Mathematical proof

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A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. Mathematical proof_sentence_0

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, along with the accepted rules of inference. Mathematical proof_sentence_1

Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Mathematical proof_sentence_2

Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. Mathematical proof_sentence_3

An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Mathematical proof_sentence_4

Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. Mathematical proof_sentence_5

In most mathematical literature, proofs are written in terms of rigorous informal logic. Mathematical proof_sentence_6

Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. Mathematical proof_sentence_7

The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. Mathematical proof_sentence_8

The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language. Mathematical proof_sentence_9

History and etymology Mathematical proof_section_0

See also: History of logic Mathematical proof_sentence_10

The word "proof" comes from the Latin probare (to test). Mathematical proof_sentence_11

Related modern words are English "probe", "probation", and "probability", Spanish probar (to smell or taste, or sometimes touch or test), Italian provare (to try), and German probieren (to try). Mathematical proof_sentence_12

The legal term "probity" means authority or credibility, the power of testimony to prove facts when given by persons of reputation or status. Mathematical proof_sentence_13

Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. Mathematical proof_sentence_14

It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, which originated in practical problems of land measurement. Mathematical proof_sentence_15

The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of its greatest achievements. Mathematical proof_sentence_16

Thales (624–546 BCE) and Hippocrates of Chios (c. 470–410 BCE) gave some of the first known proofs of theorems in geometry. Mathematical proof_sentence_17

Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Mathematical proof_sentence_18

Aristotle (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. Mathematical proof_sentence_19

Mathematical proof was revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today. Mathematical proof_sentence_20

It starts with undefined terms and axioms, propositions concerning the undefined terms which are assumed to be self-evidently true (from Greek "axios", something worthy). Mathematical proof_sentence_21

From this basis, the method proves theorems using deductive logic. Mathematical proof_sentence_22

Euclid's book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Mathematical proof_sentence_23

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. Mathematical proof_sentence_24

Further advances also took place in medieval Islamic mathematics. Mathematical proof_sentence_25

While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs with no dependence on geometric intuition. Mathematical proof_sentence_26

In the 10th century CE, the Iraqi mathematician Al-Hashimi worked with numbers as such, called "lines" but not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of irrational numbers. Mathematical proof_sentence_27

An inductive proof for 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. Mathematical proof_sentence_28

Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Mathematical proof_sentence_29

Modern proof theory treats proofs as inductively defined data structures, not requiring an assumption that axioms are "true" in any sense. Mathematical proof_sentence_30

This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example Axiomatic set theory and Non-Euclidean geometry. Mathematical proof_sentence_31

Nature and purpose Mathematical proof_section_1

As practiced, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement. Mathematical proof_sentence_32

The standard of rigor is not absolute and has varied throughout history. Mathematical proof_sentence_33

A proof can be presented differently depending on the intended audience. Mathematical proof_sentence_34

In order to gain acceptance, a proof has to meet communal standards of rigor; an argument considered vague or incomplete may be rejected. Mathematical proof_sentence_35

The concept of proof is formalized in the field of mathematical logic. Mathematical proof_sentence_36

A formal proof is written in a formal language instead of a natural language. Mathematical proof_sentence_37

A formal proof is a sequence of formulas in a formal language, starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones. Mathematical proof_sentence_38

This definition makes the concept of proof amenable to study. Mathematical proof_sentence_39

Indeed, the field of proof theory studies formal proofs and their properties, the most famous and surprising being that almost all axiomatic systems can generate certain undecidable statements not provable within the system. Mathematical proof_sentence_40

The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. Mathematical proof_sentence_41

The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. Mathematical proof_sentence_42

However, outside the field of automated proof assistants, this is rarely done in practice. Mathematical proof_sentence_43

A classic question in philosophy asks whether mathematical proofs are analytic or synthetic. Mathematical proof_sentence_44

Kant, who introduced the analytic–synthetic distinction, believed mathematical proofs are synthetic, whereas Quine argued in his 1951 "Two Dogmas of Empiricism" that such a distinction is untenable. Mathematical proof_sentence_45

Proofs may be admired for their mathematical beauty. Mathematical proof_sentence_46

The mathematician Paul Erdős was known for describing proofs which he found to be particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. Mathematical proof_sentence_47

The book Proofs from THE BOOK, published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing. Mathematical proof_sentence_48

Methods Mathematical proof_section_2

Direct proof Mathematical proof_section_3

Main article: Direct proof Mathematical proof_sentence_49

In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. Mathematical proof_sentence_50

For example, direct proof can be used to prove that the sum of two even integers is always even: Mathematical proof_sentence_51

Mathematical proof_description_list_0

  • 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.Mathematical proof_item_0_0

