11 of 11 people found the following review helpful
on 12 September 2008
Alfred North Whitehead's book is not intended to be an introduction to any particular field of mathematics. Although mathematics is a very large discipline, there are modes of thought that run through all of its branches and that are essential to it. I found that these modes of thought were not discussed explicitly enough when I was taught mathematics in school, despite the fact that they are essential to all mathematical reasoning and demonstration/proof. In 'An Introduction to Mathematics' Whitehead describes these modes of thought, which include generalisation, abstraction, and precision. He discusses the relationship that mathematics has with the sciences, and he makes some interesting philosophical observations about the subject which are reminscient of Gottlob Frege (see 'Foundations of Arithmetic'), and Bertrand Russell (see 'Introduction to Mathematical Philosophy'). He also introduces many important mathematical concepts from different branches of the discipline, thereby giving a conceptual overview of the subject. Rather than giving detailed technical explanations of these concepts, Whitehead gives general explanations of their nature and importance; and this is fitting for an introduction.
Whitehead was well qualified to write this book, as evidenced by the fact that he co-authored the very important 'Principia Mathematica' with Bertrand Russell; a three volume book which attempted to set Arithmetic on the firmest logical foundations. Whitehead was a gifted writer and in 'An Introduction to Mathematics' he conveys his thoughts very clearly and intelligibly. The book is written in a mature style. However, it is still suitable for children to read (say, 12 years or older?). A knowledge of arithmetic, elementary algebra, and elementary geometry (i.e. the stuff you are taught in school) is all that is needed to be able to fully engage with this book.
This book is also ideal for anyone who has studied elementary mathematics but wants to understand it more deeply. For, in order to deeply understand mathematics, one has to understand the logic which underlies its demonstrations, and this book introduces the modes of thought which are needed in all mathematical demonstrations. I study mathematics as a hobby, and I found that this book helped my understanding when I came to study axiom systems such as Euclid's 'Elements of Geometry', and Charles S. Peirce's 'New Elements of Mathematics' (both also brilliant books).
I would recommend this book to any beginner of the subject, especially beginners with a knowledge of arithmetic, elementary algebra, and elementary geometry who are seeking a deeper understanding of the subject.