In mathematics, constructive analysis is mathematical analysis done according to the principles of constructivist mathematics. This contrasts with classical analysis, which (in this context) simply means analysis done according to the (ordinary) principles of classical mathematics. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... Classical mathematics, as a term of art in mathematical logic, refers generally to mathematics constructed and proved on the basis of classical logic and ZFC set theory, i. ...

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 approximations. Furthermore, many classical theorems can be stated in ways that are logically equivalent according to classical logic, but not all of these forms will be valid in constructive analysis, which uses intuitionistic logic. In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ... It has been suggested that this article or section be merged with estimation. ... In logic, statements p and q are logically equivalent if they have the same logical content. ... Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...

Examples

The intermediate value theorem

For a simple example, consider the intermediate value theorem (IVT). In classical analysis, IVT says that, given any continuous function f from a closed interval [a,b] to the real line R, if f(a) is negative while f(b) is positive, then there exists a real number c in the interval such that f(c) is exactly zero. In constructive analysis, this does not hold, because the constructive interpretation of existential quantification ("there exists") requires one to be able to construct the real number c (in the sense that it can be approximated to any desired precision by a rational number). But if f hovers near zero during a stretch along its domain, then this cannot necessarily be done. In analysis, the intermediate value theorem is either of two theorems of which an account is given below. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... In mathematics, the real line is simply the set of real numbers. ... A negative number is a number that is less than zero, such as −3. ... A negative number is a number that is less than zero, such as −3. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ... 0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. ... In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

However, constructive analysis provides several alternative formulations of IVT, all of which are equivalent to the usual form in classical analysis, but not in constructive analysis. For example, under the same conditions on f as in the classical theorem, given any natural number n (no matter how large), there exists (that is, we can construct) a real number c_n in the interval such that the absolute value of f(c_n) is less than 1/n. That is, we can get as close to zero as we like, even if we can't construct a c that gives us exactly zero. In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...

Alternatively, we can keep the same conclusion as in the classical IVT -- a single c such that f(c) is exactly zero -- while strengthening the conditions on f. We require that f be locally non-zero, meaning that given any point x in the interval [a,b] and any natural number m, there exists (we can construct) a real number y in the interval such that |y - x| < 1/m and |f(y)| > 0. In this case, the desired number c can be constructed. This is a complicated condition, but there are several other conditions which imply it and which are commonly met; for example, every analytic function is locally non-zero (assuming that it already satisfies f(a) < 0 and f(b) > 0). In mathematics, an analytic function is a function that is locally given by a convergent power series. ...

For another way to view this example, notice that according to classical logic, if the locally non-zero condition fails, then it must fail at some specific point x; and then f(x) will equal 0, so that IVT is valid automatically. Thus in classical analysis, which uses classical logic, in order to prove the full IVT, it is sufficient to prove the constructive version. From this perspective, the full IVT fails in constructive analysis simply because constructive analysis does not accept classical logic. Conversely, one may argue that the true meaning of IVT, even in classical mathematics, is the constructive version involving the locally non-zero condition, with the full IVT following by "pure logic" afterwards. Some logicians, while accepting that classical mathematics is correct, still believe that the constructive approach gives a better insight into the true meaning of theorems, in much this way. Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ...

The least upper bound principle and compact sets

Another difference between classical and constructive analysis is that constructive analysis does not accept the least upper bound principle, that any subset of the real line R has a least upper bound (or supremum), possibly infinite. However, as with the intermediate value theorem, an alternative version survives; in constructive analysis, any located subset of the real line has a supremum. (Here a subset S of R is located if, whenever x < y are real numbers, either there exists an element s of S such that x < s, or y is an upper bound of S.) Again, this is classically equivalent to the full least upper bound principle, since every set is located in classical mathematics. And again, while the definition of located set is complicated, nevertheless it is satisfied by several commonly studied sets, including all intervals and compact sets. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... OR logic gate. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ...

Closely related to this, in constructive mathematics, fewer characterisations of compact spaces are constructively valid -- or from another point of view, there are several different concepts which are classically equivalent but not constructively equivalent. Indeed, if the interval [a,b] were sequentially compact in constructive analysis, then the classical IVT would follow from the first constructive version in the example; one could find c as a cluster point of the infinite sequence (c_n)_n. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ... This is a page about mathematics. ...