In mathematics, a theorem is a statement that
has been proven on the basis of previously
established statements, such as other theorems,
and generally accepted statements, such as
axioms. A theorem is a logical consequence
of the axioms. The proof of a mathematical
theorem is a logical argument for the theorem
statement given in accord with the rules of
a deductive system. The proof of a theorem
is often interpreted as justification of the
truth of the theorem statement. In light of
the requirement that theorems be proved, the
concept of a theorem is fundamentally deductive,
in contrast to the notion of a scientific
law, which is experimental.Many mathematical
theorems are conditional statements. In this
case, the proof deduces the conclusion from
conditions called hypotheses or premises.
In light of the interpretation of proof as
justification of truth, the conclusion is
often viewed as a necessary consequence of
the hypotheses, namely, that the conclusion
is true in case the hypotheses are true, without
any further assumptions. However, the conditional
could be interpreted differently in certain
deductive systems, depending on the meanings
assigned to the derivation rules and the conditional
symbol.
Although they can be written in a completely
symbolic form, for example, within the 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 organized and clearly worded
informal arguments, intended to convince readers
of the truth of the statement of the theorem
beyond any doubt, and from which a formal
symbolic proof can in principle 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
judgments 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 stated
simply, 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.
== Informal account of theorems ==
Logically, many theorems are of the form of
an indicative conditional: if A, then B. Such
a theorem does not assert B, only that B is
a necessary consequence of A. In this case
A is called the hypothesis of the theorem
("hypothesis" here is something very different
from a conjecture) and B the conclusion (formally,
A and B are termed the antecedent and consequent).
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 "n is an
even natural number" and the conclusion is
"n/2 is also a natural number".
To be proved, 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 can
produce a formal statement from the informal
one.
It is common in mathematics to choose a number
of hypotheses within a given language and
declare that the theory consists of all statements
provable from these hypotheses. These hypotheses
form the foundational basis of the theory
and are called axioms or postulates. The field
of mathematics known as proof theory studies
formal languages, axioms and the structure
of proofs.
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", because
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.
Other theorems have a known proof that 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 that is then verified
by a computer program. Initially, many mathematicians
did not accept this form of proof, but it
has become more widely accepted. 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.
Many mathematical theorems can be reduced
to more straightforward computation, including
polynomial identities, trigonometric identities
and hypergeometric identities.
== Provability and theoremhood ==
To establish a mathematical statement as a
theorem, a proof is required, that is, a line
of reasoning from axioms in the system (and
other, already established theorems) to the
given statement must be demonstrated. However,
the proof is usually considered as separate
from the theorem statement. Although more
than one proof may be known for a single theorem,
only one proof is required to establish the
status of a statement as a theorem. The Pythagorean
theorem and the law of quadratic reciprocity
are contenders for the title of theorem with
the greatest number of distinct proofs.
== Relation with scientific theories ==
Theorems in mathematics and theories in science
are fundamentally different in their epistemology.
A scientific theory cannot be proved; 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
proved.
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
that 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 less
than 10 to the power 100 (a googol), there
is no hope to find an explicit counterexample
by exhaustive search.
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 ==
A number of different terms for mathematical
statements exist; these terms indicate the
role statements play in a particular subject.
The distinction between different terms is
sometimes rather arbitrary and the usage of
some terms has evolved over time.
An axiom or postulate is a statement that
is accepted without proof and regarded as
fundamental to a subject. Historically these
have been regarded as "self-evident", but
more recently they are considered assumptions
that characterize the subject of study. In
classical geometry, axioms are general statements,
while postulates are statements about geometrical
objects. A definition is also accepted without
proof since it simply gives the meaning of
a word or phrase in terms of known concepts.An
unproved statement that is believed 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.
On the other hand, Fermat's Last Theorem has
always been known by that name, even before
it was proved; it was never known as "Fermat's
conjecture".
A proposition is a theorem of lesser importance.
This term sometimes connotes a statement with
a simple proof, while the term theorem is
usually reserved for the most important results
or those with long or difficult proofs. Some
authors never use "proposition", while some
others use "theorem" only for fundamental
results. In classical geometry, this term
was used differently: In Euclid's Elements
(c. 300 BCE), all theorems and geometric constructions
were called "propositions" regardless of their
importance.
A lemma is a "helping theorem", a proposition
with little applicability except that it forms
part of the proof of a larger theorem. In
some cases, as the relative importance of
different theorems becomes more clear, what
was once considered a lemma is now considered
a theorem, though the word "lemma" remains
in the name. Examples include Gauss's lemma,
Zorn's lemma, and the fundamental lemma.
A corollary is a proposition that follows
with little proof from another theorem or
definition. Also a corollary can be a theorem
restated for a more restricted special case.
For example, the theorem that all angles in
a rectangle are right angles has as corollary
that all angles in a square (a special case
of a rectangle) are right angles.
A converse of a theorem is a statement formed
by interchanging what is given in a theorem
and what is to be proved. For example, the
isosceles triangle theorem states that if
two sides of a triangle are equal then two
angles are equal. In the converse, the given
(that two sides are equal) and what is to
be proved (that two angles are equal) are
swapped, so the converse is the statement
that if two angles of a triangle are equal
then two sides are equal. In this example,
the converse can be proved as another theorem,
but this is often not the case. For example,
the converse to the theorem that two right
angles are equal angles is the statement that
two equal angles must be right angles, and
this is clearly not always the case.
