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huge mess. also working back from Computing#Computation



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  • 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.


  • European Digital Mathematics Library (EuDML) - mathematics literature available online in the form of an enduring digital collection, developed and maintained by a network of institutions.
  • MatematicasVisuales - visual expositions of mathematical concepts. p]


  • Mathway provides students with the tools they need to understand and solve their math problems. With hundreds of millions of problems already solved, Mathway is the #1 problem solving resource available for students, parents, and teachers.
  • Webmath is a math-help web site that generates answers to specific math questions and problems, as entered by a user, at any particular moment.


  • A Prayer for Archimedes - A long-lost text by the ancient Greek mathematician shows that he had begun to discover the principles of calculus.





  • zbMATH (Zentralblatt MATH) is the world’s most comprehensive and longest-running abstracting and reviewing service in pure and applied mathematics. It is produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructure GmbH (FIZ Karlsruhe). Editors are the European Mathematical Society (EMS), FIZ Karlsruhe, and the Heidelberg Academy of Sciences and Humanities. zbMATH is distributed by Springer. The zbMATH database contains more than 3.5 million bibliographic entries with reviews or abstracts currently drawn from more than 3,000 journals and serials, and 170,000 books. The coverage starts in the 18th century and is complete from 1868 to the present by the integration of the “Jahrbuch über die Fortschritte der Mathematik” database. About 7,000 active expert reviewers from all over the world contribute reviews to zbMATH.

zbMATH provides easy access to bibliographic data, reviews and abstracts from all areas of pure mathematics as well as applications, in particular to the natural sciences, computer science, economics and engineering. It also covers history and philosophy of mathematics and university education. All entries are classified according to the Mathematics Subject Classification Scheme (MSC 2010) and are equipped with keywords in order to characterize their particular content. zbMATH covers all available published and peer-reviewed articles, books, conference proceedings as well as other publication formats pertaining to the scope given above. For the list of journals and book series covered see the Journals search.


Formal systems


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  • - or logical calculus is any well-defined system of abstract thought based on the model of mathematics. A formal system need not be mathematical as such; for example, Spinoza's Ethics imitates the form of Euclid's Elements. Spinoza employed Euclidiean elements such as "axioms" or "primitive truths", rules of inferences etc. so that a calculus can be built using these. For nature of such primitive truths, one can consult Tarski's "Concept of truth for a formalized language". Some theorists use the term formalism as a rough synonym for formal system, but the term is also used to refer to a particular style of notation, for example, Paul Dirac's bra–ket notation.

  • - steps in reasoning, moving from premises to conclusions. Charles Sanders Peirce divided inference into three kinds: deduction, induction, and abduction. Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular premises to a universal conclusion. Abduction is inference to the best explanation.

  • - also deductive logic or logical deduction or, informally, "top-down" logic is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion. Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.

Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from initial information. As a result, induction can be used even in an open domain, one where there is epistemic uncertainty. Note, however, that the inductive reasoning mentioned here is not the same as induction used in mathematical proofs – mathematical induction is actually a form of deductive reasoning.

  • - sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well. Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference. Hilbert systems can be characterised by the choice of a large number of schemes of logical axioms and a small set of rules of inference. Systems of natural deduction take the opposite tack, including many deduction rules but very few or no axiom schemes. The most commonly studied Hilbert systems have either just one rule of inference — modus ponens, for propositional logics — or two — with generalisation, to handle predicate logics, as well — and several infinite axiom schemes. Hilbert systems for propositional modal logics, sometimes called Hilbert-Lewis systems, are generally axiomatised with two additional rules, the necessitation rule and the uniform substitution rule.

A characteristic feature of the many variants of Hilbert systems is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if we are interested only in the derivability of tautologies, no hypothetical judgments, then we can formalize the Hilbert system in such a way that its rules of inference contain only judgments of a rather simple form. The same cannot be done with the other two deductions systems: as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided — not even if we want to use them just for proving derivability of tautologies.

  • - 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.

  • - a set of production rules for strings in a formal language. The rules describe how to form strings from the language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form. Formal language theory, the discipline that studies formal grammars and languages, is a branch of applied mathematics. Its applications are found in theoretical computer science, theoretical linguistics, formal semantics, mathematical logic, and other areas.

  • - an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory or other formal system, the role of a primitive notion is analogous to that of axiom. In axiomatic theories, the primitive notions are sometimes said to be "defined" by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress.

  • - or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.' The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is simply a premise or starting point for reasoning.

As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), often shown in symbolic form, while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory.

  • - any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory typically means an axiomatic system, for example formulated within model theory. A formal proof is a complete rendition of a mathematical proof within a formal system.

  • - a formula in the metalanguage of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term.
  • - In many popular versions of axiomatic set theory the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set. Some mathematicians call it the axiom schema of comprehension, although others use that term for unrestricted comprehension, discussed below. Because restricting comprehension solved Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory.
  • - a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF. The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining formulas.

  • - a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.
  • - the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths. The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and the study of deductive systems is the branch that is known as proof theory.

  • Law of excluded middle and double negative elimination
  • Law of noncontradiction, and the principle of explosion
  • Monotonicity of entailment and idempotency of entailment
  • Commutativity of conjunction
  • De Morgan duality: every logical operator is dual to another

While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics. In other words, the overwhelming majority of time spent studying classical logic has been spent studying specifically propositional and first-order logic, as opposed to the other more obscure variations of classical logic.Classical logic is a bivalent logic, meaning it accepts only two possible truth values: true and false.

  • - also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century. This entry is an introduction to the term logic needed to understand philosophy texts written before predicate logic came to be seen as the only formal logic of interest. Readers lacking a grasp of the basic terminology and ideas of term logic can have difficulty understanding such texts, because their authors typically assumed an acquaintance with term logic.



"Inference rules are rules that describe when one can validly infer a conclusion from a set of premises. Replacement rules are rules of what one can replace and still have a wff with the same truth-value; in other words, they are a list of logical equivalencies. The main difference between the two set of rules is that the first kind allows valid inference from the first set of propositions (premises) to the second set (the conclusion), but with replacement rules we can validly infer both ways."

