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| In the [[philosophy of mathematics]], '''constructivism''' asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. When one assumes that an object does not exist and [[reductio ad absurdum|derives a contradiction from that assumption]], one still has not found the object and therefore not proved its existence, according to constructivism. This viewpoint involves a verificational interpretation of the existence quantifier, which is at odds with its classical interpretation.
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| There are many forms of constructivism.<ref>Troelstra 1977a:974</ref> These include the program of [[intuitionism]] founded by [[Luitzen Egbertus Jan Brouwer|Brouwer]], the [[finitism]] of [[David Hilbert|Hilbert]] and [[Paul Bernays|Bernays]], the constructive recursive mathematics of [[Nikolai Aleksandrovich Shanin|Shanin]] and [[Andrey Markov (Soviet mathematician)|Markov]], and [[Errett Bishop|Bishop]]'s program of [[constructive analysis]]. Constructivism also includes the study of [[constructive set theory|constructive set theories]] such as [[IZF]] and the study of [[topos theory]].
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| Constructivism is often identified with intuitionism, although intuitionism is only one constructivist program. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity.<ref>Troelstra 1977b:1</ref> Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics.
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| == Constructive mathematics ==
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| Much constructive mathematics uses [[intuitionistic logic]], which is essentially [[classical logic]] without the [[law of the excluded middle]]. This law states that, for any proposition, either that proposition is true or its negation is. This is not to say that the law of the excluded middle is denied entirely; special cases of the law will be provable. It is just that the general law is not assumed as an [[axiom]]. The [[law of non-contradiction]] (which states that contradictory statements cannot both at the same time be true) is still valid.
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| For instance, in [[Heyting arithmetic]], one can prove that for any proposition ''p'' which ''does not contain [[quantifiers]]'', <math>\forall x,y,z,\ldots \in \mathbb{N} : p \vee \neg p</math> is a theorem (where ''x'', ''y'', ''z'' ... are the [[free variables]] in the proposition ''p''). In this sense, propositions restricted to the [[finite set|finite]] are still regarded as being either true or false, as they are in classical mathematics, but this [[law of bivalence|bivalence]] does not extend to propositions which refer to [[Infinity|infinite]] collections.
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| In fact, [[L.E.J. Brouwer]], founder of the intuitionist school, viewed the law of the excluded middle as abstracted from finite experience, and then applied to the infinite without [[theory of justification|justification]]. For instance, [[Goldbach's conjecture]] is the assertion that every even number (greater than 2) is the sum of two [[prime number]]s. It is possible to test for any particular even number whether or not it is the sum of two primes (for instance by exhaustive search), so any one of them is either the sum of two primes or it is not. And so far, every one thus tested has in fact been the sum of two primes.
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| But there is no known proof that all of them are so, nor any known proof that not all of them are so. Thus to Brouwer, we are not justified in asserting "either Goldbach's conjecture is true, or it is not." And while the conjecture may one day be solved, the argument applies to similar unsolved problems; to Brouwer, the law of the excluded middle was tantamount to assuming that ''every'' mathematical problem has a solution.
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| With the omission of the law of the excluded middle as an axiom, the remaining [[logical system]] has an [[existence property]] which classical logic does not: whenever <math>\exists_{x\in X} P(x)</math> is proven constructively, then in fact <math>P(a)</math> is proven constructively for (at least) one particular <math>a\in X</math>, often called a witness. Thus the proof of the existence of a mathematical object is tied to the possibility of its [[construction]].
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| === Example from real analysis ===
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| In classical [[real analysis]], one way to [[construction of the real numbers|define a real number]] is as an [[equivalence class]] of [[Cauchy sequence]]s of [[rational number]]s.
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| In constructive mathematics, one way to construct a real number is as a [[function (mathematics)|function]] ''ƒ'' that takes a positive integer <math>n</math> and outputs a rational ''ƒ''(''n''), together with a function ''g'' that takes a positive integer ''n'' and outputs a positive integer ''g''(''n'') such that
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| :<math>\forall n\ \forall i,j \ge g(n)\quad |f(i) - f(j)| \le {1 \over n}</math>
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| so that as ''n'' increases, the values of ''ƒ''(''n'') get closer and closer together. We can use ''ƒ'' and ''g'' together to compute as close a rational approximation as we like to the real number they represent.
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| Under this definition, a simple representation of the real number [[Euler's number|''e'']] is:
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| :<math>f(n) = \sum_{i=0}^n {1 \over i!}, \quad g(n) = n.</math>
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| This definition corresponds to the classical definition using Cauchy sequences, except with a constructive twist: for a classical Cauchy sequence, it is required that, for any given distance, [[existence theorem|there exists (in a classical sense)]] a member in the sequence after which all members are closer together than that distance. In the constructive version, it is required that, for any given distance, it is possible to actually specify a point in the sequence where this happens (this required specification is often called the [[modulus of convergence]]). In fact, the [[BHK interpretation|standard constructive interpretation]] of the mathematical statement
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| :<math>\forall n : \exists m : \forall i,j \ge m: |f(i) - f(j)| \le {1 \over n}</math> | |
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| is precisely the existence of the function computing the modulus of convergence. Thus the difference between the two definitions of real numbers can be thought of as the difference in the interpretation of the statement "for all... there exists..."
