Foundations and Fundamental Concepts of Mathematics (ISBN 048669609X)

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 or implicitly defined. Unlike Euclid whose use of diagrams pitted him towards unconsciously making numerous hidden assumptions, mathematicians such as Pasch, Peano, and Pieri are credited with trying to make geometry more formalistic and thus protecting it from any such intuitions. Eves claims: "Here we have the mathematician's ultimate cloak of protection from the pitfall of overfamiliarity with his subject matter" (81). Eves spends a couple of chapters on the further formal axiomatizations of geometry and also of algebra as they progressed to modern times. Historically these developments are important in that they shape the modern axiomatic approach to mathematics in general. "The discovery of non-Euclidean geometry and, not long after, of non-commutative algebra led to a deeper study and refinement of axiomatic procedure; thus, from the material axiomatics of the ancient Greeks evolved the formal axiomatics of the twentieth century" (147). Eves makes an important observation here. The formalization of axiomatic systems brought a key distinction to the table that had never been made, at least explicitly, before. The abstract development of some branch of pure mathematics came to be recognized as formal axiomatics, whereas the concrete development of a given branch of applied mathematics came to be referred to as material axiomatics. "In the former case we think of the postulates as prior to any specification of the primitive terms, and in the latter we think of the objects and concepts that interpret the primitive terms as being prior to the postulates" (150). The former case is the newer idea of a postulate as a basic assumption about primitive terms. The latter case is the older Greek view of a postulate. The Greeks thought they were dealing with the unique structure of space and time. "But from the modern point of view, geometry is a purely abstract study devoid of any physical meaning or imagery" (150). But of course formalization brings with it its own set of issues that need to be addressed. Here Eves does an admirable job in Chapter 6 of explaining the important properties of axiomatic systems: equivalence, consistency, independence, completeness, and categoricalness. It is a lucid and crystal introduction. Eves spend another whole chapter on the subject of the real number system and its importance for the foundation of analysis. This issue became important with the initial development of calculus by Leibniz and Newton. This newly developed branch of mathematics became an astounding tool for scientific use. But it was a tool, though powerful, that had not been thoroughly examined before its use. Eves claims, "It was more exciting to apply the marvelous new tool than to examine its logical soundness, for, after all, the processes employed justified themselves to the researchers in view of the fact that they worked" (175). So, "Attracted by the powerful applicability of the subject, and lacking a real understanding of the foundations on which the subject must rest, mathematicians manipulated analytical processes in an almost blind manner, often being guided only by a native intuition of what was felt must be valid" (176). So attempts at rigorizing calculus were begun by many notable mathematicians such as Euler and Lagrange. The most successful of these attempts was by Karl Weierstrass. Weierstrass realized that the number of problems that were being found in trying to establish certain branches of mathematics, like calculus, were endemic to properties belonging to the real number system. "Accordingly, Weierstrass advocated a program wherein the real number system itself should first be rigorized; then all the basic concepts of analysis should be derived from this number system" (178). Weierstrass and his followers eventually realized the so-called arithmetization of analysis. So "today it can be fairly said that classical analysis has been firmly established on the real number system as a foundation" (178). But the consistency of the real number system ends up depending on a more fundamental system, the natural numbers. Eves shows how the real number system is obtained from the natural number system in a purely definitional way. For the not-so-mathematically-inclined this section is highly formal. But Eves does a good job of making the formalization as clear as possible. Also notable in this chapter, Eves shows how the rational and complex numbers are extensions of the natural number system. The next chapter deals with what most books that I have read start with when addressing the issues of axiomatic systems and formalizations: set theory. Though Mary Tiles and Stephen Pollard do much better book-length treatments of the subject, it is a good introduction to set theory if you are unfamiliar with it. In the last chapter Eves looks at logic and philosophy showing how symbolic logic has benefited from the axiomatizations that mathematics has experienced in its developmental history. He addresses the big three philosophies of mathematics: logicism, formalism, and intuitionism, showing a surprising attraction, at least for a mathematician, to the last one. This is a good introductory chapter as well but better book-length treatments, like Stewart Shapiro and Stephan Körner, have also been given to this subject. Eves' book also has an excellent appendix that contains some explicit proofs and has some smaller discussions that Eves evidently felt were not directly needed for the general text of his volume. In addition each chapter ends with exercises where one can become more proficient and familiar with the ideas presented in each section. There are some answers, though not all, provided in the back of the book. As a non-mathematician I found the exercises extremely difficult with the exception of the logic chapter. Like the title says, Eves' text is a historical analysis and introduction to the foundations and fundamental concepts of mathematics. If you are interested in an historical overview of how modern mathematics (or logic in my case) has developed, I cannot recommend this book enough.

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.

Fundamental yes.... but is a very demanding introduction:
Back in the days when I thought that mathematics could be understood by a through understanding of fundamental logic, I thought this book would help me on that.. well it did, by showing me that is not the path to take... the book is a good historical developments of fundamental concepts with many exercises which makes you review your calculus again.

No India China:
The reviewer who said see Page 2 is really hitting below the belt. Page 2 says nothing can be said about indian and chinese maths as they wrote on perishable items. This is a 'mention' about india and china on maths? I am ok with all but surely not this much bias..not needed... euro centric wins hands down .. the giants of maths come from europe and no grudging it... but be fair to some extent....

The author lacks basic knowledge about mathematical history:
The author reviews mathematical history but mentions no India nor China. He presented a biased view of mathematical history. The books is misleading in that regard.