A mathematical proof is an inferential

^{2} = ''a''^{2}. Since 2 divides the expression on the left, 2 must also divide the equal expression on the right. That is, ''a''^{2} is even, which implies that ''a'' must also be even, as seen in the proposition above (in #Proof by contraposition). So we can write ''a'' = 2''c'', where ''c'' is also an integer. Substitution into the original equation yields 2''b''^{2} = (2''c'')^{2} = 4''c''^{2}. Dividing both sides by 2 yields ''b''^{2} = 2''c''^{2}. But then, by the same argument as before, 2 divides ''b''^{2}, so ''b'' must be even. However, if ''a'' and ''b'' are both even, they have 2 as a common factor. This contradicts our previous statement that ''a'' and ''b'' have no common factor, so we must conclude that $\backslash sqrt$ is an irrational number.
To paraphrase: if one could write $\backslash sqrt$ as a

Steven G. Krantz. 1. February 5, 2007 However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first proof of the four color theorem is an example of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time error in its calculations calls the validity of such computer-assisted proofs into question. In practice, the chances of an error invalidating a computer-assisted proof can be reduced by incorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs. Errors can never be completely ruled out in case of verification of a proof by humans either, especially if the proof contains natural language and requires deep mathematical insight to uncover the potential hidden assumptions and fallacies involved.

File:Chinese pythagoras.jpg, Visual proof for the (3,4,5) triangle as in the
Some illusory visual proofs, such as the missing square puzzle, can be constructed in a way which appear to prove a supposed mathematical fact but only do so under the presence of tiny errors (for example, supposedly straight lines which actually bend slightly) which are unnoticeable until the entire picture is closely examined, with lengths and angles precisely measured or calculated.

Proofs in Mathematics: Simple, Charming and Fallacious

* A lesson about proofs, in a course from

argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...

for a mathematical statement, showing that the stated assumptions logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

ally guarantee the conclusion. The argument may use other previously established statements, such as theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the ...

s; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...

which establish logical certainty, to be distinguished from empirical
Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...

arguments or non-exhaustive inductive reasoning
Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' rea ...

which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

, or a hypothesis if frequently used as an assumption for further mathematical work.
Proofs employ logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

expressed in mathematical symbols, along with natural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural language ...

which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic
Informal logic encompasses the principles of logic and logical thought outside of a formal setting (characterized by the usage of particular statements). However, the precise definition of "informal logic" is a matter of some dispute. Ralph H ...

. Purely formal proof
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequ ...

s, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...

is concerned with the role of language and logic in proofs, and mathematics as a language.
History and etymology

The word "proof" comes from the Latin ''probare'' (to test). Related modern words are English "probe", "probation", and "probability", Spanish ''probar'' (to smell or taste, or sometimes touch or test), Italian ''provare'' (to try), and German ''probieren'' (to try). The legal term "probity" means authority or credibility, the power of testimony to prove facts when given by persons of reputation or status. Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection withgeometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

, which originated in practical problems of land measurement. The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of its greatest achievements. Thales
Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded hi ...

(624–546 BCE) and Hippocrates of Chios (c. 470–410 BCE) gave some of the first known proofs of theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...

(384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known.
Mathematical proof was revolutionized by Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

(300 BCE), who introduced the axiomatic method still in use today. It starts with undefined terms and axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s, propositions concerning the undefined terms which are assumed to be self-evidently true (from Greek "axios", something worthy). From this basis, the method proves theorems using deductive logic
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...

. Euclid's book, the ''Elements'', was read by anyone who was considered educated in the West until the middle of the 20th century. In addition to theorems of geometry, such as the Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

, the ''Elements'' also covers number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...

, including a proof that the square root of two is irrational and a proof that there are infinitely many prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s.
Further advances also took place in medieval Islamic mathematics. While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...

and algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...

by Islamic mathematicians allowed more general proofs with no dependence on geometric intuition. In the 10th century CE, the Iraqi mathematician Al-Hashimi Al-Hashimi, also transliterated Al-Hashemi ( ar, الهاشمي), Hashemi, Hashimi or Hashmi ( fa, هاشمی) is an Arabic, Arabian, and Persian surname.irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

s. An inductive proof for arithmetic sequences was introduced in the ''Al-Fakhri'' (1000) by Al-Karaji
( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works ar ...

, who used it to prove the binomial theorem and properties of Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...

. Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segment ...

