View source for Mathematical proof

{{Short description|Reasoning for mathematical statements}}
{{more citation needed|reason=there are too many unsourced sections|date=February 2025}}
{{Use mdy dates|date=December 2022}}
[[File:P. Oxy. I 29.jpg|thumb|upright=1.3|[[Papyrus Oxyrhynchus 29|P. Oxy. 29]], one of the oldest surviving fragments of [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.<ref>{{cite web
|url=http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html
|title=One of the Oldest Extant Diagrams from Euclid
|author=Bill Casselman
|author-link=Bill Casselman (mathematician)
|publisher=University of British Columbia
|access-date=September 26, 2008
}}</ref>]]
A '''mathematical proof''' is a [[deductive reasoning|deductive]] [[Argument-deduction-proof distinctions|argument]] for a [[Proposition|mathematical statement]], showing that the stated assumptions [[logic]]ally guarantee the conclusion. The argument may use other previously established statements, such as [[theorem]]s; but every proof can, in principle, be constructed using only certain basic or original assumptions known as [[axiom]]s,<ref>{{cite book |author1=Clapham, C.  |author2=Nicholson, J.N.  |name-list-style=amp | title = The Concise Oxford Dictionary of Mathematics, Fourth edition |quote = A statement whose truth is either to be taken as self-evident or to be assumed. Certain areas of mathematics involve choosing a set of axioms and discovering what results can be derived from them, providing proofs for the theorems that are obtained.}}</ref><ref name="nutsandbolts">{{cite book|title=The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs |last=Cupillari |first=Antonella|author-link= Antonella Cupillari |edition=Third |year=2005 |orig-year=2001 |publisher=[[Academic Press]] |isbn=978-0-12-088509-1 |page=3}}</ref><ref>{{cite book|title=Discrete Mathematics with Proof |date=July 2009 |first=Eric |last=Gossett |page=86 |quote=Definition 3.1. Proof: An Informal Definition |publisher=[[Wiley (publisher)|John Wiley & Sons]] |isbn=978-0470457931}}</ref> along with the accepted rules of [[inference]]. Proofs are examples of exhaustive [[deductive reasoning]] that establish logical certainty, to be distinguished from [[empirical evidence|empirical]] arguments or non-exhaustive [[inductive reasoning]] that 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]], or a hypothesis if frequently used as an assumption for further mathematical work.

Proofs employ [[logic]] expressed in mathematical symbols, along with [[natural language]] that usually admits some ambiguity. In most mathematical literature, proofs are written in terms of [[Rigour#Mathematics|rigorous]] [[informal logic]]. Purely [[formal proof]]s, written fully in [[Symbolic language (mathematics)|symbolic language]] without the involvement of natural language, are considered in [[proof theory]]. The distinction between [[Proof theory#Formal and informal proof|formal and informal proofs]] has led to much examination of current and historical [[mathematical practice]], [[quasi-empiricism in mathematics]], and so-called [[Mathematical folklore|folk mathematics]], oral traditions in the mainstream mathematical community or in other cultures. The [[philosophy of mathematics]] is concerned with the role of language and logic in proofs, and [[mathematics as a language]].

==History and etymology==
{{See also|History of logic}}

The word ''proof'' derives from the Latin {{lang|la|probare}} 'to test'; related words include English ''probe'', ''probation'', and ''probability'', as well as Spanish {{lang|es|probar}} 'to taste' (sometimes 'to touch' or 'to test'),<ref>"proof" New Shorter Oxford English Dictionary, 1993, OUP, Oxford.</ref> Italian {{tlit|it|provare}} 'to try', and German {{lang|de|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.<ref>{{cite book|title = The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference|first = Ian|last = Hacking|author-link=Ian Hacking |publisher = [[Cambridge University Press]] |year= 1984|orig-year=1975 |isbn=978-0-521-31803-7}}</ref>

Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof.<ref name="Krantz"/> It is likely that the idea of demonstrating a conclusion first arose in connection with [[geometry]], which originated in practical problems of land measurement.<ref>{{cite book|title=The development of logic |first1=William |last1=Kneale |first2=Martha |last2=Kneale |author-link1=William Kneale (logician) |date=May 1985 |orig-year=1962 |page=3 |edition=New |publisher=[[Oxford University Press]] |isbn=978-0-19-824773-9}}</ref> The development of mathematical proof is primarily the product of [[Greek mathematics|ancient Greek mathematics]], and one of its greatest achievements.<ref>{{Cite web|url=https://hal.archives-ouvertes.fr/hal-01281050/document|title=The genesis of proof in ancient Greece The pedagogical implications of a Husserlian reading|last1=Moutsios-Rentzos|first1=Andreas|last2=Spyrou|first2=Panagiotis|date=February 2015|website=Archive ouverte HAL|access-date=October 20, 2019}}</ref> [[Thales]] (624–546&nbsp;BCE) and [[Hippocrates of Chios]] (c.&nbsp;470–410 BCE) gave some of the first known proofs of theorems in geometry. [[Eudoxus of Cnidus|Eudoxus]] (408–355&nbsp;BCE) and [[Theaetetus (mathematician)|Theaetetus]] (417–369&nbsp;BCE) formulated theorems but did not prove them. [[Aristotle]] (384–322&nbsp;BCE) said definitions should describe the concept being defined in terms of other concepts already known.

Mathematical proof was revolutionized by [[Euclid]] (300&nbsp;BCE), who introduced the [[axiomatic method]] still in use today. It starts with [[undefined term]]s and [[axiom]]s, propositions concerning the undefined terms which are assumed to be self-evidently true (from Greek {{tlit|grc|axios}} 'something worthy'). From this basis, the method proves theorems using [[deductive logic]]. ''[[Euclid's Elements]]'' was read by anyone who was considered educated in the West until the middle of the 20th century.<ref>{{cite book|title=An Introduction to the History of Mathematics (Saunders Series) |first=Howard W. |last=Eves |author-link=Howard Eves |edition=6th |date=January 1990 |orig-year=1962 |page=141 |quote=No work, except The Bible, has been more widely used... |publisher=Cengage |isbn=978-0030295584}}</ref> In addition to theorems of geometry, such as the [[Pythagorean theorem]], the ''Elements'' also covers [[number theory]], including a proof that the [[square root of two]] is [[irrational number|irrational]] and a proof that there are infinitely many [[prime number]]s.

Further advances also took place in [[Mathematics in medieval Islam|medieval Islamic mathematics]]. In the 10th century, the Iraqi mathematician [[Abu Ali Muhammad ibn Abd al-Aziz al-Hashimi|Al-Hashimi]] worked with numbers as such, called "lines" but not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of [[irrational number]]s.<ref>{{citation|last=Matvievskaya|first=Galina|year=1987|title=The Theory of Quadratic Irrationals in Medieval Oriental Mathematics|journal=[[New York Academy of Sciences|Annals of the New York Academy of Sciences]]|volume=500|issue=1|pages=253–277 [260]|doi=10.1111/j.1749-6632.1987.tb37206.x|bibcode=1987NYASA.500..253M|s2cid=121416910}}</ref> An [[Mathematical induction|inductive proof]] for [[arithmetic progression]]s was introduced in the ''Al-Fakhri'' (1000) by [[Al-Karaji]], who used it to prove the [[binomial theorem]] and properties of [[Pascal's triangle]].

Modern [[proof theory]] treats proofs as inductively defined [[data structure]]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]].

==Nature and purpose==

As practiced, a proof is expressed in natural language and is a rigorous [[argument]] 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. 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]].<ref>{{citation|title=Handbook of Proof Theory|volume=137|series=Studies in Logic and the Foundations of Mathematics|editor-first=Samuel R.|editor-last=Buss|editor-link=Samuel Buss|publisher=Elsevier|year=1998|isbn=978-0-08-053318-6|contribution=An introduction to proof theory|pages=1–78|first=Samuel R.|last=Buss|author-link=Samuel Buss}}. See in particular [https://books.google.com/books?id=MfTMDeCq7ukC&pg=PA3 p.&nbsp;3]: "The study of Proof Theory is traditionally motivated by the problem of formalizing mathematical proofs; the original formulation of first-order logic by Frege [1879] was the first successful step in this direction."</ref> A [[formal proof]] is written in a [[formal language]] instead of natural language. A formal proof is a sequence of [[well-formed formula|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 [[independence (mathematical logic)|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]]s, this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are [[analytic proposition|analytic]] or [[synthetic proposition|synthetic]]. [[Immanuel Kant|Kant]], who introduced the [[analytic–synthetic distinction]], believed mathematical proofs are synthetic, whereas [[Willard Van Orman Quine|Quine]] argued in his 1951 "[[Two Dogmas of Empiricism]]" that such a distinction is untenable.<ref>{{Cite web|url=https://www.theologie.uzh.ch/dam/jcr:ffffffff-fbd6-1538-0000-000070cf64bc/Quine51.pdf|title=Two Dogmas of Empiricism|last=Quine|first=Willard Van Orman|date=1961|website=Universität Zürich – Theologische Fakultät|page=12|access-date=October 20, 2019}}</ref>

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.

