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| [[File:Oxyrhynchus papyrus with Euclid's Elements.jpg|right|thumb|250px|One of the oldest surviving fragments of Euclid's ''Elements'', a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.<ref>{{cite web
| | Worth noting is that there are certain aspects of the Body mass index calculator that requirements to be investigated even whilst the calculations are being completed. To start with, you have the general Body mass index rate, that stands at 18.5-24.0. This means that for every and every individual whose calculations fall within this range they're assumed to function as the usual fat category. However, must the BMI fall under this then the individual is considered to be underweight weight.<br><br><br><br>It's certainly a big choice beginning on a modern weight loss plan. Excitement, anticipation, a little of fear and apprehension. nevertheless by following the simple steps one will reach one's objectives plus gain self respect plus confidence in the process.I hope these standard tricks might encourage you to start on a sensible weight loss program plus aid we inside generating the right decisions before beginning a weight loss program.<br><br>It is surprisingly prevalent to consider the [http://safedietplans.com/calories-burned-walking calories burned] in the healthcare underwriting of private wellness insurance plans. Providers can use a significant BMI as a cut-off point to raise insurance rates or deny coverage. Also, the BMI is frequently utilized by surgeons to determine when a individual qualifies the Gastric Band process. Typically, doctors are searching for a BMI of 35 or high whenever considering possible candidates. Every case should be considered about an individual basis, however, the calories burned walking calculator does serve because a valuable tool.<br><br>The weight loss supplements selected can have the proper ingredients. These elements are all-natural to perform a certain function. As these are all-natural elements there are no negative side effects.<br><br>Calculating a BMI is done manually utilizing a formula or it can be performed conveniently by using a handy online calories burned walking calculator that is present on many health/diet associated sites. If you like to do it manually, you are able to come up with the body mass index by using the imperial formula. Your BMI is equal to your fat in lbs x 703, then we take this quantity and separate it by a height inside inches. But, with the advantageous calories burned walking calculators online, it happens to be merely easier to login the favorite website and connect the numbers plus hit calculate.<br><br>Shun you excessive eating habit. If you have this habit cut up allowable vegetables like celery, carrot plus capsicum into sticks or chunks plus keep them in the refrigerator to munch them when we feel hungry.<br><br>The formula which is being employed to compute for a person's body mass index is universal. It can plus ought to be utilized about anyone regardless of their age or gender. This is the cause why it really is fair to believe which there is no these thing as a special sort of calculator for this kind of computation. If somebody is striving to sell we these calculators, you need to say no to them instantly because they are surely struggling to con we into shelling out some cash. |
| |url=http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html
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| |title=One of the Oldest Extant Diagrams from Euclid
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| |author=Bill Casselman
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| |date=
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| |publisher=University of British Columbia
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| |accessdate=2008-09-26
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| }}</ref>]]
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| In [[mathematics]], a '''proof''' is a deductive argument for a [[mathematical statement]]. In the [[Argument-deduction-proof distinctions| argument]], other previously established statements, such as [[theorems]], can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as [[axiom]]s.<ref>{{cite book | Author = Clapham, C. and Nicholson, JN. | 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">Cupillari, Antonella. ''The Nuts and Bolts of Proofs''. Academic Press, 2001. Page 3.</ref><ref>Gossett, Eric. ''Discrete Mathematics with Proof''. John Wiley and Sons, 2009. Definition 3.1 page 86. ISBN 0-470-45793-7</ref> Proofs are examples of [[deductive reasoning]] and are distinguished from [[inductive reasoning|inductive]] or [[empirical]] arguments; a proof must demonstrate that a statement is always true (occasionally by listing ''all'' possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproven statement that is believed true is known as a [[conjecture]].
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| Proofs employ [[logic]] but usually include some amount of [[natural language]] which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous [[informal logic]]. Purely [[formal proof]]s, written in symbolic language instead 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]] (in both senses of that term). The [[philosophy of mathematics]] is concerned with the role of language and logic in proofs, and [[mathematics as a language]].
