Maths
huge mess. also working back from Computing#Computation
General
to sort a
- https://en.wikipedia.org/wiki/Outline_of_mathematics
- https://en.wikipedia.org/wiki/Foundations_of_mathematics
- * https://en.wikipedia.org/wiki/Metamathematics
- https://en.wikipedia.org/wiki/Areas_of_mathematics
- https://en.wikipedia.org/wiki/Mathematics_Subject_Classification
- https://en.wikipedia.org/wiki/Linear_system
- https://en.wikipedia.org/wiki/Superposition_principle
- https://en.wikipedia.org/wiki/Nonlinear_system
- http://en.wikipedia.org/wiki/Mathematical_objects mathematical object] - an abstract object arising in philosophy of mathematics and mathematics. Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations. Geometry as a branch of mathematics has such objects as hexagons, points, lines, triangles, circles, spheres, polyhedra, topological spaces and manifolds. Algebra, another branch, has groups, rings, fields, group-theoretic lattices, and order-theoretic lattices. Categories are simultaneously homes to mathematical objects and mathematical objects in their own right.
Learning
History
- https://en.wikipedia.org/wiki/History_of_mathematics
- https://en.wikipedia.org/wiki/Indian_mathematics
- https://en.wikipedia.org/wiki/Mathematics_in_medieval_Islam
Philosophy
- http://en.wikipedia.org/wiki/Constructivism_(mathematics)
- http://en.wikipedia.org/wiki/Intuitionism - maths is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality
- http://en.wikipedia.org/wiki/Finitism - accepts the existence only of finite mathematical objects
- http://en.wikipedia.org/wiki/Constructive_analysis
Social
- http://michaelnielsen.org/polymath1/index.php?title=Main_Page - massively collaborative online mathematical projects
Books
People
Basics
Arithmetic
- https://en.wikipedia.org/wiki/Radix
- http://threesixty360.wordpress.com/2009/06/09/ethiopian-multiplication/ [2]
Numbers
- https://en.wikipedia.org/wiki/Natural_number
- https://en.wikipedia.org/wiki/Integer - Z in group theory
- https://en.wikipedia.org/wiki/Fraction_(mathematics)
- https://en.wikipedia.org/wiki/Real_number - includes all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2 (1.41421356… the square root of two, an irrational algebraic number) and π (3.14159265…, a transcendental number).
- https://en.wikipedia.org/wiki/Rational_numbers
- https://en.wikipedia.org/wiki/Irrational_number
- https://en.wikipedia.org/wiki/Complex_number
- https://en.wikipedia.org/wiki/Imaginary_unit
- https://en.wikipedia.org/wiki/Imaginary_number
- https://en.wikipedia.org/wiki/Algebraic_number
- https://en.wikipedia.org/wiki/Algebraic_number_theory
- https://en.wikipedia.org/wiki/Algebraic_integer
- https://en.wikipedia.org/wiki/Algebraic_number_field - vector space
- Numberphile: ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12
Prime
Geometry
- https://en.wikipedia.org/wiki/Point_(geometry) - 0D
- https://en.wikipedia.org/wiki/Line_(geometry) - 1D
- https://en.wikipedia.org/wiki/Plane_(geometry) - 2D
- https://en.wikipedia.org/wiki/Solid_geometry - 3D
- https://en.wikipedia.org/wiki/Line_segment - a line with two ends
- https://en.wikipedia.org/wiki/Vertex_(geometry) - a special kind of point that describes the corners or intersections of geometric shapes.