This proof uses the definition of even integers, the integer properties of closure under addition and multiplication, and distributivity. Mathematical proof_sentence_52

Proof by mathematical induction Mathematical proof_section_4

Main article: Mathematical induction Mathematical proof_sentence_53

Despite its name, mathematical induction is a method of deduction, not a form of inductive reasoning. Mathematical proof_sentence_54

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. Mathematical proof_sentence_55

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. Mathematical proof_sentence_56

This avoids having to prove each case individually. Mathematical proof_sentence_57

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. Mathematical proof_sentence_58

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: Let N = {1,2,3,4,...} be the set of natural numbers, and P(n) be a mathematical statement involving the natural number n belonging to N such that Mathematical proof_sentence_59

Mathematical proof_unordered_list_1

  • (i) P(1) is true, i.e., P(n) is true for n = 1.Mathematical proof_item_1_1
  • (ii) P(n+1) is true whenever P(n) is true, i.e., P(n) is true implies that P(n+1) is true.Mathematical proof_item_1_2
  • Then P(n) is true for all natural numbers n.Mathematical proof_item_1_3

For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Mathematical proof_sentence_60

Let P(n) represent "2n − 1 is odd": Mathematical proof_sentence_61

Mathematical proof_description_list_2

  • (i) For n = 1, 2n − 1 = 2(1) − 1 = 1, and 1 is odd, since it leaves a remainder of 1 when divided by 2. Thus P(1) is true.Mathematical proof_item_2_4
  • (ii) For any n, if 2n − 1 is odd (P(n)), then (2n − 1) + 2 must also be odd, because adding 2 to an odd number results in an odd number. But (2n − 1) + 2 = 2n + 1 = 2(n+1) − 1, so 2(n+1) − 1 is odd (P(n+1)). So P(n) implies P(n+1).Mathematical proof_item_2_5
  • Thus 2n − 1 is odd, for all positive integers n.Mathematical proof_item_2_6

The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction". Mathematical proof_sentence_62

Proof by contraposition Mathematical proof_section_5

Main article: Contraposition Mathematical proof_sentence_63

Proof by contraposition infers the statement "if p then q" by establishing the logically equivalent contrapositive statement: "if not q then not p". Mathematical proof_sentence_64

Proof by contradiction Mathematical proof_section_6

Main article: Proof by contradiction Mathematical proof_sentence_65

Proof by construction Mathematical proof_section_7

Main article: Proof by construction Mathematical proof_sentence_66

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. Mathematical proof_sentence_67

Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example. Mathematical proof_sentence_68

It can also be used to construct a counterexample to disprove a proposition that all elements have a certain property. Mathematical proof_sentence_69

Proof by exhaustion Mathematical proof_section_8

Main article: Proof by exhaustion Mathematical proof_sentence_70

In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. Mathematical proof_sentence_71

The number of cases sometimes can become very large. Mathematical proof_sentence_72

For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. Mathematical proof_sentence_73

This proof was controversial because the majority of the cases were checked by a computer program, not by hand. Mathematical proof_sentence_74

The shortest known proof of the four color theorem as of 2011 still has over 600 cases. Mathematical proof_sentence_75

Probabilistic proof Mathematical proof_section_9

Main article: Probabilistic method Mathematical proof_sentence_76

A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. Mathematical proof_sentence_77

Probabilistic proof, like proof by construction, is one of many ways to show existence theorems. Mathematical proof_sentence_78

In the probabilistic method, one seeks an object having a given property, starting with a large set of candidates. Mathematical proof_sentence_79

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. Mathematical proof_sentence_80

This does not specify which candidates have the property, but the probability could not be positive without at least one. Mathematical proof_sentence_81

A probabilistic proof is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. Mathematical proof_sentence_82

The work on the Collatz conjecture shows how far plausibility is from genuine proof. Mathematical proof_sentence_83

While most mathematicians do not think that probabilistic evidence for the properties of a given object counts as a genuine mathematical proof, a few mathematicians and philosophers have argued that at least some types of probabilistic evidence (such as Rabin's probabilistic algorithm for testing primality) are as good as genuine mathematical proofs. Mathematical proof_sentence_84

Combinatorial proof Mathematical proof_section_10

Main article: Combinatorial proof Mathematical proof_sentence_85

A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Mathematical proof_sentence_86

Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Mathematical proof_sentence_87

Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal. Mathematical proof_sentence_88

Nonconstructive proof Mathematical proof_section_11

Main article: Nonconstructive proof Mathematical proof_sentence_89

Statistical proofs in pure mathematics Mathematical proof_section_12

Main article: Statistical proof Mathematical proof_sentence_90

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. Mathematical proof_sentence_91

It is less commonly used to refer to a mathematical proof in the branch of mathematics known as mathematical statistics. Mathematical proof_sentence_92

