théorème

In mathematics, a theorem is a statement that has been proven to be true based on axioms and previously established theorems. The proof establishes the logical necessity of the statement within the given formal system. Theorems are not empirical observations; their truth is derived deductively from earlier accepted assumptions.

Etymology: The word théorème comes from the Greek theorema, "a thing to be seen" or "that which is observed," through Latin and French usage. The term has been used in mathematical literature since ancient times to denote a proposition established by reasoning rather than experimentation.

Notes: A theorem may rely on lemmas (auxiliary results) and corollaries (statements that follow readily from it). Theorems are distinguished from conjectures (unproven statements) and from axioms (self-evident assumptions). A proof is a logical argument that demonstrates the theorem's truth under the axioms; a proof must be verifiable by others within the same system.

Examples: The Pythagorean theorem states that in a right triangle, a^2 + b^2 = c^2. Fermat's Last Theorem asserts that there are no nontrivial integer solutions to x^n + y^n = z^n for n > 2. The Fundamental Theorem of Calculus links differentiation and integration, establishing that they are inverse processes.

Impact: Theorems form the core of mathematical knowledge, enabling the structured development of theories across disciplines. They are widely stated with their assumptions clearly specified and, when possible, with explicit proofs.