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|Waismann I 66
Induction/Poincaré: One can easily pass from one statement to the other, and surrender to the imagination that the legitimacy of the recursive method has been proved. But one will always arrive at an unprovable axiom.
RussellVsPoincaré: Induction is a definition and not a principle. There are certain numbers for which it applies, others not (Cantor's infinite cardinal numbers).
Waismann (example from Wittgenstein) e.g. Division 1:3 with recurring rest.
We conclude that it always goes on like this. But does it really result in the calculation? Every calculation breaks down after a finite number of places. On the other hand, the first step already shows the return.
E.g. fiction: tribe, which possesses our decimal system, but without infinite decimal fractions. Those people break off after the 5th digit. Let us suppose one day somebody discovered that division 1:3 continues.
What would be his discovery? One might think first of all that the return of rest was the first thing he noticed. Then one had asked one who did not yet know periodical division, "is the rest equal to the dividend in this division?" He would have said yes. But with this, he would not have necessarily noticed the periodicity.
We may perhaps wish to say: Whoever discovers the periodicity sees the division differently from the one who does not know it; he sees an infinite possibility in it. But this sounds as if it were a psychological thing.
In reality, the discovery of periodicity is the construction of a new calculus. You can mark them with a line.
This is not a pure outwardness, it points to the law of division.
The way in which he draws attention to the periodicity gives the new sign. Once we have discovered the periodicity, we have discovered a new law. The dots do not represent, in a shadowy manner, the digits which are not written in the absence of ink, but are themselves a full-fledged sign in the calculus.
A proof by induction is something quite different from what else is called "proof" in the calculation of letters.
The induction proof does not lead to the formula to be proved.
Is induction only the indication that the sentence applies to all signs? The fact that the sentence applies for y + 1 if it aplies for a does not explain the meaning of the sentence.
It gives us no answer to the question, how is this sentence used? What is the criterion of its truth?
We cannot go through all numbers, not because we have too little time and paper, but because it is nothing, because it is logically impossible. In fact, the proof by induction is the only criterion we have._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
Einführung in das mathematische Denken Darmstadt 1996
Logik, Sprache, Philosophie Stuttgart 1976