In January 2003, Sweden's minister of education created a Mathematics Delegation to review mathematics education in Sweden and decide if changes were in order. At the time Dreams of Calculus was published, the Mathematics Delegation had not yet submitted its report, but public statements given by the delegation's chairman expressed the view that
Hoffman, Johnson, and Logg are experts in computational mathematics who believe there is a crisis and there is a change in paradigm going on because of the computer. In Dreams of Calculus, they argue their case and ask readers to take a stand on this issue. The authors are not newcomers in this discussion. They are involved in a reform project at Chalmers University of Technology in Göteborg, Sweden that has produced the three-volume text Applied Mathematics: Body and Soul.
In the first section of the book, entitled Perspectives, the authors discuss a variety of topics including the differences between pure mathematics (mathematics without a computer) and computational mathematics (mathematics with a computer), the history of mathematics education, and several philosophical issues regarding the nature of mathematics and science in general. Throughout this section, they make the case for the study of computational mathematics and its use in solving deep questions in science. Their Body and Soul text approaches mathematics from this viewpoint.
The authors point out that in much of pure mathematics, there is a set of known tricks that students must master so they can use them when the need arises. Certainly, computational mathematics has its share of tricks, but there is more opportunity for experimentation and improvisation. Also, since pure analytical solutions to many (most?) important questions in science are extremely difficult to find, it makes sense to look for computational methods that can produce approximate solutions.
In the second section, Essence, the authors give a very brief description of calculus, discuss computational approaches to some famous problems from science, and then return to more philosophical matters. In this section, the material is more technical and is intended to demonstrate how mathematics can be approached in a way that combines analytical and computational techniques. Some of the chapters in this section are reproductions of material from the Body and Soul text.
I recommend this book for anyone interested in computers and the teaching of calculus. Although I have not seen the Body and Soul text, I like Dreams of Calculus and the approach it suggests. I believe that significant problems in physical science are most likely to be solved by people who are well versed in both analytical and computational mathematics. Furthermore, I like the Body and Soul project because I feel that many teachers of calculus overemphasize the use of the computer in demonstrating concepts and underemphasize the use of the computer in solving problems. The recent trend of offshoring computer programming jobs to Asia has affected our students' desire to study programming and computing in general. I hope that projects such as Body and Soul will help remind students and teachers that professionals who are employed as scientists (not programmers) may need to write programs to solve problems.
At times, however, the authors seem a bit out of touch with the current state of affairs — at least in the United States. On page 12 the authors say that "not even an arts student at an American college may get away without a calculus course," and on page 16 they claim that "both calculus and linear algebra are still presented as if the computer does not exist."
There are some grammatical/typographical errors and awkward sentences that detract from the writing. For example, on pages 12 and 30 there are sentences that begin with lower case letters, and on pages 103 and 132 there are sentences where verb and subject do not agree. The book is easy to read, but I think Springer could have done a better job of editing the manuscript.
The Mathematics Delegation's final report (including a summary in English) can be found at http://www.regeringen.se/content/1/c6/03/03/48/6a32d1c0.pdf.