  • Category Theory for the Working Hacker - Philip Wadler on why category theory is relevant for developers, discussing the principle of Propositions as Types connecting propositions and proofs in logic, and types and programs in computing. [20]

A first-order term is recursively constructed from constant symbols, variables and function symbols. An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation. For example, (x+1)*(x+1) is a term built from the constant 1, the variable x, and the binary function symbols + and *; it is part of the atomic formula (x+1)*(x+1) which evaluates to true for each real-numbered value of x.

Besides in logic, terms play important roles in universal algebra, and rewriting systems.

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)

  • - 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. The last sentence in the sequence is a theorem of a formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concept of natural deduction is a generalization of the concept of proof.

  • - also known simply as an atom is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives.

The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, the atomic formulas are the propositional variables. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. In model theory, atomic formula are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given model.

  • - an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning (or "reasonable expectation"). A proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture.

Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

  • - a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually axioms, existing lemmas and theorems, without making any further assumptions. In order to directly prove a conditional statement of the form "If p, then q", it suffices to consider the situations in which the statement p is true. Logical deduction is employed to reason from assumptions to conclusion. The type of logic employed is almost invariably first-order logic, employing the quantifiers for all and there exists. Common proof rules used are modus ponens and universal instantiation.

  • - a form of proof, and more specifically a form of indirect proof, that establishes the truth or validity of a proposition. It starts by assuming that the opposite proposition is true, and then shows that such an assumption leads to a contradiction. Proof by contradiction is also known as indirect proof, apagogical argument, proof by assuming the opposite, and reductio ad impossibilem. It is a particular kind of the more general form of argument known as reductio ad absurdum. G. H. Hardy described proof by contradiction as "one of a mathematician's finest weapons", saying "It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game."

  • - a conclusion or proposition based on incomplete information, for which no proof has been found. Conjectures such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (which was a conjecture until proven in 1995) have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.

  • - a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.

  • - (plural lemmata or lemmas) from the Greek λῆμμα (lemma, “anything which is received, such as a gift, profit, or a bribe”) or helping theorem is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself. There is no formal distinction between a lemma and a theorem, only one of intention – see Theorem terminology. However, a lemma can be considered a minor result whose sole purpose is to help prove a theorem - a step in the direction of proof, so to speak.

  • - the theorem considered central to a field of mathematics. The naming of such a theorem is not necessarily based on how often it is used or the difficulty of its proofs. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus, which are two distinct branches that were not obviously related. The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory.


  • - an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics. An interpretation often (but not always) provides a way to determine the truth values of sentences in a language. If a given interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentence or theory.

  • - modus tollendo tollens and also denying the consequent) (Latin for "the way that denies by denying") is a valid argument form and a rule of inference. It is an application of the general truth that if a statement is true, then so is its contra-positive.

  • - states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic. In formal logic, the principle of bivalence becomes a property that a semantics may or may not possess. It is not the same as the law of excluded middle, however, and a semantics may satisfy that law without being bivalent.

The principle of bivalence is studied in philosophical logic to address the question of which natural-language statements have a well-defined truth value. Sentences which predict events in the future, and sentences which seem open to interpretation, are particularly difficult for philosophers who hold that the principle of bivalence applies to all declarative natural-language statements. Many-valued logics formalize ideas that a realistic characterization of the notion of consequence requires the admissibility of premises which, owing to vagueness, temporal or quantum indeterminacy, or reference-failure, cannot be considered classically bivalent. Reference failures can also be addressed by free logics.

  • - one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence.

Logical truths (including tautologies) are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and no situation could arise which would cause us to reject a logical truth. It must be true in every sense of intuition, practices, and bodies of beliefs. However, it is not universally agreed that there are any statements which are necessarily true.

  • - or untrue is the state of possessing negative truth value or a nullary logical connective. In a truth-functional system of propositional logic it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol .

Another approach is used for several formal theories (for example, intuitionistic propositional calculus) where the false is a propositional constant (i.e. a nullary connective) ⊥, the truth value of this constant being always false in the sense above.

  • - 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 value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective.

The most common logical connectives are binary connectives (also called dyadic connectives) which join two sentences which can be thought of as the function's operands. Also commonly, negation is considered to be a unary connective.

Logical connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic. Semantics of a logical connective is often, but not always, presented as a truth function. A logical connective is similar to but not equivalent to a conditional operator.

  • - not, also called logical complement, is an operation that takes a proposition p to another proposition "not p", written ¬p, which is interpreted intuitively as being true when p is false, and false when p is true. Negation is thus a unary (single-argument) logical connective. It may be applied as an operation on propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition p is the proposition whose proofs are the refutations of p.

  • - also known as material implication, material consequence, or simply implication, implies, or conditional, is a logical connective (or a binary operator) that is often symbolized by a forward arrow "". The material conditional is used to form statements of the form p→q (termed a conditional statement) which is read as "if p then q". Unlike the English construction "if...then...", the material conditional statement p→q does not specify a causal relationship between p and q. It is merely to be understood to mean "if p is true, then q is also true" such that the statement p→q is false only when p is true and q is false.

 The material conditional only states that q is true when (but not necessarily only when) p is true, and makes no claim that p causes q. The material conditional is also symbolized using:

  • 𝑝 ⊃ 𝑞 (Although this symbol may be used for the superset symbol in set theory.);
  • 𝑝 ⇒ 𝑞 (Although this symbol is often used for logical consequence (i.e., logical implication) rather than for material conditional.)
  • C𝑝𝑞 (using Łukasiewicz notation)

  • - or, the truth-functional operator of (inclusive) disjunction, also known as alternation; the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is typically written as or +.

  • - or is the truth-functional operator of (inclusive) disjunction, also known as alternation; the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is typically written as or +. "A or B" is true if A is true, or if B is true, or if both A and B are true. An operand of a disjunction is called a disjunct.
  • - or exclusive disjunction is a logical operation that outputs true only when inputs differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR (/ˌɛks ˈɔːr/), EOR, EXOR, , , , and . The negation of XOR is logical biconditional, which outputs true only when both inputs are the same. It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator excludes that case. This is sometimes thought of as "one or the other but not both". This could be written as "A or B, but not, A and B".

