Philosophy of mathematics: Making a fresh start

The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the effectiveness of mathematics in natural science.

► Two kinds of mathematics, natural and artificial, are distinguished.
► It is argued that both of them are based on the analytic method.
► This yields a new approach to mathematical demonstration.
► Also, a new approach to mathematical objects, mathematical definition and diagrams.
► Also, a new approach to the effectiveness of mathematics in natural science.