The Principle of Complete Induction
This sense of definition allowed Poincaré to argue with Bertrand Russell over Giuseppe Peano's axiomatic theory of natural numbers.
Peano's fifth axiom states:
- Allow that; zero has a property P;
- And; if every natural number less than a number x has the property P then x also has the property P.
- Therefore; every natural number has the property P.
This is the principle of complete induction, which establishes the property of induction as necessary to the system. Since Peano's axiom is as infinite as the natural numbers, it is difficult to prove that the property of P does belong to any x and also x+1. What one can do is say that, if after some number n of trials that show a property P conserved in x and x+1, then we may infer that it will still hold to be true after n+1 trials. But this is itself induction. And hence the argument is a vicious circle.
From this Poincaré argues that if we fail to establish the consistency of Peano's axioms for natural numbers without falling into circularity, then the principle of complete induction is not provable by general logic.
Thus arithmetic and mathematics in general is not analytic but synthetic. Logicism thus rebuked and Intuition is held up. What Poincaré and the Pre-Intuitionists shared was the perception of a difference between logic and mathematics which is not a matter of language alone, but of knowledge itself.
Read more about this topic: Preintuitionism
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