Editing Pre-intuitionism (section)

==The principle of complete induction==
This sense of definition allowed [[Henri Poincaré|Poincaré]] to argue with [[Bertrand Russell]] over [[Giuseppe Peano| Giuseppe Peano's]] [[Peano axioms|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 [[mathematical induction|induction]] as necessary to the system. Since Peano's axiom is as [[infinity|infinite]] as the [[natural number]]s, it is difficult to prove that the property of ''P'' does belong to any ''x'' and also ''x''&nbsp;+&nbsp;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''&nbsp;+&nbsp;1, then we may infer that it will still hold to be true after ''n''&nbsp;+&nbsp;1 trials. But this is itself induction. And hence the argument [[begging the question|begs the question]].

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 [[logic|general logic]].

Thus arithmetic and mathematics in general is not [[analytic proposition|analytic]] but [[synthetic proposition|synthetic]]. [[Logicism]] thus rebuked and [[intuitionism|Intuition]] is held up. What Poincaré and the Pre-Intuitionists shared was the perception of a difference between logic and mathematics that is not a matter of [[language]] alone, but of [[knowledge]] itself.