The Heritage Of Thales

The presentation emphasizes the structural elements over the computational - for example by connecting matrices to linear transformations from the outset - and prepares the student for further study of abstract mathematics.Uniquely among algebra texts at this level, it introduces group theory early in the discussion, as an example of the rigorous development of informal axiomatic systems. Based on lectures given at Claremont McKenna College, this text constitutes a substantial, abstract introduction to linear algebra.

Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus.

In Chapter 1, the basic principles of set theory are developed in a 'naive' manner.

. The appendix provides a brief survey of the group theory needed. that famous pedagogical method whereby one begins with the general and proceeds to the particular only after the student is too confused to understand even that anymore.

The ideas of probability are all around us.

On the other hand, it is designed to help philosophy, history, and education students come to a deeper understanding of the mathematical side of culture by means of writing short essays. This is a concise introductory textbook for a one-semester (40-class) course in the history and philosophy of mathematics. The way I myself teach the material, stu­ dents are given a choice between mathematical assignments, and more his­ torical or philosophical assignments. (Some sample assignments and tests are found in an appendix to this book.

Variational Calculus and Optimal Control Although the calculus of variations has ancient origins in questions of Ar­ istotle and Zenodoros, its mathematical principles first emerged in the post­ calculus investigations of Newton, the Bernoullis, Euler, and Lagrange. The original applications, including the Bernoulli problem of finding the brachistochrone, require opti­ mizing (maximizing or minimizing) the mass, force, time, or energy of some physical system under various constraints.

This is intended as a textbook on the history, philosophy and foundations of mathematics, primarily for students specializing in mathematics, but we also wish to welcome interested students from the sciences, humanities and education. We have tried to give ref­ cited facts and opinions. The Heritage of Thales Since neither of the authors is a professional historian, we have made liberal use of secondary sources. Philosophy Both authors having wide philosophical interests, this text contains perhaps more philosophical asides than other books on the history of mathematics.

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History We must emphasize that this is not a scholarly account of the history of mathematics, but rather an attempt to teach some good mathematics in a historical context. The Heritage of Thales Since neither of the authors is a professional historian, we have made liberal use of secondary sources.As to the mathematicians listed in our account, we admit to being colour and gender blind; we have not attempted a balanced distribution of the mathematicians listed to meet today's standards of political correctness. We have tried to give ref­ cited facts and opinions.