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Part I Introduction.md

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Part I Introduction

1. What is Mathematics About?

“What is mathematics?” The approach of this book is not to try.
Nevertheless, to make sense of all this information it is useful to be able to classify it somehow.

1.1 Algebra, Geometry, and Analysis

Although any classification of the subject matter of mathematics must immediately be hedged around with qualifications, there is a crude division that undoubtedly works well as a first approximation, namely the division of mathematics into algebra, geometry, and analysis.

1.2 The Main Branches of Mathematcis

  • Algebra: Algebraic structures
  • Number Theory: properties of the set of positive integers
  • Geometry: manifold
  • Algebraic Geomety: singularities
  • Analysis: partial differential equations, dynamics
  • Logic: Set Theory, Category Theory, Model Theory
  • Combinatorics:
  • Theoretical Computer Science:
  • Probability:
  • Mathematical Physics:

2. The Language and Grammar of Mathematics

2.1 Introduction

The main reason for using mathematical grammar is that the statements of mathematics are supposed to be completely precise, and it is not possible to achieve complete precision unless the language one uses is free of many of the vaguenesses and ambiguities of ordinary speech.

2.2 Four Basic Concepts

  • Sets: $ i\in S, \empty $
  • Functions: mathematical transformation
    • domain, range, image
    • injection, surjection, bijection
  • Relations
    • equivalence relations: reflexive, symmetric, transitive
  • Binary Operations
    • commutative
    • associative
    • identity
    • inverse

2.3 Some Elementary Logic

  • Logical Connectives: $ \land, \lor, \implies $
  • Quantifiers: $ \forall, \exists$
  • Negation: $\lnot $
  • Free and Bound Variables

2.4 Levels of Formality

3. Some Fundamental Mathematical Definitions

3.1 The Main Number Systems

  • The Natural Numbers $\N$
  • The Integers $\Z$
  • The Rational Numbers $Q$
  • The Real Numbers $\R$
  • THe Complex Numbers $C$

3.2 Four Important Algebraic Structures

  • Groups
  • Fields
  • Vector Spaces
  • Rings

3.3 Creating New Structures Out of Old Ones

  • Substructures
  • Products
  • Quotients

3.4 Functions between Algebraic Structures

  • Homomorphisms, Isomorphisms, and Automorphisms
  • Linear Maps and Matrices
  • Eigenvalues and Eigenvectors

3.5 Basic Concepts of Mathematical Analysis

  • Limits
  • Continuity
  • Differentiation
  • Partial Differential Equations
  • Integration
  • Holomorphic Functions

3.6 What is Geometry?

  • Geometry and Symmetry Groups
  • Euclidean Geometry
  • Affine Geometry
  • Topology
  • Spherical Geometry
  • Hyperbolic Geometry
  • Projective Geometry
  • Lorentz Geometry
  • Manifolds and Differential Geometry
  • Riemannian Metrics

4. The General Goals of Mathematical Research