The mathematical constructivism is a sense of philosophy of mathematics, which represents the ontological position that the existence of mathematical objects must be justified by their construction. Constructivism is an objectivist ( a mathematical object exists independent of thought, its existence is but only by its construction justified) and a subjectivist form taking ( a mathematical object is created as a product of constructing intuition of the mathematician and is the first place made of him here, intuitionism ). Mathematical statements of the form "There is ... " will be refused and - if possible - replaced by sentences of the form "We can construct ... " (eg, "There are irrational numbers, so that rational. " Vs. "We can such. numbers, construct ").
First approaches to constructive mathematics come from the intuitionism of LEJ Brouwer. Other approaches have been developed by Hermann Weyl, Andrei Nikolaevich Kolmogorov and Errett Bishop, Arend Heyting, Solomon Feferman, Paul Lorenzen, Michael J. Beeson and Anne Troelstra Sjerp. The represented especially by Weyl Constructivism was one of the positions that were facing at the beginning of the 20th century in the basic argument of mathematics, but he could not enforce it.
In a constructive proof of the mathematical objects and solutions of problems are actually constructed. Joseph Liouville delivered as a transcendent figure in his evidence that there are transcendental numbers.
The constructive mathematics avoids explicitly non- constructive proofs and comes out with the intuitionistic logic that does not allow non- constructive proofs. If, for example ( as in an indirect proof ) inferred from the falsity of a negated assertion that assertion itself, so this is a logical conclusion mold is used which does not force the construction. The main crux of constructivism, then, is to formulate only those records whose objects ( and solutions ) are constructible. This claim leads to reject applications of the law of excluded middle and the axiom of choice, since both sets statements about mathematical objects ( or solutions ) can be derived without specifying how they are constructed.
In arithmetic can always both perform constructive proofs and non- constructive proofs. The actual discussion of the foundations of mathematics occurs only in the Analysis:
Real numbers can be attributed to the convergence theory for rational numbers based defined as equivalence classes of a suitably chosen equivalence relation on the rational Cauchy sequences. An irrational number is then therefore, similar to the underlying rational numbers a lot.
The result is a rational number sequence no limit. But it is a Cauchy sequence. The set of rational to Cauchy equivalent, will be referred to with the symbol only once without the root would have a meaning. Then be for the equivalence classes of the linkages and introduced and it is shown that actually applies.
So can determine all the necessary real numbers as a basis for a constructivist analysis. Since a lot with exclusively constructed real numbers can never contain all real numbers, constructivists consider only constructible subsets of the set of all real numbers or use indefinite quantifiers (the word all is then not used as in the constructive logic ) to determine.
Since each design statement is a finite sequence of instructions from a finite set, there is a bijective function. (This is the set of all words over. ) So this constructivist sets of reals are countable. From Cantor's diagonal proof follows that the respective amount constructivist- real numbers a lower cardinality than the set of all real numbers, and thus is a proper subset of it. Constructivists argue that one needs only constructible real numbers for applications and summarize the cantor between Diagonal arguments as design provision to expand sets of real numbers is countable.
Standpoint of the mathematician
Traditionally, most mathematicians are suspicious, if not critical of the mathematical constructivism, mostly because of the restrictions, which calls for the constructive analysis. These views were clearly stated by David Hilbert for language: "From the paradise that Cantor created us, to us, no one can sell. " With the paradise set theory was meant, despite their paradoxes a special addition to the theoretical mathematics represented and includes aspects, which created a far beyond the constructive mathematics beyond the foundation of modern mathematics. The constructivist mathematics does not contain the entire set inventory of mathematics.
In particular, in the analysis hold most mathematicians the limitations of constructive mathematics unnecessary and have the end of the 20th century extended the analysis to a number of areas that are not constructive reconstructed. Mathematical constructivists such as Bishop and Lorenzen tried, however, to save possible the constructive analysis. Errett Bishop sought in 1967 in his book Foundations of Constructive Analysis, to refute doubts through the development of a large part of the analysis according to constructivist principles. Also, the German mathematician and philosopher Paul Lorenzen recorded classical analysis by constructive. Lorenzen began with an operational mathematics and logic, and later led the so-called abstraction in building ( Hugo Dingler ) of mathematics. Terms are, for example, according to Lorenzen by abstraction (of the differences ) to functions. This mathematical- philosophical approach, which eventually applied in the Erlanger constructivism, it is about a structure of mathematics by the executive and kalkulierende activity of the mathematician.
Regardless of constructive elaborations, however, almost all mathematicians see no need to limit themselves to constructivist methods.