Real proofs only exist within a closed mathematical system. Outside mathematics itself, mathematics is used for modelling, but doesn't prove anything. Also note that not everything is necessarily provable within a closed mathematical system, as a spoil-sport called Gödel proved.
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Actually, Gödel proved that a system can not be both complete (closed) and consistent, i.e. logical fallacies necessarily arise in a complete system.
The best one could hope for is to be incomplete and consistent solely because inconsistency would be madness!