History of Mathematics: From a Mathematician's Vantage Point

This book offers a very interesting panorama of the development of mathematics from the ancient Babylonians and Greeks to the present. It is written in a lucid style with very readable mathematical content. Understanding the material requires some broad mathematical education, but not a lot of specialized knowledge. One of the strongest sections deals with the accomplishments of the Greeks. The author clearly explains the problems tackled in ancient Greece, places them in context, outlines the accomplishments (some with concise proofs), and compares these with our present understanding of the subject. He also places the mathematical achievements of ancient Greece in the context of early Ionian Philosophy, Platonism, Aristotelism, or in the mindset of the Alexandrians. The chapters on the seventeenth and eighteenth centuries are presented clearly with emphasis on the great figures of these two centuries. Mathematics of the nineteenth and twentieth centuries are presented more thematically than chronologically. Analysis, in particular functional analysis, receives a very good overview. An appendix contains a transcript of the talk by Laurent Schwartz on the historical roots and basic notions of the theory of distributions. Other chapters discuss topics such as modern algebra, set theory, logic, group representations, calculus of variations, celestial mechanics, fractals, the fast Fourier transform, and wavelets. The book is supplemented with a chronological table and two detailed indices. It is suitable for graduate students, research mathematicians, and those interested in the history of mathematics and related areas.

Table of Contents:
Mathematics of the Egyptians; Mathematics of the Babylonians; Prime numbers; Mathematics of the Greeks-The beginnings; Pythagoras-The Pythagorean school; Heraclitus, Parmenides, Zeno, Empedocles, and Democritus; The mathematical school of Athens; Plato and Aristotle; Groups; The impossibility of solving the three classical problems of antiquity; Euclid; Hilbert's foundation of Euclidean geometry; Basic properties of axiomatic systems; Spaces and geometries; Non-Euclidean geometries; The geometry of experience; Aristarchus, Archimedes, Apollonius, and Eratosthenes; The period from 200 B.C. to 500 A.D. in Alexandria; A brief review of the history of Greek mathematics; Mathematics of China and India; Mathematics of the Arabs; Europe during the middle ages; Renaissance (1400-1600); The seventeenth century; The eighteenth century; Revival of synthetic geometry; The system of real numbers; The system of complex numbers and the quaternions; The fundamental theorem of algebra; Set theory; Logic; Functional analysis; Topology; Functions of real variables; Abstract algebra; Categories and functors; Recent discoveries and achievements; Language; The fast Fourier transform; The theory of wavelets; A short curriculum vitae of the author; Bibliography; Chronological table; Subject index; Name index