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.
