 2014
By Carl Beard - Published on Amazon.com
Format: Kindle Edition Verified Purchase
Nice blend of historical development of mathematics and going through the concepts themselves. Easy to read. Suitable as an entry-level textbook or for someone who is just interested in the subject.
16 of 16 people found the following review helpful
4.0 out of 5 stars Foundations and Fundamentals concepts of Mathematics 16 Mar. 2006
By James L. Mchard - Published on Amazon.com
Format: Paperback Verified Purchase
Very readable. Although, I expected something a little more technical. I wanted to see something on conceptual proof and derivation of theorems. But this is more like a history. Heavy on geometry (the author is a geometer), very light on analysis, especially calculus concepts, such as infinitesimals and limits. The Riemann, Galois, Cauchy, Weirstrass and Poincaré contributions are glossed over a bit; admittedly it's pretty deep stuff. Overall, though, it's a good historical survey.

James L. McHard
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