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Idea of the Proof

The article explores the informal side of the idea of the proof. The word "idea" is used in a sense dating back to Plato. Proof is understood as a precisely established connection of precisely formulated and objectively existing ideas. This connection of ideas belongs to the realm of the possible and can be present in some possible worlds and absent in others. Attempts to interpret the proof as a procedure of convincing argumentation are criticized. It is shown that the proof is not necessarily convincing, and that persuasiveness may be inherent outside the proof-based forms of argument. The genesis of the idea of the proof is briefly traced from its origins to the present day. Arguments are made in favor of the thesis of the idea of the proof in Pythagorean philosophy. The question of why the idea of the proof has not been rediscovered anywhere and never is discussed. The problem of the time gap between the appearance of the proof and the exact definition of the concept of the proof in modern logic is considered. An accessible example substantiates the inseparability of the idea of the proof from its formal presentation in one or another logic. A list of some basic informal predicates of the proof is given and a brief description is given.