Mathematical Vistas by Peter Hilton, Derek Holton and Jean Pedersen, Springer-Verlag, 拢49, ISBN 0387950648
The Essential John Nash edited by Harold Kuhn and Sylvia Nasar, Princeton University Press, 拢19.95, ISBN 0691095272
The Algorithmic Beauty of Seaweeds, Sponges and Corals by Jaap Kaandorp and Janet K眉bler, Springer-Verlag, 卢49.95, ISBN 3540677003
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ONCE proven, a mathematical idea remains unarguably true. If Leonhard Euler, one of history鈥檚 most prolific and gifted mathematicians, travelled forward in time from the 18th century to study mathematics today he would find much that鈥檚 familiar 鈥� and some elements named in his honour. This contrasts sharply with other sciences, where a new observation often results in a theory being adapted or abandoned.
So the sheer number of 鈥渢rue鈥� ideas in mathematics causes problems: which do you teach undergraduates and how do you cover enough material to make more recent developments accessible?
One solution is the introduction of a degree called the MMath/MSci, which requires a further year of study on top of the normal three-year undergraduate degree. An alternative, largely untried idea is to move away from the content-driven approach delivered through formal lectures (popularly described as passing information from lecturer to student without it passing through the minds of either) and allow students to direct their learning to a greater extent. This group of books could provide some inspiration.
Mathematical Vistas covers a single mathematical problem in each of its nine sections, including gems such as the Four-Colour Problem and Fermat鈥檚 Last Theorem. The authors don鈥檛 attempt to cover each comprehensively, but describe the ideas involved while encouraging the reader to investigate it through regular 鈥渂reaks鈥�. References provide pointers to more detailed information. Such an approach seems well suited to undergraduate teaching, giving students an overview of topics, then enabling them to investigate further the ones that interest them most. Even if they鈥檙e not using this approach, mathematics students will enjoy this book.
Mathematics has always had a natural affinity with physics and astronomy. Other sciences have not made such happy bedfellows. One reason for this is that the models used often result in problems that cannot be solved exactly. Faster computers and good mathematical software packages have made these problems accessible, so courses such as 鈥渕athematical biology鈥� are making their way into mathematics programmes. Jaap Kaandorp and Janet K眉bler鈥檚 book The Algorithmic Beauty of Seaweeds, Sponges and Corals covers the modelling of the growth and form of some organisms. Lots of detail is provided for the biology, but I found it more sketchy on the maths. That said, there is enough information to encourage investigations, and the many wonderful illustrations help to spur on the reader.
Remarkably, the social sciences were once a maths-free zone. This changed significantly in the second half of the 20th century with the advent of game theory, which uses ideas based on recreational games to analyse the way individuals, corporations or species interact. There is no doubt that John Nash鈥檚 work in this area has been the most influential. The Essential John Nash, edited by Harold Kuhn and Sylvia Nasar, contains Nash鈥檚 most important work, so makes a welcome addition to the game theory library. But if you鈥檝e just seen Russell Crowe in the film A Beautiful Mind and you want to know more, then Nasar鈥檚 biography (5 September 1998, p 48) of the same name is the place to start.
Neither The Essential John Nash nor The Algorithmic Beauty of Seaweeds, Sponges and Corals is suitable for teaching a course. Both could, however, form the basis for an undergraduate project, which would enable students to research a problem in greater depth, give them more idea of what mathematicians do, and probably be more fun than going to lectures.
Onlookers are still baffled by what mathematicians actually do. I fear the movie A Beautiful Mind will only serve to reinforce the idea that we are slightly unhinged individuals with few social skills, blathering incomprehensibly while scribbling strange hieroglyphs. Of course, this stereotype is largely untrue and most of us are fairly normal, though the students who have had to sit through my lectures might disagree.