Mathematics: Form and Function
This is out-of-print book that is hard to find now. I was very keen and fortunate to buy it from a 3rd-party seller a few years ago and immediately read it then. It gives great overview of modern mathematics from the perspective of the founder of category theory. The similar overview of modern mathematics can be found in The Road to Reality, 2 volumes of Comprehensive Mathematics for Computer Scientists and the recent 1000 page “The Princeton Companion to Mathematics”. One note that I didn’t like is the following passage from page 439:
“Not all outside influences are really fruitful. For example, one engineer came up with the notion of a fuzzy set…”
Mathematics: Form and Function
Here is the table of contents showing the breadth of material:
CHAPTER I
Origins of Formal Structure
The Natural Numbers
Infinite Sets
Permutations
Time and Order
Space and Motion
Symmetry
Transformation Groups
Boolean Algebra
Calculus, Continuity and Topology
Human Activity and Ideas
Mathematical Activities
Axiomatic Structure
Chapter II From Whole Numbers to Rational Numbers
Properties of Natural Numbers
The Peano Postulates
Natural Numbers Described by Recursion
Number Theory
Integers
Rational Numbers
Congruence
Cardinal Numbers
Ordinal Numbers
What Are Numbers?
Chapter III Geometry
Spatial Activities
Proofs without Figures
The Parallel Axiom
Hyperbolic Geometry
Elliptic Geometry
Geometric Magnitude
Geometry by Motion
Orientation
Groups in Geometry
Geometry by Groups
Solid Geometry
Is geometry a Science?
Chapter IV Real Numbers
Measures of Magnitude
Magnitude as a Geometric Measure
Manipulations of Magnitudes
Comparison of Magnitudes
Axioms for the Reals
Vector Geometry
Analytic Geometry
Trigonometry
Complex Numbers
Stereographic Projection and Infinity
Are Imaginary Numbers Real?
Abstract Algebra Revealed
The Quaternions - and Beyond
Summary
Chapter V Functions, Transformations, and Groups
Types of Functions
Maps
Whats is a Function?
Functions as Sets of Pairs
Transformation Groups
Groups
Galois Theory
Construction of Groups
Simple Groups
Summary: Ideas of Image and Composition
Chapter VI Concepts of Calculus
Origins
Integration
Derivatives
The Fundamental Theorem of the Integral Calculus
Kepler’s Laws and Newton’s Laws
Differential Equations
Foundations of Calculus
Approximations and Taylor’s Series
Partial Derivatives
Differential Forms
Calculus Becomes Analysis
Interconnections of the Concepts
Chapter VII Linear Algebra
Sources of Linearity
Transformations versus Matrices
Eigenvalues
Dual Spaces
Inner Product Spaces
Orthogonal Matrices
Adjoints
The Principal Axis Theorem
Bilinearity and Tenso Products
Collapse by Quotients
Exterior Algebra and Differential Forms
Similarity and Sums
Summary
Chapter VIII Forms of Space
Curvature
Gaussian Curvature for Surfaces
Arc Length and Intrinsic geometry
Many-Valued Functions and Riemann Surfaces
Examples of Manifolds
Intrinsic Surfaces and Topological Spaces
Manifolds
Smooth Manifolds
Paths and Quantities
Riemann Metrics
Sheaves
What Is Geometry?
Chapter IX Mechanics
Kepler’s Laws
Momentum, Work, and Energy
Lagrange’s Equations
Velocities and Tangent Bundles
Mechanics in Mathematics
Hamilton’s Principle
Hamilton’s Equations
Tricks versus Ideas
The Principal Function
The Hamilton-Jacobi Equation
The Spinning Top
The Form of Mechanics
Quantum Mechanics
Chapter X Complex Analysis and Topology
Functions of a Complex Variable
Pathological Functions
Complex Derivatives
Complex Integration
Paths in the Plane
The Cauchy Theorem
Uniform Convergence
Power Series
The Cauchy Integral Formula
Singularities
Riemann Surfaces
Germs and Sheaves
Analysis, Geometry, and Topology
Chapter XI Sets, Logic, and Categories
The Hierarchy of Sets
Axiomatic Set Theory
The Propositional Calculus
First Order Language
The Predicate Calculus
Precision and Understanding
Godel Incompleteness Theorems
Independence Results
Categories and Functions
Natural Transformations
Universals
Axioms on Functions
Intuitionistic Logic
Independence by Means of Sheaves
Foundation or Organization?
Chapter XII The Mathematical Network
The Formal
Ideas
The Network
Subjects, Specialties, and Subdivisions
Problems
Understanding Mathematics
Generalization and Abstraction
Novelty
Is Mathematics True?
Platonism
Preferred Directions for Research
Summary
- Dmitry Vostokov @ LiterateScientist.com -
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