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Mathematical proof
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{{use mdy datesdate=June 2019}}File:Oxyrhynchus papyrus with Euclid's Elements.jpgrightthumb250pxP. Oxy. 29, one of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proofwriting techniques. The diagram accompanies Book II, Proposition 5.WEB,weblink One of the Oldest Extant Diagrams from Euclid the content below is remote from Wikipedia
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Bill Casselman (mathematician)>Bill CasselmanBill Casselman (mathematician)>Bill CasselmanA mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms,BOOK, Clapham, C., Nicholson, JN., yes, The Concise Oxford Dictionary of Mathematics, Fourth edition, A statement whose truth is either to be taken as selfevident 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., BOOK, The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs, Cupillari, Antonella, Third, 2005, 2001, Academic Press, 9780120885091, 3, BOOK, Discrete Mathematics with Proof, July 2009, Eric, Gossett, 86, Definition 3.1. Proof: An Informal Definition, Wiley (publisher), John Wiley & Sons, 9780470457931, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning or exhaustive inductive reasoning which establish logical certainty, and to be distinguished from empirical arguments or nonexhaustive inductive reasoning which establish "reasonable expectation". Enumerating many confirmatory cases is not enough for a proof, which must demonstrate that the statement is always true (occasionally by listing all possible cases and showing that it holds in each). An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if the proposition is frequently used as an assumption to build upon similar mathematical work.WEB,weblink The Definitive Glossary of Higher Mathematical Jargon, 20190801, Math Vault, enUS, live, 20191020, Proofs employ logic expressed in mathematical symbols, along with some amount of natural language which usually admits some ambiguity. The vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, 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, quasiempiricism in mathematics, and socalled 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.History and etymology{{See alsoHistory of logic}}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),"proof" New Shorter Oxford English Dictionary, 1993, OUP, Oxford. 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.BOOK, The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference, Ian, Hacking, Ian Hacking, Cambridge University Press, 1984, 1975, 9780521318037,weblink 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 with geometry, which originally meant the same as "land measurement".BOOK, The development of logic, William, Kneale, Martha, Kneale, William Kneale (logician), May 1985, 1962, 3, New, Oxford University Press, 9780198247739, The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof.WEB,weblink The genesis of proof in ancient Greece The pedagogical implications of a Husserlian reading, MoutsiosRentzos, Andreas, Spyrou, Panagiotis, February 2015, Archive ouverte HAL, live, October 20, 2019, Thales (624â€“546 BCE) and Hippocrates of Chios (c. 470â€“410 BCE) proved some theorems in geometry. Eudoxus (408â€“355 BCE) and Theaetetus (417â€“369 BCE) formulated theorems but did not prove them. Aristotle (384â€“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 terms and axioms (propositions regarding the undefined terms assumed to be selfevidently true from the Greek "axios" meaning "something worthy"), and used these to prove theorems using deductive logic. His book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century.BOOK, An Introduction to the History of Mathematics (Saunders Series), Howard W., Eves, Howard Eves, 6th, January 1990, 1962, 141, No work, except The Bible, has been more widely used..., Cengage, Brooks/Cole, 9780030295584, 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 and a proof that there are infinitely many prime numbers.Further advances also took place in 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 mathematician AlHashimi 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 numbers.{{citationlast=Matvievskayafirst=Galinayear=1987title=The Theory of Quadratic Irrationals in Medieval Oriental Mathematicsjournal=Annals of the New York Academy of Sciencesvolume=500issue=1pages=253â€“77 [260]doi=10.1111/j.17496632.1987.tb37206.xbibcode=1987NYASA.500..253M}} An inductive proof for arithmetic sequences was introduced in the AlFakhri (1000) by AlKaraji, 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 parallel postulate.