P. Oxy. 29, one of the oldest surviving fragments ofEuclid'sElements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.[1]
Amathematical proof is adeductiveargument for amathematical statement, showing that the stated assumptionslogically guarantee the conclusion. The argument may use other previously established statements, such astheorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known asaxioms,[2][3][4] along with the accepted rules ofinference. Proofs are examples of exhaustivedeductive reasoning that establish logical certainty, to be distinguished fromempirical arguments or non-exhaustiveinductive 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 inall possible cases. A proposition that has not been proved but is believed to be true is known as aconjecture, or a hypothesis if frequently used as an assumption for further mathematical work.
The wordproof derives from the Latinprobare 'to test'; related words include Englishprobe,probation, andprobability, as well as Spanishprobar 'to taste' (sometimes 'to touch' or 'to test'),[5] Italianprovare 'to try', and Germanprobieren 'to try'. The legal termprobity means authority or credibility, the power of testimony to prove facts when given by persons of reputation or status.[6]
Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof.[7] It is likely that the idea of demonstrating a conclusion first arose in connection withgeometry, which originated in practical problems of land measurement.[8] The development of mathematical proof is primarily the product ofancient Greek mathematics, and one of its greatest achievements.[9]Thales (624–546 BCE) andHippocrates of Chios (c. 470–410 BCE) gave some of the first known proofs of theorems in geometry.Eudoxus (408–355 BCE) andTheaetetus (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 proof was revolutionized byEuclid (300 BCE), who introduced theaxiomatic method still in use today. It starts withundefined terms andaxioms, propositions concerning the undefined terms which are assumed to be self-evidently true (from Greekaxios 'something worthy'). From this basis, the method proves theorems usingdeductive logic.Euclid's Elements was read by anyone who was considered educated in the West until the middle of the 20th century.[10] In addition to theorems of geometry, such as thePythagorean theorem, theElements also coversnumber theory, including a proof that thesquare root of two isirrational and a proof that there are infinitely manyprime numbers.
Modernproof theory treats proofs as inductively defineddata structures, 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 exampleaxiomatic set theory andnon-Euclidean geometry.
As practiced, a proof is expressed in natural language and is a rigorousargument 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 ofmathematical logic.[12] Aformal proof is written in aformal language instead of natural language. A formal proof is a sequence offormulas 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 ofproof theory studies formal proofs and their properties, the most famous and surprising being that almost all axiomatic systems can generate certainundecidable 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 automatedproof assistants, this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs areanalytic orsynthetic.Kant, who introduced theanalytic–synthetic distinction, believed mathematical proofs are synthetic, whereasQuine argued in his 1951 "Two Dogmas of Empiricism" that such a distinction is untenable.[13]
Proofs may be admired for theirmathematical beauty. The mathematicianPaul 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 bookProofs from THE BOOK, published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.
In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.[14] For example, direct proof can be used to prove that the sum of twoevenintegers is always even:
Consider two even integersx andy. Since they are even, they can be written asx = 2a andy = 2b, respectively, for some integersa andb. Then the sum isx + y = 2a + 2b = 2(a+b). Thereforex+y has 2 as afactor 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 ofclosure under addition and multiplication, and thedistributive property.
Despite its name, mathematical induction is a method ofdeduction, not a form ofinductive reasoning. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary caseimplies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usuallyinfinitely many) cases are provable.[15] This avoids having to prove each case individually. A variant of mathematical induction isproof by infinite descent, which can be used, for example, to prove theirrationality 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 allnatural numbers:[16]LetN = {1, 2, 3, 4, ...} be the set of natural numbers, and letP(n) be a mathematical statement involving the natural numbern belonging toN such that
(i)P(1) is true, i.e.,P(n) is true forn = 1.
(ii)P(n+1) is true wheneverP(n) is true, i.e.,P(n) is true implies thatP(n+1) is true.
ThenP(n) is true for all natural numbersn.
For example, we can prove by induction that all positive integers of the form2n − 1 areodd. LetP(n) represent "2n − 1 is odd":
(i) Forn = 1,2n − 1 = 2(1) − 1 = 1, and1 is odd, since it leaves a remainder of1 when divided by2. ThusP(1) is true.
(ii) For anyn, if2n − 1 is odd (P(n)), then(2n − 1) + 2 must also be odd, because adding2 to an odd number results in an odd number. But(2n − 1) + 2 = 2n + 1 = 2(n+1) − 1, so2(n+1) − 1 is odd (P(n+1)). SoP(n) impliesP(n+1).
Thus2n − 1 is odd, for all positive integersn.
The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction".[17]
For example, contraposition can be used to establish that, given an integer, if is even, then is even:
Suppose is not even. Then is odd. The product of two odd numbers is odd, hence is odd. Thus is not even. Thus, ifis even, the supposition must be false, so has to be even.
In proof by contradiction, also known by the Latin phrasereductio ad absurdum (by reduction to the absurd), it is shown that if some statement is assumed true, alogical contradiction occurs, hence the statement must be false. A famous example involves the proof that is anirrational number:
Suppose that were a rational number. Then it could be written in lowest terms as wherea andb are non-zero integers withno common factor. Thus,. Squaring both sides yields 2b2 =a2. Since the expression on the left is an integer multiple of 2, the right expression is by definition divisible by 2. That is,a2 is even, which implies thata must also be even, as seen in the proposition above (in#Proof by contraposition). So we can writea = 2c, wherec is also an integer. Substitution into the original equation yields 2b2 = (2c)2 = 4c2. Dividing both sides by 2 yieldsb2 = 2c2. But then, by the same argument as before, 2 dividesb2, sob must be even. However, ifa andb are both even, they have 2 as a common factor. This contradicts our previous statement thata andb have no common factor, so we must conclude that is an irrational number.
