Now that practical skills have developed enough to provide adequately for material needs, one of these sciences which are not devoted to utilitarian ends has been able to arise in Egypt, the priestly caste there having the leisure necessary for disinterested research.
-- Aristotle

Mathematics is a language invented by humans to describe certain objects and processes. Evolving from counting, simple arithmetic, and geometry, it has blossomed most wonderfully into set theory, boolean logic, calculus, and many odder disciplines. From the beginning, math has been fueled by smart, obsessive, and bored people who had enough leisure time to discover that universe was filled with odd regularities.

The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.
-- Aristotle

Math has the distinction of being the only a priori science (unless you consider philosophy to be a science). Visual aids and experimentation can certainly help, but most math is done on paper. None the less, it has great worldly benefit, from architecture to zoology. It's fun to play with -- magic squares, sudoku, the Prisoner's Dilemma, the Monty Hall Problem. It's important to our lives -- it programs our computers, launches our spaceships, and encrpts our emails.

The whole is more than the sum of its parts.
-- Aristotle

In theory, math can be applied to describe any observable phenomenon that exists, and many that don't. But Godel's theorem shows that it can't actually describe every observable phenomenon, because any given system can't describe itself. This has not yet caused math to fall out of fashion.

To Thales the primary question was not what do we know, but how do we know it.
-- Aristotle

There is, to my knowledge, no formal categorization of the subfields of mathematics, and no clear hierarchical structure between the 'main fields' of mathematical study and the 'sub-fields'. But here is a list of some of the main branches of mathematics.

• Arithmetic
• Algebra
• Calculus
• Combinatorics
• Discrete mathematics
• Geometry
• Logic
• Mathematical analysis
• Number theory
• Set theory
• Statistics
• Topology

Needless to say, you should learn more math. You should node more math. Unfortunately, while E2 is an excellent place to node math, it doesn't actually do a very good job of teaching it, unless you've gotten a good foundation elsewhere. Fortunately, your local library will have the primers to get you started.

Mathematics is a "formal" discipline formed around a human cognitive mechanism which accounts for perception of discrete entities (and by extension, quantitative relationships). It should be noted that discrete entities and quantitative relationships do not actually exist in nature per se; rather, we accept the notion that they do as part of the natural reductive process of our consciousness (a sometimes-convenient compromise). In this context, Mathematics is a (characteristically exhaustive and rigorous) compendium of metaphors based exclusively on discrete and quantifiable subject matter (i.e. numbers), where that description is itself, as much as possible, discrete and quantifiable.

Kurt Godel's Incompleteness Theorems (published 1931) correctly identify that any formal system (i.e. Mathematics) is inherently flawed, specifically "in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system." To restate, Mathematics is effectively a study of the consequences of the particular compromises inherent in human cognition. Caveat emptor.

I believe that mathematics can best be defined as the symbolic manipulation of ideas. This definition could be interpreted to include such disciplines as literature, art, and philosophy. At its core, mathematics has more in common with literature, art, and philosophy than it does with the sciences, with which it is more often linked. This can best be explained by the fact that the qualities that make one poem better than another are the same as the qualities that make one proof better than another. The ultimate goal of all poems and proofs is to say more with less.

A Civilization advance.
Rudimentary arithmetic first gained wide use due to farmers' and traders' need to keep track of quantities, accounts, and measurements. Eventually, clever philosophers built on this mundane base to conceive an abstract theory of numbers, which they called mathematics. Soon after, military leaders found ways to use mathematics in the design of weaponry.
Prerequisites: Alphabet and Masonry.
Allows for: The University, Physics, Computers, and Astronomy.

Mathematics is mathematics.

Please /msg eien_meru if you have any additions, broken links, or comments. This list will probably always be incomplete, despite all efforts to the contrary.