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| This then opens the question as to what sort of [[function (mathematics)|function]] from a [[countable]] [[Set (mathematics)|set]] to a countable set, such as ''f'' and ''g'' above, can actually be constructed. Different versions of constructivism diverge on this point. Constructions can be defined as broadly as [[free choice sequences]], which is the intuitionistic view, or as narrowly as algorithms (or more technically, the [[computable function]]s), or even left unspecified. If, for instance, the algorithmic view is taken, then the reals as constructed here are essentially what classically would be called the [[computable number]]s.
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| === Cardinality ===
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| To take the algorithmic interpretation above would seem at odds with classical notions of [[cardinal number|cardinality]]. By enumerating algorithms, we can show classically that the computable numbers are countable. And yet [[Cantor's diagonal argument]] shows that real numbers have higher cardinality. Furthermore the diagonal argument seems perfectly constructive. To identify the real numbers with the computable numbers would then be a contradiction.
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| And in fact, Cantor's diagonal argument ''is'' constructive, in the sense that given a [[bijection]] between the real numbers and natural numbers, one constructs a real number which doesn't fit, and thereby proves a contradiction. We can indeed enumerate algorithms to construct a function ''T'', about which we initially assume that it is a function from the natural numbers [[onto]] the reals. But, to each algorithm, there may or may not correspond a real number, as the algorithm may fail to satisfy the constraints, or even be non-terminating (''T'' is a [[partial function]]), so this fails to produce the required bijection. In short, one who takes the view that real numbers are effectively computable interprets Cantor's result as showing that the real numbers are not [[recursively enumerable]].
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| Still, one might expect that since ''T'' is a partial function from the natural numbers onto the real numbers, that therefore the real numbers are ''no more than'' countable. And, since every natural number can be [[Trivial (mathematics)|trivially]] represented as a real number, therefore the real numbers are ''no less than'' countable. They are, therefore ''exactly'' countable. However this reasoning is not constructive, as it still does not construct the required bijection. In fact the cardinality of sets fails to be [[total order|totally ordered]] (see [[Cantor–Bernstein–Schroeder theorem]]).
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| === Axiom of choice ===
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| The status of the [[axiom of choice]] in constructive mathematics is complicated by the different approaches of different constructivist programs. One trivial meaning of "constructive", used informally by mathematicians, is "provable in [[ZF set theory]] without the axiom of choice." However, proponents of more limited forms of constructive mathematics would not assert that ZF itself is a constructive system.
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| In intuitionistic theories of [[type theory]] (especially higher-type arithmetic), many forms of the axiom of choice are permitted. For example, the axiom AC<sub>11</sub> can be paraphrased to say that for any relation ''R'' on the set of real numbers, if you have proved that for each real number ''x'' there is a real number ''y'' such that ''R''(''x'',''y'') holds, then there is actually a function ''F'' such that ''R''(''x'',''F''(''x'')) holds for all real numbers. Similar choice principles are accepted for all finite types. The motivation for accepting these seemingly nonconstructive principles is the intuitionistic understanding of the proof that "for each real number ''x'' there is a real number ''y'' such that ''R''(''x'',''y'') holds". According to the [[BHK interpretation]], this proof itself is essentially the function ''F'' that is desired. The choice principles that intuitionists accept do not imply the law of the excluded middle.
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| However, in certain axiom systems for constructive set theory, the axiom of choice does imply the law of the excluded middle (in the presence of other axioms), as shown by the [[Diaconescu-Goodman-Myhill theorem]]. Some constructive set theories include weaker forms of the axiom of choice, such as the [[axiom of dependent choice]] in Myhill's set theory.
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| === Measure theory ===
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| Classical [[measure theory]] makes deep usage of the [[axiom of choice]], which is fundamental to, first, distinction between measurable and [[non-measurable set]]s, the existence of the latter being behind such famous results as the [[Banach–Tarski paradox]], and secondly the hierarchies of notions of measure captured by notions such as [[Borel algebra]]s, which are an important source of intuitions in [[set theory]]. Measure theory provides the foundation for the modern notion of [[integral]], the [[Lebesgue integral]].
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| It is possible to rework measure theory on the basis of the [[computable real]] line, where the set-theoretic basis for measurability is replaced by notions from [[order theory]]. This constructive measure theory provides the basis for computable analogues for Lebesgue integration.
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| == The place of constructivism in mathematics ==
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| Traditionally, some mathematicians have been suspicious, if not antagonistic, towards mathematical constructivism, largely because of limitations they believed it to pose for constructive analysis.
