One of the best descriptions of human history as well as what is yet to come is summed up in the phrase “Necessity is the mother of invention.” That saying also applies to mathematics: until quite recently, the creation of new mathematics was a reaction to the development of a new problem in human society.
Biggs opens the book with what is plausibly the first math problem that humans ever solved, that of fair division. The one thing that allowed early humans to survive against predators and hunt down powerful animals was the ability to engage in the cooperative hunt. Yet, when a wooly mammoth was taken down the next issue would be the distribution of the kill. Different parts of the animal would be used for different things. The hide would make clothing, blankets and perhaps even tents. Bones would be used for tools, blood vessels for thread and the values of the various cuts of meat would be different. Since the needs for the auxiliary parts would be different among the members of the tribe, some form of selective and fair allocation would have to be done.
The next point of mathematical discovery described by Biggs would likely have been the act of counting animals once they were domesticated. Locked up during the night as a protection against predators and theft, the animals would be let out during the day to graze. The guardian of the herd would have to have some way to tally. The act of counting, almost certainly a simple one-to-one correspondence would have been done even by nomads.
Once grains were domesticated, humans formed larger and more permanent communities. The need for more sophisticated counting mechanisms followed, specifically regarding the concepts of weight, volume and area. This led to the emergence of a leadership and defense structure, requiring the levying of taxes. Fixed communities could not produce everything they needed or desired, which led to the development of trade, requiring a medium of exchange, also known as money.
This is an excellent popular history of mathematics. Through its pages one can see how society and mathematics advance in a symbiotic manner. Many aspects of modern societies simply could not have developed without the mathematics being created to support it. Biggs does not spare the formula to spoil the reader: when a formula is needed to make the explanation it is used. With a bit of instructor support and explanation, this book is a suitable text for a liberal arts course in mathematics. Throw in a bit of historical background and the students will learn a great deal as to why math is important.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, and teaching college classes. In his spare time, he reads about these things and helps his daughter in her lawn care business.