Article

# Mathematical constructivism

Mathematical constructivism is a school of thought in the philosophy of mathematics that takes the ontological position that the existence of mathematical objects is to be justified by their construction. Constructivism can take an objectivist form (a mathematical object exists independently of thought, but its existence is only justified by its construction) and a subjectivist form (a mathematical object arises as a product of the constructing intuition of the mathematician and is produced by him in the process in the first place, intuitionism). Mathematical statements of the form “There are …” are rejected and – if possible – replaced by propositions of the form “We can construct …” (e.g. “There are irrational numbers

${displaystyle a}$

a

{displaystyle a}

,

${displaystyle b}$

b

{displaystyle b}

so that

${displaystyle a^{b}}$

a

b

{displaystyle a^{b}}

is rational.” vs. “We can use such numbers

${displaystyle a}$

a

{displaystyle a}

,

${displaystyle b}$

b

{displaystyle b}

construct”).[1]

## Development

The first approaches to constructive mathematics came from the intuitionism of L. E. J. Brouwer. Other approaches were developed by Hermann Weyl, Andrei Nikolayevich Kolmogorov and Errett Bishop, Arend Heyting, Solomon Feferman, Paul Lorenzen, Michael J. Beeson, and Anne Sjerp Troelstra. Constructivism, represented in particular by Weyl, was one of the positions opposed in the fundamental dispute in mathematics at the beginning of the 20th century, but it did not prevail.

## Theory

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

Constructive mathematics explicitly avoids non-constructive proofs and gets by with intuitionistic logic, which does not admit non-constructive proofs. For example, if (as in an indirect proof) the falsity of a negated assertion is used to infer that assertion itself, this uses a logical form of inference that does not force construction.[2] The essential crux of constructivism, then, is to formulate only those propositions whose objects (and solutions to problems) are constructible. This claim leads to rejecting applications of the theorem of the excluded third as well as the axiom of choice, since with both theorems[3] can also be used to derive statements about mathematical objects (or solutions) without specifying how they are constructed.

In arithmetic, both can always be done, constructive proofs and non-constructive proofs. The real discussion about the foundations of mathematics occurs only in calculus:

Real numbers can be defined, building on the theory of convergence for rational numbers, as equivalence classes of a suitably chosen equivalence relation on the rational Cauchy sequences. An irrational number is then a set, similar to the rational numbers on which they are based.[4]

Example:

The consequence

${displaystyle left(a_{i}right)_{iin mathbb {N} }}$

(

a

i

)

i

N

{displaystyle left(a_{i}right)_{iin mathbb {N} }}

has no limit value as a rational number sequence. But it is a Cauchy sequence. The set of numbers to be

${displaystyle left(a_{i}right)_{iin mathbb {N} }}$

(

a

i

)

i

N

{displaystyle left(a_{i}right)_{iin mathbb {N} }}

equivalent rational Cauchy sequences,

${displaystyle left[left(a_{i}right)_{iin mathbb {N} }right]}$

[

(

a

i

)

i

N

]

{displaystyle left[left(a_{i}right)_{iin mathbb {N} }right]}

, is displayed with the symbol

${displaystyle {sqrt {2}}}$

2

{displaystyle {sqrt {2}}

first without the root having any meaning. For equivalence classes, the links are then called

${displaystyle +}$

+

{displaystyle +}

and

${displaystyle cdot }$

{displaystyle cdot }

introduced and it turns out that in fact

${displaystyle {sqrt {2}}cdot {sqrt {2}}=left[2right]}$

2

2

=

[
2
]

{displaystyle {sqrt {2}}cdot {sqrt {2}}=left[2right]}

applies.

Thus, as a basis for constructivist analysis, all necessary real numbers can be determined. Since a set with exclusively constructed real numbers can never contain all real numbers, constructivists always consider only constructible subsets of the set of all real numbers

${displaystyle mathbb {R} }$

R

{displaystyle mathbb {R} }

or use indefinite quantifiers (the word all is then not used as in constructive logic) to determine

${displaystyle mathbb {R} }$

R

{displaystyle mathbb {R} }

.

Since each construction statement is a finite sequence of statements from a finite set of

${displaystyle Sigma }$

Σ

{displaystyle Sigma }

there is a bijective function

${displaystyle fcolon Sigma ^{*}rightarrow mathbb {N} }$

f
:

Σ

N

{displaystyle fcolon Sigma ^{*}rightarrow mathbb {N} }

. (Thereby

${displaystyle Sigma ^{*}}$

Σ

{displaystyle Sigma ^{*}}

the set of all words over

${displaystyle Sigma }$

Σ

{displaystyle Sigma }

.) 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 hold that only constructible real numbers are needed for applications, and take Cantor’s diagonal arguments as a construction rule to extend sets of real numbers countably.[5]

• Calculable number
• Ultrafinitism
• Erlangen constructivism

## Writings of constructive mathematicians

• Paul du Bois-Reymond: General Theory of Functions. Tübingen 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: Vorlesungen über die Theorie der einfachen und der vielfachen Integrale. Netto, Eugen, Leipzig Teubner (ed.): 1894
• P. Martin-Löf: Notes on Constructive Analysis. Almquist & Wixsell, Stockholm 1968.
• Paul Lorenzen: Measure and integral in constructive calculus. In: Mathematische Zeitung 54: 275. (online)
• Paul Lorenzen: Einführung in die operative Logik und Mathematik. 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: Lehrbuch der konstruktiven Wissenschaftstheorie. Metzler, Stuttgart 2000, ISBN 3-476-01784-2.
• Paul Lorenzen: Elementargeometrie als Fundament der Analytischen Geometrie. Mannheim/ Zurich/ Vienna 1983, ISBN 3-411-00400-2.
• Peter Zahn: A constructive path to measure theory and functional analysis. 1978, ISBN 3-534-07767-9.