Constructivism (mathematics)
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- This article is not about the philosophy of education called constructivism; see constructivism (learning theory).
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 derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence, according to constructivists. See constructive proof.
Constructivism is often confused with intuitionism, but in fact, intuitionism is only one kind of constructivism. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Constructivism does not, and is entirely consonant with an objective view of mathematics.
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Constructivist mathematics
Constructivist mathematics uses constructivist logic, which closely identifies truth with proof. To prove <math>P \or Q<math> constructively we must prove one (or both) of <math>P<math> and <math>Q<math>. To prove <math>\exists_{x\in X} P(x)<math> constructively we must present a particular <math>a\in X<math> together with a proof of <math>P(a)<math>. To prove <math>\forall_{x\in X} P(x)<math> constructively we must present an algorithm that takes any <math>a \in X<math> and outputs a proof of <math>P(a)<math>.
Constructivism also rejects the use of infinite objects, such as infinite sets and sequences.
Example from real analysis
In classical real analysis, one way to construct a real number is as a pair of Cauchy sequences of the rational numbers. This construction doesn't work in constructivist mathematics because the sequences are infinite.
Instead, we can represent a real number as an algorithm <math>f<math> that takes a positive integer <math>n<math> and outputs a pair of rationals <math>(f_\ell(n), f_r(n))<math> such that
- <math>m \le n \implies f_\ell(m) \le f_\ell(n)<math>
- <math>m \le n \implies f_r(n) \le f_r(m)<math>
- <math>0 \le f_r(n) - f_\ell(n) \le {1\over n}<math>
so that as <math>n<math> increases, the interval <math>[f_\ell(n), f_r(n)]<math> gets smaller, and the intersection of the first <math>n<math> such intervals is non-empty. We can use <math>f<math> to compute as close a rational approximation as we like to the real number it represents.
Under this definition, the real number <math>\sqrt{2}<math> can be represented by the algorithm that computes for each <math>0 \le i \le n<math> the largest integer <math>a_i<math> such that <math>a_i^2 \le 2i^2<math> and then outputs the pair <math>\left(\mathrm{max}\left\{{a_i \over i}\right\}, \mathrm{min}\left\{{a_i+1 \over i}\right\}\right)<math>.
This definition corresponds to the classical definition using Cauchy sequences, except for the requirement that the sequences are constructive: that is, we have an algorithm for computing the <math>n<math>th element in the sequence and hence an algorithm for computing an arbitrarily accurate rational approximation to <math>\sqrt{2}<math>.
Note that the constructivity requirement makes the above definition inconsistent with the usual non-constructive definitions of the reals: Since every algorithm <math>\xi<math> must necessarily be a finite sequence over a finite set of instructions <math>\Sigma<math>, there exists a bijective function <math>f: \Sigma^* \rightarrow \mathbb N<math>. Therefore the set of all algorithms has the same cardinality as the set of all naturals. When using a non-constructive definition, Cantor's diagonal argument proves that the reals have higher cardinality than the natural numbers.
Attitude of mathematicians
Traditionally, mathematicians have been suspicious, if not downright antagonistic, towards mathematical constructivism, largely because of the limitations that it poses for constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when he wrote in Die Grundlagen der Mathematik, "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" [1]. (The law of excluded middle is not valid in constructivist logic.) Errett Bishop, in his 1967 work Foundations of Constructive Analysis, worked to dispel these fears by developing a great deal of traditional analysis in a constructive framework. Nevertheless, not every mathematician accepts that Bishop did so successfully, since his book is necessarily more complicated than a classical analysis text would be. In any case, most mathematicians see no need to restrict themselves to constructivist methods, even if this can be done.
[1] Translation from the Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/mathematics-constructive/.
Mathematicians who have contributed to constructivism
Branches
See also
External links
- Computability Logic Homepage (http://www.cis.upenn.edu/~giorgi/cl.html)
- Stanford Encyclopedia of Philosophy entry (http://plato.stanford.edu/entries/mathematics-constructive/)de:Konstruktive Mathematik