TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a theorem is a
statement Statement or statements may refer to: Common uses *Statement (computer science), the smallest standalone element of an imperative programming language *Statement (logic), declarative sentence that is either true or false *Statement, a Sentence_(lin ...
that has been proved, or can be proved. The ''proof'' of a theorem is a
logical argument In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (both spellings are acceptable), intended to determine the degree of truth of another statement, the conclusion. The logical ...
that uses the inference rules of a
deductive system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentiall ...
to establish that the theorem is a
logical consequence Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies ...
of the
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

s and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of
Zermelo–Fraenkel set theory In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...
with the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

, or of a less powerful theory, such as
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have been ...
. A notable exception is
Wiles's proof of Fermat's Last Theorem Wiles's proof of Fermat's Last Theorem is a proof Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, ...
, which involves the
Grothendieck universeIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s whose existence requires to add a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...
, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them. In this context, statements become
well-formed formula In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
s of some
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...
. A
theory A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, G ...
consists of some basis statements called ''axioms'', and some ''deducing rules'' (sometimes included in the axioms). The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules. This formalization led to
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
, which allows proving general theorems about theorems and proofs. In particular,
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (nu ...
show that every
consistent In classical deductive logic Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion. Deductive reasoning goes in the same direction as that of the conditiona ...

theory containing the natural numbers has true statements on natural numbers that are not theorems of the theory (that is they cannot be proved inside the theory). As the axioms are often abstractions of properties of the
physical world The universe ( la, universus) is all of space and time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparently irreversible process, irreversible succession from the pas ...

, theorems may be considered as the expression some truth, but in contrast to the notion of a
scientific law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
, which is ''
experimental An experiment is a procedure carried out to support, refute, or validate a hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method ...

'', the justification of the truth of a theorem is purely
deductive Deductive reasoning, also deductive logic, is the process of reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making ...
.

# Theoremhood and truth

Until the end of the 19th century and the foundational crisis of mathematics, all mathematics were built from a few basic properties that were considered as self-evident; for example, the facts that every
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
has a successor, and that there is exactly one
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...
that passing through two given distinct points. Those basic properties that were not considered as absolutely evident were called
postulate An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
s; for example . All theorems were proved by using implicitly or explicitly these basic properties, and, because of the evidence of these basic properties, a proved theorem was considered as a definitive truth, unless there was an error in the proof. For example, the sum of the
interior angle 300px, Internal and external angles In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is Polygon#Convexity and non-convexity, con ...
s of a
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

equals 180°, and this was considered as an undoubtful fact. One aspect of the foundational crisis of mathematics was the discovery of
non-Euclidean geometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
that do not lead to any contradiction, although, in such geometries, the sum of the angles of a triangle is different from 180°. So, the property ''"the sum of the angles of a triangle equals 180°"'' is either true or false, depending whether Euclid's postulates is assumed. Similarly, the use of "evident" basic properties of sets leads to the contradiction of Russel's paradox. This has been resolved by elaborating the rules that are allowed for manipulating sets. This crisis has been resolved by revisiting the foundations of mathematics for making them more rigorous. In these new fundations, a theorem is a
well-formed formula In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
of a
mathematical theory A mathematical theory is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A sys ...
that can be proved from the
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

s and
inference rules A rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax In linguistics, syntax () is the set of rules, principles, and processes that govern the structu ...
of the theory. So, the above theorem on the sum of the angles of a triangle becomes: ''Under the axioms and inference rules of
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, the sum of the interior angles of a triangle equals 180°''. Similarly, Russel's paradox disappears because, in an axiomatized set theory, the ''set of all sets'' cannot be expressed with a well-formed formula. More precisely, if the set of all sets can be expressed with a well-formed formula, this implies that the theory is
inconsistent In classical deductive logic Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion. Deductive reasoning goes in the same direction as that of the conditiona ...
, and every well-formed assertion, as well as its negation, is a theorem. In this context, the validity of a theorem depends only on the correctness of its proof. It is independent from the truth, or even the significance of the axioms. This does not mean that the significance of the axioms in uninteresting, but only that the validity (truth) of a theorem is independent from the significance of the axioms. This independence may be useful by allowing the use of results of some area of mathematics in apparently unrelated areas. An important consequence of this way of thinking mathematics is that it allows defining mathematical theories and theorems as
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...
s, and to prove theorems about them. Examples are
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (nu ...
. In particular, there are well-formed assertions than can be proved to not be a theorem of the ambient theory, although they can be proved in a wider theory. An example is
Goodstein's theorem In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alg ...
, which can be stated in
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have been ...
, but is proved to be not provable in Peano arithmetics. However, it is provable in some more general theories, such as
Zermelo–Fraenkel set theory In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...
.

