Riassunto
Trying to make mathematics understandable to the general public is a very difficult task. The writer has to take into account that his reader has very little patience with unfamiliar concepts and intricate logic and this means that large parts of mathematics are out of bounds. When planning this book, I set myself an easier goal. I wrote it for those who already know some mathematics, in particular those who study the subject the first year after high school. Its purpose is to provide a historical, scientific, and cultural frame for the parts of mathematics that meet the beginning student. Nine chapters ranging from number theory to applications are devoted to this program. Each one starts with a historical introduction, continues with a tight but complete account of some basic facts and proceeds to look at the present state of affairs including, if possible, some recent piece of research. Most of them end with one or two passages from historical mathematical papers, translated into English and edited so as to be understandable. Sometimes the reader is referred back to earlier parts of the text, but the various chapters are to a large extent independent of each other. A reader who gets stuck in the middle of a chapter can still read large parts of the others. It should be said, however, that the book is not meant to be read straight through.
Contenuti
1 Models and Reality.- 1.1 Models.- 1.2 Models and reality.- 1.3 Mathematical models.- 2 Number Theory.- 2.1 The primes.- 2.2 The theorems of Fermat and Wilson.- 2.3 The Gaussian integers.- 2.4 Some problems and results.- 2.5 Documents.- 3 Algebra.- 3.1 The theory of equations.- 3.2 Rings, fields, modules, and ideals.- 3.3 Groups.- 3.4 Documents.- 4 Geometry and Linear Algebra.- 4.1 Euclidean geometry.- 4.2 Analytical geometry.- 4.3 Systems of linear equations and matrices.- 4.4 Linear spaces.- 4.5 Linear spaces with a norm.- 4.6 Boundedness, continuity, and compactness.- 4.7 Hilbert spaces.- 4.8 Adjoints and the spectral theorem.- 4.9 Documents.- 5 Limits, Continuity, and Topology.- 5.1 Irrational numbers, Dedekind’s cuts, and Cantor’s fundamental sequences.- 5.2 Limits of functions, continuity, open and closed sets.- 5.3 Topology.- 5.4 Documents.- 6 The Heroic Century.- 7 Differentiation.- 7.1 Derivatives and planetary motion.- 7.2 Strict analysis.- 7.3 Differential equations.- 7.4 Differential calculus of functions of several variables.- 7.5 Partial differential equations.- 7.6 Differential forms.- 7.7 Differential calculus on a manifold.- 7.8 Document.- 8 Integration.- 8.1 Areas, volumes, and the Riemann integral.- 8.2 Some theorems of analysis.- 8.3 Integration in Rn and measures.- 8.4 Integration on manifolds.- 8.5 Documents.- 9 Series.- 9.1 Convergence and divergence.- 9.2 Power series and analytic functions.- 9.3 Approximation.- 9.4 Documents.- 10 Probability.- 10.1 Probability spaces.- 10.2 Stochastic variables.- 10.3 Expectation and variance.- 10.4 Sums of stochastic variables, the law of large numbers, and the central limit theorem.- 10.5 Probability and statistics, sampling.- 10.6 Probability in physics.- 10.7 Document.- 11 Applications.- 11.1 Numerical computation.- 11.2 Construction of models.- 12 The Sociology, Psychology, and Teaching of Mathematics.- 12.1 Three biographies.- 12.2 The psychology of mathematics.- 12.3 The teaching of mathematics.
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