Non-classical analysis
Encyclopedia
In mathematics
, non-classical analysis is any system of analysis, other than classical real analysis
, and complex, vector, tensor, etc., analysis based upon it.
Such systems include:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, non-classical analysis is any system of analysis, other than classical real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and 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...
, and complex, vector, tensor, etc., analysis based upon it.
Such systems include:
- Abstract Stone duality, a programme to re-axiomatise general topologyGeneral topologyIn mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
directly, instead of using set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
. It is formulated in the style of type theoryType theoryIn mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...
and is in principle computable. It is currently able to characterise the categoryCategory (mathematics)In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of (not necessarily Hausdorff) computably based locally compact spaces. It allows the development of a form of constructive real analysis using topological rather than metricalCauchy sequenceIn mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
arguments. - Chainlet geometry, a recent development of geometric integration theory which incorporates infinitesimals and allows the resulting calculus to be applied to continuous domains without local Euclidean structure as well as discrete domains.
- Constructive analysis, which is built upon a foundation of constructive, rather than classical, logic and set theory.
- Intuitionistic analysis, which is developed from constructive logic like constructive analysis but also incorporates choice sequenceChoice sequenceIn intuitionistic mathematics, a choice sequence is a constructive formulation of a sequence. Since the Intuitionistic school of mathematics, as formulated by L. E. J...
s. - p-adic analysisP-adic analysisIn mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers....
. - Paraconsistent analysis, which is built upon a foundation of paraconsistentParaconsistent logicA paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...
, rather than classical, logic and set theory. - Smooth infinitesimal analysisSmooth infinitesimal analysisSmooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete...
, which is developed in a smooth topos.