.
Modern proof theory treats proofs as inductively defined data structure
In computer science, a data structure is a data organization, management, and storage format that is usually chosen for efficient access to data. More precisely, a data structure is a collection of data values, the relationships among them, a ...

s, not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example Axiomatic set theory and Non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...

.
Nature and purpose

As practiced, a proof is expressed in natural language and is a rigorousargument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...

intended to convince the audience of the truth of a statement. The standard of rigor is not absolute and has varied throughout history. A proof can be presented differently depending on the intended audience. In order to gain acceptance, a proof has to meet communal standards of rigor; an argument considered vague or incomplete may be rejected.
The concept of proof is formalized in the field of 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 ...

. A formal proof
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequ ...

is written in a formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of sym ...

instead of natural language. A formal proof is a sequence of formulas in a formal language, starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones. This definition makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties, the most famous and surprising being that almost all axiomatic systems can generate certain undecidable statements not provable within the system.
The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated proof assistant
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof edito ...

s, this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are analytic or synthetic. Kant, who introduced the analytic–synthetic distinction, believed mathematical proofs are synthetic, whereas Quine argued in his 1951 " Two Dogmas of Empiricism" that such a distinction is untenable.
Proofs may be admired for their mathematical beauty. The mathematician Paul Erdős was known for describing proofs which he found to be particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book '' Proofs from THE BOOK'', published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.
Methods of proof

Direct proof

In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to prove that the sum of two eveninteger
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...

s is always even:
:Consider two even integers ''x'' and ''y''. Since they are even, they can be written as ''x'' = 2''a'' and ''y'' = 2''b'', respectively, for some integers ''a'' and ''b''. Then the sum is ''x'' + ''y'' = 2''a'' + 2''b'' = 2(''a''+''b''). Therefore ''x''+''y'' has 2 as a factor
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, suc ...

and, by definition, is even. Hence, the sum of any two even integers is even.
This proof uses the definition of even integers, the integer properties of closure under addition and multiplication, and the distributive property.
Proof by mathematical induction

Despite its name, mathematical induction is a method of deduction, not a form ofinductive reasoning
Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' rea ...

. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually infinitely many) cases are provable. This avoids having to prove each case individually. A variant of mathematical induction is proof by infinite descent, which can be used, for example, to prove the irrationality of the square root of two.
A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...

s:
Let be the set of natural numbers, and let be a mathematical statement involving the natural number belonging to such that
* (i) is true, i.e., is true for .
* (ii) is true whenever is true, i.e., is true implies that is true.
* Then is true for all natural numbers .
For example, we can prove by induction that all positive integers of the form are odd. Let represent " is odd":
:(i) For , , and is odd, since it leaves a remainder of when divided by . Thus is true.
:(ii) For any , if is odd (), then must also be odd, because adding to an odd number results in an odd number. But , so is odd (). So implies .
:Thus is odd, for all positive integers .
The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction".
Proof by contraposition

Proof by contraposition infers the statement "if ''p'' then ''q''" by establishing the logically equivalent contrapositive statement: "if ''not q'' then ''not p''". For example, contraposition can be used to establish that, given an integer $x$, if $x^2$ is even, then $x$ is even: : Suppose $x$ is not even. Then $x$ is odd. The product of two odd numbers is odd, hence $x^2\; =\; x\backslash cdot\; x$ is odd. Thus $x^2$ is not even. Thus, if $x^2$ ''is'' even, the supposition must be false, so $x$ has to be even.Proof by contradiction

In proof by contradiction, also known by the Latin phrase ''reductio ad absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...

'' (by reduction to the absurd), it is shown that if some statement is assumed true, a logical contradiction occurs, hence the statement must be false. A famous example involves the proof that $\backslash sqrt$ is an irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

:
:Suppose that $\backslash sqrt$ were a rational number. Then it could be written in lowest terms as $\backslash sqrt\; =$ where ''a'' and ''b'' are non-zero integers with no common factor. Thus, $b\backslash sqrt\; =\; a$. Squaring both sides yields 2''b''fraction
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

, this fraction could never be written in lowest terms, since 2 could always be factored from numerator and denominator.
Proof by construction

Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. Joseph Liouville, for instance, proved the existence oftranscendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...

s by constructing an explicit example. It can also be used to construct a counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...

to disprove a proposition that all elements have a certain property.
Proof by exhaustion

In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four color theorem still has over 600 cases.Probabilistic proof

A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods ofprobability 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 expressing it through a set ...