# Epistemological considerations

Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or
premise A premise or premiss is a true or false statement that helps form the body of an argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, appl ...
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 A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentiall ...
s, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g.,
non-classical logicNon-classical logics (and sometimes alternative logics) are formal system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are ...
). Although theorems can be written in a completely symbolic form (e.g., as propositions in
propositional calculus Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...
), 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 Aesthetics, or esthetics (), is a branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of m ...
. 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 In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
is a particularly well-known example of such a theorem.

# Informal account of theorems

Logically Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, account, re ...
, many theorems are of the form of an
indicative conditional In natural language In neuropsychology Neuropsychology is a branch of psychology. It is concerned with how a person's cognition and behavior are related to the brain and the rest of the nervous system. Professionals in this branch of psych ...
: ''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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
), and ''B'' the ''conclusion'' of the theorem. The two together (without the proof) are called the ''proposition'' or ''statement'' of the theorem (e.g. "''If A, then B''" is the ''proposition''). Alternatively, ''A'' and ''B'' can be also termed the ''
antecedent An antecedent is a preceding event, condition, cause, phrase, or word. More specifically, it may refer to: * Antecedent (behavioral psychology), the stimulus that occurs before a trained behavior * Antecedent (genealogy), antonym of descendant, gen ...
'' and the ''
consequent A consequent is the second half of a hypothetical proposition In linguistics and logic, a proposition is the meaning of a declarative sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which i ...
'', respectively. The theorem "If ''n'' is an even
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
, 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
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

s or postulates. The field of mathematics known as
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
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 In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
, and there are many other examples of simple yet deep theorems in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

and
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
, 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#REDIRECT Kepler conjecture The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician A mathematician i ...
. 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 Doron Zeilberger (דורון ציילברגר, born 2 July 1950 in Haifa Haifa ( he, חֵיפָה ' ; ar, حيفا ') is the List of cities in Israel, third-largest city in Israel—after Jerusalem and Tel Aviv—with a population of in . ...
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.

# Relation with scientific theories

Theorems in mathematics and theories in science are fundamentally different in their
epistemology Epistemology (; ) is the Outline of philosophy, branch of philosophy concerned with knowledge. Epistemologists study the nature, origin, and scope of knowledge, epistemic Justification (epistemology), justification, the Reason, rationality o ...

. A scientific theory cannot be proved; its key attribute is that it is
falsifiable In the philosophy of science, a theory is falsifiable if it is contradicted by ''possible observations''—i.e., by any observations that can be described in the language of the theory, which must have a conventional empirical interpretation. ...
, that is, it makes predictions about the natural world that are testable by
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...

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. It is also possible to find a single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs. For example, both the
Collatz conjecture The Collatz conjecture is a conjecture in mathematics that concerns sequence, sequences of positive integer, positive integers. Starting from , each term is obtained from the previous one as follows: If the previous term is Parity (mathematics), ...
and the
Riemann hypothesis In mathematics, the Riemann hypothesis is a conjecture In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The
Collatz conjecture The Collatz conjecture is a conjecture in mathematics that concerns sequence, sequences of positive integer, positive integers. Starting from , each term is obtained from the previous one as follows: If the previous term is Parity (mathematics), ...
has been verified for start values up to about 2.88 × 1018. The
Riemann hypothesis In mathematics, the Riemann hypothesis is a conjecture In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
has been verified to hold for the first 10 trillion non-trivial zeroes of the
zeta function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

. Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved. Such evidence does not constitute proof. For example, the
Mertens conjecture The graph shows the Mertens function M(n) and the square roots \pm \sqrt for n \le 10,000. After computing these values, Mertens conjectured that the absolute value of M(n) is always bounded by \sqrt. This hypothesis, known as the Mertens conjectur ...
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 Exponential may refer to any of several mathematical topics related to exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and ...