  • - sometimes known as the material biconditional, is the logical connective of two statements asserting "p if and only if q", where p is an antecedent and q is a consequent. This is often abbreviated "p iff q". The operator is denoted using a doubleheaded arrow (), a prefixed E (Epq), an equality sign (=), an equivalence sign (), or EQV. It is logically equivalent to (p → q) ∧ (q → p). It is also logically equivalent to "(p and q) or (not p and not q)" (or the XNOR (exclusive nor) boolean operator), meaning "both or neither". The only difference from material conditional is the case when the hypothesis is false but the conclusion is true. In that case, in the conditional, the result is true, yet in the biconditional the result is false.

Quantity: How much?
Quality: Affirmative, negative


  • - asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In standard mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. This proof by contradiction is not constructively valid. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.

There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories such as IZF and the study of topos theory.

Constructivism is often identified with intuitionism, although intuitionism is only one constructivist program. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics.

  • - an approach to mathematical constructivism following the program of axiomatic set theory. That is, it uses the usual first-order language of classical set theory, and although of course the logic is constructive, there is no explicit use of constructive types. Rather, there are just sets, thus it can look very much like classical mathematics done on the most common foundations, namely the Zermelo–Fraenkel axioms (ZFC).

In 1973, John Myhill proposed a system of set theory based on intuitionistic logic taking the most common foundation, ZFC, and throwing away the axiom of choice (AC) and the law of the excluded middle (LEM), leaving everything else as is. However, different forms of some of the ZFC axioms which are equivalent in the classical setting are inequivalent in the constructive setting, and some forms imply LEM. The system, which has come to be known as IZF, or Intuitionistic Zermelo–Fraenkel (ZF refers to ZFC without the axiom of choice), has the usual axioms of extensionality, pairing, union, infinity, separation and power set. The axiom of regularity is stated in the form of an axiom schema of set induction. Also, while Myhill used the axiom schema of replacement in his system, IZF usually stands for the version with collection.


Univalent foundations

  • (−2)-types are the contractible ones,
  • (−1)-types are the truth values,
  • 0-types are the sets,
  • 1-types are the groupoids,
  • etc.

"So UA can be stated: 'Identity is isomorphic to isomorphism'"

"In the presence of a universe, UA is equivelent to invarience."


  • - a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance. A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.

Sometimes, a set is endowed with more than one structure simultaneously; this enables mathematicians to study it more richly. For example, an ordering imposes a rigid form, shape, or topology on the set. As another example, if a set has both a topology and is a group, and these two structures are related in a certain way, the set becomes a topological group.

Mappings between sets which preserve structures (so that structures in the source or domain are mapped to equivalent structures in the destination or codomain) are of special interest in many fields of mathematics. Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures.

  • - consists of a set along with a collection of finitary operations and relations that are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures with no relation symbols.

From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. For a given theory in model theory, a structure is called a model, if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a semantic model when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as interpretations.

In database theory, structures with no functions are studied as models for relational databases, in the form of relational models.


  • [22] - "Category theory generalises some intuitive relations, such as how numbers combine (e.g. via addition or multiplication). Instead of discussing numbers, however, category theory considers abstract 'objects'. This field of mathematics explore how object relate and compose. Some category theory concepts can be translated to code. These universal abstractions can form the basis of a powerful and concise software design vocabulary.

"The design patterns movement was an early attempt to create such a vocabulary. I think using category theory offers the chance of a better vocabulary, but fortunately, all the work that went into design patterns isn't wasted. It seems to me that some design patterns are essentially ad-hoc, informally specified, specialised instances of basic category theory concepts. There's quite a bit of overlap. This should further strengthen the argument that category theory is valuable in programming, because some of the concepts are equivalent to design patterns that have already proven useful."

Category theory is a mathematical approach to the study of algebraic structure that has become an important tool in theoretical computing science, particularly for semantics-based research. The aim of this course is to teach the basics of category theory, in a way that is accessible and relevant to computer scientists. The emphasis is on gaining a good understanding the basic definitions, examples, and techniques, so that students are equipped for further study on their own of more advanced topics if required.

  • 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.


  • - means the real numbers, or the corresponding (infinite) cardinal number, c. Georg Cantor proved that the cardinality c is larger than the smallest infinity, namely, aleph 0. He also proved that c equals 2^aleph 0, the cardinality of the power set of the natural numbers.

  • - named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axio, which states that for each set there is a Grothendieck universe it belongs to (see below). Tarski's axiom implies the existence of inaccessible cardinals, providing a richer ontology than that of conventional set theories such as ZFC. For example, adding this axiom supports category theory. The Mizar system and Metamath use Tarski–Grothendieck set theory for formal verification of proofs.

  • - the set of all subsets of S, including the empty set and S itself. The power set of a set S is variously denoted as P(S), ℘(S) (using the "Weierstrass p"), P(S), ℙ(S), or, identifying the powerset of S with the set of all functions from S to a given set of two elements, 2S. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. Any subset of P(S) is called a family of sets over S.

  • - or Gödel's constructible universe, denoted L, is a particular class of sets that can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this, he proved that the constructible universe is an inner model of ZF set theory, and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.


  • - one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in x+2=5 the letter x is unknown, but the law of inverses can be used to discover its value: x=3. In E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words.

The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.

  • - the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) and connected problems like representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic (Czelakowski 2003). Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator (Czelakowski 2003).

  • - a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers (constants), variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic symbols.

  • - a formula of the form A = B, where A and B are expressions that may contain one or several variables called unknowns, and "=" denotes the equality binary relation. Although written in the form of proposition, an equation is not a statement that is either true or false, but a problem consisting of finding the values, called solutions, that, when substituted for the unknowns, yield equal values of the expressions A and B.

"Here is a thought experiment: can you identify 4 big ideas in algebra, ideas that not only provide a powerful set of intellectual priorities for the course but that have rich connections to other fields? Doubt it. Because algebra courses, as designed, have no big ideas, as taught, just a list of topics. Look at any textbook: each chapter is just a new tool. There is no throughline to the course nor are their priority ideas that recur and go deeper, by design. In fact, no problems ever require work from many chapters simultaneously, just learning and being quizzed on each topic – a telling sign.

