Wikipedia's contents: Mathematics and logic

Mathematics is the study of topics such as quantity (numbers), structure, space, and change. It evolved through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason or principle) is the study of the principles and criteria of valid inference and demonstration. As a formal science, logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and through the study of arguments in natural language. The field of logic ranges from core topics such as the study of fallacies and paradoxes, to a specialized analysis of reasoning using probability and to arguments involving causality. Logic is also commonly used today in argumentation theory. Since the mid-nineteenth century formal logic has been studied in the context of the foundations of mathematics.

Formal science branches of knowledge that are concerned with formal systems. Unlike other sciences, the formal sciences are not concerned with the validity of theories based on observations in the real world, but instead with the properties of formal systems based on definitions and rules.
  • Mathematics study of quantity, structure, space, and change. Mathematicians seek out patterns, and formulate new conjectures. (See also: Lists of mathematics topics)
    • Arithmetic the oldest and most elementary branch of mathematics, involving the study of quantity, especially as the result of combining numbers. The simplest arithmetical operations include addition, subtraction, multiplication and division.
    • Algebra the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures.
      • Linear algebra the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices.
      • Abstract algebra the branch of mathematics concerning algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras.
        • Commutative algebra branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings.
      • Algebraic coding theory aka coding theory, is the study of the properties of codes and their respective fitness for specific applications.
      • Boolean algebra branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0, respectively. It is used for describing logical operations.
    • Analysis/Calculus the branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.
    • Category theory the branch of mathematics examining the properties of mathematical structures in terms of collections of objects and arrows
    • Discrete mathematics the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values.
      • Combinatorics the branch of mathematics concerning the study of finite or countable discrete structures.
    • Geometry this is one of the oldest branches of mathematics, it is concerned with questions of shape, size, relative position of figures, and the properties of space.
      • Algebraic geometry study of zeros of multivariate polynomials.
      • Circles geometric shapes consisting of all points in a plane that are at a given distance from a given point, the center.
      • Combinatorial computational geometry states problems in terms of geometric objects as discrete entities and hence the methods of their solution are mostly theories and algorithms of combinatorial character.
      • Computer graphics and descriptive geometry
      • Differential geometry geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.
      • Topology developed from geometry, it looks at those properties that do not change even when the figures are deformed by stretching and bending, like dimension.
        • Algebraic topology uses tools from abstract algebra to study topological spaces.
        • General topology also known as point-set topology, it deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation for most of the other branches of topology.
        • Geometric topology study of manifolds and maps between them, particularly embeddings of one manifold into another.
    • Mathematical logic study of formal logic within mathematics.
      • Set theory studies sets, which can be informally described as collections of objects.
        • Algebraic structure the sum total of all properties that arise from the inclusion of one or more operations on a set.
    • Trigonometry branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves.
      • Triangles type of polygon, with three edges and three vertices. The triangle is one of the basic shapes in geometry.
  • Logic formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science.
  • Other mathematical sciences academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper.
    • Statistics study of the collection, organization, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.
      • Regression analysis techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps one understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed.
    • Probability way of expressing knowledge or belief that an event will occur or has occurred. The concept has an exact mathematical meaning in probability theory, which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence/machine learning and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
    • Theoretical computer science a division or subset of general computer science and mathematics that focuses on more abstract or mathematical aspects of computing and includes the theory of computation.
Mathematics lists 

Probability

Basic mathematics Trigonometric identities
Algebra Algebraic structures Reciprocity laws Cohomology theories
Calculus and analysis Integrals Mathematical series Vector spaces
Geometry and topology Geometric shapes Algebraic surfaces Points
Logic First-order theories Large cardinal properties Paradoxes
Number theory Prime numbers
Differential equations Nonlinear partial differential equations
Game theory Games
Operations research Knapsack problems

Methodology Graphical methods Mathematics-based methods Rules of inference

Mathematical statements • Algorithms Axioms Conjectures Erdős conjectures Combinatorial principles Equations Formulae involving pi Mathematical identities Inequalities Lemmas Mathematical proofs NP-complete problems Statements undecidable in ZFC Mathematical symbols Undecidable problems Theorems (Fundamental theorems)

General concepts • Dualities Transforms Recursion

Mathematical objects • Mathematical examples Curves Complex reflection groups Complexity classes Examples in general topology Finite simple groups Fourier-related transforms Mathematical functions Mathematical knots and links Manifolds Mathematical shapes Matrices Numbers Polygons, polyhedra and polytopes Regular polytopes Simple Lie groups Small groups Special functions and eponyms Algebraic surfaces Surfaces Table of Lie groups
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Statistics Logic Mathematical logic Waves Information theory Fractals