Mathematical foundation

In mathematics, the term mathematical basis is sometimes used in specific mathematical fields, such as mathematical logic, axiomatic set theory, proof theory, model theory and recursion theory. But seeking the foundation of mathematics is also the central problem of mathematical philosophy: on what ultimate basis can a proposition be called truth?

At present, the dominant mathematical paradigm is based on axiomatic set theory and formal logic. In fact, almost all mathematical theorems can be expressed as theorems under set theory. In this view, the so-called truth of mathematical propositions can only be deduced from the axioms of set theory by formal logic.

This formal method can't explain some problems: why should we follow the existing axioms instead of others', why should we follow the existing logical rules instead of others', and why the "true" mathematical proposition (for example, piano's axiom in the field of arithmetic) seems to be true in the physical world. This is called "unreasonable validity of mathematics in natural science" by eugene wigner in 1960.

The formal authenticity mentioned above may also be completely meaningless: it is possible that all propositions, including contradictory propositions, can be derived from the axioms of set theory. Moreover, as a result of Godel's second incompleteness theorem, we can never rule out this possibility.

In mathematical realism (sometimes called Platonism), the existence of a mathematical object world independent of human beings is taken as a basic assumption; The authenticity of these objects was discovered by human beings. In this view, the laws of nature and mathematics have similar status, so "effective" is no longer "unreasonable". What constitutes the basis of mathematics is not our axiom, but the real world of mathematical objects. However, the obvious question is, how do we get in touch with the world?

Some modern mathematical philosophy theories do not recognize the existence of this mathematical foundation. Some theories tend to pay attention to mathematical practice and try to describe and analyze the practical work of mathematicians as a social group. There are also some theories that try to create a mathematical cognitive science and attribute the reliability of mathematics in the "real world" to human cognition. These theoretical suggestions are only found in human thinking.