A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967
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Formalism: the formalist sees arithmetic and logic as complementary.
A certain agreement between the two doctrines results from the impossibility of defining the number and, in particular, the whole number (VsFrege). The formalists, however, assert an indirect possibility on the basis of axioms.
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Formalism/Frege, cf. >
Formalism/Heyting.
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Intuitionism/formalism/d'Abro: The intuitionist is a rigorist, insofar as he considers definitions and proofs accepted by the formalist to be inadequate. It should be admitted that they are not given by logic, but by intuition.
E.g. Zermelo's (formalist) proof that the continuum is an ordered set. I.e., the points can be placed one after the other, with a successor for each point.
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Intuitionism.
PoincaréVsZermelo: he invented a typical argument: the pragmatist rejected Zermelo's proof because it would take too much time to carry it out, and the number of operations to be performed would be even greater than Aleph
0, not to be expressed with a finite number of words. The pragmatist will conclude that the theorem is pointless.
Camps: Formalists: Cantor, Hilbert, Zermelo, Russell - Intuitionists: Poincaré, Weyl
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G. Cantor, >
D. Hilbert, >
E. Zermelo, >
B. Russell, >
H. Poincaré.
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According to Weyl, the concept of the irrational number must either be abandoned, or thoroughly modified.
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Irrational numbers.
Brouwer: when dealing with infinite quantities, the law of the excluded middle does not apply.
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Excluded MIddle.
The intuitionists assert with Poincaré that antinomies without any infinity are lopish.
Poincaré: The antinomies of certain logicians are simply circular.
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Paradoxes, >
Circularity.
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Formalism/d'Abro: E.g. d'Abro sees no obstacle to define x in the following way:
(a) x has this and this relation to all members of type G.
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(b) x is a term of G.
For an intuitionist, according to Poincaré, such a definition is circular.
For example, controversy about definitions that cannot be expressed in a finite number of words. It is refused by the intuitionists.
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Definitions, >
Definability.
1 + 1/2 + 1/4 + 1/8...
This series, according to the intuitionists, is capable of being expressed in a finite number of words, since a rule can be formulated.
It should be noted that the difference is theoretical and not practically important, a proof that e.g. could be formulated in a trillion words would be acceptable.