=={{Anchor|Types of proof}}Methods of proof==

===Direct proof===
{{Main|Direct proof}}

In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.<ref>Cupillari, p. 20.</ref> For example, direct proof can be used to prove that the sum of two [[parity (mathematics)|even]] [[integer]]s is always even:

:Consider two even integers ''x'' and ''y''. Since they are even, they can be written as ''x''&nbsp;=&nbsp;2''a'' and ''y''&nbsp;=&nbsp;2''b'', respectively, for some integers ''a'' and ''b''. Then the sum is ''x''&nbsp;+&nbsp;''y''&nbsp;= 2''a''&nbsp;+&nbsp;2''b''&nbsp;= 2(''a''+''b''). Therefore ''x''+''y'' has 2 as a [[divisor|factor]] 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 (mathematics)|closure]] under addition and multiplication, and the [[distributive property]].

===Proof by mathematical induction===
{{Main|Mathematical induction}}
Despite its name, mathematical induction is a method of [[Deductive reasoning|deduction]], not a form of [[inductive reasoning]]. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case [[Material conditional|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 [[Infinite set|infinitely]] many) cases are provable.<ref>Cupillari, p. 46.</ref> 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 [[Square root of 2#Proofs of irrationality|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]]s:<ref>[http://zimmer.csufresno.edu/~larryc/proofs/proofs.mathinduction.html Examples of simple proofs by mathematical induction for all natural numbers]</ref>
Let {{math|1='''N''' = {1, 2, 3, 4, ...}}} be the set of natural numbers, and let {{math|''P''(''n'')}} be a mathematical statement involving the natural number {{math|''n''}} belonging to {{math|'''N'''}} such that
* '''(i)''' {{math|''P''(1)}} is true, i.e., {{math|''P''(''n'')}} is true for {{math|1=''n'' = 1}}.
* '''(ii)''' {{math|''P''(''n''+1)}} is true whenever {{math|''P''(''n'')}} is true, i.e., {{math|''P''(''n'')}} is true implies that {{math|''P''(''n''+1)}} is true.
* '''Then {{math|''P''(''n'')}} is true for all natural numbers {{math|''n''}}.'''

For example, we can prove by induction that all positive integers of the form {{math|2''n''&nbsp;−&nbsp;1}} are [[parity (mathematics)|odd]].  Let {{math|''P''(''n'')}} represent "{{math|2''n''&nbsp;−&nbsp;1}} is odd":
:'''(i)''' For {{math|1=''n'' = 1}}, {{math|1=2''n''&nbsp;−&nbsp;1&nbsp;=&nbsp;2(1)&nbsp;−&nbsp;1&nbsp;=&nbsp;1}}, and {{math|1}} is odd, since it leaves a remainder of {{math|1}} when divided by {{math|2}}. Thus {{math|''P''(1)}} is true.
:'''(ii)''' For any {{math|''n''}}, if {{math|2''n''&nbsp;−&nbsp;1}} is odd ({{math|''P''(''n'')}}), then {{math|(2''n''&nbsp;−&nbsp;1)&nbsp;+&nbsp;2}} must also be odd, because adding {{math|2}} to an odd number results in an odd number.  But {{math|1=(2''n''&nbsp;−&nbsp;1)&nbsp;+&nbsp;2&nbsp;=&nbsp;2''n''&nbsp;+&nbsp;1&nbsp;=&nbsp;2(''n''+1)&nbsp;−&nbsp;1}}, so {{math|1=2(''n''+1)&nbsp;−&nbsp;1}} is odd ({{math|''P''(''n''+1)}}).  So {{math|''P''(''n'')}} implies {{math|''P''(''n''+1)}}.
:'''Thus''' {{math|2''n''&nbsp;−&nbsp;1}} is odd, for all positive integers {{math|''n''}}.