- https://en.wikipedia.org/wiki/Differential_geometry
- https://en.wikipedia.org/wiki/Discrete_geometry
- https://en.wikipedia.org/wiki/Convex_geometry
Shapes
- https://en.wikipedia.org/wiki/Platonic_solid
- http://www.mathsisfun.com/geometry/platonic-solids-why-five.html
- https://en.wikipedia.org/wiki/Polytope - geometric solid object with flat sides and straight edges
- https://en.wikipedia.org/wiki/Polygon - polytope in two dimensions
- https://en.wikipedia.org/wiki/Simple_polygon
- https://en.wikipedia.org/wiki/Regular_polygon
- https://en.wikipedia.org/wiki/Equiangular_polygon
- https://en.wikipedia.org/wiki/Convex_and_concave_polygons
- https://en.wikipedia.org/wiki/Quadrilateral - polygon with four sides (or edges) and four vertices or corners
- https://en.wikipedia.org/wiki/Polyhedron - polytope in three dimensions
- https://en.wikipedia.org/wiki/Regular_polyhedron
- https://en.wikipedia.org/wiki/Polychoron - polytope in four dimensions
- https://en.wikipedia.org/wiki/Flag_(geometry) - sequence of faces of a polytope, each contained in the next, with just one face from each dimension.
Tiles
Algebra
- http://www.cut-the-knot.org/ctk/between.shtml
- http://profkeithdevlin.org/2011/11/20/what-is-algebra/
- https://en.wikipedia.org/wiki/Map_(mathematics)
- https://en.wikipedia.org/wiki/Function_(mathematics)
- https://en.wikipedia.org/wiki/Morphism
- https://en.wikipedia.org/wiki/Homomorphism - structure-preserving map between two algebraic structures
- https://en.wikipedia.org/wiki/Isomorphism - homomorphism that admits an inverse
- https://en.wikipedia.org/wiki/Endomorphism - morphism (or homomorphism) from a mathematical object to itself.
- https://en.wikipedia.org/wiki/Automorphism - isomorphism from a mathematical object to itself, invertible and preserving all of its structure. set of all automorphisms of an object forms an automorphism group. loosely speaking, the symmetry group of the object.
- https://en.wikipedia.org/wiki/Partial_function
- https://en.wikipedia.org/wiki/Domain_of_a_function
- https://en.wikipedia.org/wiki/Codomain
- https://en.wikipedia.org/wiki/Category:Types_of_functions
- https://en.wikipedia.org/wiki/List_of_special_functions_and_eponyms
- https://en.wikipedia.org/wiki/Boolean_algebra
- https://en.wikipedia.org/wiki/Boolean_algebras_canonically_defined
Universal algebra
Linear algebra
- https://en.wikipedia.org/wiki/Linear_algebra
- https://en.wikipedia.org/wiki/List_of_linear_algebra_topics
- http://minireference.com/blog/linear-algebra-tutorial/
- http://betterexplained.com/articles/linear-algebra-guide/
- https://news.ycombinator.com/item?id=7280980
Abstract algebra
- https://en.wikipedia.org/wiki/Homological_algebra
- https://en.wikipedia.org/wiki/Homology_(mathematics)
Algebraic geometry
- https://en.wikipedia.org/wiki/Algebraic_variety
- https://en.wikipedia.org/wiki/Dimension_of_an_algebraic_variety
- https://en.wikipedia.org/wiki/Elliptic_curve
- http://jeremykun.com/2014/02/10/elliptic-curves-as-elementary-equations/
Group theory
Ring theory
Scheme theory
Topology
- https://en.wikipedia.org/wiki/Neighbourhood_(mathematics) - a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.