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| ==History and etymology==
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| {{See also|History of logic}}
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| The word "proof" comes from the Latin ''probare'' meaning "to test". Related modern words are the English "probe", "probation", and "probability", the Spanish ''probar'' (to smell or taste, or (lesser use) touch or test),<ref>New Shorter Oxford English Dictionary, 1993, OUP, Oxford.</ref> Italian ''provare'' (to try), and the German ''probieren'' (to try). The early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony.<ref>The Emergence of Probability, Ian Hacking</ref>
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| 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 originally meant the same as "land measurement".<ref>Kneale, p. 2</ref> The development of mathematical proof is primarily the product of [[Greek mathematics|ancient Greek mathematics]], and one of its greatest achievements. [[Thales]] (624–546 BCE) proved some theorems in geometry. [[Eudoxus of Cnidus|Eudoxus]] (408–355 BCE) and [[Theaetetus (mathematician)|Theaetetus]] (417–369 BCE) formulated theorems but did not prove them. [[Aristotle]] (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. Mathematical proofs were revolutionized by [[Euclid]] (300 BCE), who introduced the [[axiomatic method]] still in use today, starting with [[undefined term]]s and [[axioms]] (propositions regarding the undefined terms assumed to be self-evidently true from the Greek "axios" meaning "something worthy"), and used these to prove theorems using [[deductive logic]]. His book, the [[Euclid's Elements|''Elements'']], was read by anyone who was considered educated in the West until the middle of the 20th century.<ref>Howard Eves, ''An Introduction to the History of Mathematics'', Saunders, 1990, ISBN 0-03-029558-0 p. 141: "No work, except [[The Bible]], has been more widely used...."</ref> In addition to the familiar theorems of geometry, such as the [[Pythagorean theorem]], the ''Elements'' includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.
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| Further advances took place in [[Mathematics in medieval Islam|medieval Islamic mathematics]]. While earlier Greek proofs were largely geometric demonstrations, the development of [[arithmetic]] and [[algebra]] by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the [[Iraqi people|Iraqi]] mathematician Al-Hashimi provided general proofs for numbers (rather than geometric demonstrations) as he considered multiplication, division, etc. for "lines." He used this method to provide a proof of 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|pages=253–277 [260]|doi=10.1111/j.1749-6632.1987.tb37206.x}}</ref> An [[Mathematical induction|inductive proof]] for [[Arithmetic progression|arithmetic sequences]] was introduced in the ''Al-Fakhri'' (1000) by [[Al-Karaji]], who used it to prove the [[binomial theorem]] and properties of [[Pascal's triangle]]. Alhazen also developed the method of [[proof by contradiction]], as the first attempt at proving the [[Euclidean geometry|Euclidean]] [[parallel postulate]].<ref>{{Citation |last=Eder |first=Michelle |year=2000 |title=Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam |url=http://www.math.rutgers.edu/~cherlin/History/Papers2000/eder.html |publisher=[[Rutgers University]] |accessdate=2008-01-23 }}</ref>
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| Modern [[proof theory]] treats proofs as inductively defined data structures. There is no longer an assumption that axioms are "true" in any sense; this allows for parallel mathematical theories built on alternate sets of axioms (see [[Axiomatic set theory]] and [[Non-Euclidean geometry]] for examples).
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| == Nature and purpose ==
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| As practised, 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. In order to gain acceptance, a proof has to meet communal statements of rigor; an argument considered vague or incomplete may be rejected.
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| The concept of a proof is formalized in the field of mathematical logic.<ref>Buss, 1997, p. 3</ref> A [[formal proof]] is written in a [[formal language]] instead of a natural language. A formal proof is defined as sequence for formulas in a formal language in which each formula is a logical consequence of preceding formulas. Having a definition of formal proof makes the concept of proof amenable to study. Indeed, the field of [[proof theory]] studies formal proofs and their properties, for example, the property that a statement has a formal proof. An application of proof theory is to show that certain [[independence (mathematical logic)|undecidable statement]]s are not provable.
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| 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 assistants, 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. <!-- [[Willard Van Orman Quine]] argued that mathematical proofs are analytic expressions, relying on no empirical observations or facts.<ref> Quine, Two Dogmas of Empiricism</ref> -->
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| Proofs may be viewed as aesthetic objects, admired for their [[mathematical beauty]]. The mathematician [[Paul Erdős]] was known for describing proofs he found 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.
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| ==Methods of proof==
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| ===Direct proof===
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| {{Main|Direct proof}}
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| In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.<ref>Cupillari, page 20.</ref> For example, direct proof can be used to establish that the sum of two [[Even and odd numbers|even]] [[integer]]s is always even:
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| :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 integers ''a'' and ''b''. Then the sum ''x'' + ''y'' = 2''a'' + 2''b'' = 2(''a''+''b''). Therefore ''x''+''y'' has 2 as a factor and, by definition, is even. Hence the sum of any two even integers is even.
