Theorem is a statement that is proved to be true after logical restriction.

Law is the expression of objective facts and the conclusion drawn through a large number of specific objective facts.

Axiom refers to the basic proposition that is based on the self-evident basic facts of human reason and has been tested by human practice for a long time without further proof.

2, the difference:

Law is an expression or word that describes the changing law of the objective world.

Axiom does not need certification, everyone recognizes it and can be used directly.

Theorem needs to be proved to be correct before it can be used.

3. Axiom

The propositions and principles that have been repeatedly tested by human practice for a long time are true and universally recognized, and do not need to be proved by other judgments, nor can they be proved by other judgments. Some disciplines are based on such axioms.

Axiom 1: You can draw a straight line from any point to any other point.

Axiom 2: Finite line segments can extend continuously.

Axiom 3: You can draw a circle with any point as the center and any distance.

Axiom 4: All right angles are equal.

Axiom 5: One straight line intersects the other two straight lines in the same plane. If the sum of two internal angles of a certain side is less than the sum of two right angles, then two straight lines extend indefinitely and then intersect at that side.

For example, if the traditional formal logic syllogism is about what a class of things is or is not, then some of these things are or are not, that is, if all of a class of things are judged, then some of them are judged, which is axiomatic.

However, this does not mean that the axiom must be right, and human understanding of the world is limited. This recognized and self-evident axiom may cause problems. Making mistakes is not necessarily a bad thing, but it promotes human beings to understand the world step by step. For example, the fifth axiom of Euclidean geometry cannot be said to be wrong, but several other geometries-elliptic geometry, Euclidean geometry and hyperbolic geometry can be obtained through different assumptions.

So it can be concluded that this foundation is not unbreakable, but relatively correct in human cognitive system.

4. theorem

A proposition or formula that is proved to be correct and can be used as a principle or law, such as a geometric theorem. Theorem is a true proposition (axiom or other theorems that have been proved), and it is proved to be a correct conclusion after logical restriction, that is, another true proposition. For example, "the opposite sides of a parallelogram are equal" is a theorem in plane geometry.

Generally speaking, only important or interesting statements are called theorems in mathematics. Proving theorem is the central activity of mathematics.

A mathematical statement that is considered true but not proved is a conjecture. When it is proved to be true, it is a theorem. It is the source of the theorem, but it is not the only source. A mathematical statement derived from other theorems can become a guessing process and an unproven theorem.

That is, theorems are propositions or formulas derived from axioms or theorems. The derivation method depends on human logic.

5. Law

Law is proved by practice and facts, which reflects the objective law of development and change of things under certain conditions, and is a conclusion accumulated and summarized through a large number of specific objective factual experiences. Such as Newton's law of motion, law of conservation of energy, ohm's law, etc.

Law is a theoretical model used to describe the real world in a specific situation and scale, which may be invalid or inaccurate in other scales. No theory can describe all the situations in the universe, and no theory can be completely correct.

In short, the law is verified by people's guesses and proved by countless practices. Generally, there is a special deduction, and the overall judgment has a partial deduction. Many scientific and philosophical developments are based on this.

What I want to point out is the limitations of the law. It is the inductive assumption of things in the case of poverty, which is not necessarily correct, and of course it is impossible to exhaust all situations.

Therefore, we can know three possible sources of human cognitive system errors: First, the laws summed up by practice are not comprehensive enough to cover all situations. Second, these axioms are self-evident. The third is the logic used to judge deduction. (Of course, this can be included in article 12. )

I think human's understanding of the world is only a small part, and the part that has been recognized is so fragile. But I am an optimist, I believe in the knowability of the world, I believe that one day mankind will know everything about this world, and I hope to see the unity of all this in my own lifetime.