Index

00 - General Mathematics
01 - Biography
02 - History
03 - Books, Papers, and Essays
04 - Mathematical Logic and Foundations
05 - Combinatorics
06 - Ordered Algebraic Structures
07 - Rings and Algebras
08 - Number Theory
09 - Field Theory and Polynomials
10 - Algebraic Geometry
11 - Linear Algebra and Matrix Theory
12 - Category Theory
13 - Group Theory
14 - Real Analysis
15 - Complex Analysis
16 - Measure and Integration
17 - Ordinary Differential Equations
18 - Partial Differential Equations
19 - Dynamical Systems
20 - Sequences and Series
21 - Approximations and Expansions
22 - Fourier Analysis
23 - Harmonic Analysis
24 - Functional Analysis
25 - Calculus of Variations
26 - Geometry
27 - Differential Geometry
28 - Topology
29 - Manifolds
30 - Probability Theory and Statistics
31 - Numerical Analysis
32 - Physics, Biology, and Natural Science
33 - Computer Science
34 - Game Theory, Economics, and Social Science
99 - Other

00 - General Mathematics

• ⊕, ⋅, (really) big numbers
• 0!, 1 = -1, 1 is not prime, 1/x = x-1, 1=2, 2^.5 = 2, 3-4-5 triangle
• absolute value, addition, All bases are base 10, angle, area, arithmetic, arrow notation, average
• balanced ternary, base 8, base 10, base 16, base 36, base 60, base 112, big number, binary, binomial, binomial theorem, bit
• ceiling function, Coefficient, common logarithm, common mathematical abbreviations, complex number, constant, coordinate, cosecant, cosine, cotangent, cubic equation
• de Moivre's Theorem, decimal, decimal to binary, difference, Digits of Pi, discriminant, division
• e, Egyptian fractions, elementary, equation, equation of a line, Euclid's algorithm, Euclid's Limerick, exponents, exponential
• false mathematical proofs, Fibonacci base, floor function, fractal, fraction, function, Fundamental theorem of algebra
• Gaussian formula, golden ratio, greatest common divisor
• Half-angle Trigonometric Functions, hexadecimal, How to find out if ANY number is divisible by eleven, hyperbolic function, hypothesis
• i, imaginary, imaginary number, improper fraction, indiscrete mathematics, inf, integer, interesting number, interesting number bases, irrational, irrational number, irrational-base number system
• The Law of Cosines, logarithm
• magic square, maximum, mean, median, minima and maxima, minimum, mixed number, mode, modulo, modulus, multiplication, Multiplication tables
• natural logarithm, natural number, negative number, Node of Numbers
• octal
• Pascal's Triangle, pi, polynomial, polynomial equation, positional notation, prime, prime number, probability puzzles, proof, Pythagorean Theorem, Pythagorean Triple
• quadratic equation, quadratic formula
• radix, rational number, real number, recreational mathematics, reduced fraction, rule of three
• scientific notation, secant, send more money solution, sexagesimal, sexagesimal numbers, significant digits, sine, Slide rule, slope, soh cah toa, subtraction, sum, sup, surface area, surface area to volume ratio
• tangent, The square root of -1, There are as many numbers between 0 and 1 as there are in the set of all real numbers, transcendental function, transcendental number, triangular number, trichotomy law, trigonometric function, trigonometry, trinary, twos complement
• variable, Venn Diagram
• weighted average, whole number
• The zero/infinity paradox

01 - Biography

• Alexander Abian, Archimedes, Aristotle
• Charles Babbage, Jakob Bernoulli, George Boole, Nicholas Bourbaki
• Georg Cantor, Girolamo Cardano, Lewis Carroll, Condorcet, John Conway
• Rene Descartes, A.K. Dewdney, Diophantus, Paul Dirac
• Albert Einstein, Eratosthanes, Pal Erdos, Euclid, Leonhard Euler
• Pierre de Fermat, Gottlob Frege
• Galileo Galilei, Martin Gardner, Kurt Godel, George Green
• David Hilbert, Douglas Hofstadter, Christiaan Huygens
• Johannes Kepler, David Knuth, Kazimierz Kuratowski
• Joseph Lagrange, Pierre Laplace, Tom Lehrer, Gottfried Leibnitz, Ada Lovelace
• Benoit Mandelbrot
• John Von Neumann, Isaac Newton, Emmy Noether
• Blaise Pascal, Pythagoras
• Srinivasa Ramanujan, Bertrand Russell
• Raymond Smullyan
• Alan Turing
• Alfred North Whitehead, Norbert Wiener, Andrew Wiles

02 - History

• analytical engine
• Erlanger Programm
• Hardy-Ramanujan number, Hilbert Problems, How Gauss quickly added up the numbers 1 to 100, how Wiener found his way home
• Lives of the mathematicians
• new math
• "true" story about Poincaré's baker
• Why is there no Nobel Prize in Mathematics?

