To develop a critical and conscious attitude towards different views of mathematics. Reasons for, analysis, and relevance of the mathematical notions.
Gödel's incompleteness theorems. Greek viewpoint of mathematics. Mathematics as a prototype of deductive science. The problem of infinity. First crisis of the foundations of mathematics. The problem of the consistency of theories. Reduction of a theory to another. Different viewpoints of mathematics: neo-platonic, logicist, formalist, constructivist. Second crisis of the foundations of mathematics. Effective computability. The problem of communicating mathematical concepts through the language. Beyond the language. How to capture and communicate concepts which are non definable through the language. The problems of objectivity, certainty and relevance of the mathematical knowledge.
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