"Here’s a simple test, if you doubt this point, to give to algebra students in order to see if they have any understanding of what they have learned:

  • What can algebra do that arithmetic cannot do or does very inefficiently?
  • Is the order of operations a matter of core truth or convention? How does that compare with the Associative Property? What is and isn’t arbitrary in algebra?
  • Why, mathematically speaking, are imaginary numbers and the inability to divide by zero wise premises?
  • What is an equation that states the proper relationship between feet and yards? (60-80% of students will wrongly write: 3F = Y, showing that they have failed to understand the difference between English and algebra)"

  • - an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

  • - a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression. In the latter case, the variables appearing in the coefficients are often called parameters, and must be clearly distinguished from the other variables.

For example, in 7x^{2}-3xy+1.5+y, the first two terms respectively have the coefficients 7 and −3. The third term 1.5 is a constant. The final term does not have any explicitly written coefficient, but is considered to have coefficient 1, since multiplying by that factor would not change the term.

  • - The variables appearing in the coefficients are often called parameters. Mathematical functions have one or more arguments that are designated in the definition by variables. A function definition can also contain parameters, but unlike variables, parameters are not listed among the arguments that the function takes. When parameters are present, the definition actually defines a whole family of functions, one for every valid set of values of the parameters.

  • - or polynomial equation is an equation of the form P = Q where P and Q are polynomials with coefficients in some field, often the field of the rational numbers. For most authors, an algebraic equation is univariate, which means that it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called multivariate and the term polynomial equation is usually preferred to algebraic equation.

Abstract algebra

Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex algebraic structures include vector spaces, modules, and algebras.

The properties of specific algebraic structures are studied in abstract algebra. The general theory of algebraic structures has been formalized in universal algebra. The language of category theory is used to express and study relationships between different classes of algebraic and non-algebraic objects. This because it is sometimes possible to find strong connections between some classes of objects, sometimes of different kinds. For example, Galois theory establishes a connection between certain fields and groups: two algebraic structures of different kinds.

Universal algebra
  • - sometimes called general algebra is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes "the theory of groups" as an object of study.

Boolean algebra / propositional logic/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.

Propositional calculus is about the simplest kind of logical calculus in current use. It can be extended in several ways. (Aristotelian "syllogistic" calculus, which is largely supplanted in modern logic, is in some ways simpler – but in other ways more complex – than propositional calculus.) The most immediate way to develop a more complex logical calculus is to introduce rules that are sensitive to more fine-grained details of the sentences being used.

  • - atomic formulas are called propositional variables. which can either be true or false. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher logics. In a sense, these are nullary (i.e. 0-arity) predicates.

Predicate / first-order 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.

With the tools of first-order logic it is possible to formulate a number of theories, either with explicit axioms or by rules of inference, that can themselves be treated as logical calculi. Arithmetic is the best known of these; others include set theory and mereology.

  • - commonly understood to be a Boolean-valued function P: X→ {true, false}, called the predicate on X; a statement that may be true or false depending on the values of its variables. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. So, for example, when a theory defines the concept of a relation, then a predicate is simply the characteristic function or the indicator function of a relation. However, not all theories have relations, or are founded on set theory, and so one must be careful with the proper definition and semantic interpretation of a predicate.

  • - a formal system used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus', the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. This distinguishes it from propositional logic which does not use quantifiers (except possibly over truth values or propositions).

  • - a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a propositional function can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.

It is usually denoted by the turned A () logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("∀x", "∀(x)", or sometimes by "(x)" alone). Universal quantification is distinct from existential quantification ("there exists"), which asserts that the property or relation holds only for at least one member of the domain.

  • - a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". Some sources use the term existentialization to refer to existential quantification. It is usually denoted by the turned E () logical operator symbol, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)"). Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain.

  • - a notation that specifies places in an expression where substitution may take place. Some older books use the terms real variable and apparent variable for free variable and bound variable. The idea is related to a placeholder (a symbol that will later be replaced by some literal string), or a wildcard character that stands for an unspecified symbol.

In computer programming, the term free variable refers to variables used in a function that are not local variables nor parameters of that function. The term non-local variable is often a synonym in this context. A bound variable is a variable that was previously free, but has been bound to a specific value or set of values.

  •ödel%27s_completeness_theorem - a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems.

Other logics

  • Second-order and Higher-order Logic - Second-order logic is an extension of first-order logic where, in addition to quantifiers such as “for every object (in the universe of discourse),” one has quantifiers such as “for every property of objects (in the universe of discourse).” This augmentation of the language increases its expressive strength, without adding new non-logical symbols, such as new predicate symbols. For classical extensional logic (as in this entry), properties can be identified with sets, so that second-order logic provides us with the quantifier “for every set of objects.”

Categorical logic

Proof calculus

  • - corresponds to a family of formal systems that use a common style of formal inference for its inference rules. The specific inference rules of a member of such a family characterize the theory of a logic.

Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determining and can be used for radically different logics.

Loosely speaking, a proof calculus is a template or design pattern, characterized by a certain style of formal inference, that may be specialized to produce specific formal systems, namely by specifying the actual inference rules for such a system. There is no consensus among logicians on how best to define the term.

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.

Linear logic

Sequential logic

  • - a type of logic circuit whose output depends not only on the present value of its input signals but on the sequence of past inputs, the input history. This is in contrast to combinational logic, whose output is a function of only the present state of input. That is, sequential logic has state (memory) while combinational logic does not. Or, in other words, sequential logic is combinational logic with memory.

Combinatorial logic

Model theory

  • - the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a formal language. We call a theory a set of sentences in a formal language, and model of a theory a structure (e.g. an interpretation) that satisfies the sentences of that theory.

Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang and Keisler (1990):

  • universal algebra + logic = model theory.

Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997):

  • model theory = algebraic geometry − fields

although model theorists are also interested in the study of fields. Other nearby areas of mathematics include combinatorics, number theory, arithmetic dynamics, analytic functions, and non-standard analysis.

In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. The most prominent professional organization in the field of model theory is the Association for Symbolic Logic.

