# Mathematical object A mathematical object is an abstract object arising in mathematics. The concept is studied in philosophy of mathematics. Key topics **Mathematical object** is anything that can be formally defined and used in deductive reasoning and mathematical proofs. In modern math, most mathematical objects are defined in terms of sets. Mathematical objects: * permutations * combinations * partitions * relations * functions * proofs * theorems **Discrete Structures** aid us in modelling the world in a way that enables us to think about it rigorously and computationally. **Mathematical statements** have their own moderately complex taxonomy, being divided into: * axioms * conjectures * theorems * lemmas * corollaries An **axiom** or **postulate** is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. A **conjecture** is a conclusion or proposition based on incomplete information, for which no proof or disproof has yet been found. Conjectures such as the `Riemann hypothesis` (still a conjecture) or `Fermat's Last Theorem` (which was a conjecture until proven in 1995 by Andrew Wiles) have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. A **theorem** is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. A **lemma** is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself. A **corollary** is a statement that readily follows from a previous statement or proof. MATHEMATICAL OBJECTS: Introduction [https://web.archive.org/web/20081014205857/http://www.abstractmath.org/MM//MMMathObj.htm](https://web.archive.org/web/20081014205857/http://www.abstractmath.org/MM/MMMathObj.htm)