User:Kku/Nonexistence

A statement about nonexistence is a statement that either
 * 1) something (possessing certain properties) does not exist (within a certain scope) or
 * 2) that a nothing exists or
 * 3) that there is a reality to non-existent objects or
 * 4) that nothing exists (at all).

In case 1, the negation or falsifiability of existence can be proven, especially when the scope of the question of nonexistence is limited, as in a finite set or collection. Showing that a certain element x is not part of a finite set is trivial. This latter fact can be expressed in the Frege-Quine-tradition with the negated existential quantifier $$\nexists{x}$$. Case 4 has first been phrased by Gorgias.

Mathematics


Non-existence in mathematics refers to the nonexistence of solutions to a mathematical formula or problem (e.g. Fermat's last theorem, Doubling the cube).

The constructive method to prove "there does not exist an object with property P" is to assume there is an object x with property P and derive a contradiction. \nexists{x}{\in}\mathbf{X}\, P(x) \equiv \lnot\ \exists{x}{\in}\mathbf{X}\, P(x) \equiv\ \forall{x}{\in}\mathbf{X}\, \lnot P(x) $$ \exists{x}{\in}\mathbf{X}\, P(x) $$
 * 1) Proof by contradiction

Physics
The nonexistence of certain phenomena, like Electric monopole radiation can be proven mathematically and correspond to the nonexistence of a solution to a (physical) formula.

Non-existence as such can be approximated by the concept of emptiness - more precisely: a vacuum. James Clark Maxwell defined it as the When physical entities or laws become undefined in singularities like the initial singularity when our universe was non-existent, physicists will (jokingly) refer to a 'nothing'.

Philosophy
Much that can be said about existence or nonexistence in philosophy rests on logic with a good deal of mathematical logic.

In the Frege-Quine-tradition, both “there is” and “exists” are expressed by means of the “existential quantifier” (“∃”). A distinction can be made by expressing existence with the stronger "E!", as it used in free logic which allows for non-existence of an object p to be expressed as ¬E!p.

Linguistics
Some languages have particles or suffixes to express non-existence, like Turkish copula yok or the infix -me- or Hungarian nincs. Of particular philosophical importance is the concept of mu in various Asian languages.