Mathematics: Fundamentals and Subdisciplines

The major subdisciplines of modern mathematics mentioned in the text are number theory, algebra, geometry, and analysis.

The process of mathematical proof involves a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and, in the case of abstraction from nature, some basic properties that are considered true starting points of the theory under consideration.

Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. It is extensively used for modeling phenomena in these fields. For example, in medicine, mathematical models are used to simulate the spread of diseases, and in finance, mathematical tools are employed to analyze and predict market trends.

The fundamental truths of mathematics are independent from any scientific experimentation. They consist of abstract objects and the use of pure reason to prove them. These objects can be abstractions from nature or entities stipulated to have certain properties, called axioms.

Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. These areas are extensively used in fields like economics, social sciences, and decision-making processes to analyze and interpret data, outcomes, and strategies.