. Probabilistic proof, like proof by construction, is one of many ways to prove existence theorems.
In the probabilistic method, one seeks an object having a given property, starting with a large set of candidates. One assigns a certain probability for each candidate to be chosen, and then proves that there is a non-zero probability that a chosen candidate will have the desired property. This does not specify which candidates have the property, but the probability could not be positive without at least one.
A probabilistic proof is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work on the Collatz conjecture shows how far plausibility is from genuine proof. While most mathematicians do not think that probabilistic evidence for the properties of a given object counts as a genuine mathematical proof, a few mathematicians and philosophers have argued that at least some types of probabilistic evidence (such as Rabin's probabilistic algorithm for testing primality) are as good as genuine mathematical proofs.
Combinatorial proof

A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often abijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.
Nonconstructive proof

A nonconstructive proof establishes that a mathematical object with a certain property exists—without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. The following famous example of a nonconstructive proof shows that there exist twoirrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

s ''a'' and ''b'' such that $a^b$ is a rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...

. This proof uses that $\backslash sqrt$ is irrational (an easy proof is known since Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

), but not that $\backslash sqrt^$ is irrational (this true, but the proof is not elementary).
:Either $\backslash sqrt^$ is a rational number and we are done (take $a=b=\backslash sqrt$), or $\backslash sqrt^$ is irrational so we can write $a=\backslash sqrt^$ and $b=\backslash sqrt$. This then gives $\backslash left\; (\backslash sqrt^\backslash right\; )^=\backslash sqrt^=2$, which is thus a rational number of the form $a^b.$
Statistical proofs in pure mathematics

The expression "statistical proof" may be used technically or colloquially in areas of pure mathematics, such as involvingcryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or '' -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...

, chaotic series, and probabilistic number theory or analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...

. It is less commonly used to refer to a mathematical proof in the branch of mathematics known as mathematical statistics. See also the " Statistical proof using data" section below.
Computer-assisted proofs

Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity.The History and Concept of Mathematical ProofSteven G. Krantz. 1. February 5, 2007 However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first proof of the four color theorem is an example of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time error in its calculations calls the validity of such computer-assisted proofs into question. In practice, the chances of an error invalidating a computer-assisted proof can be reduced by incorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs. Errors can never be completely ruled out in case of verification of a proof by humans either, especially if the proof contains natural language and requires deep mathematical insight to uncover the potential hidden assumptions and fallacies involved.

Undecidable statements

A statement that is neither provable nor disprovable from a set ofaxiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s is called undecidable (from those axioms). One example is the parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segment ...

, which is neither provable nor refutable from the remaining axioms of Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...

.
Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see List of statements undecidable in ZFC.
Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements.
Heuristic mathematics and experimental mathematics

While early mathematicians such asEudoxus of Cnidus
Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments are ...

did not use proofs, from Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

to the foundational mathematics developments of the late 19th and 20th centuries, proofs were an essential part of mathematics. With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects outside of the proof-theorem framework, in experimental mathematics
Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with th ...

. Early pioneers of these methods intended the work ultimately to be embedded in a classical proof-theorem framework, e.g. the early development of fractal geometry, which was ultimately so embedded.
Related concepts

Visual proof

Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a " proof without words". The left-hand picture below is an example of a historic visual proof of thePythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

in the case of the (3,4,5) triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...

.
Zhoubi Suanjing
The ''Zhoubi Suanjing'' () is one of the oldest Chinese mathematical texts. "Zhou" refers to the ancient Zhou dynasty (1046–256 BCE); "Bì" literally means " thigh", but in the book refers to the gnomon of a sundial. The book is dedicated to ...

500–200 BCE.
File:Pythagoras-2a.gif, Animated visual proof for the Pythagorean theorem by rearrangement.
File:Pythag anim.gif, A second animated proof of the Pythagorean theorem.
Elementary proof

An elementary proof is a proof which only uses basic techniques. More specifically, the term is used innumber theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...

to refer to proofs that make no use of complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

. For some time it was thought that certain theorems, like the prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...

, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.
Two-column proof

A particular way of organising a proof using two parallel columns is often used as a mathematical exercise in elementary geometry classes in the United States. The proof is written as a series of lines in two columns. In each line, the left-hand column contains a proposition, while the right-hand column contains a brief explanation of how the corresponding proposition in the left-hand column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The left-hand column is typically headed "Statements" and the right-hand column is typically headed "Reasons".Colloquial use of "mathematical proof"

The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with mathematical objects, such as numbers, to demonstrate something about everyday life, or when data used in an argument is numerical. It is sometimes also used to mean a "statistical proof" (below), especially when used to argue fromdata
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpret ...

.
Statistical proof using data

"Statistical proof" from data refers to the application ofstatistics
Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...

, data analysis
Data analysis is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. Data analysis has multiple facets and approaches, enc ...

, or Bayesian analysis to infer propositions regarding the probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...

of data
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpret ...

. While ''using'' mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that the ''assumptions'' from which probability statements are derived require empirical evidence from outside mathematics to verify. In physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rela ...

, in addition to statistical methods, "statistical proof" can refer to the specialized '' mathematical methods of physics'' applied to analyze data in a particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and ...

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 cause-and-effect by demonstrating what outcome occurs when ...

or observational study
In fields such as epidemiology, social sciences, psychology and statistics, an observational study draws inferences from a sample to a population where the independent variable is not under the control of the researcher because of ethical conc ...

in physical cosmology
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of ...

. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as scatter plot
A scatter plot (also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram) is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data ...

s, when the data or diagram is adequately convincing without further analysis.
Inductive logic proofs and Bayesian analysis

Proofs using inductive logic, while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner toprobability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...

, and may be less than full certainty
Certainty (also known as epistemic certainty or objective certainty) is the epistemic property of beliefs which a person has no rational grounds for doubting. One standard way of defining epistemic certainty is that a belief is certain if and ...

. Inductive logic should not be confused with mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...

.
Bayesian analysis uses Bayes' theorem to update a person's assessment of likelihoods of hypotheses when new evidence
Evidence for a proposition is what supports this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field.
In epistemology, evide ...

or information
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely rando ...

is acquired.
Proofs as mental objects

Psychologism views mathematical proofs as psychological or mental objects. Mathematicianphilosopher
A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...

s, such as Leibniz, Frege, and Carnap have variously criticized this view and attempted to develop a semantics for what they considered to be the language of thought, whereby standards of mathematical proof might be applied to empirical science
In philosophy, empiricism is an epistemological theory that holds that knowledge or justification comes only or primarily from sensory experience. It is one of several views within epistemology, along with rationalism and skepticism. Empiri ...

.
Influence of mathematical proof methods outside mathematics

Philosopher-mathematicians such asSpinoza
Baruch (de) Spinoza (born Bento de Espinosa; later as an author and a correspondent ''Benedictus de Spinoza'', anglicized to ''Benedict de Spinoza''; 24 November 1632 – 21 February 1677) was a Dutch philosopher of Portuguese-Jewish origin, ...

have attempted to formulate philosophical arguments in an axiomatic manner, whereby mathematical proof standards could be applied to argumentation in general philosophy. Other mathematician-philosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at statements outside of mathematics, but having the certainty
Certainty (also known as epistemic certainty or objective certainty) is the epistemic property of beliefs which a person has no rational grounds for doubting. One standard way of defining epistemic certainty is that a belief is certain if and ...

of propositions deduced in a mathematical proof, such as Descartes' ''cogito'' argument.
Ending a proof

Sometimes, the abbreviation ''"Q.E.D."'' is written to indicate the end of a proof. This abbreviation stands for ''"quod erat demonstrandum"'', which isLatin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of ...

for ''"that which was to be demonstrated"''. A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a " tombstone" or "halmos" after its eponym
An eponym is a person, a place, or a thing after whom or which someone or something is, or is believed to be, named. The adjectives which are derived from the word eponym include ''eponymous'' and ''eponymic''.
Usage of the word
The term ''epon ...

Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operato ...

. Often, "which was to be shown" is verbally stated when writing "QED", "□", or "∎" during an oral presentation. Unicode explicitly provides the "end of proof" character, U+220E (∎) (220E(hex) = 8718(dec)).
See also

References

Further reading

* . * . * . * * . * .External links

*Proofs in Mathematics: Simple, Charming and Fallacious

* A lesson about proofs, in a course from

Wikiversity
Wikiversity is a Wikimedia Foundation project that supports learning communities, their learning materials, and resulting activities. It differs from Wikipedia in that it offers tutorials and other materials for the fostering of learning, rather ...

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