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|Waismann I 70
Induction/Brouwer/Poincaré/Waismann: the power of induction: it is not a conclusion that carries to infinity. The set 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, we begin with the formulation of the formulas
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 that 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".
Einführung in das mathematische Denken Darmstadt 1996
Logik, Sprache, Philosophie Stuttgart 1976