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[geoffreycaveney]

If a mathematician brings forward a theorem, everybody can follow the proof for himself.

Anton, unfortunately this statement is false. It's even more false for physics and completely false for linguistics, biology, history, etc.

It depends on the theorem, the mathematician, and the existing knowledge of the specialized area of mathematics among other qualified mathematicians.

When Andrew Wiles originally proposed his proof of Fermat's Last Theorem, there were only a small handful of specialists in the field qualified to study it and evaluate it, but nevertheless they did so, and one colleague found an error in his argumentation within about a year or less. Fortunately for Wiles, he successfully corrected the error himself, and within about a year he completed a revised proof that was soon accepted as correct by all specialists in that particular branch of mathematics.

When Grigori Perelman posted his series of papers on the arXiv website outlining his proof of the Poincaré Conjecture, specialists in the field were quickly able to verify that his proof was correct and complete. The details that he omitted were generally agreed by specialists in the field to be obvious to knowledgeable mathematicians in this field. 

On the other hand, when Shinichi Mochizuki published a series of papers claiming to elaborate a proof of the so-called "abc conjecture" in number theory, the novelty, difficulty, and sheer length of his arguments has made it challenging for other mathematicians to confirm or reject the validity of his proof. For a summary of the history of discussion and reception of Mochizuki's proposed proof, see You are not allowed to view links. Register or Login to view..

Claimed proofs of the Riemann Hypothesis have all been fairly clearly rejected as incorrect to date.