- https://en.wikipedia.org/wiki/Manifold - a topological space that resembles Euclidean space near each point
- https://en.wikipedia.org/wiki/Scheme_(mathematics)
- https://en.wikipedia.org/wiki/Glossary_of_scheme_theory
Combinatorics
Trigonometry
- https://en.wikipedia.org/wiki/Trigonometry - studies relationships involving lengths and angles of triangles
Calculus
- https://en.wikipedia.org/wiki/Calculus - the mathematical study of change, or any method or system of calculation guided by the symbolic manipulation of expressions
- https://en.wikipedia.org/wiki/History_of_calculus
- https://en.wikipedia.org/wiki/Infinitesimal_calculus
- https://en.wikipedia.org/wiki/Differential_calculus
- https://en.wikipedia.org/wiki/Integral_calculus
Logic
meeeess
- https://en.wikipedia.org/wiki/Term_logic - also known as traditional logic or Aristotelian logic
- https://en.wikipedia.org/wiki/Organon - Aristotle's six works on logic
- https://en.wikipedia.org/wiki/Proposition - refers to the meaning, basic semantics
- https://en.wikipedia.org/wiki/Antecedent_(logic) - first half of a proposition
- https://en.wikipedia.org/wiki/Consequent - second half of a proposition
- https://en.wikipedia.org/wiki/Syllogism - 2 premises and a conclusion
All S are P. (A form) All S are not P. (E form) Some S are P. (I form) Some S are not P. (O form)
- https://en.wikipedia.org/wiki/Syntax_(logic)
- https://en.wikipedia.org/wiki/Argument#Formal_and_informal_arguments
- https://en.wikipedia.org/wiki/Premise
- https://en.wikipedia.org/wiki/Statement_(logic) - refers to the wording
- https://en.wikipedia.org/wiki/Sentence_(mathematical_logic) - grammar
- https://en.wikipedia.org/wiki/Logical_form
- https://en.wikipedia.org/wiki/Square_of_opposition
- https://en.wikipedia.org/wiki/Validity - valid form
- https://en.wikipedia.org/wiki/Soundness - valid form and true proposition
- https://en.wikipedia.org/wiki/Propositional_calculus - logic of sentences. propositional calculus or logic (also called sentential calculus or sentential logic) is a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of inference rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions.
- https://en.wikipedia.org/wiki/Tautology_(logic) - always true
- https://en.wikipedia.org/wiki/Tautology_(rule_of_inference)
- https://en.wikipedia.org/wiki/Inference
- https://en.wikipedia.org/wiki/Interpretation_(logic)
- https://en.wikipedia.org/wiki/Logical_consequence - conclusion
- https://en.wikipedia.org/wiki/Validity
- https://en.wikipedia.org/wiki/False_(logic)
- https://en.wikipedia.org/wiki/Truth_value
- https://en.wikipedia.org/wiki/Logical_truth
- https://en.wikipedia.org/wiki/Truth_table
- http://jamie-wong.com/experiments/truthtabler/SLR1/
Bacon;
Leibniz;
- https://en.wikipedia.org/wiki/Characteristica_universalis
- https://en.wikipedia.org/wiki/Calculus_ratiocinator
- https://en.wikipedia.org/wiki/Algebraic_logic
- https://en.wikipedia.org/wiki/Boolean_algebra - first algebra of logic. primary: sets, secondary;
- https://en.wikipedia.org/wiki/Principle_of_bivalence - states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false
- https://en.wikipedia.org/wiki/Variable_(mathematics)
- https://en.wikipedia.org/wiki/Free_variables_and_bound_variables - symbol that will later be replaced by some literal string
- https://en.wikipedia.org/wiki/Domain_of_discourse
- https://en.wikipedia.org/wiki/Universe_(mathematics)
- https://en.wikipedia.org/wiki/Gottlob_Frege - sense and reference, connotation and denotation
- https://en.wikipedia.org/wiki/Principia_Mathematica - Russell and Whitehead
- https://en.wikipedia.org/wiki/Formal_language - well-defined vocabulary and grammar
- https://en.wikipedia.org/wiki/Formal_system
- https://en.wikipedia.org/wiki/Axiom
- https://en.wikipedia.org/wiki/Axiomatic_system
- https://en.wikipedia.org/wiki/Formal_methods
- YouTube: Introduction to Logic series
- Chico Jones - Logic
- https://en.wikipedia.org/wiki/Propositional_function - a statement expressed in a way that would assume the value of true or false, except that within the statement is a variable (x) that is not defined or specified, which leaves the statement undetermined
- https://en.wikipedia.org/wiki/Propositional_formula - formal expression that denotes a proposition
- https://en.wikipedia.org/wiki/Propositional_variable
Quantity: How much? Quality: Affirmative, negative
- https://en.wikipedia.org/wiki/Formal_proof - or derivation, is a finite sequence of sentences, each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference
- https://en.wikipedia.org/wiki/Well-formed_formula - proof in a formal system
- https://en.wikipedia.org/wiki/Predicate_(mathematical_logic) - a statement that may be true or false depending on the values of its variables.
