Pure mathematics studies the internal relations of mathematical laws abstracted from the objective world, and it can also be said that it is the study of the laws of mathematics itself. It can be roughly divided into three categories, namely

In the 20th century, mathematicians began to use axiomatic systems under the influence of Hilbert. Russell established the logical formula of "pure mathematics" in the form of quantitative proposition. With the axiomatization of mathematics, these formulas become more and more abstract, and "strict proof" becomes a simple standard. In fact, "strictness" is nothing new in "proof". According to bourbaki Group, pure mathematics has been proved.