{{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=January 23, 2008 }}Modern proof theory treats proofs as inductively defined data structures, so that there is no longer an assumption that axioms are "true" in any sense, thereby allowing parallel mathematical theories to be built based on alternate sets of axioms (for more examples, see Axiomatic set theory and NonEuclidean geometry).Nature and purposeAs 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. In order to gain acceptance, a proof has to meet communal statements of rigor; an argument that is considered vague or incomplete may be rejected.The concept of a proof is formalized in the field of mathematical logic.{{citationtitle=Handbook of Proof Theoryvolume=137series=Studies in Logic and the Foundations of Mathematicseditorfirst=Samuel R.editorlast=Busseditorlink=Samuel Busspublisher=Elsevieryear=1998isbn=9780080533186contribution=An introduction to proof theorypages=1â€“78first=Samuel R.last=Bussauthorlink=Samuel Buss}}. See in particular p. 3: "The study of Proof Theory is traditionally motivated by the problem of formalizing mathematical proofs; the original formulation of firstorder logic by Frege [1879] was the first successful step in this direction." A formal proof is written in a formal language instead of a natural language. A formal proof is defined as sequence of 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 (e.g., the property that a statement has a formal proof), and an application of proof theory is to show that certain undecidable statements are not provable.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 or synthetic. Kant, who introduced the analyticâ€“synthetic distinction, believed mathematical proofs are synthetic, whereas Quine argued in his 1951 "Two Dogmas of Empiricism" that such distinction is untenable.WEB,weblink Two Dogmas of Empiricism, Quine, Willard Van Orman, 1961, UniversitÃ¤t ZÃ¼rich â€” Theologische FakultÃ¤t, 12, live, October 20, 2019, Proofs may be viewed as aesthetic objects, admired for their mathematical beauty. The mathematician Paul ErdÅ‘s 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.{{AnchorTypes of proof}}MethodsDirect proofIn direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.Cupillari, p. 20. For example, direct proof can be used to prove that the sum of two even integers is always even:
Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b, respectively, for integers a and b. Then the sum x + y = 2a + 2b = 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.
This proof uses the definition of even integers, the integer properties of closure under addition and multiplication, and distributivity.Proof by mathematical inductionDespite its name, mathematical induction is a method of 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 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.Cupillari, p. 46. 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 numbers:Examples of simple proofs by mathematical induction for all natural numbersLet {{math1=N = {1,2,3,4,...}}} be the set of natural numbers, and {{mathP(n)}} be a mathematical statement involving the natural number {{mathn}} belonging to {{mathN}} such that
(i) For {{math1=n = 1}}, {{math1=2n âˆ’ 1 = 2(1) âˆ’ 1 = 1}}, and {{math1}} is odd, since it leaves a remainder of {{math1}} when divided by {{math2}}. Thus {{mathP(1)}} is true.
(ii) For any {{mathn}}, if {{math2n âˆ’ 1}} is odd ({{mathP(n)}}), then {{math(2n âˆ’ 1) + 2}} must also be odd, because adding {{math2}} to an odd number results in an odd number. But {{math1=(2n âˆ’ 1) + 2 = 2n + 1 = 2(n+1) âˆ’ 1}}, so {{math1=2(n+1) âˆ’ 1}} is odd ({{mathP(n+1)}}). So {{mathP(n)}} implies {{mathP(n+1)}}.
Thus {{math2n âˆ’ 1}} is odd, for all positive integers {{mathn}}.
The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction".Proof by induction, University of Warwick Glossary of Mathematical TerminologyProof by contrapositionProof 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 = xcdot 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 contradictionIn 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 involves the following proof that sqrt{2} is an irrational number:
Suppose that sqrt{2} were a rational number. This number could be written in lowest terms as sqrt{2} = {aover b} where a and b are nonzero integers with no common factor. Thus, bsqrt{2} = a. Squaring both sides yields 2b2 = a2. Since 2 divides the expression on the left, 2 must also divide the equal expression on the right. That is, a2 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 = 2c, where c is also an integer. Substitution into the original equation yields 2b2 = (2c)2 = 4c2. Dividing both sides by 2 yields b2 = 2c2. But then, by the same argument as before, 2 divides b2, so b must be even. However, if a and b are both even, they have a common factor, namely 2. This contradicts our initial supposition, so we are forced to conclude that sqrt{2} is an irrational number.
