Mathematics is a collection of formal theories, each with a symbolic system of objects and consistent rules for manipulation.
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.
Abstract objects in mathematics:
Topics:
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 | $\langle \cdot, \cdot\rangle_g, d_g, \omega_g, Rm$ |
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