- https://en.wikipedia.org/wiki/Rule_of_inference - the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions)
- https://en.wikipedia.org/wiki/Modus_ponens - "P implies Q; P is asserted to be true, so therefore Q must be true."
- "P implies Q; P is asserted to be true, so therefore Q must be true."
- https://en.wikipedia.org/wiki/List_of_logic_symbols
- https://en.wikipedia.org/wiki/Turnstile_(symbol)
- https://en.wikipedia.org/wiki/Logical_connective - or logical operator, a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the sense of the compound sentence produced depends only on the original sentences.
- https://en.wikipedia.org/wiki/Material_conditional - precisely or exactly implies
- https://en.wikipedia.org/wiki/If_and_only_if - equivalent (or materially equivalent) implication - ↔ ⇔ ≡
- https://en.wikipedia.org/wiki/Corresponding_conditional - a valid/truthful argument. negation of its corresponding conditional is a contradiction.
- https://en.wikipedia.org/wiki/Negation - not, logical compliment
- https://en.wikipedia.org/wiki/Logical_conjunction - and
- https://en.wikipedia.org/wiki/Logical_disjunction - or
- https://en.wikipedia.org/wiki/Exclusive_or
- https://en.wikipedia.org/wiki/Assignment_(mathematical_logic)
- https://en.wikipedia.org/wiki/T-schema
- https://en.wikipedia.org/wiki/Predicate_logic - logic of objects. generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic, or infinitary logic. formal system is distinguished from other systems in that its formulae contain variables which can be quantified.
- https://en.wikipedia.org/wiki/Zeroth-order_logic - first-order logic without quantifiers. finitely axiomatizable zeroth-order logic is isomorphic to a propositional logic.
- https://en.wikipedia.org/wiki/Second-order_logic - see frege's first predicate logic. quantifies over relations.
- https://en.wikipedia.org/wiki/Higher-order_logic - from first-order logic by additional quantifiers and a stronger semantics
- https://en.wikipedia.org/wiki/Many-sorted_logic
- https://en.wikipedia.org/wiki/Quantification#Logic
- https://en.wikipedia.org/wiki/Universal_quantification - all or any - universal - ∀
- https://en.wikipedia.org/wiki/Existential_quantification - there exists - particular - ∃
- https://en.wikipedia.org/wiki/Hypostatic_abstraction
- https://en.wikipedia.org/wiki/Continuous_predicate
- https://en.wikipedia.org/wiki/Interpretation_(logic)
- https://en.wikipedia.org/wiki/Formal_semantics_(logic)
- https://en.wikipedia.org/wiki/Categorical_logic - represents both syntax and semantics by a category, and an interpretation by a functor
- https://en.wikipedia.org/wiki/Mathematical_induction - an inference rule, not inductive reasoning
- https://en.wikipedia.org/wiki/Category:Systems_of_formal_logic
- https://en.wikipedia.org/wiki/List_of_logic_systems
- https://en.wikipedia.org/wiki/Combinatorial_logic - a notation to eliminate the need for variables in mathematical logic
- https://en.wikipedia.org/wiki/Natural_deduction - a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with the axiomatic systems which instead use axioms as much as possible to express the logical laws of deductive reasoning.
- https://en.wikipedia.org/wiki/Sequent_calculus
- Interactive Tutorial of the Sequent Calculus - This interactive tutorial will teach you how to use the sequent calculus, a simple set of rules with which you can use to show the truth of statements in first order logic. It is geared towards anyone with some background in writing software for computers, with knowledge of basic boolean logic.
- https://en.wikipedia.org/wiki/Cut-elimination_theorem
- https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof
- http://plus.maths.org/content/goumldel-and-limits-logic
- What Logic Is Not
- http://merrigrove.blogspot.co.uk/2013/12/what-heck-is-relation-from-tables-to.html
- http://www.linusakesson.net/programming/pipelogic/index.php
- http://plato.stanford.edu/entries/logic-combining/
See also Computing#Computation, Semantic web
Non-classical
all nc?
