# Dictionary Definition

theorem

### Noun

1 a proposition deducible from basic
postulates

2 an idea accepted as a demonstrable truth

# User Contributed Dictionary

## English

### Etymology

Greek θεώρημα, "speculation", "proposition to be proved" (Euclid), from θεωρειν, from θεωρός, "spectator". From vulgar Latin theōrēma. Confer theory, and theater.### Noun

- A mathematical statement of some importance that has been proven to be true. Minor theorems are often called propositions. Theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called lemmas
- A mathematical statement that is expected to be true; as, Fermat's Last Theorem (as which it was known long before it was proved in the 1990s.)

#### Synonyms

#### Derived terms

#### Related terms

#### Translations

proved mathematical statement

mathematical statement that is expected to be
true

- Finnish: olettamus

# Extensive Definition

In mathematics, a theorem is a
statement proven
on the basis of previously accepted or established statements. In
mathematical
logic, theorems are modeled as
formulas that can be derived
according to the derivation
rules of a fixed formal
system.

In formal settings, an essential property of
theorems is that they are derivable using a fixed set of inference
rules and axioms
without any additional assumptions. This is not a matter of the
semantics of the
language: the expression that results from a derivation is a
syntactic
consequence of all the expressions that precede it. In
mathematics, the derivation of a theorem is often interpreted as a
proof of the truth of the resulting expression, but different
deductive
systems can yield other interpretations, depending on the
meanings of the derivation rules.

The proofs of theorems have two components,
called the hypotheses
and the conclusions.
The proof of a mathematical theorem is a logical argument
demonstrating that the conclusions are a necessary consequence of
the hypotheses, in the sense that if the hypotheses are true then
the conclusions must also be true, without any further assumptions.
The concept of a theorem is therefore fundamentally deductive, in contrast to the
notion of a scientific theory, which is empirical.

Although they can be written in a completely
symbolic form using, for example, propositional
calculus, theorems are often expressed in a natural language
such as English. The same is true of proofs, which are often
expressed as logically organised and clearly worded informal
arguments intended to demonstrate that a formal symbolic proof can
be constructed. Such arguments are typically easier to check than
purely symbolic ones — indeed, many mathematicians would express a
preference for a proof that not only demonstrates the validity of a
theorem, but also explains in some way why it is obviously true. In
some cases, a picture alone may be sufficient to prove a
theorem.

Because theorems lie at the core of mathematics,
they are also central to its aesthetics. Theorems are often
described as being "trivial", or "difficult", or "deep", or even
"beautiful". These subjective judgements vary not only from person
to person, but also with time: for example, as a proof is
simplified or better understood, a theorem that was once difficult
may become trivial. On the other hand, a deep theorem may be simply
stated, but its proof may involve surprising and subtle connections
between disparate areas of mathematics. Fermat's
last theorem is a particularly well-known example of such a
theorem.

## Formal and informal notions

Logically most
theorems are of the form of an indicative
conditional: if A, then B. Such a theorem does not state that B
is always true, only that B must be true if A is true. In this case
A is called the hypothesis of the theorem
(note that "hypothesis" here is something very different from a
conjecture) and B the
conclusion. The theorem "If n is an even natural
number then n/2 is a natural number" is a typical example in
which the hypothesis is that n is an even natural number and the
conclusion is that n/2 is also a natural number.

In order to be proven, a theorem must be
expressible as a precise, formal statement. Nevertheless, theorems
are usually expressed in natural language rather than in a
completely symbolic form, with the intention that the reader will
be able to produce a formal statement from the informal one. In
addition, there are often hypotheses which are understood in
context, rather than explicitly stated.

It is common in mathematics to choose a number of
hypotheses that are assumed to be true within a given theory, and
then declare that the theory consists of all theorems provable
using those hypotheses as assumptions. In this case the hypotheses
that form the foundational basis are called the axioms (or postulates) of the
theory. The field of mathematics known as proof theory
studies formal axiom systems and the proofs that can be performed
within them.

Some theorems are "trivial," in the sense that
they follow from definitions, axioms, and other theorems in obvious
ways and do not contain any surprising insights. Some, on the other
hand, may be called "deep": their proofs may be long and difficult,
involve areas of mathematics superficially distinct from the
statement of the theorem itself, or show surprising connections
between disparate areas of mathematics. A theorem might be simple
to state and yet be deep. An excellent example is Fermat's
Last Theorem, and there are many other examples of simple yet
deep theorems in number
theory and combinatorics, among other
areas.

There are other theorems for which a proof is
known, but the proof cannot easily be written down. The most
prominent examples are the Four
color theorem and the Kepler
conjecture. Both of these theorems are only known to be true by
reducing them to a computational search which is then verified by a
computer program. Initially, many mathematicians did not accept
this form of proof, but it has become more widely accepted in
recent years. The mathematician Doron
Zeilberger has even gone so far as to claim that these are
possibly the only nontrivial results that mathematicians have ever
proved.http://www.math.rutgers.edu/~zeilberg/Opinion51.html
Many mathematical theorems can be reduced to more straightforward
computation, including polynomial identities, trigonometric
identities and hypergeometric identities.