The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction".<ref>[http://www.warwick.ac.uk/AEAhelp/glossary/glossaryParser.php?glossaryFile=Proof%20by%20induction.htm Proof by induction] {{Webarchive|url=https://web.archive.org/web/20120218033011/http://www.warwick.ac.uk/AEAhelp/glossary/glossaryParser.php?glossaryFile=Proof%20by%20induction.htm |date=February 18, 2012 }}, University of Warwick Glossary of Mathematical Terminology</ref>

===Proof by contraposition===
{{Main|Contraposition}}
[[Proof by contrapositive|Proof by contraposition]] [[Rule of inference|infers]] the statement "if ''p'' then ''q''" by establishing the [[logically equivalent]] [[contrapositive|contrapositive statement]]: "if ''not q'' then ''not p''".

For example, contraposition can be used to establish that, given an integer <math>x</math>, if <math> x^2 </math> is even, then <math>x</math> is even:

: Suppose <math>x</math> is not even. Then <math>x</math> is odd. The product of two odd numbers is odd, hence <math> x^2  = x\cdot x </math> is odd. Thus <math> x^2 </math> is not even. Thus, if <math> x^2 </math> ''is'' even, the supposition must be false, so <math> x </math> has to be even.

===Proof by contradiction===
{{Main|Proof by contradiction}}
In proof by contradiction, also known by the Latin phrase ''[[reductio ad absurdum]]'' (by reduction to the absurd), it is shown that if some statement is assumed true, a [[contradiction|logical contradiction]] occurs, hence the statement must be false.  A famous example involves the proof that <math>\sqrt{2}</math> is an [[irrational number]]:

:Suppose that <math>\sqrt{2}</math> were a rational number. Then it could be written in lowest terms as <math>\sqrt{2} = {a\over b}</math> where ''a'' and ''b'' are non-zero integers with [[coprime|no common factor]]. Thus, <math>b\sqrt{2} = a</math>. Squaring both sides yields 2''b''<sup>2</sup> = ''a''<sup>2</sup>. Since the expression on the left is an integer multiple of 2, the right expression is by definition divisible by 2. That is, ''a''<sup>2</sup> 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''<sup>2</sup> = (2''c'')<sup>2</sup> = 4''c''<sup>2</sup>. Dividing both sides by 2 yields ''b''<sup>2</sup> = 2''c''<sup>2</sup>. But then, by the same argument as before, 2 divides ''b''<sup>2</sup>, 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 <math>\sqrt{2}</math> is an irrational number.

To paraphrase: if one could write <math>\sqrt{2}</math> as a [[fraction]], this fraction could never be written in lowest terms, since 2 could always be factored from numerator and denominator.

===Proof by construction===
{{Main|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 of [[transcendental number]]s by constructing an [[Liouville number|explicit example]].  It can also be used to construct a [[counterexample]] to disprove a proposition that all elements have a certain property.

===Proof by exhaustion===
{{Main|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.<ref>See [[Four color theorem#Simplification and verification]].</ref>

=== Closed chain inference ===
{{Main|Closed chain inference}}

A closed chain inference shows that a collection of statements are pairwise equivalent.

In order to prove that the statements <math>\varphi_1,\ldots,\varphi_n</math> are each pairwise equivalent, proofs are given for the implications <math>\varphi_1\Rightarrow\varphi_2</math>, <math>\varphi_2\Rightarrow\varphi_3</math>, <math>\dots</math>, <math>\varphi_{n-1}\Rightarrow\varphi_n</math> and <math>\varphi_{n}\Rightarrow\varphi_1</math>.<ref>{{Cite book |last1=Plaue |first1=Matthias |url=https://books.google.com/books?id=-WCHDwAAQBAJ |title=Mathematik für das Bachelorstudium I: Grundlagen und Grundzüge der linearen Algebra und Analysis |last2=Scherfner |first2=Mike |date=2019-02-11 |publisher=Springer-Verlag |isbn=978-3-662-58352-4 |pages=26 |language=de |trans-title=Mathematics for the Bachelor's degree I: Fundamentals and basics of linear algebra and analysis}}</ref><ref>{{Cite book |last1=Struckmann |first1=Werner |url=https://books.google.com/books?id=1epNDQAAQBAJ |title=Mathematik für Informatiker: Grundlagen und Anwendungen |last2=Wätjen |first2=Dietmar |date=2016-10-20 |publisher=Springer-Verlag |isbn=978-3-662-49870-5 |pages=28 |language=de |trans-title=Mathematics for Computer Scientists: Fundamentals and Applications}}</ref>

The pairwise equivalence of the statements then results from the [[Transitive relation|transitivity]] of the [[material conditional]].

