A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967
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Existence/d'Abro: Boundaries of the Axiomatic Method: one of the goals of mathematicians is to establish so-called existence theorems that are to prove that the solution we are looking for actually exists.
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Proofs, >
Provability, >
Theorems.
Non-existence/Meinong/d'Abro: since we can truthfully say "something like a round square does not exist," there must be something like a round square, albeit as a non-existent object. At first Russell had not been able to escape this, but in 1905 he discovered a theory of representation, according to which the round square seems to be mentioned when one says: "A round square does not exist." (Principia Mathematica)
(1).
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Non-existence, >
Round square.
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Existence/d'Abro: in Meinong "exist" and "there is" are used synonymously, but they are not synonyms: exist in the mathematical sense means to contain no contradiction.
If one takes Meinong seriously, this is evidence of the inability to think clearly, as in the joke: "Where does the light go when it goes out?".
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"There is".
Thus, a proof of existence for a solution is the finding that no contradiction arises from the assumption of a solution, even if the solution is not yet known.
Cf. >
Assertion of existence, >
Contradictions, >
Consistency.
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The famous Dirichlet problem is an existence theorem. The question is 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. ((s) It does not solve the problem for non-mathematical objects.)
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.
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Hilbert avoids this difficulty by saying that the system is proved to be consistent when it is possible to prove the existence of a model which confirms the system. So existence equals 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.
1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press.
