In what way intuitionism is unique in the constructive approach

Question

I am writing a paper on the subjects of constructivism and intuitionism. While I do know that intuitionism is a part of constructivism; it is also written that a lot of logic in intuitionism is unique and not shared with other constructivist approaches (such as finitism) and I can hardly understand what are the differences between the two (constructivism and intuitionism).

Thanks

Answer 1

There are different "flavours" and schools of Constructivism in mathematics.

From a "logical point of view" we may classify the different approaches as follows :

• finitism : only natural numbers, using computable functions and quantifier-free methods of proof.

See Thoralf Skolem (1923), The foundations of elementary arithmetic and Reuben Goodstein : Recursive number theory (1957) and Recursive Analysis (1961).

• intuitionism : quantifiers are admitted but Excluded Middle is rejected; this is reflected into the rejection of the inference from the absurdity of a general statement : $\forall x \lnot A(x) \to \bot$ to the existence of a witness : $\exists x A(x)$.

See Brouwer and Intuitionism in the Philosophy of Mathematics

• predicativism : quantifiers and the general law of excluded middle are permitetd, but impredicative definitions are forbidden.

See Principia Mathematica and Ramified Type Theory and Hermann Weyl (1918) Das Kontinuum.

More recently : Solomon Fefereman (2013), Predicative Foundations of Analysis.

We may add also :

• ultrafinitism with Alexander Esenin-Volpin (1970), The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics. Only "manageable" numbers are intuitively evident, and not all finite numbers are manageable.

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From a "more mathematical" point of view, at least four varieties of Constructive Mathematics are available:

• Intuitionistic Mathematics. See at least Arend Heyting (1956), Intuitionism : An introduction and A.S.Troelstra and D.van Dalen (1988), Constructivism in Mathematics: An Introduction, 2 volls.

See also : John Lane Bell, Intuitionistic Set Theory (2014).

• Recursive Constructive Mathematics: starting with the Russian school of A.A.Markov (jr). See Oliver Aberth : (1980) Computable Analysis and (2001) Computable Calculus.

• Errett Bishop's Constructive Mathematics : Foundations of Constructive Analysis (1967).

See also Errett Bishop & Douglas Bridges, Constructive Analysis (1985).

• Per Martin-Löf's Constructive Type Theory : Notes on Constructive Mathematics (1968).

See also : Giovanni Sambin & Jan Smith (editors), Twenty-five years of constructive type theory (1998).

In general, see also :

• Fred Richman & Douglas Bridges, Varieties of Constructive Mathematics (1987)

• Michael Beeson Foundations of Constructive Mathematics (1985)

• Giovanni Sommaruga (editor), Foundational theories of classical and constructive mathematics (2011).