===Probabilistic proof===
{{Main|Probabilistic method}}
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of [[probability theory]]. Probabilistic proof, like proof by construction, is one of many ways to prove [[existence theorem]]s.

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 toward the [[Collatz conjecture]] shows how far plausibility is from genuine proof, as does the disproof of the [[Mertens conjecture]]. 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 [[primality test|testing primality]]) are as good as genuine mathematical proofs.<ref>Davis, Philip J. (1972), "Fidelity in Mathematical Discourse: Is One and One Really Two?" ''American Mathematical Monthly'' 79:252–63.</ref><ref>Fallis, Don (1997), "The Epistemic Status of Probabilistic Proof." ''Journal of Philosophy'' 94:165–86.</ref>

===Combinatorial proof===
{{Main|Combinatorial proof}}
A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a [[bijection]] between two [[set (mathematics)|sets]] is used to show that the expressions for their two sizes are equal. Alternatively, a [[double counting (proof technique)|double counting argument]] provides two different expressions for the size of a single set, again showing that the two expressions are equal.

===Nonconstructive proof===
{{Main|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 two [[irrational number]]s ''a'' and ''b'' such that <math>a^b</math> is a [[rational number]]. This proof uses that <math>\sqrt{2}</math> is irrational (an easy proof is known since [[Euclid]]), but not that <math>\sqrt{2}^{\sqrt{2}}</math> is irrational (this is true, but the proof is not elementary).
:Either <math>\sqrt{2}^{\sqrt{2}}</math> is a rational number and we are done (take <math>a=b=\sqrt{2}</math>), or <math>\sqrt{2}^{\sqrt{2}}</math> is irrational so we can write <math>a=\sqrt{2}^{\sqrt{2}}</math> and <math>b=\sqrt{2}</math>. This then gives <math>\left (\sqrt{2}^{\sqrt{2}}\right )^{\sqrt{2}}=\sqrt{2}^{2}=2</math>, which is thus a rational number of the form <math>a^b.</math>

===Statistical proofs in pure mathematics===
{{Main| Statistical proof}}
The expression "statistical proof" may be used technically or colloquially in areas of [[pure mathematics]], such as involving [[cryptography]], [[chaotic series]], and [[probabilistic|probabilistic number theory]] or [[analytic number theory]].<ref>"in number theory and commutative algebra... in particular the statistical proof of the lemma." [https://www.jstor.org/pss/2686395]</ref><ref>"Whether constant π (i.e., pi) is normal is a confusing problem without any strict theoretical demonstration except for some ''statistical'' proof"" (Derogatory use.)[https://doi.org/10.1007%2F978-3-540-74282-1_78]</ref><ref>"these observations suggest a statistical proof of Goldbach's conjecture with very quickly vanishing probability of failure for large E" [http://people.web.psi.ch/gassmann/eneseminare/abstracts/Goldbach1.pdf]</ref> It is less commonly used to refer to a mathematical proof in the branch of mathematics known as [[mathematical statistics]].  See also the "[[#Colloquial use, Statistical proof using data|Statistical proof using data]]" section below.

===Computer-assisted proofs===
{{Main|Computer-assisted proof }}
Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity.<ref name="Krantz">[http://www.math.wustl.edu/~sk/eolss.pdf The History and Concept of Mathematical Proof], Steven G. Krantz. 1. February 5, 2007</ref> 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 of [[axiom]]s is called undecidable (from those axioms). One example is the [[parallel postulate]], which is neither provable nor refutable from the remaining axioms of [[Euclidean geometry]].

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 incompleteness theorem|Gödel's (first) incompleteness theorem]] shows that many axiom systems of mathematical interest will have undecidable statements.

