Mathematics and philosophy are both related and different: because they are both reactions to objective things, mathematics and philosophy are both discoveries of the material world and must be related; And there are differences between them, because objective things are developing, and the appearances of objective things are not only the same, so they must be different in mathematics and philosophy.
The research contents of mathematical philosophy mainly include:
① The relationship between mathematics and the real world, the development of mathematical theory and practice;
(2) The dialectical relationship between mathematical concepts and mathematical operations, and the internal logic of the development of mathematical concepts;
(3) Dialectical unity of mathematical categories, such as constant and variable, finite and infinite, straight line and curve, continuity and discontinuity.
Extended data:
Due to the different philosophical positions, logicism and intuitionism have gradually formed in the basic research of modern mathematics. As the embodiment of their mathematical philosophy, these schools all put forward specific plans for the basic research of mathematics.
The so-called formal axiomatization of a mathematical theory is to purify all the contents and explanations of mathematical objects that have nothing to do with form, so that mathematics can form a pure formal deduction system from a set of axioms. In this system, the propositions as the starting point are axioms or basic assumptions, and all other propositions or theorems can be deduced one by one by following the formal rules and symbolic logic rules of some assumptions.
Baidu Encyclopedia-Mathematical Philosophy