Abstract

The philosophy of mathematics provides a severe test for a materialist explanation of science. This is because mathematics is mostly abstract and mathematical theory is rarely tested directly in practice. All the main schools of the philosophy of mathematics — platonism, logicism, intuitionism, formalism — are varieties of idealism. Nevertheless all human ideas, including mathematical ideas, originate from our experience of the world and are rooted in reality. In the history of mathematics it can be seen that problems facing society have given a great impetus to the development of the subject. Thus the rise of trade and changes in technology have each led to great advances, though purely internal contradictions within mathematics have also been of considerable importance. Formal mathematical reasoning, like classical logic, has been highly successful in the evolution of science, but is inadequate for reasoning about indeterministic processes in a state of change. Thus formal logic and dialectics should be thought of as complementary and both are required for a fully scientific understanding of the world.