, a theorem is a non-self-evident statement
that has been proven
to be true, either on the basis of generally accepted statements such as axiom
s or on the basis of previously established statements such as other theorems. A theorem is hence a logical consequence
of the axioms, with a proof
of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system
. As a result, 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, whose proofs deduce conclusions from conditions known as hypotheses or premise
s. 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 also be interpreted differently in certain deductive system
s, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic
Although theorems can be written in a completely symbolic form (e.g., as propositions in propositional calculus
), they are often expressed informally in a natural language such as English for better readability. 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.
In addition to the better readability, informal 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, one might even be able to substantiate a theorem by using a picture as its proof.
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 and culture: for example, as a proof is obtained, 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
, 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 means something very different from a conjecture
), and ''B'' the conclusion of the theorem. Alternatively, ''A'' and ''B'' can be also termed the ''antecedent
'' and the ''consequent
'', respectively. 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".
In order for a theorem be proved, it must be in principle expressible as a precise, formal statement. However, theorems are usually expressed in natural language rather than in a completely symbolic form—with the presumption that a formal statement can be derived 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
, 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 valid line of reasoning from the axioms and other already-established theorems to the given statement must be demonstrated. In general, the proof is considered to be separate from the theorem statement itself. This is in part because while 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 experiment
s. 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
(see mathematical 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.
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 yet another form of statement that 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 angle
s has as corollary that all angles in a square
(a special case
of a rectangle) are right angle
* 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 the values being used for any variables
s appearing in the expressions (as long as they are within the range of validity). 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 paradox
ical in the sense that it contradicts common intuitions about volume in three-dimensional space.
A theorem and its proof are typically laid out as follows:
:Theorem (name of the person who proved it, along with year of discovery or publication of the proof).
:''Statement of theorem (sometimes called the ''proposition'').''
:''Description of proof.''
The end of the proof may be signaled by the letters Q.E.D.
(''quod erat demonstrandum'') or by one of the tombstone
marks, such as "□" or "∎", meaning "End of Proof", introduced by Paul Halmos
following their use in magazines to mark the end of an article.
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 definition
s 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.
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 among the longest known proofs of a theorem whose statement can be easily understood by a layman.
Theorems in logic
, especially in the field of proof theory
, considers theorems as statements (called formula
s or well formed formula
s) 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 theorems
; 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
is often used to indicate that
is a theorem.
Formal theorems consist of formulas
of a formal language and the transformation rule
s 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 interesting is that they can be interpreted
as true proposition
s 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 (as opposed to ''of'' 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
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 (for example, in classical propositional logic, validities are tautologies
). A formal system is considered semantically complete
when all of its theorems are also tautologies.
Derivation of a theorem
The notion of a theorem is very closely connected to its formal proof (also called a "derivation"). As an illustration, consider a very simplified formal system
whose alphabet consists of only two symbols , and whose formation rule for formulas is:
:Any string of symbols of
that is at least three symbols long, and is not infinitely long, is a formula. Nothing else is a formula.
The single axiom of
The only rule of inference
(transformation rule) for
:Any occurrence of "A" in a theorem may be replaced by an occurrence of the string "AB" and the result is a theorem.
are defined as those formulas that have a derivation ending with it. 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
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
:Every theorem begins with "A".
:Every theorem has exactly two "A"s.
Interpretation of a formal theorem
Theorems and theories
* List of theorems
* Toy theorem
*Theorem of the Day