A generalization is a theorem which includes
a previously proved theorem as a special case
and hence as a corollary.There are other terms,
less commonly used, that are conventionally
attached to proved statements, so that certain
theorems are referred to by historical or
customary names. For example:
An identity is an equality, contained in a
theorem, between two mathematical expressions
that holds regardless of what values are used
for any variables or parameters appearing
in the expressions. Examples include Euler's
formula and Vandermonde's identity.
A rule is a theorem, such as Bayes' rule and
Cramer's rule, that establishes a useful formula.
A law or a principle is a theorem that applies
in a wide range of circumstances. Examples
include the law of large numbers, the law
of cosines, Kolmogorov's zero–one law, Harnack's
principle, the least-upper-bound principle,
and the pigeonhole principle.A few well-known
theorems have even more idiosyncratic names.
The division algorithm (see Euclidean division)
is a theorem expressing the outcome of division
in the natural numbers and more general rings.
Bézout's identity is a theorem asserting
that the greatest common divisor of two numbers
may be written as a linear combination of
these numbers. 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.
== 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 (sometimes called the
proposition).
Proof.
Description of proof.
EndThe end of the proof may be signalled by
the letters Q.E.D. (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 depends 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 that 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 several ongoing projects hope to shorten
and simplify this proof. Another theorem of
this type is the four color theorem whose
computer generated proof is too long for a
human to read. It is certainly the longest
known proof of a theorem whose statement can
be easily understood by a layman.
== 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.
The statements of the language are strings
of symbols and may be broadly divided into
nonsense and well-formed formulas. A set of
deduction rules, also called transformation
rules or rules of inference, must be provided.
These deduction rules tell exactly when a
formula can be derived from a set of premises.
The set of well-formed formulas may be broadly
divided into theorems and non-theorems. However,
according to Hofstadter, a formal system often
simply defines all its well-formed formula
as theorems.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.
A theorem may be expressed in a formal language
(or "formalized"). A formal theorem is the
purely formal analogue of a theorem. In general,
a formal theorem is a type of well-formed
formula that satisfies certain logical and
syntactic conditions. The notation
⊢
{\displaystyle \vdash }
S
{\displaystyle S}
is often used to indicate that
S
{\displaystyle S}
is a theorem.
Formal theorems consist of formulas of a formal
language and the transformation rules of a
formal system. Specifically, a formal theorem
is always the last formula of a derivation
in some formal system each formula of which
is a logical consequence of the formulas that
came before it in the derivation. The initially
accepted formulas in the derivation are called
its axioms, and are the basis on which the
theorem is derived. A set of theorems is called
a theory.
What makes formal theorems useful and of interest
is that they can be interpreted as true propositions
and their derivations may be interpreted as
a proof of the truth of the resulting expression.
A set of formal theorems may be referred to
as a formal theory. A theorem whose interpretation
is a true statement about a formal system
is called a metatheorem.
=== Syntax and semantics ===
The concept of a formal theorem is fundamentally
syntactic, in contrast to the notion of a
true proposition, which introduces semantics.
Different deductive systems can yield other
interpretations, depending on the presumptions
of the derivation rules (i.e. belief, justification
or other modalities). The soundness of a formal
system depends on whether or not all of its
theorems are also validities. A validity is
a formula that is true under any possible
interpretation, e.g. in classical propositional
logic validities are tautologies. A formal
system is considered semantically complete
when all of its tautologies are also theorems.
=== Derivation of a theorem ===
The notion of a theorem is very closely connected
to its formal proof (also called a "derivation").
To illustrate how derivations are done, we
will work in a very simplified formal system.
Let us call ours
F
S
{\displaystyle {\mathcal {FS}}}
Its alphabet consists only of two symbols
{ A, B } and its formation rule for formulas
is:
Any string of symbols of
F
S
{\displaystyle {\mathcal {FS}}}
that is at least three symbols long, and is
not infinitely long, is a formula. Nothing
else is a formula.The single axiom of
F
S
{\displaystyle {\mathcal {FS}}}
is:
ABBA.The only rule of inference (transformation
rule) for
F
S
{\displaystyle {\mathcal {FS}}}
is:
Any occurrence of "A" in a theorem may be
replaced by an occurrence of the string "AB"
and the result is a theorem.Theorems in
F
S
{\displaystyle {\mathcal {FS}}}
are defined as those formulae that have a
derivation ending with that formula. For example,
ABBA (Given as axiom)
ABBBA (by applying the transformation rule)
ABBBAB (by applying the transformation rule)is
a derivation. Therefore, "ABBBAB" is a theorem
of
F
S
.
{\displaystyle {\mathcal {FS}}\,.}
The notion of truth (or falsity) cannot be
applied to the formula "ABBBAB" until an interpretation
is given to its symbols. Thus in this example,
the formula does not yet represent a proposition,
but is merely an empty abstraction.
Two metatheorems of
F
S
{\displaystyle {\mathcal {FS}}}
are:
Every theorem begins with "A".
Every theorem has exactly two "A"s.
=== Interpretation of a formal theorem ===
=== 
Theorems and theories ===
== See also ==
Inference
List of theorems
Toy theorem
Metamath – a language for developing strictly
formalized mathematical definitions and proofs
accompanied by a proof checker for this language
and a growing database of thousands of proved
theorems
== Notes
