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173 of 175 people found the following review helpful.
Excellent Overview. Belongs on Your Bookshelf.
By Michael Wischmeyer
Howard Eves presents this five-star story of mathematics as two intertwined threads: one describes the growing content of mathematics and the other the altering nature of mathematics. In exploring these two elements, Eves has invented a outstanding book for the layman. I find myself returning to his book again and again.

My few semesters of calculus, differential equations, and other used math failed to formally introduce me to abstract algebras, non-Euclidian geometries, projective geometry, symbolic logic, and mathematical philosophy. I in general considered algebra and geometry to be singular nouns. Howard Eves corrected my grammar.

"Foundations and Fundamental Concepts" is not a traditionalisti 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 calling for significant mathematics by the reader. Calculus is not required.

The introductory 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 eclipsed mathematics for closely 2000 years.

In chapter three Eves introduces non-Euclidian geometry. Mathematics is transformed from an empirical method focalized on describing our real, three-dimensional world to a originative endeavor that manufactures new, abstract geometries.

This discussion of geometries, as opposed to geometry, proceeds in chapter four. The key topics include Hilbert's highly influential work that placed Euclidian geometry on a firm (but more abstract) postulational basis, Poincaire's model and the consistency of Lobachevskian geometry, the principle of duality in projective geometry, and Decartes development of analytic geometry. For the non-initiated these topics may seem daunting, but Eves' approach is clear and rather fascinating.

Chapter five, which might have been titled "The Liberation of Algebra", may at introductory be a bit overpowering to those incognizant of algebraic structures like groups, rings, and fields. But take solace as even mathematicians in the early nineteenth century still considered algera to be little more than symbolized arithmetic. As Eves says, non-Euclidian geometry freed the "invisible shackles of Euclidian geometry". Likewise, abstract algebra produced a parallel revolution. (Again, don't be intimidated by the terminology. Eves is rather good.)

The remaining four chapters look at the axiomatic foundation of progressed mathematics, the real number system, set theory, and at long last mathematical logic and philosophy. Eves concludes with the surprising invention of contradictions within Cantor's set theory as well as Hilbert's not successful venture to define procedures to keep away from inconsistencies or contradictions within an axiomatic system.

Eves mentions Godel's rudimentary contribution to mathematical logic, but stops short of delving into Godel's Proof. For further and added reading I highly commend "Godel's Proof" by Ernest Nagel and James R. Newman.

I likewise highly commend Richard Courant's and Herbert Robbins' classic, "What is Mathematics?", a more elaborate examination of the development of rudimentary ideas and methods underlying mathematics. I would suggest that most readers, peculiarly non-math majors, firstborn read Eves and later tackle Courant and Robbins.

I have read "Foundations and Fundamentals of Mathematics" at least twice. I gave my son a copy for Christmas. He says that the book is outstanding and he even claims to be reading it as he walks throughout his campus amid classes. The price is great. It belongs in your book collection.

149 of 151 humans found the following review helpful.
Ecellent description of the history of mathematical thinking
By Juergen Kahrs
There are various books available on the history of mathematics. Some give an account on the development of a sure area, others focus on a group of people and a lot of do scarcely 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 humans thought it is and must be). Howard Eves' book is the original book I purchased 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 actually 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 amazed to see how much these troubles disclose of the epoch, it is difficulties 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 a lot of interesting appendices on specific troubles like the initial propositions of Euclid, nonstandard analysis and even Gödel's incompleteness theorem. Bibliography, solutions to chosen difficultnesses and an index are cautiously prepared to round up an magnificent book.

Should you buy this book ? Yes. What kind of fault may you make in spending US$ 12.95 on a book that has withstood the test of time through three editions (each with a dissimilar publisher). I have not finished reading the book yet, but I do not regret having purchased it.

56 of 61 people found the following review helpful.
'Swiss Army Knife' of Upper Level Mathematics
By A
I wholly agree with the former two reviewers on what they had to say regarding this fantasti book. However, I did want to briefly note that -- beyond plainly being a arousing and attention holding overview of the development of beyond-calculus mathematics -- it is also a great resource for persons requiring to look up or review topics in progressed mathematics (especially mathematical logic). Again, to repeat what the others have said, buy this book if you have ANY interest in mathematics. You won't regret it.

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