Proof by constructionProof 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 numbers by constructing an explicit example. It can also be used to construct a counterexample to disprove a proposition that all elements have a certain property.Proof by exhaustionIn 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 of2011lc=on}} still has over 600 cases.See Four color theorem#Simplification and verification.Probabilistic proofA 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 theorems.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.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.Davis, Philip J. (1972), "Fidelity in Mathematical Discourse: Is One and One Really Two?" American Mathematical Monthly 79:252â€“63.Fallis, Don (1997), "The Epistemic Status of Probabilistic Proof." Journal of Philosophy 94:165â€“86.Combinatorial proofA combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a bijection 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 proofA nonconstructive proof establishes that a mathematical object with a certain property exists â€” without explaining how such an object are to 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. A famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that a^b is a rational number:
Either sqrt{2}^{sqrt{2}} is a rational number and we are done (take a=b=sqrt{2}), or sqrt{2}^{sqrt{2}} is irrational so we can write a=sqrt{2}^{sqrt{2}} and b=sqrt{2}. This then gives left (sqrt{2}^{sqrt{2}}right )^{sqrt{2}}=sqrt{2}^{2}=2, which is thus a rational of the form a^b.
Statistical proofs in pure mathematicsThe 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."in number theory and commutative algebra... in particular the statistical proof of the lemma." weblink"Whether constant Ï€ (i.e., pi) is normal is a confusing problem without any strict theoretical demonstration except for some statistical proof"" (Derogatory use.)weblink"these observations suggest a statistical proof of Goldbach's conjecture with very quickly vanishing probability of failure for large E" weblink It is less commonly used to refer to a mathematical proof in the branch of mathematics known as mathematical statistics. See also "Statistical proof using data" section below.Computerassisted proofsUntil 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 Proof, 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 computerassisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a runtime error in its calculations calls the validity of such computerassisted proofs into question. In practice, the chances of an error invalidating a computerassisted proof can be reduced by incorporating redundancy and selfchecks 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 statementsA 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.Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermeloâ€“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.GÃ¶del's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements.Heuristic mathematics and experimental mathematicsWhile 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.BOOK, Indra's Pearls: The Vision of Felix Klein,weblink Cambridge University Press, Mumford, David B., David Mumford, Series, Caroline, Caroline Series, Wright, David, David Wright (arranger), 2002, 9780521352536, 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., With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects outside of the prooftheorem framework,WEB,weblink A Note on the History of Fractals,weblink February 15, 2009, dead, 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., in experimental mathematics. Early pioneers of these methods intended the work ultimately to be embedded in a classical prooftheorem framework, e.g. the early development of fractal geometry,BOOK, Introducing Fractal Geometry, LesmoirGordon, Nigel, Introducing... (book series), Icon Books, 2000, 9781840461237, ...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'..., which was ultimately so embedded.Related conceptsVisual proofAlthough not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "proof without words". The lefthand picture below is an example of a historic visual proof of the Pythagorean theorem in the case of the (3,4,5) triangle.Image:Chinese pythagoras.jpgVisual proof for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500â€“200 BCE.File:Pythagoras2a.gifAnimated visual proof for the Pythagorean theorem by rearrangement.File:Pythag anim.gifA second animated proof of the Pythagorean theorem.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.Elementary proofAn 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.Twocolumn proof(File:twocolumnproof.pngthumbrightA twocolumn proof published in 1913)A particular way of organising a proof using two parallel columns is often used in elementary geometry classes in the United States.JOURNAL, Patricio G., Herbst, Establishing a Custom of Proving in American School Geometry: Evolution of the TwoColumn Proof in the Early Twentieth Century, Educational Studies in Mathematics, 49, 3, 2002, 283â€“312, 10.1023/A:1020264906740,weblink The proof is written as a series of lines in two columns. In each line, the lefthand column contains a proposition, while the righthand column contains a brief explanation of how the corresponding proposition in the lefthand column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The lefthand column is typically headed "Statements" and the righthand column is typically headed "Reasons".WEB,weblink Introduction to the TwoColumn Proof, Dr. Fisher Burns, onemathematicalcat.org, 15 October 2009,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 from data.Statistical proof using data"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 plots, when the data or diagram is adequately convincing without further analysis.Inductive logic proofs and Bayesian analysisProofs 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 assessment of likelihoods of hypotheses when new evidence or information is acquired.Proofs as mental objectsPsychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, 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.{{citation neededdate=November 2014}}Influence of mathematical proof methods outside mathematicsPhilosophermathematicians 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 mathematicianphilosophers 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 argument.Ending a proofSometimes, 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 â–¡ or âˆŽ, known as a "tombstone" or "halmos" after its eponym Paul Halmos. Often, "which was to be shown" is verbally stated when writing "QED", "â–¡", or "âˆŽ" during an oral presentation.See also{hide}columnslistcolwidth=30em
References{{Reflist}}Further reading
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