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Encyclopedia > Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true, within the accepted standards of the field. A proof is a logically deduced argument, not an empirical one. That is, the proof must demonstrate that a proposition is true in all cases to which it applies, without a single exception. An unproven proposition believed or strongly suspected to be true is known as a conjecture. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This article documents the word proposition as it is used in logic, philosophy, and linguistics. ... Deductive reasoning is reasoning whose conclusions are intended to necessarily follow from its premises. ... A central concept in science and the scientific method is that all evidence must be empirical, or empirically based, that is, dependent on evidence or consequences that are observable by the senses. ... In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ...

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 informal logic. Purely formal proofs are considered in proof theory. 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 (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. Logic (from Classical Greek Î»ÏŒÎ³Î¿Ï‚ logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... In the philosophy of language, a natural language (or ordinary language) is a language that is spoken, written, or signed by humans for general-purpose communication, as distinguished from formal languages (such as computer-programming languages or the languages used in the study of formal logic, especially mathematical logic) and... Informal logic is the study of arguments as presented in ordinary language, as contrasted with the presentations of arguments in an artificial (technical) or formal language (see formal logic). ... ... Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... In the philosophy of mathematics, mathematical practice is used to distinguish the working practices of professional mathematicians (eg. ... Quasi-empiricism in mathematics is the movement in the philosophy of mathematics to direct philosophers attention to mathematical practice, in particular, relations with physics and social sciences, rather than the foundations problem in mathematics. ... As the term is understood by mathematicians, folk mathematics or mathematical folklore means theorems, definitions, proofs, or mathematical facts or techniques that circulate among mathematicians by word-of-mouth but have not appeared in print, either in books or in scholarly journals. ... // Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. ... The central question involved in discussing mathematics as a language can be stated as follows : What do we mean when we talk about the language of mathematics ? To what extent does mathematics meet generally accepted criteria of being a language ? A secondary question is : If it is valid to consider...

Regardless of one's attitude to formalism, the result that is proved to be true is a theorem; in a completely formal proof it would be the final line, and the complete proof shows how it follows from the axioms alone by the application of the rules of inference. Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a lemma if it is used as a stepping stone in the proof of a theorem. The axioms are those statements one cannot, or need not, prove. These were once the primary study of philosophers of mathematics. Today focus is more on practice, i.e. acceptable techniques. A mathematical picture paints a thousand words: the Pythagorean theorem. ... This article is about a logical statement. ... In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. ... In the philosophy of mathematics, mathematical practice is used to distinguish the working practices of professional mathematicians (eg. ...

### Direct proof

Main article: Direct proof

In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to establish that the sum of two even integers is always even: In mathematics, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions. ... In mathematics, the parity of an object refers to whether it is even or odd. ... Not to be confused with Natural number. ...

For any two even integers x and y we can write x = 2a and y = 2b for some integers a and b, since both x and y are multiples of 2. But the sum x + y = 2a + 2b = 2(a + b) is also a multiple of 2, so it is therefore even by definition.

This proof uses definition of even integers, as well as distribution law. In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

### Proof by induction

In proof by induction, first a "base case" is proved, and then an "induction rule" is used to prove a (often infinite) series of other cases. 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 irrationality of the square root of two. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction. ... The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ...

The principle of mathematical induction states that: 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 (i) P(1) is true, ie, P(n) is true for n = 1 (ii) P(m + 1) is true whenever P(m) is true, ie, P(m) is true implies that P(m + 1) is true. Then P(n) is true for the set of natural numbers N.

### Proof by transposition

Main article: Transposition (logic)

Proof by Transposition establishes the conclusion "if p then q" by proving the equivalent contrapositive statement "if not q then not p". In the methods of deductive reasoning in classical logic, transposition is the rule of inference that permits one to infer from the truth of A implies B the truth of Not-B implies not-A, and conversely.[1] Its symbolic expression is: (P â†’ Q) â†” (Â¬Q â†’ Â¬P)[2] The â†’ is the... In predicate logic, the contrapositive (or transposition) of the statement p implies q is not-q implies not-p. ...

In proof by contradiction (also known as reductio ad absurdum, Latin for "reduction into the absurd"), it is shown that if some statement were false, a logical contradiction occurs, hence the statement must be true. This method is perhaps the most prevalent of mathematical proofs. A famous example of a proof by contradiction shows that $sqrt{2}$ is irrational: Reductio ad absurdum (Latin: reduction to the absurd) also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument, derives an absurd or ridiculous outcome, and then concludes that the original assumption... In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ...

Suppose that $sqrt{2}$ is rational, so $sqrt{2} = {aover b}$ where a and b are non-zero integers with no common factor (definition of rational number). Thus, $bsqrt{2} = a$. Squaring both sides yields 2b2 = a2. Since 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So a2 is even, which implies that a must also be even. 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 share a factor, namely 2. This contradicts our assumption, so we are forced to conclude that $sqrt{2}$ is irrational.

### Proof by construction

Main article: Proof by construction

Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example. In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object. ... Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In number theory, a Liouville number is a real number x with the property that, for any positive integer n, there exist integers p and q with q > 1 and such that 0 < |x âˆ’ p/q| < 1/qn. ...

### Proof by exhaustion

Main article: Proof by exhaustion

In Proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four colour 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 colour theorem today still has over 600 cases. Proof by exhaustion, also known as the brute force method or case analysis, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases, and each case is proved separately. ... Example of a four color map The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same colour. ...