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| These views were forcefully expressed by [[David Hilbert]] in 1928, when he wrote in <cite>''[[Die Grundlagen der Mathematik]]''</cite>, "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists".<ref>Translation from the [[Stanford Encyclopedia of Philosophy]][http://plato.stanford.edu/entries/mathematics-constructive/].</ref>
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| [[Errett Bishop]], in his 1967 work <cite>''Foundations of Constructive Analysis''</cite>, worked to dispel these fears by developing a great deal of traditional analysis in a constructive framework. Nevertheless, some mathematicians{{Who|date=October 2009}} do not accept that Bishop did so successfully, since his book is necessarily more complicated than a classical analysis text would be.
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| Even though most mathematicians do not accept the constructivist's thesis, that only mathematics done based on constructive methods is sound, constructive methods are increasingly of interest on non-ideological grounds. For example, constructive proofs in analysis may ensure [[witness extraction]], in such a way that working within the constraints of the constructive methods may make finding witnesses to theories easier than using classical methods. Applications for constructive mathematics have also been found in [[typed lambda calculus|typed lambda calculi]], [[History of topos theory|topos theory]] and [[categorical logic]], which are notable subjects in foundational mathematics and [[computer science]]. In algebra, for such entities as [[topos]]es and [[Hopf algebra]]s, the structure supports an [[internal language]] that is a constructive theory; working within the constraints of that language is often more intuitive and flexible than working externally by such means as reasoning about the set of possible concrete algebras and their [[homomorphism]]s.
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| Physicist [[Lee Smolin]] writes in [[Three Roads to Quantum Gravity]] that topos theory is "the right form of logic for cosmology" (page 30) and "In its first forms it was called 'intuitionistic logic'" (page 31). "In this kind of logic, the statements an observer can make about the universe are divided into at least three groups: those that we can judge to be true, those that we can judge to be false and those whose truth we cannot decide upon at the present time" (page 28).
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| == Constructive mathematicians who have made major contributions to constructivism ==
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| * [[Leopold Kronecker]] (old constructivism, semi-intuitionism)
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| * [[Luitzen Egbertus Jan Brouwer|L. E. J. Brouwer]] (forefather of intuitionism)
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| * [[Andrey Markov (Soviet mathematician)|A. A. Markov]] (forefather of Russian school of constructivism)
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| * [[Arend Heyting]] (formalized intuitionistic logic and theories)
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| * [[Per Martin-Löf]] (founder of constructive type theories)
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| * [[Errett Bishop]] (promoted a version of constructivism which is consistent with classical mathematics)
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| == Branches ==
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| * [[Constructive logic]]
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| * [[Constructive type theory]]
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| * [[Constructive analysis]]
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| * [[Constructive non-standard analysis]]
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| == See also ==
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| * [[Computability theory]]
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| * [[Constructive proof]]
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| * [[Finitism]]
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| * [[Game semantics]]
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| * [[Intuitionism]]
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| * [[Intuitionistic type theory]]
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| == Notes ==
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| <references/> | |
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| == References ==
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| *[[Solomon Feferman]] (1997), ''Relationships between Constructive, Predicative and Classical Systems of Analysis'', http://math.stanford.edu/~feferman/papers/relationships.pdf.
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| * [[A. S. Troelstra]] (1977a), "Aspects of constructive mathematics", ''Handbook of Mathematical Logic, pp. 973–1052.
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| *[[A. S. Troelstra]] (1977b), ''Choice sequences'', Oxford Logic Guides. ISBN 0-19-853163-X
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| *[[A. S. Troelstra]] (1991), "A History of Constructivism in the 20th Century", University of Amsterdam, ITLI Prepublication Series ML-91-05, http://staff.science.uva.nl/~anne/hhhist.pdf,
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| *[[Harold Edwards (mathematician)|H. M. Edwards]] (2005), ''Essays in Constructive Mathematics'', Springer-Verlag, 2005, ISBN 0-387-21978-1
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| * [[Douglas Bridges]], [[Fred Richman]], "Varieties of Constructive Mathematics", 1987.
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| * [[Michael J. Beeson]], "Foundations of constructive mathematics: metamathematical studies", 1985.
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| * [[Anne Sjerp Troelstra]], [[Dirk van Dalen]], "Constructivism in Mathematics: An Introduction, Volume 1", 1988
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| * [[Anne Sjerp Troelstra]], [[Dirk van Dalen]], "Constructivism in Mathematics: An Introduction, Volume 2", 1988
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| == External links == | |
| * {{IEP|con-math|Constructive Mathematics}}
| |
| * [http://plato.stanford.edu/entries/mathematics-constructive/ Stanford Encyclopedia of Philosophy entry]
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| {{logic}}
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| {{DEFAULTSORT:Constructivism (Mathematics)}}
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| [[Category:Epistemology]]
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| [[Category:Constructivism (mathematics)| ]]
| |
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