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 A googol is the large number Numbers that are significantly larger than those typically used in everyday life, for instance in simple counting or in monetary transactions, appear frequently in fields such as mathematics Mathematics (fro ...

), there is no hope to find an explicit counterexample by
exhaustive search In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algor ...
. The word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example,
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
(see
mathematical theory A mathematical theory is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A sys ...
). 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 An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

'' or ''postulate'' is a fundamental assumption regarding the object of study, that is accepted without proof. A related concept is that of a ''
definition A definition is a statement of the meaning of a term (a word In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language ...

'', which gives the meaning of a word or a phrase in terms of known concepts. Classical geometry discerns between axioms, which are general statements; and postulates, which are statements about geometrical objects. Historically, axioms were regarded as "
self-evident In epistemology Epistemology (; ) is the Outline of philosophy, branch of philosophy concerned with knowledge. Epistemologists study the nature, origin, and scope of knowledge, epistemic Justification (epistemology), justification, the Re ...
"; today they are merely ''assumed'' to be true. * A ''
conjecture In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
'' is an unproved statement that is believed to be true. Conjectures are usually made in public, and named after their maker (for example,
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician ...
and
Collatz conjecture The Collatz conjecture is a conjecture in mathematics that concerns sequence, sequences of positive integer, positive integers. Starting from , each term is obtained from the previous one as follows: If the previous term is Parity (mathematics), ...
). The term ''hypothesis'' is also used in this sense (for example,
Riemann hypothesis In mathematics, the Riemann hypothesis is a conjecture In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
), which should not be confused with "hypothesis" as the premise of a proof. Other terms are also used on occasion, for example ''problem'' when people are not sure whether the statement should be believed to be true.
Fermat's Last Theorem In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
was historically called a theorem, although, for centuries, it was only a conjecture. * A ''theorem'' is a statement that has been proven to be true based on axioms and other theorems. * A ''
proposition In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
'' is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should not be confused with "proposition" as used in
propositional logic Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...
. In classical geometry the term "proposition" was used differently: in
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

's ''Elements'' (), all theorems and geometric constructions were called "propositions" regardless of their importance. * A ''
lemma Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered * Headword, under which a set of related dict ...
'' is an "accessory proposition" - a proposition with little applicability outside its use in a particular proof. Over time a lemma may gain in importance and be considered a ''theorem'', though the term "lemma" is usually kept as part of its name (e.g. Gauss's lemma,
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...
, and the fundamental lemma). * A ''
corollary In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
'' is a proposition that follows immediately from another theorem or axiom, with little or no required proof. A corollary may also be a restatement of a theorem in a simpler form, or for a
special case In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
: for example, the theorem "all internal angles in a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

are
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

s" has a corollary that "all internal angles in a ''
square In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

'' are
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

s" - a square being a special case of a rectangle. * A ''
generalization A generalization is a form of abstraction Abstraction in its main sense is a conceptual process where general rules and concept Concepts are defined as abstract ideas or general notions that occur in the mind, in speech, or in thought. They ...

'' of a theorem is a theorem with a similar statement but a broader scope, from which the original theorem can be deduced as a
special case In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
(a ''corollary''). Other terms may also be used for historical or customary reasons, for example: * An ''
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
'' is a theorem stating an equality between two expressions, that holds for any value within its
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
(e.g.
Bézout's identity In elementary number theory, Bézout's identity (also called Bézout's lemma) is the following theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical ...
and
Vandermonde's identity In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: :=\sum_^r for any nonnegative integers ''r'', ''m'', ''n''. The identity is named after Alexandre-Théophile Vandermonde ...
). * A ''rule'' is a theorem that establishes a useful formula (e.g.
Bayes' rule In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expr ...
and
Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical s ...
). * A ''
law Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its bounda ...
'' or ''
principle A principle is a proposition or value that is a guide for behavior or evaluation. In law, it is a Legal rule, rule that has to be or usually is to be followed. It can be desirably followed, or it can be an inevitable consequence of something, suc ...

