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A Mathematical Bridge: An Intuitive Journey in Higher Mathematics

Stephen Fletcher Hewson
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
David Roberts
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How do we help mathematically talented undergraduates become mathematicians? We math professors have a double viewpoint on this question. Each of us, somehow, has made the journey ourselves. Each of us is entrusted with helping some members of younger generations make their own journeys.

Stephen Fletcher Hewson says we could be doing a better job. The literature, according to Hewson's preface, is bipolar. "On the one hand there are the standard textbooks" which are "essential" but "very dense, difficult and compact affairs." They have a tendency to dissolve for students into "masses of symbols." On the other hand, there are "popular" math books. While these are "great sources of long-term inspiration", "little mathematical skill is required or imparted by such books."

Hewson aims to bridge the gap by the book under review, a "hybrid popular-textbook." Hewson aims for a "real mathematics book", but one which is "more conversational, intuitive and accessible" than standard texts. He writes that he wants to provide others with "the book I would have loved to own before beginning my career as a mathematician." Specifically, the goal of the book is to present "the core elements and highlights of a typical mathematics degree course" in an accessible way.

The preface just summarized prepares the reader well for the book as a whole. The book indeed surveys an entire undergraduate math major, with a slant towards Hewson's research interest, mathematical physics. There are six chapters: Numbers, Analysis, Algebra, Calculus and Differential Equations, Probability, and Theoretical Physics. Each chapter contains more than one might guess from the title. For example, the first chapter has substantial discussion of infinite cardinals, complex numbers, and modular arithmetic. The last has about ten to fifteen pages each on mechanics, electricity and magnetism, relativity, and quantum mechanics. Throughout the book, specific topics typically hold center stage for only a page or two, but nonetheless there is a serious attempt made at depth as well as breadth. Calculations are typically presented, rather than just talked about. There are even a number of proofs and an eighty page exercise set at the end.

Unfortunately, there are a fair number of places where the exposition suddenly stumbles. For example, "The space of everywhere differentiable functions...[has a basis consisting of]...{xn, n=0,1,...}" on page 162 is rather jarring. And what about "We may only expand a solution to a differential equation about a given point if all of the derivatives of that function are bounded by some fixed constant" on page 246? These stumbles are all local and could be removed in a second edition.

The disconcerting stumbles aside, the text generally moves along quite well. For example, the somewhat dry technique of solving partial differential equations by separation of variables is nicely made more interesting by an application to solar heating. Readers see that mathematics confirms their intuition by saying that the effect of the seasons dampens as one goes deeper underground. But readers also see that the same mathematics reveals an increasing time delay with complete inversion of the seasons obtained at a depth of only six meters. There are many places where the virtues of a "more conversational, intuitive and accessible" approach are apparent.

Undergraduates thinking about going to math graduate school will find Hewson's book interesting outside reading. It should help them synthesize what they are learning in their various courses. However they should keep a somewhat skeptical mind-set as they read, as indicated above.

Professionals thinking of writing their own texts may want to reflect on Hewson's criticism of the current textbook situation. Are the textbooks you use really what you would have loved to learn from as an undergraduate? I agree with Hewson that we would serve aspiring mathematicians better if our textbooks moved a step or two in the direction of the popular literature.

David Roberts is associate professor of mathematics at the University of Minnesota, Morris.

  • Numbers
  • Analysis
  • Algebra
  • Calculus and Differential Equations
  • Probability
  • Theoretical Physics
  • Appendices:
  • Exercises for the Reader
  • Further Reading
  • Basic Mathematical Background
  • Dictionary of Symbols