Proof theory

  • - a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applications in computer science, linguistics, and philosophy.

to sort

  • - a formal system with a set of logical rules for reasoning rigorously about the correctness of computer programs. It was proposed in 1969 by the British computer scientist and logician C. A. R. Hoare, and subsequently refined by Hoare and other researchers. The original ideas were seeded by the work of Robert Floyd, who had published a similar system for flowcharts.

See also Computing#Computation, Semantic web

Non-classical logic

all nc?

Fuzzy logic

Modal logic

to sort

to sort

  • - dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean spaaaaaace of dimension n that preserve a fixed point, where the group operation is given by composing transformations. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, and an orthogonal matrix is a real matrix whose inverse equals its transpose.


  • - a finite number of "places". In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple. For a given set of k-tuples, a truth value is assigned to each k-tuple according to whether the property does or does not hold.

  • - used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in arithmetic, "is congruent to" in geometry, "is adjacent to" in graph theory, "is orthogonal to" in linear algebra and many more. The concept of function is defined as a special kind of binary relation. A binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.
  • - a category that has some of the structure of the category of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in this sense the theory of allegories is a generalization of relation algebra to relations between different sorts. Allegories are also useful in defining and investigating certain constructions in category theory, such as exact completions.

  • - or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place.

Just as a binary relation is formally defined as a set of pairs, i.e. a subset of the Cartesian product A × B of some sets A and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product A × B × C of three sets A, B and C.

  • - a binary relation R over a set X is reflexive if every element of X is related to itself. Formally, this may be written ∀x ∈ X : x R x. An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.

  • - a binary relation that is at the same time a reflexive relation, a symmetric relation and a transitive relation. As a consequence of these properties an equivalence relation provides a partition of a set into equivalence classes.

Operations and functions

  • - a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function.

  • - or function symbol, is a logical symbol that may be applied to an object term to produce another object term. Functional predicates are also sometimes called mappings, but that term has other meanings as well.

Specifically, the symbol F in a formal language is a functional symbol if, given any symbol X representing an object in the language, F(X) is again a symbol representing an object in that language. In typed logic, F is a functional symbol with domain type T and codomain type U if, given any symbol X representing an object of type T, F(X) is a symbol representing an object of type U. One can similarly define function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in zero variables is simply a constant symbol.

  • - a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space (including metric spaces), a vector space, or both. Namely, if Y is a field, functions have inherent vector structure with two operations of pointwise addition and multiplication to a scalar. Topological and metrical structures of function spaces are more diverse.
  • - or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still have an exponential law.

  • - an operation that takes a finite number of input values to produce an output, like those of arithmetic. Operations on infinite numbers of input values are called infinitary.

  • - or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them. This concept is used in algebraic structures such as groups. The term identity element is often shortened to identity (as will be done in this article) when there is no possibility of confusion.

  • - refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.

The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are structure-preserving functions. In category theory, morphisms are sometimes also called arrows.

  • - a morphism (or homomorphism) from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map, f: V → V, and an endomorphism of a group, G, is a group homomorphism f: G → G. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to itself. In any category, the composition of any two endomorphisms of X is again an endomorphism of X. It follows that the set of all endomorphisms of X forms a monoid.

  • - create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function h : G → H is a group homomorphism if whenever a ∗ b = c we have h(a) ⋅ h(b) = h(c). In other words, the group H in some sense has a similar algebraic structure as G and the homomorphism h preserves that.

  • - a homomorphism or morphism (i.e. a mathematical mapping) that admits an inverse. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences. For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective.

In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called diffeomorphisms.

A canonical isomorphism is a canonical map that is an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space V to its second dual space is a canonical isomorphism; on the other hand, V is isomorphic to its dual space but not canonically in general.

Isomorphisms are formalized using category theory. A morphism f : X → Y in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism g : Y → X in that category such that gf = 1X and fg = 1Y, where 1X and 1Y are the identity morphisms of X and Y, respectively.[

f :: a -> b
g :: b -> a
g.f = id^a
f.g = id^b

  • - an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

The identity morphism (identity mapping) is called the trivial automorphism in some contexts. Respectively, other (non-identity) automorphisms are called nontrivial automorphisms.

  • - refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function. There are also a few, less common uses in logic and graph theory. Mapping is sometimes used for non sets of numbers.

  • - or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence (a.k.a. bijective function), which uniquely maps all elements in both domain and codomain to each other, (see figures).

  • - or onto, or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y. It is not required that x is unique; the function f may map one or more elements of X to the same element of Y. Image overs the whole codomain.
  • - categorical analogues of surjective functions (and in the category of sets the concept corresponds to the surjective functions), but it may not exactly coincide in all contexts. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category C is a monomorphism in the dual category Cop).

  • - total function between the elements of two sets, where every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set. injective and surjective

  • - from X to Y (written as f: X ↛ Y) is a function f: X ′ → Y, for some subset X ′ of X. It generalizes the concept of a function f: X → Y by not forcing f to map every element of X to an element of Y (only some subset X ′ of X). If X ′ = X, then f is called a total function and is equivalent to a function. Partial functions are often used when the exact domain, X, is not known (e.g. many functions in computability theory).

  • - a calculation from zero or more input values (called operands) to an output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations of arity 2, such as addition and multiplication, and unary operations of arity 1, such as additive inverse and multiplicative inverse. An operation of arity zero, or 0-ary operation is a constant. The mixed product is an example of an operation of arity 3, or ternary operation. Generally, the arity is supposed to be finite, but infinitary operations are sometimes considered. In this context, the usual operations, of finite arity are also called finitary operations.

It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, or noncommutative operations. The idea that simple operations, such as multiplication and addition of numbers, are commutative was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized. A corresponding property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of the order of the two.

  • - the pointwise application of one function to the result of another to produce a third function. For instance, the functions f : X → Y and g : Y → Z can be composed to yield a function which maps x in X to g(f(x)) in Z. The composition of functions is a special case of the composition of relations, so all properties of the latter are true of composition of functions. The composition of functions has some additional properties.

  • - of a function or operation is the number of arguments or operands that the function takes. The arity of a relation (or predicate) is the dimension of the domain in the corresponding Cartesian product. (A function of arity n thus has arity n+1 considered as a relation.) The term springs from words like unary, binary, ternary, etc. Unary functions or predicates may be also called "monadic"; similarly, binary functions may be called "dyadic".