==Heuristic mathematics and experimental mathematics==
{{Main|Experimental mathematics}}
While early mathematicians such as [[Eudoxus of Cnidus]] did not use proofs, from [[Euclid]] to the [[foundational mathematics]] developments of the late 19th and 20th centuries, proofs were an essential part of mathematics.<ref>{{cite book|title=[[Indra's Pearls (book)|Indra's Pearls: The Vision of Felix Klein]] |publisher=[[Cambridge University Press]] |last1=Mumford |first1=David B. |author1-link=David Mumford |last2=Series |first2=Caroline |author2-link=Caroline Series |last3=Wright |first3=David  |author3-link=David Wright (arranger) |year=2002 |isbn=978-0-521-35253-6 |quote=What to do with the pictures? Two thoughts surfaced: the first was that they were unpublishable in the standard way, there were no theorems only very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, but theorems were coins of the realm and the conventions of that day dictated that journals only published theorems.}}</ref> With the increase in computing power in the 1960s, significant work began to be done investigating [[mathematical object]]s beyond the proof-theorem framework,<ref>{{cite web|url=https://home.att.net/~fractalia/history.htm |title=A Note on the History of Fractals |archive-url=https://web.archive.org/web/20090215114618/https://home.att.net/~fractalia/history.htm |archive-date=February 15, 2009 |url-status=dead |quote=Mandelbrot, working at the IBM Research Laboratory, did some computer simulations for these sets on the reasonable assumption that, if you wanted to prove something, it might be helpful to know the answer ahead of time.}}</ref> in [[experimental mathematics]]. Early pioneers of these methods intended the work ultimately to be resolved into a classical proof-theorem framework, e.g. the early development of [[fractal geometry]],<ref>{{cite book |title=Introducing Fractal Geometry |last=Lesmoir-Gordon |first=Nigel |publisher=[[Introducing... (book series)|Icon Books]] |year=2000 |isbn=978-1-84046-123-7 |quote=...brought home again to Benoit [Mandelbrot] that there was a 'mathematics of the eye', that visualization of a problem was as valid a method as any for finding a solution. Amazingly, he found himself alone with this conjecture. The teaching of mathematics in France was dominated by a handful of dogmatic mathematicians hiding behind the pseudonym 'Bourbaki'... |url-access=registration |url=https://archive.org/details/introducingfract0000lesm }}</ref> which was ultimately so resolved.

==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 the [[Pythagorean theorem]] in the case of the (3,4,5) [[triangle]].
<gallery>
File:Chinese pythagoras.jpg|Visual proof for the (3,4,5) triangle as in the [[Zhoubi Suanjing]] 500–200&nbsp;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.
</gallery>

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 by neglecting 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.

===Elementary proof===
{{Main|Elementary proof}}
An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in [[number theory]] to refer to proofs that make no use of [[complex analysis]]. For some time it was thought that certain theorems, like the [[prime number theorem]], could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.

===Two-column proof===
[[File:twocolumnproof.png|thumb|right|A two-column proof published in 1913]]

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.<ref>{{cite journal |first=Patricio G. |last=Herbst |title=Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century |journal=[[Educational Studies in Mathematics]] |volume=49 |issue=3 |year=2002 |pages=283–312 |doi=10.1023/A:1020264906740 |hdl=2027.42/42653 |s2cid=23084607 |url=https://deepblue.lib.umich.edu/bitstream/2027.42/42653/1/10649_2004_Article_5096042.pdf |hdl-access=free }}</ref> 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".<ref>{{cite web|url=https://www.onemathematicalcat.org/Math/Geometry_obj/two_column_proof.htm |title=Introduction to the Two-Column Proof |author=Dr. Fisher Burns |website=onemathematicalcat.org |access-date=October 15, 2009}}</ref>

===Colloquial use of "mathematical proof"===
The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with [[mathematical object]]s, 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 from data.

===Statistical proof using data===
{{Main|Statistical proof}}
"Statistical proof" from data refers to the application of statistics, [[data analysis]], or [[Bayesian analysis]] to infer propositions regarding the [[probability]] of data. 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, in addition to statistical methods, "statistical proof" can refer to the specialized ''[[mathematical methods of physics]]'' applied to analyze data in a [[particle physics]] experiment or [[observational study]] in [[physical cosmology]]. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as [[scatter plot]]s, when the data or diagram is adequately convincing without further analysis.