## Relation to proof

The notion of a theorem is deeply intertwined
with the concept of proof. Indeed, theorems are true precisely in
the sense that they possess proofs. Therefore, to establish a
mathematical statement as a theorem, the existence of a line of
reasoning from axioms in the system (and other, already established
theorems) to the given statement must be demonstrated.

Although the proof is necessary to produce a
theorem, it is not usually considered part of the theorem. And even
though more than one proof may be known for a single theorem, only
one proof is required to establish the theorem's validity. The
Pythagorean
theorem and the law of quadratic
reciprocity are contenders for the title of theorem with the
greatest number of distinct proofs.

## Theorems in logic

Logic, especially in
the field of proof
theory, considers theorems as statements (called formulas or well
formed formulas) of a formal
language. A set of deduction rules, also called transformation
rules or a formal
grammar, must be provided. These deduction rules tell exactly
when a formula can be derived from a set of premises.

Different sets of derivation rules give rise to
different interpretations of what it means for an expression to be
a theorem. Some derivation rules and formal languages are intended
to capture mathematical reasoning; the most common examples use
first-order
logic. Other deductive systems describe term
rewriting, such as the reduction rules for λ
calculus.

The definition of theorems as elements of a
formal language allows for results in proof theory that study the
structure of formal proofs and the structure of provable formulas.
The most famous result is
Gödel's incompleteness theorem; by representing theorems about
basic number theory as expressions in a formal language, and then
representing this language within number theory itself, Gödel
constructed examples of statements that are neither provable nor
disprovable from axiomatizations of number theory.

## Relation with scientific theories

Theorems in mathematics and theories in science are
fundamentally different in their epistemology. A scientific
theory cannot be proven; its key attribute is that it is falsifiable, that is, it
makes predictions about the natural world that are testable by
experiments. Any
disagreement between prediction and experiment demonstrates the
incorrectness of the scientific theory, or at least limits its
accuracy or domain of validity. Mathematical theorems, on the other
hand, are purely abstract formal statements: the proof of a theorem
cannot involve experiments or other empirical evidence in the same
way such evidence is used to support scientific theories.

Nonetheless, there is some degree of empiricism
and data collection involved in the discovery of mathematical
theorems. By establishing a pattern, sometimes with the use of a
powerful computer, mathematicians may have an idea of what to
prove, and in some cases even a plan for how to set about doing the
proof. For example, the Collatz
conjecture has been verified for start values up to about
2.88 × 1018. The Riemann
hypothesis has been verified for the first 10 trillion zeroes
of the zeta
function. Neither of these statements is considered to be
proven.

Such evidence does not constitute proof. For
example, the Mertens
conjecture is a statement about natural numbers that is now
known to be false, but no explicit counterexample (i.e., a natural
number n for which the Mertens function M(n) equals or exceeds the
square root of n) is known: all numbers less than 1014 have the
Mertens property, and the smallest number which does not have this
property is only known to be less than the exponential
of 1.59 × 1040, which is approximately 10 to the
power 4.3 × 1039. Since the number of particles
in the universe is generally considered to be less than 10 to the
power 100 (a googol),
there is no hope to find an explicit counterexample by exhaustive
search at present.

Note that the word "theory" also exists in
mathematics, to denote a body of mathematical axioms, definitions
and theorems, as in, for example, group
theory. There are also "theorems" in science, particularly
physics, and in engineering, but they often have statements and
proofs in which physical assumptions and intuition play an
important role; the physical axioms on which such "theorems" are
based are themselves falsifiable.

## Terminology

Theorems are often indicated by several other terms: the actual label "theorem" is reserved for the most important results, whereas results which are less important, or distinguished in other ways, are named by different terminology.- A proposition is a statement not associated with any particular theorem. This term sometimes connotes a statement with a simple proof.
- A lemma is a "pre-theorem", a statement that forms part of the proof of a larger theorem. The distinction between theorems and lemmas is rather arbitrary, since one mathematician's major result is another's minor claim. Gauss's lemma and Zorn's lemma, for example, are interesting enough that some authors present the nominal lemma without going on to use it in the proof of a theorem.
- A corollary is a proposition that follows with little or no proof from one other theorem or definition. That is, proposition B is a corollary of a proposition A if B can readily be deduced from A.
- A claim is a necessary or independently interesting result that may be part of the proof of another statement. Despite the name, claims must be proved.

There are other terms, less commonly used, which
are conventionally attached to proven statements, so that certain
theorems are referred to by historical or customary names. For
examples:

- Identity, used for theorems which state an equality between two mathematical expressions. Examples include Euler's identity and Vandermonde's identity.
- Rule, used for certain theorems such as Bayes' rule and Cramer's rule, that establish useful formulas.
- Law. Examples include the law of large numbers, the law of cosines, and Kolmogorov's zero-one law.
- Principle. Examples include Harnack's principle, the least upper bound principle, and the pigeonhole principle.
- A Converse is a reverse theorem. For example, If a theorem states that A is a related to B, its converse would state that B is related to A. The converse of a theorem need not be always true.