To paraphrase: if one could write as afraction, this fraction could never be written in lowest terms, since 2 could always be factored from numerator and denominator.
Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists.Joseph Liouville, for instance, proved the existence oftranscendental numbers by constructing anexplicit example. It can also be used to construct acounterexample to disprove a proposition that all elements have a certain property.
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 thefour 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.[18]
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods ofprobability theory. Probabilistic proof, like proof by construction, is one of many ways to proveexistence theorems.
In the probabilistic method, one seeks an object having a given property, starting with a large set of candidates. One assigns a certain probability for each candidate to be chosen, and then proves that there is a non-zero probability that a chosen candidate will have the desired property. This does not specify which candidates have the property, but the probability could not be positive without at least one.
A probabilistic proof is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work toward theCollatz conjecture shows how far plausibility is from genuine proof, as does the disproof of theMertens 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'sprobabilistic algorithm fortesting primality) are as good as genuine mathematical proofs.[21][22]
A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often abijection between twosets is used to show that the expressions for their two sizes are equal. Alternatively, adouble counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.
A nonconstructive proof establishes that amathematical object with a certain property exists—without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. The following famous example of a nonconstructive proof shows that there exist twoirrational numbersa andb such that is arational number. This proof uses that is irrational (an easy proof is known sinceEuclid), but not that is irrational (this is true, but the proof is not elementary).
Either is a rational number and we are done (take), or is irrational so we can write and. This then gives, which is thus a rational number of the form
Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity.[7] 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 thefour 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.
A statement that is neither provable nor disprovable from a set ofaxioms is called undecidable (from those axioms). One example is theparallel postulate, which is neither provable nor refutable from the remaining axioms ofEuclidean geometry.
While early mathematicians such asEudoxus of Cnidus did not use proofs, fromEuclid to thefoundational mathematics developments of the late 19th and 20th centuries, proofs were an essential part of mathematics.[26] With the increase in computing power in the 1960s, significant work began to be done investigatingmathematical objects beyond the proof-theorem framework,[27] inexperimental mathematics. Early pioneers of these methods intended the work ultimately to be resolved into a classical proof-theorem framework, e.g. the early development offractal geometry,[28] which was ultimately so resolved.
Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "proof without words". The left-hand picture below is an example of a historic visual proof of thePythagorean theorem in the case of the (3,4,5)triangle.
Visual proof for the (3,4,5) triangle as in theZhoubi Suanjing 500–200 BCE.
Animated visual proof for the Pythagorean theorem by rearrangement.
A second animated proof of the Pythagorean theorem.
Some illusory visual proofs, such as themissing 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.
An elementary proof is a proof which only uses basic techniques. More specifically, the term is used innumber theory to refer to proofs that make no use ofcomplex analysis. For some time it was thought that certain theorems, like theprime number theorem, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.
A particular way of organising a proof using two parallel columns is often used as amathematical exercise in elementary geometry classes in the United States.[29] 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".[30]
The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing withmathematical 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" from data refers to the application of statistics,data analysis, orBayesian analysis to infer propositions regarding theprobability of data. Whileusing mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that theassumptions 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 specializedmathematical methods of physics applied to analyze data in aparticle physics experiment orobservational study inphysical cosmology. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such asscatter plots, when the data or diagram is adequately convincing without further analysis.
Proofs usinginductive logic, while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner toprobability, and may be less than fullcertainty. Inductive logic should not be confused withmathematical induction.
Psychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, such asLeibniz,Frege, andCarnap have variously criticized this view and attempted to develop a semantics for what they considered to be thelanguage of thought, whereby standards of mathematical proof might be applied toempirical science.
Influence of mathematical proof methods outside mathematics
Philosopher-mathematicians such asSpinoza 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 thecertainty of propositions deduced in a mathematical proof, such asDescartes'cogito argument.
Sometimes, the abbreviation"Q.E.D." is written to indicate the end of a proof. This abbreviation stands for"quod erat demonstrandum", which isLatin for"that which was to be demonstrated". A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a "tombstone" or "halmos" after itseponymPaul 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 (∎)(220E(hex) = 8718(dec)).
^Clapham, C. & Nicholson, J.N.The Concise Oxford Dictionary of Mathematics, Fourth edition.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.
^Eves, Howard W. (January 1990) [1962].An Introduction to the History of Mathematics (Saunders Series) (6th ed.). Cengage. p. 141.ISBN978-0030295584.No work, except The Bible, has been more widely used...
^Buss, Samuel R. (1998), "An introduction to proof theory", inBuss, Samuel R. (ed.),Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, vol. 137, Elsevier, pp. 1–78,ISBN978-0-08-053318-6. See in particularp. 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."
^Quine, Willard Van Orman (1961)."Two Dogmas of Empiricism"(PDF).Universität Zürich – Theologische Fakultät. p. 12. RetrievedOctober 20, 2019.
^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.
^"in number theory and commutative algebra... in particular the statistical proof of the lemma."[1]
^"Whether constant π (i.e., pi) is normal is a confusing problem without any strict theoretical demonstration except for somestatistical proof"" (Derogatory use.)[2]
^"these observations suggest a statistical proof of Goldbach's conjecture with very quickly vanishing probability of failure for large E"[3]
^"A Note on the History of Fractals". Archived fromthe original on February 15, 2009.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.
^Lesmoir-Gordon, Nigel (2000).Introducing Fractal Geometry.Icon Books.ISBN978-1-84046-123-7....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'...
