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|Waismann I 70
Proof/Induction/Intuitionism/Brouwer/Waismann: If it is said that the proof applies to all numbers, one has to be clear that one only determines by the proof the meaning of the word "all".
And this meaning is different than e.g. "All the chairs in this room are made of wood". For when I deny the last statement, this means that there is at least one that is not made of wood.
If, however, I deny "A applies to all natural numbers", that means only: One of the equations in the proof of A is false, but not, there is a number for which A does not apply.
The general formula in mathematics and the existence statement do not belong to the same logical system. (Brouwer: the incorrectness of a statement does not mean the existence of a counterexample).
Now the performance of the induction becomes clear: 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 little as 0.333 ... is 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.
|Brouwer, L. E. J.
Einführung in das mathematische Denken Darmstadt 1996
Logik, Sprache, Philosophie Stuttgart 1976