A few well-known theorems have even more
idiosyncratic names. The division
algorithm is a theorem expressing the outcome of division in
the natural numbers and more general rings. The
Banach–Tarski paradox is a theorem in measure
theory that is paradoxical in the sense that it
contradicts common intuitions about volume in three-dimensional
space.

An unproven statement that is believed to be true
is called a conjecture (or sometimes a
hypothesis, but with
a different meaning from the one discussed above). To be considered
a conjecture, a statement must usually be proposed publicly, at
which point the name of the proponent may be attached to the
conjecture, as with Goldbach's
conjecture. Other famous conjectures include the Collatz
conjecture and the Riemann
hypothesis.

## Layout

A theorem and its proof are typically laid out as
follows:

- Theorem (name of person who proved it and year of discovery, proof or publication).
- Statement of theorem.
- Proof.
- Description of proof.

The end of the proof may be signalled by the
letters Q.E.D. meaning
"quod
erat demonstrandum" or by one of the tombstone
marks "" or "" meaning "End of Proof", introduced by Paul Halmos
following their usage in magazine articles.

The exact style will depend on the author or
publication. Many publications provide instructions or macros
for typesetting in the house
style.

It is common for a theorem to be preceded by
definitions
describing the exact meaning of the terms used in the theorem. It
is also common for a theorem to be preceded by a number of propositions or lemmas
which are then used in the proof. However, lemmas are sometimes
embedded in the proof of a theorem, either with nested proofs, or
with their proofs presented after the proof of the theorem.

Corollaries to a
theorem are either presented between the theorem and the proof, or
directly after the proof. Sometimes corollaries have proofs of
their own which explain why they follow from the theorem.

## Lore

It has been estimated that over a quarter of a
million theorems are proved every year.

The well-known aphorism, "A mathematician is
a device for turning coffee into theorems", is probably due to
Alfréd
Rényi, although it is often attributed to Rényi's colleague
Paul
Erdős (and Rényi may have been thinking of Erdős), who was
famous for the many theorems he produced, the number of
his collaborations, and his coffee drinking.

The
classification of finite simple groups is regarded by some to
be the longest proof of a theorem; it comprises tens of thousands
of pages in 500 journal articles by some 100 authors. These papers
are together believed to give a complete proof, and there are
several ongoing projects to shorten and simplify this proof.

## See also

- Metatheorem
- List of theorems
- Gödel's incompleteness theorem, that establishes very general conditions under which a formal system will contain a true statement for which there exists no proof within the system.
- Inference
- Toy theorem

## Notes

## References

## External links

portalpar Logictheorem in Arabic: مبرهنة

theorem in Bengali: উপপাদ্য

theorem in Belarusian (Tarashkevitsa):
Тэарэма

theorem in Bosnian: Teorem

theorem in Bulgarian: Теорема

theorem in Catalan: Teorema

theorem in Czech: Matematická věta

theorem in Danish: Sætning (matematik)

theorem in German: Theorem

theorem in Estonian: Teoreem

theorem in Spanish: Teorema

theorem in Esperanto: Teoremo

theorem in Persian: قضیه

theorem in French: Théorème

theorem in Scottish Gaelic: Teòirim

theorem in Galician: Teorema

theorem in Korean: 정리

theorem in Hindi: प्रमेय

theorem in Croatian: Teorem

theorem in Ido: Teoremo

theorem in Indonesian: Teorema

theorem in Icelandic: Setning (stærðfræði)

theorem in Italian: Teorema

theorem in Hebrew: משפט (מתמטיקה)

theorem in Georgian: თეორემა

theorem in Kazakh: Теорема

theorem in Lithuanian: Teorema

theorem in Macedonian: Теорема

theorem in Dutch: Stelling (wiskunde)

theorem in Japanese: 定理

theorem in Norwegian: Teorem

theorem in Polish: Twierdzenie

theorem in Portuguese: Teorema

theorem in Romanian: Teoremă

theorem in Russian: Теорема

theorem in Simple English: Theorem

theorem in Slovak: Teoréma

theorem in Slovenian: Izrek

theorem in Serbian: Теорема

theorem in Serbo-Croatian: Teorem

theorem in Finnish: Teoreema

theorem in Swedish: Teorem

theorem in Vietnamese: Định lý toán học

theorem in Turkish: Teorem

theorem in Ukrainian: Теорема

theorem in Chinese: 定理

# Synonyms, Antonyms and Related Words

a priori principle, a priori truth, affirmation, apriorism, assertion, assumed position,
assumption, axiom, basis, brocard, categorical
proposition, conjecture, data, deduction, dictate, dictum, first principles,
formula, foundation, fundamental, golden rule,
ground, hypothesis, hypothesis ad
hoc, law, lemma, major premise, minor
premise, philosopheme, philosophical
proposition, position,
postulate, postulation, postulatum, premise, presupposition, principium, principle, proposition, propositional
function, rule,
self-evident truth, settled principle, statement, sumption, supposal, thesis, truism, truth, truth table,
truth-function, truth-value, universal truth