Mathematical constructivism

The mathematical constructivism is a direction of the philosophy of mathematics that the ontological represents view that the existence of mathematical objects to by their construction justify is. Constructivism can take an objectivist form (a mathematical object exists independently of thinking, but its existence is only established through its construction) and a subjectivist form (a mathematical object arises as a product of the mathematician's constructive intuition and is created by him in the first place. 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 . Is rational" vs. "We can such numbers , construct "). ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a^{b}}$ ${\displaystyle a}$ ${\displaystyle b}$

development

The first approaches to constructive mathematics come from the intuitionism of LEJ Brouwer . Further approaches were developed by Hermann Weyl , Andrei Nikolajewitsch Kolmogorow and Errett Bishop , Arend Heyting , Solomon Feferman , Paul Lorenzen , Michael J. Beeson and Anne Sjerp Troelstra . Constructivism, particularly represented by Weyl, was one of the positions that faced each other in the fundamental dispute of mathematics at the beginning of the 20th century , but it was unable to prevail.

theory

In a constructive proof , the mathematical objects and solutions to problems are actually constructed.

The constructive mathematics avoids explicitly non-constructive proofs and comes with the intuitionistic logic from that allows any non-constructive proofs. For example, if (as in an indirect proof ) the falsehood of a negated assertion is used to infer this assertion itself, a logical conclusion is used that does not force construction. The essential core of constructivism is to formulate only those sentences whose objects (and problem solutions) can be constructed. This claim leads to rejecting applications of the theorem of the excluded third party and of the axiom of choice , since statements about mathematical objects (or solutions) can also be derived with both theorems without specifying how these are constructed.

In arithmetic , both constructive proofs and non-constructive proofs can always be carried out. The actual discussion about the fundamentals of mathematics only arises in analysis :

Based on the convergence theory for rational numbers , real numbers can be defined as equivalence classes of a suitably chosen equivalence relation on the rational Cauchy sequences . An irrational number is then, like the rational numbers on which it is based, a quantity.

Example:

${\displaystyle a_{1}:=1,\quad a_{i+1}:={\frac {a_{i}}{2}}+{\frac {1}{a_{i}}}}$

The result is a rational number sequence no limit. But it is a Cauchy line. The set of equivalent rational Cauchy sequences,, is denoted by the symbol , initially without the root having any meaning. The links and are then introduced for equivalence classes and it is shown that it actually applies. ${\displaystyle \left(a_{i}\right)_{i\in \mathbb {N} }}$ ${\displaystyle \left(a_{i}\right)_{i\in \mathbb {N} }}$ ${\displaystyle \left[\left(a_{i}\right)_{i\in \mathbb {N} }\right]}$ ${\displaystyle {\sqrt {2}}}$ ${\displaystyle +}$ ${\displaystyle \cdot }$ ${\displaystyle {\sqrt {2}}\cdot {\sqrt {2}}=\left[2\right]}$

In this way, all necessary real numbers can be determined as the basis for a constructivist analysis. Since a set with exclusively constructed real numbers can never contain all real numbers, constructivists always only consider constructible subsets of the set of all real numbers or use indefinite quantifiers (the word all is then not used as in constructive logic) to determine . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$

Since every construction instruction is a finite sequence of instructions from a finite set , there is a bijective function . ( The set of all words is over .) So these constructivist sets of real numbers are countable. From Cantor's diagonal proof it follows that the respective set of constructivist-real numbers has a lower cardinality than the set of all real numbers and is therefore a real subset of it. Constructivists take the position that one only needs constructible real numbers for applications, and take Cantor's diagonal arguments as a construction rule to extend sets of real numbers countably. ${\displaystyle \Sigma }$ ${\displaystyle f\colon \Sigma ^{*}\rightarrow \mathbb {N} }$ ${\displaystyle \Sigma ^{*}}$ ${\displaystyle \Sigma }$

See also

• Calculable number
• Ultrafinitism
• Erlangen constructivism

Writings of constructive mathematicians

• Paul du Bois-Reymond: General function theory . Tuebingen 1882.
• Michael Beeson: Foundations of Constructive Mathematics . Springer-Verlag, Heidelberg 1985.
• Errett Bishop: Foundations of Constructive Analysis . McGraw-Hill, New York 1967.
• D. Bridges, F. Richman: Varieties of Constructive Mathematics . London Math. Soc. Lecture Notes 97, Cambridge: Cambridge University Press 1987.
• Leopold Kronecker: Lectures on the theory of simple and multiple integrals. Netto, Eugen, Leipzig Teubner (Ed.): 1894
• P. Martin-Löf: Notes on Constructive Analysis . Almquist & Wixsell, Stockholm 1968.
• Paul Lorenzen: Measure and Integral in Constructive Analysis. In: Mathematische Zeitung 54: 275. (online)
• Paul Lorenzen: Introduction to Operational Logic and Mathematics . Berlin / Göttingen / Heidelberg 1955.
• Paul Lorenzen: Metamathematics . Mannheim 1962.
• Paul Lorenzen: Differential and Integral. A constructive introduction to classical analysis . Frankfurt 1965.
• Paul Lorenzen: Constructive philosophy of science . Frankfurt 1974.
• Paul Lorenzen: Textbook of the constructive philosophy of science. Metzler, Stuttgart 2000, ISBN 3-476-01784-2 .
• Paul Lorenzen: Elementary Geometry as the Foundation of Analytical Geometry . Mannheim / Zurich / Vienna 1983, ISBN 3-411-00400-2 .
• Peter Zahn: A constructive way to measure theory and functional analysis. 1978, ISBN 3-534-07767-9 .

literature

• JJ O'Connor, EF Robertson: Errett Albert Bishop . MacTutor History of Mathematics . November 2004
• Constructivism. In: Michel Serres, Nayla Farouki: Thesaurus of the exact sciences . Zweiausendeins, Frankfurt am Main 2001, pp. 508–509.
• Eric Schechter: Constructivism is difficult (PDF; 74 kB). In: American Mathematical Monthly. 108, 2001, pp. 50-54.
• Solomon Feferman : Relationships between Constructive, Predicative and Classical Systems of Analysis. ( Memento of March 11, 2017 in the Internet Archive ) (PDF; 199 kB). In: ibid. Pp. 221-236.

Web links

• Douglas Bridges :  Constructive Mathematics. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .
• Maarten McKubre-Jordens:  Constructive Mathematics. In: Internet Encyclopedia of Philosophy .
• The concept of sets in mathematics - discussion of the classical and constructivist standpoints (PDF; 173 kB)

Individual evidence

1. ^ Douglas Bridges:  Constructive Mathematics. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .
2. ↑ On the relationship between constructivism and intuitionism see also Matthias Baaz, Rosalie Iemhoff: Konstruktivismus und Intuitionismus ( Memento of April 24, 2008 in the Internet Archive ) (PostScript; 267 kB) In: Internationale Mathematische Nachrichten 201 (2006; PDF; 1.3 MB )
3. ↑ See: Bridges , l. c.! In any case, the axiom of choice already uses the theorem of excluded third parties: ND Goodman, J. Myhill: Choice Implies Excluded Middle. In: Journal for Mathematical Logic and Fundamentals of Mathematics. 24, 1978, p. 461.
4. ↑ With Paul Lorenzen, on the other hand, an irrational number is determined by abstraction. The differences between two sequences are ignored if their difference is a zero sequence.
5. ↑ compare: Paul Lorenzen , Elementarge Geometry