See also "Statistical proof using data" section below. Mathematical proof_sentence_93

Computer-assisted proofs Mathematical proof_section_13

Main article: Computer-assisted proof Mathematical proof_sentence_94

Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity. Mathematical proof_sentence_95

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. Mathematical proof_sentence_96

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. Mathematical proof_sentence_97

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. Mathematical proof_sentence_98

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. Mathematical proof_sentence_99

Undecidable statements Mathematical proof_section_14

A statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). Mathematical proof_sentence_100

One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry. Mathematical proof_sentence_101

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. Mathematical proof_sentence_102

Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements. Mathematical proof_sentence_103

Heuristic mathematics and experimental mathematics Mathematical proof_section_15

Main article: Experimental mathematics Mathematical proof_sentence_104

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. Mathematical proof_sentence_105

With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects outside of the proof-theorem framework, in experimental mathematics. Mathematical proof_sentence_106

Early pioneers of these methods intended the work ultimately to be embedded in a classical proof-theorem framework, e.g. the early development of fractal geometry, which was ultimately so embedded. Mathematical proof_sentence_107

Related concepts Mathematical proof_section_16

Visual proof Mathematical proof_section_17

Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "proof without words". Mathematical proof_sentence_108

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. Mathematical proof_sentence_109

Mathematical proof_unordered_list_3

  • Mathematical proof_item_3_7
  • Mathematical proof_item_3_8
  • Mathematical proof_item_3_9

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. Mathematical proof_sentence_110

Elementary proof Mathematical proof_section_18

Main article: Elementary proof Mathematical proof_sentence_111

An elementary proof is a proof which only uses basic techniques. Mathematical proof_sentence_112

More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Mathematical proof_sentence_113

For some time it was thought that certain theorems, like the prime number theorem, could only be proved using "higher" mathematics. Mathematical proof_sentence_114

However, over time, many of these results have been reproved using only elementary techniques. Mathematical proof_sentence_115

Two-column proof Mathematical proof_section_19

A particular way of organising a proof using two parallel columns is often used in elementary geometry classes in the United States. Mathematical proof_sentence_116

The proof is written as a series of lines in two columns. Mathematical proof_sentence_117

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. Mathematical proof_sentence_118

The left-hand column is typically headed "Statements" and the right-hand column is typically headed "Reasons". Mathematical proof_sentence_119

Colloquial use of "mathematical proof" Mathematical proof_section_20

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. Mathematical proof_sentence_120

It is sometimes also used to mean a "statistical proof" (below), especially when used to argue from data. Mathematical proof_sentence_121

Statistical proof using data Mathematical proof_section_21

Main article: Statistical proof Mathematical proof_sentence_122

"Statistical proof" from data refers to the application of statistics, data analysis, or Bayesian analysis to infer propositions regarding the probability of data. Mathematical proof_sentence_123

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. Mathematical proof_sentence_124

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. Mathematical proof_sentence_125

"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. Mathematical proof_sentence_126

Inductive logic proofs and Bayesian analysis Mathematical proof_section_22

Main articles: Inductive logic and Bayesian analysis Mathematical proof_sentence_127

Proofs using inductive logic, while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to probability, and may be less than full certainty. Mathematical proof_sentence_128

Inductive logic should not be confused with mathematical induction. Mathematical proof_sentence_129

Bayesian analysis uses Bayes' theorem to update a person's assessment of likelihoods of hypotheses when new evidence or information is acquired. Mathematical proof_sentence_130

Proofs as mental objects Mathematical proof_section_23

Main articles: Psychologism and Language of thought Mathematical proof_sentence_131

Psychologism views mathematical proofs as psychological or mental objects. Mathematical proof_sentence_132

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. Mathematical proof_sentence_133

Influence of mathematical proof methods outside mathematics Mathematical proof_section_24

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. Mathematical proof_sentence_134

Other mathematician-philosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at statements outside of mathematics, but having the certainty of propositions deduced in a mathematical proof, such as Descartes' cogito argument. Mathematical proof_sentence_135

Ending a proof Mathematical proof_section_25

Main article: Q.E.D. Mathematical proof_sentence_136

Sometimes, the abbreviation "Q.E.D." Mathematical proof_sentence_137

is written to indicate the end of a proof. Mathematical proof_sentence_138

This abbreviation stands for "quod erat demonstrandum", which is Latin for "that which was to be demonstrated". Mathematical proof_sentence_139

A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a "tombstone" or "halmos" after its eponym Paul Halmos. Mathematical proof_sentence_140

Often, "which was to be shown" is verbally stated when writing "QED", "□", or "∎" during an oral presentation. Mathematical proof_sentence_141

See also Mathematical proof_section_26

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: proof.