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|Induction||Waismann I 70
Induction/Brouwer/Poincaré/Waismann: the power of induction: it is not a conclusion that carries to infinity. The sentence a + b = b + a is not an abbreviation for infinitely many individual equations, as well as 0.333 ... is not an abbreviation, and the inductive proof is not the abbreviation for infinitely many syllogisms (VsPoincaré).
In fact, with the formulation of the formulas we begin
a+b = b+a
a+(b+c) = (a+b)+c
a whole new calculus, which cannot be inferred from the calculations of arithmetic in any way. But:
Principle/Induction/Calculus/Definition/Poincaré/Waismann: ... this is the correct thing in Poincaré's assertion: the principle of induction cannot be proved logically. VsPoincaré: But he does not represent, as he thought, a synthetic judgment a priori; it is not a truth at all, but a determination: If the formula f(x) applies for x = 1, and f(c + 1) follows from f(c), let us say that "the formula f(x) is proved for all natural numbers".
A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967
Induction/PoincaréVsHilbert: in some of his demonstrations, the principle of induction is used and he asserts that this principle is the expression of an extra-logical view of the human mind. Poincaré concludes that the geometry cannot be derived in a purely logical manner from a group of postulates.
Induction is continually applied in mathematics, inter alia also in Euclid's proof of the infinity of the prime numbers.
Induction principle/Poincaré: it cannot be a law of logic, for it is quite possible to construct a mathematics in which the principle of induction is denied. Hilbert, too, does not postulate it among his postulates, so he also seems to be of the opinion that it is not a pure postulate.
Einführung in das mathematische Denken Darmstadt 1996
Logik, Sprache, Philosophie Stuttgart 1976