The history of mathematics is very linear, with new ideas being proposed, debated, verified and accepted before being used as the foundation for the next idea. That is the theme of this book, as the author takes you through the origins of mathematics up through calculus. Written in an entertaining, fresh style, the topics are clearly and completely explained.
There are many candidates for the major ideas of mathematics, and the first two chapters of the book deal with the two that I consider most essential. The first is the abstraction of the concept of a number. It is not hard to see how using the one-to-one correspondence of objects was used as an accounting tool. But somewhere in the distant past, humans took the revolutionary step of creating a symbol that can be used to represent a certain amount of any type of object. This is clearly a major point in the history of humans as it is arguably the first and foremost trait that separates humans from other earthly creatures. As Clawson points out, much of the first writing used in civilization was simply marking down how many items of various types are present.
The events described in the second chapter are those that get my vote for the most significant advancement in the history of mathematics. Aptly titled, "The Amazing Greeks," it is a description of how there emerged among the citizens of the Greek city states the most significant abstract idea humans have ever developed. The Greek intellectuals began working with objects such as circles that existed nowhere but in their minds. A fact that they were well aware of and accepted as important but not critical to understanding. For the first time, it was proper to deal with things that did not exist except in some pure abstract form. This is an event that is very unappreciated and it should get more time in mathematical education. Chapter two was clearly the best in the book, and is extended into the next chapter, which is a description of the idea of a proof.
It was at this point where humans began to understand the concept of absolute truth. For example, when it was proven that there are an infinite number of primes the problem was solved and the result established for all time. No political change, fickle act of a god or even the elimination of humans from the universe will ever be able to change that fact. The remainder of the book deals with additional mathematical topics that have evolved from these old, but powerful ideas. Conic sections, infinite series, the formalization of functions, transformations of space, and the development of calculus round out the book. While not quite as captivating as the first three chapters, they still provide some interesting reading.
Clawson is clearly enthusiastic about mathematics, being a true believer in the power it has to excite and enthrall while it helps us understand the physical world we find ourselves in. I have read many math books in the past decades, but none where the authors enthusiasm for the subject comes across any stronger than it does in this one. I have recommended that the library purchase the book and also that it be used in courses in mathematical history.
Charles Ashbacher (firstname.lastname@example.org) is co-editor of Journal of Recreational Mathematics, a part-time instructor at Kirkwood Community College and President/CEO of Charles Ashbacher Technologies.