Mathematics

Mathematics is a collection of formal theories, each with a symbolic system of objects and consistent rules for manipulation.

Abstract objects and their structures. Figure: Abstract objects $(X, \cdots)$ and their structures: (A) topological and algebraic structures; (B) set-theoretic structures. Mathematical structures in (A) and (B) are independent, but can be related via $L^p_\mu$.

Primitive. Abstract, representation. Set, structure. Object, mapping. Intrinsic (本征), invariant; Extrinsic (表征), reference.

Mathematical structures include relation, algebraic operation, vector space, norm, inner product, topology, smoothness, geometry, distance, set algebra, measure. Each structure has been developed with minimum dependence on others to maximize generality. Structures are often combined to attain more interesting properties.

Foundations

• Mathematical logic
• Set theory: formal theories of sets, in varying consistency strengths.
• Category theory: algebraic properties of collections of transformations between mathematical objects of the same type.

Pure Mathematics

Abstract objects in mathematics:

• Algebra: scalar, vector, tensor.
  • Universal Algebra: group, ring, field; module;
  • Linear Algebra: vector space, basis, linear operator;
  • Multilinear Algebra: tensor;
• Analysis: smooth function, differentiation, integration, approximation.
  • Dynamical System: vector field, flow; iteration;
• Topology: open/closed, convergent, continuous, connected, compact, manifold.
  • Metric Space: distance, complete;
  • Normed Space: length, Banach Space;
  • Inner Product Space: angle, orthogonal, Hilbert Space, generalized Fourier series;
• Measure Space: volume, density, integration;
• Geometry: curvature, distance, length, angle, volume.
  • Axiomatic geometry: Euclidean, elliptic, and hyperbolic geometries.
  • Differential Geometry: Riemannian, Lorentz, symplectic, and contact geometries.
• Number theory: properties of integers, especially primes and prime factorization.
• Discrete math: graph; combinatorics;

Applied Mathematics

• Probability and Statistics;
• Optimization and Game theory;
• Numerical methods;
• Computation theory: tasks that are theoretically possible with computing machines, and their relative difficulty and complexity;

Miscellaneous

Topics:

• Mathematical Characters
• Principle of Duality

Table: Development of Common Mathematical Objects

Set	New Structure	Origin/Motivation	Feature; Abstraction
$\mathbb{N}$	$+$, $*$	counting
$\mathbb{Z}$	$-$	closed inversion of $+$
$\mathbb{Q}$	$/$	closed inversion of $*$	rational function
$\mathbb{R}$	^	complete metric	analysis; metric, topology
$\mathbb{C}$		root of negative numbers	harmonic analysis
$\mathbb{F}^n$	$\cdot$	product space	inner product, norm
$\mathbb{F^N}$		discrete process/function
$L^p$		approximation of functions

Table: Constructs on Topological, Smooth, and Riemannian Manifolds

$(M, \cdots)$	$\mathcal{T}$	$\mathcal{A}$	$g$
Topology	$\lim, C(M)$; $d, \mathcal{A}$
Dynamical System		$\theta$; $\mathfrak{X}(M), (\partial/\partial x^i)$
Differentiation		$d$; $\Omega^∗(M), (dx^I)$	$\text{grad}, \text{div}, \Delta$
Integration		$\int_M \omega,\int_S \iota^∗\omega;\int_M\mu$	$\int_M f\omega_g;\int_M f\mu_g$
Measure			$\mu = f \mu_g$
Geometry			$\langle \cdot, \cdot\rangle_g, d_g, \omega_g, Rm$

Resources:

• Wolfram MathWorld;
• Encyclopaedia of Mathematics;

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🏷 Category=Topics