UNDERSTANDING OF THE MATHEMATICAL PROOF: LOGIC vs. MATHEMATICS
Vitaliy Valentinovich Tselishchev, Aleksandr Valerievich Khlebalin
Institute of Philosophy and Law, Siberian Branch of the Russian Academy of Sciences, 8 Nikolaeva str., Novosibirsk, 630090, Russia
Keywords: mathematical proof, understanding, logic, mathematics, language of mathematics
Abstract
The article analyzes the opposition of the understanding of the mathematical proof and its formalization. It is shown that two problems are mixed in this opposition; those are the relation between mathematical logic and mathematical proof practice and the one between mathematical and formal proofs. It is demonstrated that the mathematical proof being meaningful aims at explaining, while the formal proof is meaningless and confines itself to representing the rules. The strategy of D. Macbeth, which claims to overcome the mentioned opposition, is analyzed. According to this strategy, the integral system of signs is correlated with mathematical ideas that are expressed in ordinary language and the logical reconstruction of the translation of these ideas into a series of manipulations with signs loses sight of the content of the ideas. The vulnerability of such a position is shown and instead an interpretation of the intensional content of mathematical discourse as a result of the translation of a mathematical statement into a formal system is proposed.
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