'' is a theorem with wide applicability (e.g. the
law of large numbers In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

,
law of cosines In trigonometry Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focu ...
,
Kolmogorov's zero–one law In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...
,
Harnack's principleIn complex analysis, Harnack's principle or Harnack's theorem is one of several closely related theorems about the convergence of sequences of harmonic functions, that follow from Harnack's inequality. If the function (mathematics), functions u_1(z ...
, the least-upper-bound principle, and the
pigeonhole principle In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
). A few well-known theorems have even more idiosyncratic names, for example, the
division algorithm A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software. Divisi ...
,
Euler's formula Euler's formula, named after Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

, and the
Banach–Tarski paradox The Banach–Tarski paradox is a theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, ...
.

# Layout

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'')'' :''Proof'' :''Description of proof'' :''End'' The end of the proof may be signaled by the letters
Q.E.D. Q.E.D. or QED is an initialism An acronym is a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical mean ...
(''quod erat demonstrandum'') or by one of the tombstone marks, such as "□" or "∎", meaning "end of proof", introduced by
Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a HungarianHungarian may refer to: * Hungary, a country in Central Europe * Kingdom of Hungary, state of Hungary, existing between 1000 and 1946 * Hungarians, ethnic ...
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 A definition is a statement of the meaning of a term (a word In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language ...

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.

# Lore

It has been estimated that over a quarter of a million theorems are proved every year. The well-known
aphorism An aphorism (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...
, "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 Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a renowned Hungarian mathematician In this page we keep the names in Hungarian order (family name first). {{compact ToC , short1, side=yes A * Alexits György (1899–1 ...

(and Rényi may have been thinking of Erdős), who was famous for the many theorems he produced, the
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...
of his collaborations, and his coffee drinking. The
classification of finite simple groups In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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

In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...
, a formal theory is a set of sentences within a
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...
. A sentence is a
well-formed formula In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its theorems. Usually a theory is understood to be closed under the relation of
logical consequence Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies ...
. Some accounts define a theory to be closed under the
semantic consequence Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies ...
relation ($\models$), while others define it to be closed under the syntactic consequence, or derivability relation ($\vdash$).van Dalen, p. 104. For a theory to be closed under a derivability relation, it must be associated with a
deductive system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentiall ...
that specifies how the theorems are derived. The deductive system may be stated explicitly, or it may be clear from the context. The closure of the empty set under the relation of logical consequence yields the set that contains just those sentences that are the theorems of the deductive system. In the broad sense in which the term is used within logic, a theorem does not have to be true, since the theory that contains it may be relative to a given semantics, or relative to the standard
interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event i ...
of the underlying language. A theory that is
inconsistent In classical deductive logic Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion. Deductive reasoning goes in the same direction as that of the conditiona ...
has all sentences as theorems. The definition of theorems as sentences of a formal language is useful within
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
, which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
, which is concerned with the relationship between formal theories and structures that are able to provide a semantics for them through
interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event i ...
. Although theorems may be uninterpreted sentences, in practice mathematicians are more interested in the meanings of the sentences, i.e. in the propositions they express. What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be interpreted as a proof of their truth. A theorem whose interpretation is a true statement ''about'' a formal system (as opposed to ''within'' a formal system) is called a ''
metatheorem In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
''. Some important theorems in mathematical logic are: * Compactness of first-order logic * Completeness of first-order logic * Gödel's incompleteness theorems of first-order arithmetic * Consistency of first-order arithmetic * Tarski's undefinability theorem * Church-Turing theorem of undecidability * Löb's theorem * Löwenheim–Skolem theorem * Lindström's theorem * Craig's theorem * Cut-elimination theorem

## 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, Theory of justification, justification or other Modal logic, modalities). The soundness of a formal system depends on whether or not all of its theorems are also Validity (logic), validities. A validity is a formula that is true under any possible interpretation (for example, in classical propositional logic, validities are tautology (logic), tautologies). A formal system is considered completeness (logic), semantically complete when all of its theorems are also tautologies.

## Theorems and theories

* List of theorems *Fundamental theorem *Formula *Inference * Toy theorem

# References

* * * * * * * * * * * * * * *