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Introduction to the Foundations of Mathematics

Raymond L. Wilder
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Michael Berg
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It really does bear repeating that Dover Publications has long been, and still is, a real blessing on the mathematical community. This is doubly so in this day and age when a few huge publication houses exert a near-stranglehold on mathematics publishing, making for higher and higher book prices, while, at the same time, the sheer number of offerings in any given field is more and more daunting. Once upon a time, the phrase “undergraduate level real analysis,” for example, triggered a pretty terse response: blue Rudin, Protter, or perhaps Caspar Goffman’s (wonderful) book — there might have been others, but certainly not many. In sharp contrast, bringing this theme to that most terrible of cyber-demons, Google, produces a yield of 2060 titles! To be sure, only a proper subset actually qualifies — after all, it is only a search engine — but still… Indeed, times have changed.

With this bit of historical kvetching in place, let’s say that the books offered by Dover have a great deal to offer under the heading of a return to sanity. They are often reprints of classics, marketed at a very low price; these contributions to the mathematical literature (and pedagogy, in the old-fashioned meaning of the word) go back to those halcyon times when the game was played differently. As illustrated above, there were far fewer books per subject, vetted very carefully, and, to be sure, the ones surviving for any reasonable span of time did so in a Darwinian environment. The book under review is an example of this feature: it is still very fit and deserves to keep surviving.

The author, R. L. Wilder, was a student of R. L. Moore, and, as such, a “Texas topologist”; therefore, in a sense (in a very good sense!), the book resonates with the pedagogical approach crystallized in the Moore Method: it is ultimately all about discovery and owning technique. In this spirit, Wilder’s style of presentation is, first of all, far from terse; it is rather different from, say, the telegraph style of mathematical prose championed by Edmund Landau; Wilder’s narrative is much more discursive than that, even as Introduction to the Foundations of Mathematics is filled to the brim with definitions, theorems and proofs. It is by no means the case that the book is chatty or rambling.

Wilder notes in the Preface that “[t]he reason for … the course [corresponding to the book] was simply the conviction that it was not good to have teachers, actuaries, statisticians, and others who had specialized in undergraduate mathematics, and who were to base their life’s work on mathematics, leave the university without some knowledge of modern mathematics and its foundations.” He then goes on to stress that he seeks to address the problem that his chosen charges’ training would typically include only “pre-twentieth century and, and, in large part, … pre-Cantorian ideas and methods.” He adds, too, that “a course in Foundations at about the senior level might serve to unify and extend the material covered in the traditional … curriculum.”

Well, plus ça change, plus c’est la meme chose. But it must be added that in one obvious and undeniable respect, mathematico-pedagogical circumstances are actually very different now than in 1952. Which of us would not rejoice at having before us a class of senior-level university students with the mindset and depth (as opposed to breadth) of that generation? This contrast, too, is discernible from the pages of this Introduction to the Foundations of Mathematics. The presentation is, so to speak, for adults, not children. The reader is supposed to be, if not as mathematically mature as his classmate who is headed for a PhD in mathematics, then certainly as intellectually mature: able to read, work, follow mathematical discussion, and even take part in the process of discovery of the workings of deeper mathematical themes by joining Wilder in his explications: the pencil and note sheet are non-negotiable.

Regarding the book’s subject matter per se, suffice it to say that it’s a smorgasbord of great stuff: in his first part of two, Wilder starts with a discussion of the axiomatic method, hits set theory with zest (and talks about “the Russell contradiction”), develops Cantorian set theory, talks about well-ordering, and then heads the reals à la Peano, followed by a discussion of “groups and their significance for the foundations.” Then, in his second part, Wilder turns to a more philosophical theme, for lack of a better word: he discusses, e.g., the views and work on foundational matters on the parts of Zermelo, Poincaré, Frege, Russell (with Principia Mathematica highlighted). Brouwer’s intuitionism, Russell’s theory of types, and Hilbert’s proof theory are discussed, with the latter a prelude to nothing less than Gödel’s incompleteness theorem (and how could it be otherwise?). Wilder then goes on to finish the book with a chapter on “the cultural setting of mathematics.”

What then is the role of such a book in this day and age? The answer is clear: although the audience Wilder wrote the book for sixty years ago no longer exists, there is another audience in evidence today, when even future professional mathematicians (i.e. graduate school bound majors) are deficient in any real understanding of the bigger picture as far as mathematics goes: a transition course to bridge the gap from calculus to ostensibly upper division mathematics is pretty much all that’s on the menu these days, at least presque partout. Any real discussion on foundations is relegated to courses in the history of mathematics (light on proof and necessarily broad and not deep), Peano and Zermelo are strangers to most students, and Gödel is a curio of sorts. For any one who finds this state of affairs lamentable or even unacceptable, Introduction to the Foundations of Mathematics is a fabulous antidote.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Part One: Fundamental Concepts and Methods of Mathematics
I. The Axiomatic Method
II. Analysis of the Axiomatic Method
III Theory of Sets
III Theory of Sets
IV. Infinite Sets
V. Well-Ordered Sets; Ordinal Numbers
VI. The Linear Continuum and the Real Number System
VII. Groups and Their Significance for the Foundations
Part Two: Development of Various Viewpoints on Foundations
VIII. The Early Developments
IX. The Frege-Russell Thesis: Mathematics an Extension of Logic
X. Intuitionism
XI. Formal Systems; Mathematical Logic
XII. The Cultural Setting of Mathematics
Index of Symbols
Index of Topics and Technical Terms
Index of Names