The word "theorem" comes from the Ancient Greek word "thelemma," which means "something hidden." When mathematicians prove a theorem, they are showing that something is hidden in the abstract world of numbers, lines and other entities.
There are two types of proofs: constructive and destructive. In constructive proofs, all results are derived from axioms or basic properties, and no assumptions are required. In destructive proofs, one or more assumptions are made to start with and then shown to lead to a contradiction.
The majority of mathematical work entails using pure reasoning to determine and support the qualities of abstract objects. These objects are either abstractions from nature, such as natural numbers or lines, or — in modern mathematics — entities that are stipulated with certain properties, called axioms. A proof consists of a succession of applications of some deductive rules to already known results, including previously proved theorems, axioms and (in case of abstraction from nature) some essential characteristics that are thought of as the theory's real beginning points. A theorem is the conclusion of a proof.
Mathematics is the process of reasoning systematically, using abstract concepts and logical inference. It is based on the idea that knowledge can be organized and formalized, so that it can be expressed in a finite set of general rules. Mathematics has applications in many fields, including engineering, physics, computer science, statistics and finance.
