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
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
done according to some principles of
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 th ...
.
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 space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of th ...
s; 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
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
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
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
.
Examples
The intermediate value theorem
For a simple example, consider the
intermediate value theorem (IVT).
In classical analysis, IVT implies that, given any
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
''f'' from a
closed interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
'a'',''b''to the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
''R'', if ''f''(''a'') is
negative while ''f''(''b'') is
positive
Positive is a property of positivity and may refer to:
Mathematics and science
* Positive formula, a logical formula not containing negation
* Positive number, a number that is greater than 0
* Plus sign, the sign "+" used to indicate a posi ...
, then there exists a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
''c'' in the interval such that ''f''(''c'') is exactly
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
.
In constructive analysis, this does not hold, because the constructive interpretation of
existential quantification
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, ...
("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
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
).
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
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
''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
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
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.
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 that imply it and that are commonly met; for example, every
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. Functions of each type are infinitely differentiable, but complex an ...
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 (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 ...
, 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
In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if ev ...
, that any
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of the real line R has a
least upper bound
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
(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
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an eleme ...
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 many commonly studied sets, including all
intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets
* A statistical level of measurement
* Interval e ...
and all
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
s.
Closely related to this, in constructive mathematics, fewer characterisations of
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
s are constructively valid—or from another point of view, there are several different concepts that are classically equivalent but not constructively equivalent.
Indeed, if the interval
'a'',''b''were
sequentially compact
In mathematics, a topological space ''X'' is sequentially compact if every sequence of points in ''X'' has a convergent subsequence converging to a point in X.
Every metric space is naturally a topological space, and for metric spaces, the notio ...
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
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
of the
infinite sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
(''c''
''n'')
''n''∈N.
Uncountability of the real numbers
The diagonal construction in
Cantors theorem is
intuitionistically valid.
Indeed, the constructive component of the diagonal argument already appeared in Cantor's work.
Akihiro Kanamori
is a Japanese-born American mathematician. He specializes in set theory and is the author of the monograph on large cardinal property, large cardinals, ''The Higher Infinite''. He has written several essays on the history of mathematics, especia ...
, "The Mathematical Development of Set Theory from Cantor to Cohen", '' Bulletin of Symbolic Logic'' / Volume 2 / Issue 01 / March 1996, pp 1-71 According to
Kanamori, ''a historical misrepresentation has been perpetuated that associates diagonalization with non-constructivity''. As a result, the real numbers
are uncountable in any constructive system.
In some
models,
is
subcountable.
A variant found in constructive analysis textbooks may go as follows:
"Let be a sequence of real numbers. Let ''x''
0 and ''y''
0 be real numbers, ''x''
0 < ''y''
0. Then there exists a real number ''x'' with ''x''
0 ≤ ''x'' ≤ ''y''
0 and ''x'' ≠''a''
''n'' (''n'' ∈ N) . . .
The proof is essentially Cantor's '
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δΠ...
' proof." (Theorem 1 in
Errett Bishop
Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an Americans, American mathematician known for his work on analysis. He expanded constructive analysis in his 1967 ''Foundations of Constructive Analysis'', where he Mathematical proof, p ...
, ''Foundations of Constructive Analysis'', 1967, page 25.)
Sequences of reals appear commonly in analysis. Forms of constructive analysis that reject not just the
law of 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 noncontradic ...
but also the
limited principle of omniscience and even
Markov's principle
Markov's principle, named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below.
The principle is logically valid classically, but not in intuitionistic constructive m ...
may make use of the
axiom of dependent choice In mathematics, the axiom of dependent choice, denoted by \mathsf , is a weak form of the axiom of choice ( \mathsf ) that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores whi ...
for sequences of reals.
See also
*
Computable analysis
In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functional analysis that can be carried out in a co ...
*
Indecomposability (constructive mathematics)
In intuitionistic analysis and in computable analysis, indecomposability or indivisibility (german: Unzerlegbarkeit, from the adjective ''unzerlegbar'') is the principle that the continuum cannot be partitioned into two nonempty pieces. This pr ...
*
Constructive nonstandard analysis In mathematics, constructive nonstandard analysis is a version of Abraham Robinson's nonstandard analysis, developed by Moerdijk (1995), Palmgren (1998), Ruokolainen (2004). Ruokolainen wrote:
:The possibility of constructivization of nonstanda ...
References
Further reading
*
{{DEFAULTSORT:Constructive Analysis
*
*
Intuitionism