Saunders Mac Lane (1909–2005)

In this book, Saunders Mac Lane attempts to answer some questions about mathematics; it's origins, how it is organized, whether or not mathematics has formalisms that come from facts, how mathematics develops, how does someone evaluate the depth and importance of mathematical research, what should be the standards of rigor and finally, why is mathematics effective?

That final question goes a lot deeper than some would think. It ties itself right into epistemology, and how we know things that we know. At the same time, it covers a great deal of mathematical ideas. Starting at counting it goes into Peano axioms and other things. It structures itself differently from other books that survey mathematics in that it seems to organize the information by subject rather than chronologically.

At the end of the book, Mac Lane answers the questions he put forth in the introduction of the book. All in all it was a satisfying and interesting read.… (plus d'informations)

Indeholder "Preface to the Second Edition", "List of Symbols", "Chapter 1. Sets, Functions, and Integers", " 1. Sets", " 2. Functions", " 3. Relations and Binary Operations", " 4. The Natural Numbers", " 5. Addition and Multiplication", " 6. Inequalities", " 7. The Integers", " 8. The Integers Modulo n", " 9. Equivalence Relations and Quotient Sets", " 10. Morphisms", " 11. Semigroups and Monoids", "Chapter 2. Groups", " 1. Groups and Symmetry", " 2. Rules of Calculation", " 3. Cyclic Groups", " 4. Subgroups", " 5. Defining Relations", " 6. Symmetric and Alternating Groups", " 7. Transformation Groups", " 8. Cosets", " 9. Kernel and Image", " 10. Quotient Groups", "Chapter 3. Rings", " 1. Axioms for Rings", " 2. Constructions for Rings", " 3. Quotient Rings", " 4. Integral Domains and Fields", " 5. The Field of Quotients", " 6. Polynomials", " 7. Polynomials as Functions", " 8. The Division Algorithm", " 9. Principal Ideal Domains", " 10. Unique Factorization", " 11. Prime Fields", " 12. The Euclidean Algorithm", " 13. Commutative Quotient Rings", "Chapter 4. Universal Constructions", " 1. Examples of Universals", " 2. Functors", " 3. Universal Elements", " 4. Polynomials in Several Variables", " 5. Categories", " 6. Posets and Lattices", " 7. Contravariance and Duality", " 8. The Category of Sets", " 9. The Category of Finite Sets", "Chapter 5. Modules", " 1. Sample Modules", " 2. Linear Transformations", " 3. Submodules", " 4. Quotient Modules", " 5. Free Modules", " 6. Biproducts", " 7. Dual Modules", "Chapter 6. Vector Spaces", " 1. Bases and Coordinates", " 2. Dimension", " 3. Constructions for Bases", " 4. Dually Paired Vector Spaces", " 5. Elementary Operations", " 6. Systems of Linear Equations", "Chapter 7. Matrices", " 1. Matrices and Free Modules", " 2. Matrices and Biproducts", " 3. The Matrix of a Map", " 4. The Matrix of a Composite", " 5. Ranks of Matrices", " 6. Invertible Matrices", " 7. Change of Bases", " 8. Eigenvectors and Eigenvalues", "Chapter 8. Special Fields", " 1. Ordered Domains", " 2. The Ordered Field Q", " 3. Polynomial Equations", " 4. Convergence in Ordered Fields", " 5. The Real Field R", " 6. Polynomials over R", " 7. The Complex Plane", " 8. The Quaternions", " 9. Extended Formal Power Series", " 10. Valuations and p-adic Numbers", "Chapter 9. Determinants and Tensor Products", " 1. Multilinear and Alternating Functions", " 2. Determinants of Matrices", " 3. Cofactors and Cramer's Rule", " 4. Determinants of Maps", " 5. The Characteristic Polynomial", " 6. The Minimal Polynomial", " 7. Universal Bilinear Functions", " 8. Tensor Products", " 9. Exact Sequences", " 10. Identities on Tensor Products", " 11. Change of Rings", " 12. Algebras", "Chapter 10. Bilinear and Quadratic Forms", " 1. Bilinear Forms", " 2. Symmetric Matrices", " 3. Quadratic Forms", " 4. Real Quadratic Forms", " 5. Inner Products", " 6. Orthonormal Bases", " 7. Orthogonal Matrices", " 8. The Principal Axis Theorem", " 9. Unitary Spaces", " 10. Normal Matrices", "Chapter 11. Similar Matrices and Finite Abelian Groups", " 1. Noetherian Modules", " 2. Cyclic Modules", " 3. Torsion Modules", " 4. The Rational Canonical Form for Matrices", " 5. Primary Modules", " 6. Free Modules", " 7. Equivalence of Matrices", " 8. The Calculation of Invariant Factors", "Chapter 12. Structure of Groups", " 1. Isomorphism Theorems", " 2. Group Extensions", " 3. Characteristic Subgroups", " 4. Conjugate Classes", " 5. The Sylow Theorems", " 6. Nilpotent Groups", " 7. Solvable Groups", " 8. The Jordan-Hölder Theorem", " 9. Simplicity of A_n", "Chapter 13. Galois Theory", " 1. Quadric and Cubic Equations", " 2. Algebraic and Transcendental Elements", " 3. Degrees", " 4. Ruler and Compass", " 5. Splitting Fields", " 6. Galois Group of Polynomials", " 7. Separable Polynomials", " 8. Finite Fields", " 9. Normal Extensions", " 10. The Fundamental Theorem", " 11. The Solution of Equations by Radicals", "Chapter 14. Lattices", " 1. Posets: Duality Principle", " 2. Lattice Identities", " 3. Sublattices and Products of Lattices", " 4. Modular Lattices", " 5. Jordan-Hölder-Dedekind Theorem", " 6. Distributive Lattices", " 7. Rings of Sets", " 8. Boolean Algebras", " 9. Free Boolean Algebras", "Chapter 15. Categories and Adjoint Functors", " 1. Categories", " 2. Functors", " 3. Contravariant Functors ", " 4. Natural Transformations", " 5. Representable Functors and Universal Elements", " 6. Adjoint Functors", "Chapter 16. Multilinear Algebra", " 1. Iterated Tensor Products", " 2. Spaces of Tensors", " 3. Graded Modules", " 4. Graded Algebras", " 5. The Graded Tensor Algebra", " 6. The Exterior Algebra of a Module", " 7. Determinants by Exterior Algebras", " 8. Subspaces by Exterior Algebra", " 9. Duality in Exterior Algebra", " 10. Alternating Forms and Skew-Symmetric Tensors", "Bibliography", "Index".

En rimelig kompakt udgave af i alt fald første og andet år af et matematik-bifag. Hård kost som selvstudie.… (plus d'informations)