<2014 10 01> mathematical basics Wikipedia

Mathematical Basics

In mathematics,Mathematical BasicsA word is sometimes used in a specific field of mathematics, such as mathematical logic, public-Physical-chemical set theory, proof theory, model theory, and progressive theory. However, finding the foundation of mathematics is also the central issue of mathematics philosophy: What is the ultimate foundation can be calledTrue?

Currently, the dominant mathematical paradigm is based on the theory of physical and chemical sets and formal logic. In fact, almost all current mathematical theorems can be expressed as theorems under the set theory. Under this viewpoint, the authenticity of the so-called mathematical proposition, however, is that the proposition can be derived from the formal logic used by the theory of set.

This formal method cannot explain some problems: why should we follow the current principle rather than others, why should we follow the current logic rules rather than others, and why "true" mathematical propositions (for example, the piano justice in the field of arithmetic) seems to be true in the physical world. This was called by Eugene vergena in 1960 "the no-justification effectiveness of mathematics in Natural Science "(The unreasonable extends tiveness of mathematics in the natural sciences).

The above formal Authenticity may also be completely meaningless: it is possibleAllPropositions, including self-contradictory ones, can all be derived from the set theory principle. Furthermore, as a result of geder's second incomplete theorem, we will never be able to exclude this possibility.

In mathematical realism (sometimes Plato), the existence of a world independent of human mathematical objects is considered as a basic assumption. the authenticity of these objects is determined by the authenticity of human beings.Found. In this view, the law of nature is similar to the law of mathematics, so "effectiveness" is no longer "no reason ". Not our principle, but the real world of the object of mathematics forms the foundation of mathematics. But the obvious question is, how do we get into this world?

Some modern theories of mathematical philosophy do not recognize the existence of this mathematical foundation. Some theories tend to focus on mathematical practices and try to describe and analyze the real work of mathematicians as a social group. There are also theories that try to create a mathematical cognitive science and attribute the reliability of mathematics in the "real world" to human cognition. These theoretical recommendations only lay the foundation in human thinking, rather than any "objective" external structure. This topic has always been highly controversial.

<2014 10 01> mathematical basics Wikipedia