### Probabilistic proof

Main article: Probabilistic method

A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. This is not be confused with an argument that a theorem is 'probably' true. The latter type of reasoning can be called a 'plausibility argument' and is not a proof; in the case of the Collatz conjecture it is clear how far that is from a genuine proof.[1] Probabilistic proof, like proof by construction, is one of many ways to show existence theorems. This article is not about probabilistic algorithms, which give the right answer with high probability but not with certainty, nor about Monte Carlo methods, which are simulations relying on pseudo-randomness. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... The Collatz conjecture is an unsolved conjecture in mathematics. ... In mathematics, an existence theorem is a theorem with a statement beginning there exist(s) .., or more generally for all x, y, ... there exist(s) .... That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. ...

### Combinatorial proof

Main article: Combinatorial proof

A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Usually a bijection is used to show that the two interpretations give the same result. A combinatorial proof is a method of proving a statement, usually a combinatorics identity, by counting some carefully chosen object in different ways to obtain different expressions in the statement (see also double counting). ... A bijective function. ...

### Nonconstructive proof

Main article: Nonconstructive proof

A nonconstructive proof establishes that a certain mathematical object must exist (e.g. "Some X satisfies f(X)"), 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 nonconstructive proof shows that there exist two irrational numbers a and b such that ab is a rational number: In mathematics, a nonconstructive proof, is a mathematical proof that purports to demonstrate the existence of something, but which does not say how to construct it. ... In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...

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 ab

### Proof nor disproof

There is a class of mathematical statements for which neither a proof nor disproof exists, using only ZFC, the standard form of axiomatic set theory. Examples include the continuum hypothesis; see further List of statements undecidable in ZFC. Under the assumption that ZFC is consistent, the existence of such statements follows from Gödel's (first) incompleteness theorem. Whether a particular unproven proposition can be proved or disproved using a standard set of axioms is not always obvious, and can be extremely technical to determine. The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematics, axiomatic set theory is a rigorous reformulation of set theory in first-order logic created to address paradoxes in naive set theory. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... The following is a list of mathematical statements that are undecidable in ZFC (the Zermelo-Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. ... Consistency has three technical meanings: In mathematics and logic, as well as in theoretical physics, it refers to the proposition that a formal theory or a physical theory contains no contradictions. ... In mathematical logic, GÃ¶dels incompleteness theorems are two celebrated theorems proven by Kurt GÃ¶del in 1931. ...

### Elementary proof

Main article: Elementary proof

An elementary proof is (usually) a proof which does not use 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. In mathematics a proof is said to be elementary if uses only ideas from within its field and closely related issues. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

## End of a proof

Main article: Q.E.D.

Sometimes, the abbreviation "Q.E.D." is written to indicate the end of a proof. This abbreviation stands for "Quod Erat Demonstrandum", which is Latin for "that which was to be demonstrated". An alternative is to use a square or a rectangle, such as or , known as a "tombstone" or "halmos". This article is about Latin phrase Q.E.D., as used in proofs. ... For other uses, see Latins and Latin (disambiguation). ... The tombstone, or halmos, symbol â€” (Unicode U+220E) â€” is used in mathematics to denote the end of a proof. ...

Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... A computer-assisted proof is a mathematical proof that has been generated by computer. ... Automated theorem proving (ATP) or automated deduction, currently the most well-developed subfield of automated reasoning (AR), is the proving of mathematical theorems by a computer program. ... In mathematics, there are a variety of spurious proofs of obvious contradictions. ... In mathematics, a nonconstructive proof, is a mathematical proof that purports to demonstrate the existence of something, but which does not say how to construct it. ... Wikipedia contains a number of articles with mathematical proofs: Theorems of which articles are primarily devoted to proving them Bertrands postulate and a proof Erdős-Ko-Rado theorem Estimation of covariance matrices Fermats little theorem and some proofs Gödels completeness theorem and its original proof Mathematical induction... Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and GÃ¼nter M Ziegler. ...

## References

1. ^ 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.

A randomized algorithm is an algorithm which is allowed to flip a truly random coin. ...

## Sources

• Polya, G. Mathematics and Plausible Reasoning. Princeton University Press, 1954.
• Fallis, Don (2002) “What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians.” Logique et Analyse 45:373-88.
• Franklin, J. and Daoud, A. Proof in Mathematics: An Introduction. Quakers Hill Press, 1996. ISBN 1-876192-00-3
• Solow, D. How to Read and Do Proofs: An Introduction to Mathematical Thought Processes. Wiley, 2004. ISBN 0-471-68058-3
• Velleman, D. How to Prove It: A Structured Approach. Cambridge University Press, 2006. ISBN 0-521-67599-5

Results from FactBites:

 Metamath Site Selection (984 words) ...let's look at why mathematical proofs are so difficult to understand for most people...any realistic mathematical proof will leave out a great many steps, which are considered to be the "required background knowledge" for anyone who wants to understand the proof. By the way, a very interesting project called the Metamath project is trying to create an online archive of mathematical proofs which are specified all the way to the bottom, starting from set theory. But this is a very rare exception to the general rule.
 Mathematical proof - Wikipedia, the free encyclopedia (581 words) In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language. Proof by contradiction (also known as reductio ad absurdum): where it is shown that if some statement were false, a logical contradiction occurs, hence the statement must be true.
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