Constructivist analysis
Encyclopedia
In mathematics
, constructive analysis is mathematical analysis
done according to the principles of constructive mathematics.
This contrasts with classical analysis, which (in this context) simply means analysis done according to the (ordinary) principles of classical mathematics
.
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
s.
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
.
(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.
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 cn in the interval such that the absolute value
of f(cn) 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.
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).
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.
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.
Closely related to this, in constructive mathematics, fewer characterisations of compact space
s 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 (cn)n.
' proof." (Theorem 1 in Errett Bishop
, Foundations of Constructive Analysis, 1967, page 25.)
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...
, constructive analysis is mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
done according to the principles of constructive mathematics.
This contrasts with classical analysis, which (in this context) simply means analysis done according to the (ordinary) principles of classical mathematics
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 mathematics or predicative mathematics; theories other...
.
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
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
s.
Furthermore, many classical theorems can be stated in ways that are logically equivalent according to classical logic
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
, but not all of these forms will be valid in constructive analysis, which uses intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
.
The intermediate value theorem
For a simple example, consider the intermediate value theoremIntermediate value theorem
In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....
(IVT).
In classical analysis, IVT says that, given any continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
f from a closed interval [a,b] to the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R, if f(a) is negative while f(b) is positive, then there exists a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
c in the interval such that f(c) is exactly zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
.
In constructive analysis, this does not hold, because the constructive interpretation of existential quantification
Existential quantification
In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. It is denoted by the logical operator symbol ∃ , which is called the existential quantifier...
("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
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
).
But if f hovers near zero during a stretch along its domain, then this cannot necessarily be done.
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
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
n (no matter how large), there exists (that is, we can construct) a real number cn in the interval such that the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
of f(cn) 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.
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
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
is locally non-zero (assuming that it already satisfies f(a) < 0 and f(b) > 0).
For another way to view this example, notice that according to classical logic
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
, 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.
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 subsetSubset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
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
Logical disjunction
In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...
y is an upper bound
Upper bound
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
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
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
and compact sets.
Closely related to this, in constructive mathematics, fewer characterisations of compact space
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
s 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 (cn)n.
Uncountability of the real numbers
A constructive version of "the famous theorem of Cantor, that the real numbers are uncountable" is: "Let {an} be a sequence of real numbers. Let x0 and y0 be real numbers, x0 < y0. Then there exists a real number x with x0 ≤ x ≤ y0 and x ≠ an (n ∈ Z+) . . . The proof is essentially Cantor's 'diagonalCantor's diagonal argument
Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural...
' proof." (Theorem 1 in Errett Bishop
Errett Bishop
Errett Albert Bishop was an American mathematician known for his work on analysis. He is the father of constructive analysis, because of his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.-Life:Errett Bishop's...
, Foundations of Constructive Analysis, 1967, page 25.)
See also
- Computable analysisComputable analysisIn mathematics, computable analysis is the study of which parts of real analysis and functional analysis can be carried out in a computable manner. It is closely related to constructive analysis.- Basic results :...
- IndecomposabilityIndecomposabilityIn constructive mathematics, indecomposability or indivisibility is the principle that the continuum cannot be partitioned into two nonempty pieces. This principle was established by Brouwer in 1928 using intuitionistic principles, and can also be proven using Church's thesis...