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| This proof uses the definition of even integers, the integer properties of [[Closure (mathematics)|closure]] under addition and multiplication, and [[distributivity]].
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| ===Proof by mathematical induction===
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| {{Main|Mathematical induction}}
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| Mathematical induction is not a form of [[inductive reasoning]]. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved, which establishes that a certain case [[Material conditional|implies]] the next case. Applying the induction rule repeatedly, starting from the independently proved base case, proves many, often [[Infinite set|infinitely]] many, other cases.<ref>Cupillari, page 46.</ref> Since the base case is true, the infinity of other cases must also be true, even if all of them cannot be proved directly because of their infinite number. A subset of induction is [[infinite descent]]. Infinite descent can be used to prove the [[Square root of 2#Proofs of irrationality|irrationality of the square root of two]].
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| A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all natural numbers:<ref>[http://zimmer.csufresno.edu/~larryc/proofs/proofs.mathinduction.html Examples of simple proofs by mathematical induction for all natural numbers]</ref>
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| Let {{math|1='''N''' = {1,2,3,4,...}}} be the set of natural numbers, and {{math|''P''(''n'')}} be a mathematical statement involving the natural number {{math|''n''}} belonging to {{math|'''N'''}} such that
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| * '''(i)''' {{math|''P''(1)}} is true, i.e., {{math|''P''(''n'')}} is true for {{math|1=''n'' = 1}}.
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| * '''(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.
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| * '''Then {{math|''P''(''n'')}} is true for all natural numbers {{math|''n''}}.'''
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| For example, we can prove by induction that all integers of the form {{math|2''n'' + 1}} are odd:
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| :'''(i)''' For {{math|1=''n'' = 1}}, {{math|1=2''n'' + 1 = 2(1) + 1 = 3}}, and {{math|3}} is odd. Thus {{math|''P''(1)}} is true.
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| :'''(ii)''' For {{math|2''n'' + 1}} for some {{math|''n''}}, {{math|1=2(''n''+1) + 1 = (2''n''+1) + 2}}. If {{math|2''n'' + 1}} is odd, then {{math|(2''n''+1) + 2}} must also be odd, because adding {{math|2}} to an odd number results in an odd number. So {{math|''P''(''n''+1)}} is true if {{math|''P''(''n'')}} is true.
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| :'''Thus''' {{math|2''n'' + 1}} is odd, for all natural numbers {{math|''n''}}.
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| It is common for the phrase "proof by induction" to be used for a "proof by mathematical induction".<ref>[http://www.warwick.ac.uk/AEAhelp/glossary/glossaryParser.php?glossaryFile=Proof%20by%20induction.htm Proof by induction], University of Warwick Glossary of Mathematical Terminology</ref>
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| ===Proof by contraposition===
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| {{Main|Contraposition}}
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| [[Proof by contrapositive|Proof by contraposition]] [[Rule of inference|infers]] the conclusion "if ''p'' then ''q''" from the premise "if ''not q'' then ''not p''". The statement "if ''not q'' then ''not p''" is called the [[contrapositive]] of the statement "if ''p'' then ''q''". For example, contraposition can be used to establish that, given an integer ''x'', if ''x''² is even, then ''x'' is even:
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| : Suppose ''x'' is not even. Then ''x'' is odd. The product of two odd numbers is odd, hence ''x''² = ''x''·''x'' is odd. Thus ''x''² is not even.
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| ===Proof by contradiction===
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| {{Main|Proof by contradiction}}
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| In proof by contradiction (also known as ''[[reductio ad absurdum]]'', Latin for "by reduction to the absurd"), it is shown that if some statement were true, a logical contradiction occurs, hence the statement must be false. A famous example of proof by contradiction shows that <math>\sqrt{2}</math> is an [[irrational number]]:
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| :Suppose that <math>\sqrt{2}</math> were a rational number, so by definition <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 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So ''a''<sup>2</sup> is even, which implies that ''a'' must also be even. 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 share a factor, namely 2. This contradicts our assumption, so we are forced to conclude that <math>\sqrt{2}</math> is an irrational number.
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| ===Proof by construction===
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| {{Main|Proof by construction}}
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| 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.
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| ===Proof by exhaustion===
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| {{Main|Proof by exhaustion}}
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| 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 {{As of|2011|lc=on}} still has over 600 cases.