===Inductive logic proofs and Bayesian analysis===
{{Main|Inductive logic|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 to [[probability]], and may be less than full [[certainty]]. Inductive logic should not be confused with [[mathematical induction]].

Bayesian analysis uses [[Bayes' theorem]] to update a person's [[Bayesian probability|assessment of likelihoods]] of hypotheses when new [[evidence]] or information is acquired.

===Proofs as mental objects===
{{Main|Psychologism|Language of thought}}
Psychologism views mathematical proofs as psychological or mental objects.  Mathematician philosophers, such as [[Gottfried Wilhelm Leibniz|Leibniz]], [[Gottlob Frege|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]].

===Influence of mathematical proof methods outside mathematics===
Philosopher-mathematicians such as [[Spinoza]] 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]] of propositions deduced in a mathematical proof, such as [[Descartes]]' [[cogito ergo sum|''cogito'']] argument.

==Ending a proof==
{{Main|Q.E.D.}}
Sometimes, the abbreviation ''"Q.E.D."'' is written to indicate the end of a proof. This abbreviation stands for ''"quod erat demonstrandum"'', which is [[Latin]] 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 (typography)|tombstone]]" or "halmos" after its [[eponym]] [[Paul Halmos]]. 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 (∎) <small>(220E(hex) = 8718(dec))</small>.

==See also==
{{Portal|Philosophy|Mathematics}}
{{columns-list|colwidth=30em|
* [[Automated theorem proving]]
* [[Invalid proof]]
* [[List of incomplete proofs]]
* [[List of long proofs]]
* [[List of mathematical proofs]]
* [[Nonconstructive proof]]
* [[Proof by intimidation]]
* [[Termination analysis]]
* [[Thought experiment]]
* ''[[What the Tortoise Said to Achilles]]''
* [[Zero-knowledge proof]]
}}

==References==
{{Reflist}}

==Further reading==
* {{citation|last=Pólya|first=G.|author-link=George Pólya|title=Mathematics and Plausible Reasoning|publisher=Princeton University Press|year=1954|title-link=Mathematics and plausible reasoning|hdl=2027/mdp.39015008206248|isbn=9780691080055|hdl-access=free}}.
* {{citation|last=Fallis|first=Don|year=2002|url=http://dlist.sir.arizona.edu/1581/|title=What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians|journal=Logique et Analyse|volume=45|pages=373–88}}.
* {{citation|author1-link=James Franklin (philosopher)|last1=Franklin|first1=J.|last2=Daoud|first2=A.|url=http://www.maths.unsw.edu.au/~jim/proofs.html|title=Proof in Mathematics: An Introduction|publisher=Kew Books|year=2011|isbn=978-0-646-54509-7}}.
* {{cite book |title=Proof and Other Dilemmas: Mathematics and Philosophy |author1=Gold, Bonnie|author1-link=Bonnie Gold |author2=Simons, Rogers A. |publisher=MAA |year=2008}}
* {{citation|last=Solow|first=D.|title=How to Read and Do Proofs: An Introduction to Mathematical Thought Processes|publisher=[[Wiley Publishing|Wiley]]|year=2004|isbn=978-0-471-68058-1}}.
* {{citation|last=Velleman|first=D.|title=How to Prove It: A Structured Approach|publisher=Cambridge University Press|year=2006|isbn=978-0-521-67599-4}}.
* {{citation |last=Hammack |first=Richard |title=Book of Proof |url=https://richardhammack.github.io/BookOfProof/ |publisher= Richard Hammack|year=2018 |isbn=978-0-9894721-3-5}}.

==External links==
{{wikiquote}}
{{Wiktionary|proof}}

* {{Commons category-inline}}
* [http://www.cut-the-knot.org/proofs/index.shtml Proofs in Mathematics: Simple, Charming and Fallacious]
* A [[v:Discrete Mathematics for Computer Science/Proof|lesson]] about proofs, in a [[v:Discrete Mathematics for Computer Science|course]] from [[v:|Wikiversity]]

{{Mathematical logic}}
{{Logic}}
{{Authority control}}

{{DEFAULTSORT:Mathematical Proof}}
[[Category:Mathematical logic|Proof]]
[[Category:Mathematical terminology|Proof]]
[[Category:Mathematical proofs| ]]
[[Category:Proof theory]]
[[Category:Sources of knowledge]]