In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s) ..', or more generally 'for all x, y, ... there exist(s) ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not do so explicitly, as usually stated in standard mathematical language. For example, the statement that the sine function is continuous; or any theorem written in big O notation. The quantification can be found in the definitions of the concepts used.
A controversy that goes back to the early twentieth century concerns the issue of pure existence theorems. Such theorems may depend on non-constructive foundational material such as the axiom of infinity, the axiom of choice, or the law of excluded middle. From a constructivist viewpoint, by admitting them mathematics loses its concrete applicability (see nonconstructive proof). The opposing viewpoint is that abstract methods are far-reaching, in a way that numerical analysis cannot be.
Read more about Existence Theorem: 'Pure' Existence Results, Constructivist Ideas
Famous quotes containing the words existence and/or theorem:
“Suppose that humans happen to be so constructed that they desire the opportunity for freely undertaken productive work. Suppose that they want to be free from the meddling of technocrats and commissars, bankers and tycoons, mad bombers who engage in psychological tests of will with peasants defending their homes, behavioral scientists who can’t tell a pigeon from a poet, or anyone else who tries to wish freedom and dignity out of existence or beat them into oblivion.”
—Noam Chomsky (b. 1928)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)