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Univalent Foundations
Univalent foundations are an approach to the foundations of mathematics in which mathematical structures are built out of objects called ''types''. Types in univalent foundations do not correspond exactly to anything in set-theoretic foundations, but they may be thought of as spaces, with equal types corresponding to homotopy equivalent spaces and with equal elements of a type corresponding to points of a space connected by a path. Univalent foundations are inspired both by the old Platonic ideas of Hermann Grassmann and Georg Cantor and by " categorical" mathematics in the style of Alexander Grothendieck. Univalent foundations depart from the use of classical predicate logic as the underlying formal deduction system, replacing it, at the moment, with a version of Martin-Löf type theory. The development of univalent foundations is closely related to the development of homotopy type theory. Univalent foundations are compatible with structuralism, if an appropriate (i.e., categorica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Mathematics
Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (set, function, geometrical figure, number, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their model theory, models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematics, metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher Category Theory
In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as their fundamental weak ∞-groupoid. Strict higher categories An ordinary category has objects and morphisms, which are called 1-morphisms in the context of higher category theory. A 2-category generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to ''n''-morphisms between (''n'' − 1)-morphisms gives an ''n''-category. Just as the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms, the category ''n''-Cat of (small) ''n''-categories is actually a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thierry Coquand
Thierry Coquand (; born 18 April 1961 in Jallieu, Isère, France) is a professor in computer science at the University of Gothenburg, known for his work in constructive mathematics, especially the calculus of constructions. He received his Ph.D. under the supervision of Gérard Huet. See also * Coq * Girard's paradox External links Academic homepage * {{DEFAULTSORT:Coquand, Thierry French computer scientists École Normale Supérieure alumni 20th-century French mathematicians 21st-century French mathematicians University of Gothenburg faculty 1961 births Living people ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marc Bezem
Marc or MARC may refer to: People * Marc (given name), people with the first name * Marc (surname), people with the family name Acronyms * MARC standards, a data format used for library cataloging, * MARC Train, a regional commuter rail system of the State of Maryland, serving Maryland, Washington, D.C., and eastern West Virginia * MARC (archive), a computer-related mailing list archive * M/A/R/C Research, a marketing research and consulting firm * Massachusetts Animal Rights Coalition, a non-profit, volunteer organization * Matador Automatic Radar Control, a guidance system for the Martin MGM-1 Matador cruise missile * Mid-America Regional Council, the Council of Governments and the Metropolitan Planning Organization for the bistate Kansas City region * Midwest Association for Race Cars, a former American stock car racing organization * Revolutionary Agrarian Movement of the Bolivian Peasantry (''Movimiento Agrario Revolucionario del Campesinado Boliviano''), a defunct right- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets, there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, a set arising from choosing elements arbitrarily can be made without invoking the axiom of choice; this is, in particular, the case if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguishin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Univalence Axiom
In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies. This includes, among other lines of work, the construction of homotopical and higher-categorical models for such type theories; the use of type theory as a logic (or internal language) for abstract homotopy theory and higher category theory; the development of mathematics within a type-theoretic foundation (including both previously existing mathematics and new mathematics that homotopical types make possible); and the formalization of each of these in computer proof assistants. There is a large overlap between the work referred to as homotopy type theory, and as the univalent foundations project. Although neither is precisely delineated, and the terms are sometimes used interchangeably, the choice of usage also sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Shulman (mathematician)
Michael "Mike" Shulman (; born 1980) is an American associate professor of mathematics at the University of San Diego who works in category theory and higher category theory, homotopy theory, logic as applied to set theory, and computer science. Work Shulman did his undergraduate work at the California Institute of Technology and his postgraduate work at the University of Cambridge and the University of Chicago, where he received his Ph.D. in 2009. His doctoral thesis and subsequent work dealt with applications of category theory to homotopy theory. In 2009, he received a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship. In 2012–13, he was a visiting scholar at the Institute for Advanced Study, where he was one of the official participants in the ''Special Year on Univalent Foundations of Mathematics''. Shulman was one of the principal authors of the book ''Homotopy type theory: Univalent foundations of mathematics'', an informal expositio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Set
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a "well-behaved" topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagram (category Theory)
In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a ''function'' from a fixed index ''set'' to the class of ''sets''. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a ''functor'' from a fixed index ''category'' to some ''category''. The universal functor of a diagram is the diagonal functor; its right adjoint is the limit of the diagram and its left adjoint is the colimit. The natural transformation from the diagonal functor to some arbitrary diagram is called a cone. Definition Formally, a diagram of type ''J'' in a category ''C'' is a ( covariant) functor The category ''J'' is called the index category or the scheme of the diagram ''D''; the functor is sometimes called a ''J' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus Of Inductive Constructions
In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants. Some of its variants include the calculus of inductive constructions (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity). General traits The CoC is a higher-order typed lambda calculus, initially developed by Thierry Coquand. It is well known for being at the top of Barendregt's lambda cube. It is possible within CoC to define functions from terms to terms, as well as terms to types, types to types, and types to terms. The CoC is strongly normalizing, and hence consistent. Usage The CoC has been developed alongsid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excluded Middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law (or principle) of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'': "no third ossibilityis given". It is a tautology. The principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable now, but provable in the futu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical 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. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. 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 CZF ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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