In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, constructive analysis is
mathematical analysis done according to some principles of
constructive mathematics.
This contrasts with ''classical analysis'', which (in this context) simply means analysis done according to the (more common) principles of
classical mathematics In the foundations of mathematics, classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory. It stands in contrast to other types of mathematics such as constructive m ...
.
Generally speaking, constructive analysis can reproduce theorems of classical analysis, but only in application to
separable spaces; also, some theorems may need to be approached by
approximation
An approximation is anything that is intentionally similar but not exactly equality (mathematics), equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very ...
s.
Furthermore, many classical theorems can be stated in ways that are
logically equivalent according to
classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...
, but not all of these forms will be valid in constructive analysis, which uses
intuitionistic logic.
Examples
The intermediate value theorem
For a simple example, consider the
intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two import ...
(IVT).
In classical analysis, IVT implies that, given any