03 - Books, Papers, and Essays

• Best Math Books Ever Written
• Concrete Mathematics, Cryptonomicon
• Flatland
• Gödel, Escher, Bach: An Eternal Golden Braid, Guides to Godel
• Innumeracy
• Math is not a social construct, Math is a social construct, Mathematical Games, Metamagical Themas
• numerical recipes
• On computable numbers, with an application to the Entscheidungs, On Formally Undecidable Propositions of Principia Mathematica and Related Systems
• Principia Mathematica, Principles of Mathematical Analysis
• Winning Ways

04 - Mathematical Logic and Foundations

• σ algebra
• algebraic, analytic logic, antichain, associative, associative logic, automorphism, axiom of choice, axiom of infinity
• Banach-Tarski Theorem, A Blather of Paradoxes, Bertrand's Paradox, bijection, binary operation, boolean, boolean logic
• Cartesian product, class, commutative, commutativity, compactness theorem, conjunction, Connectivity is not expressible in first-order logic, consistent, Construction of the rational numbers, Construction of the real numbers, countable, currying functions
• density, denumerable, diagonal argument, disjoint union, disjunction, distribution, distributive, downwards Skolem-Löwenheim theorem
• empty set, epimorphism, example of an ultraproduct
• finite, finitely satisfiable, fixed point, fixpoint, for every set there exists a larger set, formula, foundations of mathematics, function, fuzzy logic
• Godel's theorem
• if and only if, iff, image, implicit function, inaccessible cardinal, infimum, infinite propositions paradox, infinite set, infinitesimal, infinity, injection, Inside every surjection is a bijection waiting to get out., intersection, Is mathematics consistent?
• kernel
• Liar Paradox, logic, logical operation
• mapping, Mapping the power set of the integers to the real numbers, Material implication, mathematical logic, monomorphism
• natural numbers as sets, Non-standard analysis, NSA: Introduction and Construction, NSA: non-standard analysis footnote 1, NSA proof that a compact field is finite, NSA: proof of Sierpinski's theorem, NSA: Robinson's "overspill" lemma, NSA: some pseudo set theory, NSA: some pseudo set theory footnote 1, NSA: What's a language?, NSA: What's a model?, null set, numeral
• one to one, onto, ordered n-tuple
• Peano postulates, perfect set, potency axiom, power set, principal ultrafilter, Proof that there is no largest prime number, proposition
• recursively enumerable, A recursively enumerable set whose complement is also recursively enumerable is recursive, reflexive, Russell's paradox,
• self-reference, set, set theory, set theory notation, Skolem-Löwenheim theorems, standard approximation, subset, successor cardinal, supremum, surreal numbers, symbol
• This statement cannot be proven, true arithmetic, truth table, Turing machine, tuple, two-way implication
• ultrafilter, ultraproduct, Unbounded, uncountable, union, upwards Skolem-Löwenheim theorem
• weakly inaccessible cardinal, Well Ordering Theorem
• ZF, ZFC, Zorn's Lemma

05 - Combinatorics

• 7 color torus
• amenable, antichain, the average number of fixed points of a permutation is 1
• biconnected component, bipartite graph
• combination, Combinatorics, Counting proof that there are infinitely many prime numbers, cut
• derangement, directed graph, digraph
• edge, edge isoperimetric constant
• factorial, fixed points of permutations, Ford-Fulkerson algorithm, Four color theorem
• generating function, generating function tricks, Golomb Ruler, grandparented tree, graph, graph theory, group acting on a set, group action
• inclusion / exclusion principle
• projective plane
• How to paint a tetrahedron
• permutation, Permutations with no fixpoints, Proof of Sperner's theorem
• spanning tree, Sperner's theorem
• transitive, Transposition, Two color theorem
• unimodular
• vertex, vertex isoperimetric constant

06 - Ordered Algebraic Structures

• Archimedean
• boolean algebra, boolean algebra duality
• complex numbers can't be ordered, connective constant
• dense
• every partial order can be extended to a total order
• maximal, Monotone function
• partial order, poset
• lattice
• order type
• total order, trichotomy law
• well ordered

