Intuitionism holds that mathematics originated from intuition.
Logicalism holds that mathematics originates from logic.
Formalism holds that both are equally important.
If only from the perspective of discipline, both of them are basic disciplines, and no one is a branch of anyone. UNESCO has a scientific classification, which lists logic, mathematics, astronomy, chemistry, life science, astronomy and geography as seven basic disciplines.
From Aristotle's logic in ancient Greece to today, logic has a history of more than two thousand years. Aristotle's logic developed more maturely in the Middle Ages, and it formed classical formal logic. By the end of 19 and the beginning of the 20th century, German mathematician and logician Frege and British mathematician and logician Russell had established mathematical logic, which was different from Aristotle's logic and was called classical logic calculus. Since Frege and Russell, we have started the development of modern logic. The classical logic calculus founded by Frege and Russell became the basis of modern logic. On the basis of classical logic calculus, it has developed into a huge discipline group of modern logic. There are two main disciplines: one is called mathematical logic and the other is called philosophical logic. Mathematical logic is the intersection of logic and mathematics and a branch of logic. The intersection of logic and philosophy leads to philosophical logic, the intersection of logic and law leads to legal logic, and so on.