Lexicon of Arguments

 

A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967

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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.
>Formalism.
E.g. Zermelo's (formalist) proof that the continuum is an ordered set. I.e., that the points can be placed one after the other, with a successor for each point.

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 zero, it cannot 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
>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.
>Irrational numbers.
Brouwer: when dealing with infinite quantities, the law of the excluded middle does not apply.
>Law of the excluded middle.
The intuitionists assert with Poincaré that antinomies without any infinity are lopish.
Poincaré: The antinomies of certain logicians are simply circular.
>Circularity, >Paradoxes, >Infinity.