In mathematics arity may also be named rank, but this word can have many other meanings in mathematics. In logic and philosophy, arity is also called adicity and degree. In linguistics, arity is usually named valency.

In computer programming, there is often a syntactical distinction between operators and functions; syntactical operators usually have arity 0, 1, or 2 (the ternary operator ?: is also common). Functions vary widely in the number of arguments, though large numbers can become unwieldy. Some programming languages also offer support for variadic functions, i.e., functions syntactically accepting a variable number of arguments.

  • - the property of certain operations in mathematics and computer science, that can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency).

  • - A set is closed under an operation if that operation returns a member of the set when evaluated on members of the set. Sometimes the requirement that the operation be valued in a set is explicitly stated, in which case it is known as the axiom of closure. For example, one may define a group as a set with a binary product operator obeying several axioms, including an axiom that the product of any two elements of the group is again an element. However the modern definition of an operation makes this axiom superfluous; an n-ary operation on S is just a subset of Sn+1. By its very definition, an operator on a set cannot have values outside the set.

Unary operation

Binary operation

  • - the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

  • - a mathematical operation that returns a set (or product set or simply product) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Although the Cartesian product is traditionally applied to sets, category theory provides a more general interpretation of the product of mathematical structures. This is distinct from, although related to, the notion of a Cartesian square in category theory, which is a generalization of the fiber product.

Order and domain
  • - a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that".

  • - or monotone function, is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
  • - a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of the orders can be obtained from the other just by renaming of elements.

  • - a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and has close relations to topology. An alternative important approach to denotational semantics in computer science is that of metric spaces.



Monoids are studied in semigroup theory, because they are semigroups with identity. Monoids occur in several branches of mathematics; for instance, they can be regarded as categories with a single object. Thus, they capture the idea of function composition within a set. In fact, all functions from a set into itself form naturally a monoid with respect to function composition. Monoids are also commonly used in computer science, both in its foundational aspects and in practical programming.

The set of strings built from a given set of characters is a free monoid. The transition monoid and syntactic monoid are used in describing finite state machines, whereas trace monoids and history monoids provide a foundation for process calculi and concurrent computing. Some of the more important results in the study of monoids are the Krohn–Rhodes theorem and the star height problem. The history of monoids, as well as a discussion of additional general properties, are found in the article on semigroups.


  • - (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure. Specifically, a magma consists of a set, M, equipped with a single binary operation, M × M → M. The binary operation must be closed by definition but no other properties are imposed.


  • - a set of elements together with an operation that combines any two of its elements to form a third element satisfying four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation; the addition of any two integers forms another integer.

The abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity.

The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups.

In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A theory has been developed for finite groups, which culminated with the classification of finite simple groups announced in 1983. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.

  • - a way of interpreting the elements of the group as "acting" on some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group.

  • - a group of which the group operation is to be thought of as addition in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.

  • - also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers.

  • - also sometimes called a noncommutative group, is a group (G, * ) in which there are at least two elements a and b of G such that a * b ≠ b * a. The term nonabelian is used to distinguish from the idea of an abelian group, where all of the elements of the group commute.

Nonabelian groups are pervasive in mathematics and physics. One of the simplest examples of a nonabelian group is the dihedral group of order 6. It is the smallest finite nonabelian group. A common example from physics is the rotation group SO(3) in three dimensions (rotating something 90 degrees away from you and then 90 degrees to the left is not the same as doing them the other way round). Both discrete groups and continuous groups may be nonabelian. Most of the interesting Lie groups are nonabelian, and these play an important role in gauge theory.

  • - In mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually represented notationally by H ≤ G, read as "H is a subgroup of G".

A proper subgroup of a group G is a subgroup H which is a proper subset of G (i.e. H ≠ G). The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. If H is a subgroup of G, then G is sometimes called an overgroup of H.

The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the ordered pair (G, ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures.

  • - describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.

The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations.

Simple groups
Group of Lie type
  • - a group closely related to the group G(k) of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type give the bulk of nonabelian finite simple groups. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups.
  • - defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces.

Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.

The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics. A few examples are the following. The rotation group SO(3) is a symmetry of Euclidean spaaaaaace and all fundamental laws of physics, the Lorentz group O(3,1) is a symmetry group of spacetime of special relativity. The special unitary group SU(3) is the symmetry group of quantum chromodynamics and the symplectic group Sp(m) finds application in hamiltonian mechanics and quantum mechanical versions of it.

Sporadic group


  • - one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

Some specific kinds of commutative rings are given with the following chain of class inclusions: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

  • - a special subset of a ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring similarly to the way that, in group theory, a normal subgroup can be used to construct a quotient group.
  • - an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x of P, which is to say the set of all elements less than or equal to x in P.

  • - also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra. One starts with a ring R and a two-sided ideal I in R, and constructs a new ring, the quotient ring R/I, whose elements are the cosets of I in R subject to special + and ⋅ operations. Quotient rings are distinct from the so-called 'quotient field', or field of fractions, of an integral domain as well as from the more general 'rings of quotients' obtained by localization.

  • - unique factorization domain (UFD) is a commutative ring in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki.

Unique factorization domains appear in the following chain of class inclusions:

Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

  • rings with multiplicative identity: unital ring, unitary ring, ring with unity, ring with identity, or ring with 1
  • rings not requiring multiplicative identity: rng or pseudo-ring.

  • - commonly known as a "whole number", is a number that can be written without a fractional component. For example, 21, 4, and −2048 are integers, while 9.75, 5½, and √2 are not. Z in group theory

Real numbers
  • - 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).

Rational numbers


  • - a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics.

The best known fields are the field of rational numbers and the field of real numbers. The field of complex numbers is also widely used, not only in mathematics, but also in many areas of science and engineering. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Finite fields are used in most cryptographic protocols used for computer security.

Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. The theory of field extensions (including Galois theory) involves the roots of polynomials with coefficients in a field; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations.

