Books on Amazon
|Postulates||A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967
Postulates are not the only elements that need to be examined; we must also consider the laws to which they are subject to.
Existence/consistency/d'Abro: E.g. the famous Dirichlet problem is an existential theorem. It is about whether or not there is always a solution for the Laplace equation satisfying certain boundary conditions.
An inconsistent model has just as little claim to mathematical existence as a round square.
A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967
The compatibility of a postulate system can only be checked if it has only a finite number of consequences. Hilbert's postulates, however, allow infinitely many conclusions.
Hilbert circumvents this difficulty by saying that the system is proved to be consistent, if it succeeds to prove the existence of a model which confirms the system. So existence equals the lack of an internal inconsistency.
Hilbert then asserts that the numerical model satisfies this requirement. He thus accepts the consistency of the arithmetic continuum. The only problem is that we are not sure about it.
Brouwer and Weyl are seriously questioning them, with the result that we can only believe the 5th Hilbertian postulate and all the models which are to be confirmed. Logic alone does not help.
Is it true that we get the Euclidean geometry only if we apply the logical rules to Hilbert's postulates? Poincaré denies this question.