07 - Rings and Algebras

• A beautiful application of ring theory, Abelian, algebra, associate
• centralizer, centre, Commutator, connection between quaternions and cross product
• divides, division ring
• Euclidean ring
• field of fractions
• ideal, infinite divisor chain, integers modulo n, integral domain
• Lie algebra
• maximal ideal, module, multiplicative group
• Noetherian
• prime, prime ideal, principal ideal domain, Proof of Wedderburn's theorem about finite division rings
• representations of Lie algebras, radical, ring homomorphism, ring theory
• simple, subring
• Quaternion, quotient ring
• unique factorization domain, unit
• Weyl algebra, Wedderburn's theorem about finite division rings

08 - Number Theory

• 1 is not prime
• All finite fields are isomorphic to GF(p^n), automorphisms of finite fields
• Brun's constant
• Chinese remainedr theorem, Collatz conjecture, composite
• Diophantine equation, Discrete Logarithm
• Euler Phi function, Euler's theorem
• Fermat's Last Theorem, Fermat's little theorem, Fibonacci number, finite field, Furstenberg's proof that there are infinitely many prime numbers
• Galois Field, Galois theory, Gaussian integer, Goldbach's conjecture, greatest common divisor
• Is pi normal?
• Kaprekar number
• lattice, Legendre symbols and Gauss' quadratic reciprocity law, Lucas number
• Mersenne prime
• normal, normal number, number theory
• p-adic integer, p-adic integers: analytic properties, p-adic norm, p-adic rational, p-adic valuation, perfect number, prime decomposition, Proof that there exist irrational a,b for which a^b is rational, proof that the multiplicative group of a finite field is cyclic, Proof that there is no largest prime number, proof of the properties of the Euler Phi function, Proth's Theorem
• relatively prime, repunit prime, Riemann zeta function, RSA
• squarefree
• Taniyama-Shimura conjecture, totient, transcendental number, twin primes
• very composite number
• Wilson's theorem

09 - Field theory and polynomials

• algebraic, algebraically closed
• characteristic, a compact field is finite, complex root of unity, cubic, cyclotomic polynomial
• Eisenstein's irreducibility criterion
• the field of real numbers has no non-trivial automorphisms, field extension, Frobenius endomorphism, fundamental theory of algebra
• Galois theory, Gauss's Lemma
• highest common factor
• irreducibility of cyclotomic polynomials, irreducible
• Jones polynomial
• Laurent polynomial
• minimal polynomial
• normal
• polynomial, primitive, prime-valued polynomial, Proof that the algebraic closure of a field is unique, proof that the field of real numbers has no non-trivial automorphisms, proof of Sturm's theorem, proof of the fundamental theorem on symmetric polynomials, Proof of the uniqueness of splitting fields
• separable, skew field, splitting field, "standard" proof that a compact field is finite, Sturm's theorem, subfield, symmetric polynomial
• quadratic, quantic, quartic, quintic
• unique factorization of polynomials
• Why the sequence of coefficients of a cyclotomic polynomial is a palindrome

10 - Algebraic Geometry

• algebraic geometry
• The Hilbert Basis Theorem, Hilbert's Nullstellensatz
• Goursat's Surface
• proof of Diophantus' theorem on Pythagorean triples, projective plane
• simple, singularity
• Tooth Surface
• Zariski topology

11 - Linear Algebra and Matrix Theory

• Annihilator
• basis for a vector space, bilinear
• Cayley-Hamilton Theorem
• Determinant, dimension of a vector space, dual of a vector space
• eigenvalue, eigenvalue multiplicity, eigenvector
• homogeneous system of linear equations
• linear, Linear Dependence and Independence
• matrix multiplication
• negative semidefinite
• orthogonal group
• permanent, positive semidefinite
• A real symmetric matrix has n eigenvalues
• square matrix, symmetric matrix
• tensor product
• vector space

12 - Category Theory

• abstract nonsense
• category theory
• exact sequence
• functor
• identity function, isomorphism theorems
• kernel
• natural transformation
• Product
• short exact sequence
• universal algebra

13 - Group Theory

• Abelian, alternating group
• Cayley graph, centralizer, Commutator, conjugacy class, coset,