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| ===Probabilistic proof===
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| {{Main|Probabilistic method}}
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| 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 show [[existence theorem]]s.
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| This 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.<ref>While most mathematicians do not think that probabilistic evidence ever 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. See, for example, Davis, Philip J. (1972), "Fidelity in Mathematical Discourse: Is One and One Really Two?" ''American Mathematical Monthly'' 79:252-63. Fallis, Don (1997), "The Epistemic Status of Probabilistic Proof." ''Journal of Philosophy'' 94:165-86.</ref>
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| ===Combinatorial proof===
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| {{Main|Combinatorial proof}}
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| A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a [[Bijective proof|bijection]] between two 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.
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| ===Nonconstructive proof===
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| {{Main|Nonconstructive proof}}
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| 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 proven to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. A famous example of a
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| nonconstructive proof shows that there exist two [[irrational number]]s ''a'' and ''b'' such that <math>a^b</math> is a [[rational number]]:
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| :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 of the form <math>a^b.</math>
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| ===Statistical proofs in pure mathematics===
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| {{Main| Statistical proof}}
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| The expression "statistical proof" may be used technically or colloquially in areas of [[pure mathematics]], such as involving [[cryptography]], [[chaotic series]], and probabilistic or analytic [[number theory]].<ref>"in number theory and commutative algebra... in particular the statistical proof of the lemma." [http://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.)[http://www.springerlink.com/content/nj34v59p71m11125/]</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 "[[#Colloquial use, Statistical proof using data|Statistical proof using data]]" section below.
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| ===Computer-assisted proofs===
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| {{Main|Computer-assisted proof }}
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| 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.
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| ==Undecidable statements==
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| A statement that is neither provable nor disprovable from a set of axioms 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]].
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| 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]].
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| [[Gödel's incompleteness theorem|Gödel's (first) incompleteness theorem]] shows that many axiom systems of mathematical interest will have undecidable statements.
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| ==Heuristic mathematics and experimental mathematics==
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| {{Main|Experimental mathematics}}
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| 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>"''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 ant the conventions of that day dictated that journals only published theorems''", David Mumford, Caroline Series and David Wright, [[Indra's Pearls (book)|Indra's Pearls]], 2002</ref> With the increase in computing power in the 1960s, significant work began to be done investigating [[mathematical objects]] outside of the proof-theorem framework,<ref>"''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.''"[http://home.att.net/~fractalia/history.htm A Note on the History of Fractals],</ref> in [[experimental mathematics]]. 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]],<ref>"''… 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'… ''", Introducing Fractal Geometry, Nigel Lesmoir-Gordon</ref> which was ultimately so embedded.
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| ==Related concepts==
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| ===Visual proof===
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| 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.
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| <gallery>
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| Image:Chinese pythagoras.jpg|Visual proof for the (3, 4, 5) triangle as in the [[Chou Pei Suan Ching]] 500–200 BC.
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| File:Pythagoras-2a.gif|Animated visual proof for the Pythagorean theorem by rearrangement.
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| File:Pythag anim.gif|A second animated proof of the Pythagorean theorem.
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| </gallery>
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| 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.
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| ===Elementary proof===
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| {{Main|Elementary proof}}
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| 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.
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| ===Two-column proof===
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| [[File:twocolumnproof.png|thumb|right|A two-column proof published in 1913]]
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| A particular way of organising a proof using two parallel columns is often used in elementary geometry classes in the United States.<ref>Patricio G. Herbst, Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century, Educational Studies in Mathematics, Vol. 49, No. 3 (2002), pp. 283-312,</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>[http://www.onemathematicalcat.org/Math/Geometry_obj/two_column_proof.htm Introduction to the Two-Column Proof], Carol Fisher</ref>
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| ===Colloquial use of "mathematical proof"===
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| 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 from [[data]].
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| ===Statistical proof using data===
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| {{Main|Statistical proof}}
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| "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 [[cosmology]]. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as [[scatter plots]], when the data or diagram is adequately convincing without further analysis.
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| ===Inductive logic proofs and Bayesian analysis===
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| {{Main|Inductive logic|Bayesian analysis}}
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| 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 one [[certainty]]. Bayesian analysis establishes assertions as to the degree of a person's [[Bayesian probability|subjective belief]]. Inductive logic should not be confused with [[mathematical induction]].
