For undergraduate math students and teachers, this introduction to the history of the subject and influential mathematicians encompasses the Greeks, Indian arithmetic, integral and differential calculus, the analytic geometry of Rene Descartes, non-commutative algebra, the arithmetization of analysis
, and the beginnings of algebra as well as figures such as Newton, Hamilton, Kepler, Fibonacci, and Euclid.
After a brief but engaging account of Frege's life and career, Noonan's introductory chapter provides a helpful sketch of the origins and development of his leading ideas in their philosophical and mathematical context--Kant's thesis that mathematics, while a priori, must be synthetic and his associated insistence on the role of intuition; the emergence of non-Euclidean geometries; and the drive for rigor and the arithmetization of analysis
by Augustine Cauchy, Karl Weierstrass, and others--followed by a concise overview of Frege's main contributions which serves as a useful background to their more detailed discussion in the chapters that follow.
The nineteenth-century arithmetization of analysis
, culminating in the work of Dedekind and Cantor, produced the familiar arithmetical continuum, the field of real numbers, which is of course Archimedean.