Foundations and Fundamental Concepts of Mathematics

Rating:
Summary: Excellent Overview. Belongs on Your Bookshelf.
Review: Howard Eves presents this five-star story of mathematics as two intertwined threads: one describes the growing content of mathematics and the other the changing nature of mathematics. In exploring these two elements, Eves has created a great book for the layman. I find myself returning to his book again and again.

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. 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 quite fascinating.

Chapter five, which might have been titled "The Liberation of Algebra", may at first be a bit overwhelming to those unaware 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 released the "invisible shackles of Euclidian geometry". Likewise, abstract algebra created a parallel revolution. (Again, don't be intimidated by the terminology. Eves is quite good.)

The remaining four chapters look at the axiomatic foundation of modern mathematics, the real number system, set theory, and finally mathematical logic and philosophy. Eves concludes with the surprising discovery of contradictions within Cantor's set theory as well as Hilbert's unsuccessful effort to define procedures to avoid inconsistencies or contradictions within an axiomatic system.

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

I also highly recommend Richard Courant's and Herbert Robbins' classic, "What is Mathematics?", a more detailed examination of the development of fundamental ideas and methods underlying mathematics. I would suggest that most readers, particularly non-math majors, first 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 great and he even claims to be reading it as he walks across his campus between classes. The price is great. It belongs in your book collection.

Rating:
Summary: 'Swiss Army Knife' of Upper Level Mathematics
Review: I totally agree with the previous two reviewers on what they had to say about this wonderful book. However, I did want to briefly note that -- beyond merely being a fascinating overview of the development of beyond-calculus mathematics -- it is also a great resource for people needing to look up or review topics in advanced 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.

Rating:
Summary: Ecellent description of the history of mathematical thinking
Review: 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. What kind of mistake can you make in spending US$ 12.95 on a book that has withstood the test of time through three editions (each with a different publisher). I havent completed reading the book yet, but I dont regret having bought it.