God Created the Integers

Description

Dedekind's Essays on the Theory of Numbers (¶61.1ff.) quoted in Jeff Kalb's Music and Measurement: An the Eidetic Principles of Harmony and Motion **(St. Cecilia's Feast Day, 2016)

Bestselling author and physicist Stephen Hawking explores the "masterpieces" of mathematics, 25 landmarks spanning 2,500 years and representing the work of 15 mathematicians.

MR2382242

This anthology includes selections from twenty-one mathematicians. With four exceptions, the English translations have previously been published; the author provides a brief biography and appreciation of the work of each mathematician. The collection ranges over a variety of topics; as well as number theory, there are works on geometry, combinatorics, algebra, analysis, foundations and applied mathematics. Some examples are substantial, the largest being Laplace's 100-page essay, Philosophical essay on probabilities.
Over a fifth of the volume is devoted to Greek mathematics. Euclid is represented by extracts from Books I, V, VII, IX and X of his Elements. Apart from the proof of Pythagoras' Theorem (I.47), the examples are essentially arithmetical and include the proof of the infinitude of primes, the description of even perfect numbers, theory of proportion, and commensurable and incommensurable magnitudes. The other classical authors are Archimedes (with four works, including the Sand reckoner and Method) and Diophantus (with selections from Books II, III and V of Arithmetica).
The rest of the volume consists of European works of the last three centuries, by Descartes (Geometry), Newton (on first and last ratios of quantities), Euler (sum of square reciprocals, seven bridges, integers as sum of four squares), Laplace (essay on probability), Fourier (trigonometric series in the study of propagation of heat in an infinite rectangular solid), Gauss (selections from Disquisitiones on residues of powers, particularly squares), Cauchy (on derivative and integral), Lobachevsky and Bolyai (on non-Euclidean geometry, with introductions by G. B. Halsted), Galois (solvability by radicals, groups and equations), Boole (Investigation of laws of thought), Riemann (definite integral, Fourier series representation, foundation of geometry, frequency of primes and the zeta function), Weierstrass (uniform continuity), Dedekind (Essays on the theory of numbers), Cantor (transfinite numbers), Lebesgue (on the integral), Gödel (On formally undecidable propositions) and Turing (on computable numbers).
The value of this particular source book is enhanced by the extensiveness of many of the entries, so that the reader can have access to mathematical developments in the words of their creators, rather than just through brief extracts and secondary sources. Reviewed by E. J. Barbeau