In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, 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 logicSymbolic logic is the area of mathematics which studies the purely formal properties of strings of symbols. The interest in this area springs from two sources. First, the symbols used in symbolic logic can be seen as representing the words used in philosophical 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
sineMaurice Sinet, known as Siné is a French cartoonist.As a young man he studied drawing and graphic arts, while earning a living as a cabaret singer. After his military service he started publishing his drawings and also worked as a photo-retoucher for porn magazines. His first published drawing...
function is
continuousIn mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous...
; or any theorem written in
big O notationIn mathematics, computer science, and related fields, big O notation describes the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions...
.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, 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 logicSymbolic logic is the area of mathematics which studies the purely formal properties of strings of symbols. The interest in this area springs from two sources. First, the symbols used in symbolic logic can be seen as representing the words used in philosophical 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
sineMaurice Sinet, known as Siné is a French cartoonist.As a young man he studied drawing and graphic arts, while earning a living as a cabaret singer. After his military service he started publishing his drawings and also worked as a photo-retoucher for porn magazines. His first published drawing...
function is
continuousIn mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous...
; or any theorem written in
big O notationIn mathematics, computer science, and related fields, big O notation describes the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions...
. 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. From a
constructivistIn the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists...
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 analysisNumerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square.Being able to compute the sides...
cannot be.
'Pure' existence results
An existence theorem may be called
pure if the proof given of it doesn't also indicate a construction of whatever kind of object the existence of which is asserted.
From a more rigorous point of view, this is a problematic concept. This is because it is a tag applied to a
theorem, but qualifying its
proof; hence,
pure is here defined in a way which violates the standard proof irrelevance of mathematical theorems. That is, theorems are statements for which the fact is that a proof exists, without any 'label' depending on the proof: they may be applied without knowledge of the proof, and indeed if that's not the case the statement is faulty. Thus, many constructivist mathematicians work in extended logics (such as
intuitionistic logicIntuitionistic logic, or constructivist logic, is the symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer's programme of intuitionism. The system preserves justification, rather than truth, across transformations yielding derived propositions...
) where pure existence statements are intrinsically weaker than their constructivist counterparts.
Such pure existence results are in any case ubiquitous in contemporary mathematics. For example, for a linear problem the set of solutions will be a
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, and some
a priori calculation of its dimension may be possible. In any case where the dimension is probably at least 1, an existence assertion has been made (that a non-zero solution exists.)
Theoretically, a proof could also proceed by way of a
metatheoremIn logic, a metatheorem is a true statement about a formal system expressed in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory.- Discussion :A...
, stating that a proof of the original theorem exists (for example, that a
proof by exhaustionProof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases, and each case is proved separately...
search for a proof would always succeed). Such theorems are relatively unproblematic when all of the proofs involved are constructive; however, the status of "pure existence metatheorems" is extremely unclear,
Constructivist ideas
From the other direction there has been considerable clarification of what constructive mathematics is; without the emergence of a 'master theory'. For example according to
Errett BishopErrett Albert Bishop was an American mathematician known for his work on analysis. He is the father of constructivist analysis, by virtue of his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.-Life:Errett Bishop's...
's definitions, the continuity of a function (such as
sin x) should be proved as a constructive bound on the
modulus of continuityIn mathematical analysis, a modulus of continuity is a functionused to measure quantitatively the uniform continuity of functions. So, a function admits as a modulus of continuity if and only iffor all and in the domain of...
, meaning that the existential content of the assertion of continuity is a promise that can always be kept. One could get another explanation from
type theoryIn mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...
, in which a proof of an existential statement can come only from a
term (which we can see as the computational content).