  • Every irreducible polynomial over k has distinct roots.
  • Every irreducible polynomial over k is separable.
  • Every finite extension of k is separable.
  • Every algebraic extension of k is separable.
  • Either k has characteristic 0, or, when k has characteristic p > 0, every element of k is a pth power.
  • Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x→xp is an automorphism of k
  • The separable closure of k is algebraically closed.
  • Every reduced commutative k-algebra A is a separable algebra; i.e., {\displaystyle A\otimes _{k}F} A\otimes _{k}F is reduced for every field extension F/k. (see below)
  • Otherwise, k is called imperfect.

In particular, all fields of characteristic zero and all finite fields are perfect.

  • - or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number.

  • - also F2, Z/2Z or Z2, is the Galois field of two elements. It is the smallest finite field. The two elements are nearly always called 0 and 1, being the additive and multiplicative identities, respectively.

  • - named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood. Originally, Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions. Further abstraction of Galois theory is achieved by the theory of Galois connections.

Natural numbers - ℕ
  • - real numbers that have no decimal and are bigger than zero. those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".
Integers - ℤ
  • - a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75,  5 1⁄2, and √2 are not. The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, …), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, …).

Rational numbers - ℚ

Irrational numbers

  • - a mathematical constant, the ratio of a circle's circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes spelled out as "pi" (/paɪ/).

Being an irrational number, π cannot be expressed exactly as a fraction (equivalently, its decimal representation never ends and never settles into a permanent repeating pattern). Still, fractions such as 22/7 and other rational numbers are commonly used to approximate π. The digits appear to be randomly distributed. In particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a transcendental number, i.e., a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.

Ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using geometrical techniques, in Chinese mathematics, and to about five digits in Indian mathematics in the 5th century AD. The historically first exact formula for π, based on infinite series, was not available until a millennium later, when in the 14th century the Madhava–Leibniz series was discovered in Indian mathematics. In the 20th and 21st centuries, mathematicians and computer scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits after the decimal point. Practically all scientific applications require no more than a few hundred digits of π, and many substantially fewer, so the primary motivation for these computations is the quest to find more efficient algorithms for calculating lengthy numeric series, as well as the human desire to break records. The extensive calculations involved have also been used to test supercomputers and high-precision multiplication algorithms.

Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its special role as an eigenvalue, π appears in areas of mathematics and the sciences having little to do with the geometry of circles, such as number theory and statistics. It is also found in cosmology, thermodynamics, mechanics, and electromagnetism. The ubiquity of π makes it one of the most widely known mathematical constants both inside and outside the scientific community; several books devoted to it have been published, the number is celebrated on Pi Day, and record-setting calculations of the digits of π often result in news headlines. Attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits.

Real numbers - ℝ
Complex numbers

  • - i, a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. Zero is considered to be both real and imaginary.

Algebraic numbers

Transcendental numbers






  • - a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli). The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. Modular arithmetic is referenced in number theory, group theory, ring theory, knot theory, abstract algebra, computer algebra, cryptography, computer science, chemistry and the visual and musical arts. It is one of the foundations of number theory, touching on almost every aspect of its study, and provides key examples for group theory, ring theory and abstract algebra.


Peano arithmetic
Number theory

  • YouTube: Elliptic curves - This lecture will explore the history of counting the number of points on elliptic curves, from ancient Greece to the present day. It will also discuss the law of quadratic reciprocity, a highlight of eighteenth century mathematics; recent controversies about cryptographic algorithms; and a unifying principle in mathematics, the Langlands program. Inaugural lecture of Professor Toby Gee, Imperial College London.

  • - the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional spaaaaaace. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional spaaaaaace or equivalently as the quotient of two vectors. Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics and computer vision. In practical applications, they can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them depending on the application.

  • - a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H. The octonions are the largest such algebra, with eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity, namely they are alternative. Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Despite this, they have some interesting properties and are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. Additionally, octonions have applications in fields such as string theory, special relativity, and quantum logic.

Linear algebra / vector spaces

  • - an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable (however, different variables may occur in different terms). A simple example of a linear equation with only one variable, x, may be written in the form: ax + b = 0, where a and b are constants and a ≠ 0. The constants may be numbers, parameters, or even non-linear functions of parameters, and the distinction between variables and parameters may depend on the problem (for an example, see linear regression).

Matrix multiplication

  • - or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context. (In some fields, such as quantum mechanics, the identity matrix is denoted by a boldface one, 1; otherwise it is identical to I.) Less frequently, some mathematics books use U or E to represent the identity matrix, meaning "unit matrix" and the German word "Einheitsmatrix", respectively.

  • - or self-adjoint matrix', is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column

Quadratic equation


Algebraic geometry

See also Physics


  • - a mathematical structure that enlarges the notion of algebraic variety to include, among other things multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety and different schemes) and "varieties" defined over rings (for example Fermat curves are defined over the integers).

Schemes were introduced by Alexander Grothendieck in 1960 in his treatise Éléments de géométrie algébrique; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. By including rationality questions inside the formalism, scheme theory introduces a strong connection between algebraic geometry and number theory, which eventually allowed Wiles' proof of Fermat's Last Theorem.

To be technically precise, a scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets. In other words, it is a locally ringed space which is locally a spectrum of a commutative ring.

  • - A scheme is a space that locally looks like a particularly simple ringed space: an affine scheme. This can be formalised either within the category of locally ringed spaces or within the category of presheaves of sets on the category of affine schemes Aff Aff.

  • - a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance.

The category of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have kernels, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems.

to sort


These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any spaaaaaace of mathematical objects that has a definition of nearness (a topological spaaaaaace) or specific distances between objects (a metric spaaaaaace).

  • - traditionally, the theory of functions of a real variable, is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.
  • - branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
  • - traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics and thermodynamics and also in engineering fields such as; nuclear, aerospace, mechanical and electrical engineering.

Murray R. Spiegel described complex analysis as "one of the most beautiful as well as useful branches of Mathematics". Complex analysis is particularly concerned with analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.


Type theory

See also Computing#Computability theory

Homotopy theory



See Computing


  • - a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th century BC).

Absolute geometery

  • - a geometry based on an axiom system for Euclidean geometry with the parallel postulate removed and none of its alternatives used in place of it. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate.

Euclidean geometry



Synthetic geometry

  • - sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates. According to Felix Klein, "Synthetic geometry is that which studies figures as such, without recourse to formulas, whereas analytic geometry consistently makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates."