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| ===Proofs as mental objects===
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| {{Main|Psychologism|Language of thought}}
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| Psychologism views mathematical proofs as psychological or mental objects. Mathematician [[philosopher]]s, such as [[Gottfried Wilhelm Leibniz|Leibniz]], [[Frege]], and [[Carnap]], have 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]].
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| ===Influence of mathematical proof methods outside mathematics===
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| 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 [[Descarte]]'s [[cogito ergo sum|''cogito'']] argument.
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| ==Ending a proof==
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| {{Main|Q.E.D.}}
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| 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 {{Unicode|□}} or {{Unicode|∎}}, 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", "{{Unicode|□}}", or "{{Unicode|∎}}" in an oral presentation on a board.
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| ==See also==
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| * [[Automated theorem proving]]
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| * [[Invalid proof]]
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| * [[List of incomplete proofs]]
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| * [[List of long proofs]]
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| * [[List of mathematical proofs]]
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| * [[Nonconstructive proof]]
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| * [[Proof by intimidation]]
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| * ''[[What the Tortoise Said to Achilles]]''
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| ==References==
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| {{Reflist}}
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| ==Sources==
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| *{{citation|last=Pólya|first=G.|authorlink=George Pólya|title=Mathematics and Plausible Reasoning|publisher=Princeton University Press|year=1954}}.
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| *{{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=373–388}}.
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| *{{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=0-646-54509-4}}.
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| *{{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=0-471-68058-3}}.
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| *{{citation|last=Velleman|first=D.|title=How to Prove It: A Structured Approach|publisher=Cambridge University Press|year=2006|isbn=0-521-67599-5}}.
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| ==External links==
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| {{Wiktionary|proof}}
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| * {{springer|title=Proof theory|id=p/p075430}}
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| * [http://www.math.uconn.edu/~hurley/math315/proofgoldberger.pdf What are mathematical proofs and why they are important?]
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| * [http://2piix.com/articles/title/Logic/ 2πix.com: Logic] Part of a series of articles covering mathematics and logic.
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| * [http://zimmer.csufresno.edu/~larryc/proofs/proofs.html How To Write Proofs] by Larry W. Cusick
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| * [http://research.microsoft.com/users/lamport/pubs/lamport-how-to-write.pdf How to Write a Proof] by [[Leslie Lamport]], and [http://research.microsoft.com/users/lamport/pubs/pubs.html#lamport-how-to-write the motivation of proposing such a hierarchical proof style].
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| * [http://www.cut-the-knot.org/proofs/index.shtml Proofs in Mathematics: Simple, Charming and Fallacious]
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| * ''[http://www.cs.ru.nl/~freek/comparison/comparison.pdf The Seventeen Provers of the World]'', ed. by Freek Wiedijk, foreword by Dana S. Scott, Lecture Notes in Computer Science 3600, Springer, 2006, ISBN 3-540-30704-4. Contains formalized versions of the proof that <math>\sqrt{2}</math> is irrational in several automated proof systems.
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| * [http://www.cut-the-knot.org/WhatIs/WhatIsProof.shtml What is Proof?] Thoughts on proofs and proving.
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| * [http://www.proofwiki.org ProofWiki.org] A wiki compendium of mathematical proofs.
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| * [http://planetmath.org planetmath.org] A wiki style encyclopedia of proofs
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| * A [[v:Discrete Mathematics for Computer Science/Proof|lesson]] about proofs, in a [[v:Discrete Mathematics for Computer Science|course]] from [[v:|Wikiversity]]
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| * [http://mzone.mweb.co.za/residents/profmd/proof.pdf The role and function of proof] by Michael de Villiers
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| * [http://mzone.mweb.co.za/residents/profmd/profmat.pdf Developing understanding of different roles of proof] by Michael de Villiers
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| * [http://www.people.vcu.edu/~rhammack/BookOfProof/index.html "Book of Proof" by Richard Hammack 2009] Part of the Open Textbook Initiative. Provides an introduction to mathematical proofs.
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| * ''[http://arxiv.org/abs/math.HO/9404236 On proof and progress in mathematics]''. [[William Thurston|Thurston, William P]]. 1994.
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| {{logic}}
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| {{DEFAULTSORT:Mathematical Proof}}
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| [[Category:Mathematical logic|Proof]]
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| [[Category:Mathematical terminology|Proof]]
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| [[Category:Mathematical proofs| ]]
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| [[Category:Sources of knowledge]]
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| {{Link FA|cs}}
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| {{Link FA|eo}}
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| {{Link GA|sk}}
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