I 59ff
Intuitionism/Heyting: Brouwer studied the conceptual mathematical construction as such, without questioning the nature of things, for example, whether these things exist independently of our knowledge of them.
>
L. Brouwer.
I 60
Sentence of the excluded middle: e.g. the invalidity of the sentence of the excluded middle: if we compare the definitions of two natural numbers, k and l then:
(A) k is the largest prime number such that k 1 is also a prime number, if there is no such number, k = 1.
(B) l is the largest prime number such that l 2 is also a prime number, if there is no such number, l = 1.
Intuitionists reject (B) as a definition of an integer. K can be really calculated (k = 3), while we have no method of determining l, since it is not known whether the sequence of the prime number twins is infinite or not. The intuitionists regard something as well-defined only when a method of determination is given.
>
Law of the Excluded Middle, >
Numbers.
Classical mathematics: one can argue that the extent of our knowledge about the existence of the last twin is purely coincidental. And completely irrelevant in questions of mathematical truth.
Existence/intuitionism/Heyting: the argument of the representative of classical mathematics is of a metaphysical kind. If existing does not mean "constructible", it must have a metaphysical meaning.
Cf. >
Constructivism.
I 61
Classical mathematics/VsIntuitionism/Heyting: assuming that on January 1st, 1970, it is proved that there are infinitely many twins, l is equal to 1. Was that not already the case before the date (Menger, 1930)?
Intuitionism/Heyting: a mathematical assertion states that a certain construction is possible. Before the construction exists, the construction is not there. Even the intuitionists are convinced that mathematics is based on eternal truths in some sense, but when one attempts to define this meaning one gets entangled in metaphysics.
>
Metaphysics.
I 62
Formalism/Carnap/Heyting: there always remains the doubt, which conclusions are correct, and which are not (Carnap, 1934
(1), S. 44; 1937
(2), S. 51).
>
Correctness.
I 63
Intuitionism: we are not interested in the formal side, but precisely in the nature of inferences in meta-mathematics. There is a fundamental ambiguity in the language.
Classical mathematics: the semanticists are even worse relativists than the formalists and intuitionists.
Cf. >
Semantic truth, >
Truth conditions.
I 65
Intuitionism: there is an intuitionist logic, e.g. transitivity. Conclusion: logic is a part of mathematics and therefore cannot be taken as its basis.
>
Blackening of the paper, >
Formalism, >
Evidence, >
Proofs, >
Provability,
>
VsFormalism, cf. >
Foundation.
1. R. Carnap, Logische Syntax der Sprache, Wien 1934, p. 44.
2. R. Carnap, Testability and Meaning, in: Philosophy of Science 4, 1937, p. 51.
3. Karl Menger. Der Intuitionismus. Blätter Für Deutsche Philosophie 4:311--325 (1930)