A defining characteristic of synthetic geometry is the use of the axiomatic method to draw conclusions and solve problems, as opposed to analytic and algebraic geometries, where one would use analysis and algebraic techniques to obtain these geometric results.

Euclidean geometry, as presented by Euclid, is the quintessential example of the use of the synthetic method. However, only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach to the subject. As a field of study, synthetic geometry was most prominent during the nineteenth century when some geometers rejected coordinate methods in establishing the foundations of projective geometry and non-Euclidean geometries.

a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set. The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them.

Analytic geometry

  • - also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.
  • - a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.

Differential geometry

  • - uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field.

  • - a surface that locally minimizes its area. This is equivalent to (see definitions below) having a mean curvature of zero. The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum.

Discrete geometry

  • - branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

Discrete geometry has large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.

Computational geometry

  • - a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry. While modern computational geometry is a recent development, it is one of the oldest fields of computing with history stretching back to antiquity. An ancient precursor is the Sanskrit treatise Shulba Sutras , or "Rules of the Chord", that is a book of algorithms written in 800 BCE. The book prescribes step-by-step procedures for constructing geometric objects like altars using a peg and chord.

Computational complexity is central to computational geometry, with great practical significance if algorithms are used on very large datasets containing tens or hundreds of millions of points. For such sets, the difference between O(n2) and O(n log n) may be the difference between days and seconds of computation.

The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (CAD/CAM), but many problems in computational geometry are classical in nature, and may come from mathematical visualization.

Other important applications of computational geometry include robotics (motion planning and visibility problems), geographic information systems (GIS) (geometrical location and search, route planning), integrated circuit design (IC geometry design and verification), computer-aided engineering (CAE) (mesh generation), computer vision (3D reconstruction). The main branches of computational geometry are:

Combinatorial computational geometry, also called algorithmic geometry, which deals with geometric objects as discrete entities. A groundlaying book in the subject by Preparata and Shamos dates the first use of the term "computational geometry" in this sense by 1975. Numerical computational geometry, also called machine geometry, computer-aided geometric design (CAGD), or geometric modeling, which deals primarily with representing real-world objects in forms suitable for computer computations in CAD/CAM systems. This branch may be seen as a further development of descriptive geometry and is often considered a branch of computer graphics or CAD. The term "computational geometry" in this meaning has been in use since 1971.


Geometric transformation

Other geometery

  • - also sometimes called Riemannian geometry, is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p, as all lines in elliptic geometry intersect. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°.

Convex geometry

  • - In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins the pair of points is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex. A convex curve forms the boundary of a convex set. The notion of a convex set can be generalized to other spaces.

  • Feeling Your Way Around in High Dimensions - The simplest objects of interest in any dimension, which are also the basis for approximating arbitrary objects, are the convex polytopes and in this column I'll explain how to begin to probe them...


  • - studies relationships involving lengths and angles of triangles. The field emerged during the 3rd century BC from applications of geometry to astronomical studies. The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are pervasive in both pure and applied mathematics: fundamental methods of analysis such as the Fourier transform, for example, or the wave equation, use trigonometric functions to understand cyclical phenomena across many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology, and biology. Trigonometry is also the foundation of surveying.

  • - a trigonometric function of an angle. The sine of an angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle (that is not the hypotenuse) to the length of the longest side of the triangle (i.e., the hypotenuse).


  • - the mathematical study of shapes and topological spaces. It is an area of mathematics concerned with the properties of space that are preserved under continuous deformations including stretching and bending, but not tearing or gluing. This includes such properties as connectedness, continuity and boundary. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. Such ideas go back to Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.

  • - a type of category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion of localization; they are in a sense a generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic.

  • - the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology. The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points; Compact sets are those that can be covered by finitely many sets of arbitrarily small size; Connected sets are sets that cannot be divided into two pieces that are far apart.
  • - branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

  • - a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.
  • - a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example in physics.

The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities.

A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. In the most general definition of a metric space, the distance between set elements can be negative. Spaces like these are important in the theory of relativity.

  • JTS Topology Suite - an API of spatial predicates and functions for processing geometry. It has the following design goals: JTS conforms to the Simple Features Specification for SQL published by the Open Geospatial Consortium. JTS provides a complete, consistent, robust implementation of fundamental algorithms for processing linear geometry on the 2-dimensional Cartesian plane. JTS is fast enough for production use JTS is written in 100% pure Java.

Reverse mathematics

Probability theory


Information theory

See also Computing, Language

Cellular automaton


SEe also JS scripts#Maths

  • 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. [77]

  • SageMath is a free open-source mathematics software system licensed under the GPL. It builds on top of many existing open-source packages: NumPy, SciPy, matplotlib, Sympy, Maxima, GAP, FLINT, R and many more. Access their combined power through a common, Python-based language or directly via interfaces or wrappers.

Mission: Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab.

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  • Idris - a general purpose pure functional programming language with dependent types. Dependent types allow types to be predicated on values, meaning that some aspects of a program’s behaviour can be specified precisely in the type. It is compiled, with eager evaluation. Its features are influenced by Haskell and ML,


  • a programming language which can be verified statically. You write programs in Albatross and prove them to be correct in the same language., What is a correct program? A program is correct if it is consistent with its specification. Specifications in Albatross are assertions which express correctness conditions. Assertions are boolean expressions in predicate logic. A verified Albatross program has as proof for each assertion. The proof is generated by the compiler.m But since assertions are expressed in predicate logic and predicate logic is not decidable for arbitrary expressions the theorem prover in the Albatross compiler cannot prove all valid assertions. Therefore the programmer has to provide the proof steps which cannot be done by the compiler automatically.



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  • 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.




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  • - a function f that maps a set X to itself, i.e. f : X → X. In other areas of mathematics, a transformation may simply be any function, regardless of domain and codomain. This wider sense shall not be considered in this article; refer instead to the article on function for that sense.

Examples include linear transformations and affine transformations, rotations, reflections and translations. These can be carried out in Euclidean space, particularly in R2 (two dimensions) and R3 (three dimensions). They are also operations that can be performed using linear algebra, and described explicitly using matrices.