Foundations and Fundamental Concepts of Mathematics

by Howard W. Eves and Howard Eves

Book Overview

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. Among the topics covered in this wide-ranging text are: mathematics before Euclid,...

• Format: Paperback
• Language: English
• ISBN: 048669609X
• ISBN13: 9780486696096
• Release Date: May 1997
• Publisher: Dover Publications
• Length: 368 Pages
• Weight: 1.10 lbs.
• Dimensions: 0.7" x 6.2" x 9.2"

Customer Reviews

Good Introduction to Mathematics, Historically and Philosophically

Though originally published in 1958, Howard Eves' book was a completely new find for me. Fortunately this classic text has found extended life through Dover Publications, which is making many great older volumes available for newer generations. I am not a mathematician by vocation or training and I am usually only interested in more philosophically focused books concerning logic or meta-logical issues. But I found this book extremely enlightening, showing the interrelations of (what had previously been to my mind) unrelated historical streams of thought. In the following I will give a brief summary and point out some of, what I consider, the highlights of Eves' volume. In the first chapter Eves gives a brief but good historical overview of mathematics in ancient civilizations. He deals with the early Egyptians, Babylonians, and of course the Greeks. This approach naturally segues into an emphasis upon Euclid and his monumental Elements. Eves pays particular attention to Euclid's methodology, the material axiomatic, discussing its origin and ensuing problems. Other texts that I have read on the subject of mathematical logic tend to give quite a bit of time to Euclid's fifth (or parallel) postulate. Not until reading Eves' book have I understood why though. Euclid's fifth postulate has the appearance of being quite different from the first four; any non-mathematician can perceive this fact from a mere browsing of the first several postulates. Euclid needed this fifth statement for his geometry; and since he could never prove it as a theorem, he made it a postulate in his system. Eves notes that a good deal of mathematical history is devoted to this same exact project that Euclid failed to accomplish. "It would be difficult to estimate the number of attempts that have been made, throughout the centuries to deduce Euclid's fifth postulate as a consequence of the other Euclidean assumptions, either explicitly stated or tacitly implied. All these attempts ended unsuccessfully, and most of them were sooner or later shown to rest on an assumption equivalent to the postulate itself" (53). Several notable mathematicians (Gauss for instance) suspected the fifth postulate was independent of Euclid's system. But these results were considered far too radical or ridiculous in their time. Eventually certain mathematicians (Saccheri, Lambert, and Legendre) did go forward with geometrical systems that excluded the fifth postulate, showing its independence; and thus non-Euclidean geometry was born. The far greater importance of this though was that geometry had been liberated from its traditional mold. What had been previously considered absolute and intuitive was shown not to be the case. Eves notes that another shortcoming of Euclid's system, besides the independence of his fifth postulate, was that some of his basic definitions proved to be circular. This problem eventually showed that some terms in a mathematical system had to be conceived of as primitive

india and china

in answer to the reviewer who stated: "The author reviews mathematical history but mentions no India nor China." this is plainly false as any reader of this review can attest to simply by clicking on the 'look inside this book' link and reading page 2 of the book.

'Swiss Army Knife' of Upper Level Mathematics

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.

Excellent Overview. Belongs on Your Bookshelf.

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 addi

Ecellent description of the history of mathematical thinking

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.