history and philosophy

Katz's book is one of the best general works on the history of Mathematics around!

Its over-riding feature is that it is a TEXTBOOK - one that can be used for systematic study of the subject. Though tilted in favour of the mathematically inclined reader, the author has managed to connect the Maths to the History. The book has numerous topical exercises, sidebars and focus essays, which makes the subject easily accessible to the student. Yet, the structure and presentation are such that they also allow the book to be used simply as a reference or one that can be read purely for interest. Each chapter is followed by Exercises to assist the student to assess their learning and copious references that can be followed up for more details.

As with most good books of this genre, Mathematical developments from the last four centuries or so are most comprehensively presented. All the material is here: the "tussle" between Algebra and Geometry, the formal beginnings of the Calculus, the growth of Analysis, the development of new Mathematical techniques to tackle problems in Physics, and Probability mathematics.

The book places these developments within the socio-political context. Each chapter and main section starts with a preamble setting out the environment, the stimuli for the mathematical development to be discussed, etc. So, important events like the Renaissance, the French Revolution, etc. are discussed. In this regard, the use of Biography boxes for the main characters in the story of Mathematics helps to render the book more accessible to readers who may not be Mathematical. For instance, the chapter on Differential Equations would be inaccessible to the non-Mathematical reader were it not for such boxes retelling the lives and times of people like Bernoulli, Euler, Lagrange, and Laplace.

The early chapters deal with Babylonian and Greek developments, the latter with well presented biographies of Aristotle, Plato, and Euclid, among others. The chapters on the mathematics of the Arabs is well balanced, whilst that on India and China is possibly the best I have seen in a "mainstream" work of this type.

Where other authors like Morris Kline have almost totally ignored the contributions of these cultures to the subject, Katz has done a fine job. To note a couple of examples:

(1) India as the rightful source of the decimal place value system;
(2) Bhramagupta's research into what it usually known as Pell's equation, some 1000 years before Pell, and,
(3) Madhava's derivation of the power series for the arcsine and his appreciation of convergence over 200 years before Gregory.

Overall, a very good book that, like Edna Kramer's work, adds to the accessibility of a stimulating subject that is at the heart of the intellectual development of mankind.

This book is excellent. It provides a general view of mathematics evolution that both discusses the mathematical formulation of the problems and the historic details. It is organized as a textbook, but it is interesting to read or to use as reference. For me, one of the most gratifying features was the cross cultural details that went beyond the so common vague and politically correct lip service and actually referred the content of often-forgotten important contributors.
It is interesting, for instance, to see correct and detailed references to Pedro Nunes and other mathematicians of the Discoveries time, and the relation between geometric developments and navigation problems.