- https://en.wikipedia.org/wiki/Fuzzy_logic - rejects the law of the excluded middle and allows as a truth value any real number between 0 and 1
- https://en.wikipedia.org/wiki/Kripke_semantics - for non-classical logic systems, first modal logics, later adapted to intuitionistic logic and others
- https://en.wikipedia.org/wiki/Modal_logic - extends classical logic with non-truth-functional ("modal") operators ("necassery", "possible")
- https://en.wikipedia.org/wiki/Deontic_logic - relating to duty, introduces operators "ought" and "can"
- https://en.wikipedia.org/wiki/Hybrid_logic - a modal logic
- https://en.wikipedia.org/wiki/Pure_type_systems
- https://en.wikipedia.org/wiki/Calculus_of_constructions
- https://en.wikipedia.org/wiki/Categorial_grammar - Lambek calculus
Software
- https://en.wikipedia.org/wiki/ATS_(programming_language)
- https://en.wikipedia.org/wiki/Agda_(theorem_prover)
- https://en.wikipedia.org/wiki/Epigram_(programming_language)
Set theory
- https://en.wikipedia.org/wiki/Category:Basic_concepts_in_set_theory
- https://en.wikipedia.org/wiki/Category:Basic_concepts_in_infinite_set_theory
- https://en.wikipedia.org/wiki/Set_(mathematics)
- https://en.wikipedia.org/wiki/Element_(mathematics) - or member, of a set is any one of the distinct objects that make up that set.
- https://en.wikipedia.org/wiki/Cardinality - a measure of the "number of elements of the set"
- https://en.wikipedia.org/wiki/Class_(set_theory) - collection of sets (or other mathematical objects) with a property that all its members share
- https://en.wikipedia.org/wiki/Continuum_(set_theory) - set of real numbers
- https://en.wikipedia.org/wiki/Cardinality_of_the_continuum - the size of the set of real numbers
- https://en.wikipedia.org/wiki/Continuum_hypothesis
- https://en.wikipedia.org/wiki/Empty_set
- https://en.wikipedia.org/wiki/Finite_set
- https://en.wikipedia.org/wiki/Infinite_set
- https://en.wikipedia.org/wiki/Aleph_number - aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets
- https://en.wikipedia.org/wiki/Countable_set - set with the same cardinality as some subset of the set of natural numbers
- https://en.wikipedia.org/wiki/Uncountable_set - cardinal number is larger than that of the set of all natural numbers
- https://en.wikipedia.org/wiki/Axiom_of_choice
- https://en.wikipedia.org/wiki/Axiom_of_power_set
- https://en.wikipedia.org/wiki/Axiom_of_infinity
- https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory
- https://math.stackexchange.com/questions/136215/difference-between-zfc-nbg
- https://en.wikipedia.org/wiki/Subset
- https://en.wikipedia.org/wiki/Power_set
- https://en.wikipedia.org/wiki/Family_of_sets
- https://en.wikipedia.org/wiki/Binary_relation
- https://en.wikipedia.org/wiki/Composition_of_relations
- https://en.wikipedia.org/wiki/Algebraic_structures
- https://en.wikipedia.org/wiki/Constructible_universe
Type theory
Homotopy theory
Field theory
- https://en.wikipedia.org/wiki/Field_theory_(mathematics)
- https://en.wikipedia.org/wiki/Field_(mathematics)
Category theory
- https://en.wikipedia.org/wiki/Category_theory - deals in an abstract way with mathematical structures and relationships between them
- https://en.wikipedia.org/wiki/Outline_of_category_theory
- https://en.wikipedia.org/wiki/Abstract_nonsense
- nLab is a wiki-lab for collaborative work on Mathematics, Physics and Philosophy — especially from the n-point of view: insofar as these subjects are usefully treated with tools and notions of category theory or higher category theory.
