Lexicon of Arguments

 

Waismann I 70
Principle/Induction/Calculus/Definition/Poincaré/Waismann: ... this is the right thing in Poincaré's assertion that the principle of induction cannot be proved logically. VsPoincaré: But he does not represent, as he claimed, 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".
>Numbers, >Induction.
I 71
But is this really just a determination? It might seem paradoxical that the associative law of addition should emerge from a mere definition (of formula D) (II 62). But the formula D is not a definition in the sense of school logic, namely, a substitution rule, but an instruction for the formation of definitions.
In the formula, there are only letters, but in the proof there are numbers! Therefore, we can predict results without performing the calculation.
The commutative law could be compared with an arrow pointing the series of numbers along into infinity.
This is not the same as saying that the law comprehends infinitely many single sentences. Example: this is similar to the sentences.
E.g., The headlight shines to infinity (true) and the headlight illuminates the infinity (impossible).
By making that convention, that is, by constructing such formulas, we adjust the calculus with letters with the calculus with numbers.
>Infinity, >Symbols, >Formalization, >Formalism.