- An Elementary Theory of the Category of Sets - alternative to ZFC
- http://golem.ph.utexas.edu/category/2014/01/an_elementary_theory_of_the_ca.html
- http://ncatlab.org/nlab/show/ETCS
- http://ncatlab.org/nlab/show/Trimble+on+ETCS+I
- https://en.wikipedia.org/wiki/Category_(category_theory)
- https://en.wikipedia.org/wiki/Concrete_category
- https://en.wikipedia.org/wiki/Functor - a type of mapping between categories
- https://en.wikipedia.org/wiki/Natural_transformation - provides a way of transforming one functor into another while respecting the internal structure (i.e. the composition of morphisms) of the categories involved. a "morphism of functors".
- https://en.wikipedia.org/wiki/Equivalence_of_categories
- https://en.wikipedia.org/wiki/Isomorphism_of_categories
- https://en.wikipedia.org/wiki/Category:Category-theoretic_categories
- https://en.wikipedia.org/wiki/Category_of_sets
- https://en.wikipedia.org/wiki/Category_of_small_categories
- https://en.wikipedia.org/wiki/Category_of_groups
- https://en.wikipedia.org/wiki/Homotopy_category
Model theory
The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory.
- universal algebra + logic = model theory
- model theory = algebraic geometry − fields
- http://en.wikipedia.org/wiki/Mathematical_structure
- http://en.wikipedia.org/wiki/Morphism
- http://en.wikipedia.org/wiki/Space_%28mathematics%29
Order theory
Domain theory
Proof theory
Analysis
Statistics
- https://github.com/mavam/stat-cookbook
- https://normaldeviate.wordpress.com/2012/11/17/what-is-bayesianfrequentist-inference
- http://www.greenteapress.com/thinkbayes
Information theory
Cellular automaton
Computing
- Mathics is a free, general-purpose online computer algebra system featuring Mathematica-compatible syntax and functions. It is backed by highly extensible Python code, relying on SymPy for most mathematical tasks and, optionally, Sage for more advanced stuff.
- http://www.p-value.info/2012/11/free-datascience-books.html
- How to implement an algorithm from a scientific paper
Visualisation
to find those prime vis things again
Gephi
- Gephi is an interactive visualization and exploration platform for all kinds of networks and complex systems, dynamic and hierarchical graphs.
- https://wiki.gephi.org
- http://en.wikipedia.org/wiki/Gephi
Fractals
- http://en.wikipedia.org/wiki/Julia_set
- http://acko.net/blog/how-to-fold-a-julia-fractal/
- http://sprott.physics.wisc.edu/carlson/dragons.html
Software
- mandelstir - Animating fractional iterations in the Mandelbrot Set and Julia Sets.
- Mandelbulber is an experimental application that helps to make rendering 3D Mandelbrot fractals much more accessible. A few of the supported 3D fractals: Mandelbulb, Mandelbox, BulbBox, JuliaBulb, Menger Sponge, Quaternion, Trigonometric, Hypercomplex, and Iterated Function Systems (IFS). All of these can be combined into infinite variations with the ability to hybridize different formulas together.
- FractalNow - A fast, advanced, multi-platform fractal generator.
- Fraqtive is an open source, multi-platform generator of the Mandelbrot family fractals. It uses very fast algorithms supporting SSE2 and multi-core processors. It generates high quality anti-aliased images and renders 3D scenes using OpenGL. It allows real-time navigation and dynamic generation of the Julia fractal preview.
- Fragmentarium is an open source, cross-platform IDE for exploring pixel based graphics on the GPU. It is inspired by Adobe's Pixel Bender, but uses GLSL, and is created specifically with fractals and generative systems in mind.
Other
Informatics
to sort
- http://mathoverflow.net/questions/156263/perfectly-centered-break-of-a-perfectly-aligned-pool-ball-rack/156407#156407 [4]
- https://en.wikipedia.org/wiki/Theory_of_relations
- http://www.wisdomandwonder.com/link/6582/some-thoughts-on-mathematics
- http://news.ycombinator.com/item?id=4839881
- http://www.evanmiller.org/mathematical-hacker.html http://www.evanmiller.org/mathematical-hacker.html]