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.

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.

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 another 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.

Methods of proof

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:

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.

Proof by mathematical induction

In proof by mathematical 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.

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 all natural numbers n.

Mathematicians often use the term "proof by induction" as shorthand for a proof by mathematical induction.^[1] However, the term "proof by induction" may also be used in logic to mean an argument that uses inductive reasoning.

Proof by transposition

Proof by Transposition establishes the conclusion "if p then q" by proving the equivalent contrapositive statement "if not q then not p".

Proof by contradiction

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:

Suppose that \sqrt{2} is rational, so \sqrt{2} = {a\over b} where a and b are non-zero integers with no common factor (definition of rational number). Thus, b\sqrt{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

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.

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.

Probabilistic proof

A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. This is not to 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.^[2] Probabilistic proof, like proof by construction, is one of many ways to show existence theorems.

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.

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 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

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.

Visual proof

Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500?200 BC.

Statistical proofs in pure mathematics

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.^[3][4][5][6] 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.http://en.wikipedia.org/wiki/Mathematical_proof#Colloquial_use.2C_Statistical_proof_using_data.

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.

Computer proofs

Until the later 20th century, the sine qua non of a mathematical proof was it being analytic apriori, relying on no empirical observations or facts.^[7]The ability of a mathematician to always check a proof was part of the distinct universality and timelessness mathematics.^[8] With the advent of high powered computing, computers became able to both prove theorems and to produce proofs that were too long for any human or team of humans to check. The theory under which computers operated was empirical, whence the proof could at best be only as certain as the merely probable empirical theory. This caused such controversy among mathematicians that the mainstream press covered it. The four color theorem is an example.

Heuristic mathematics and experimental mathematics

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 the essential in a rigid framework as to what counted as doing mathematics, with ever increasing standards of rigor.^[9] With the increase in computing ability in the 1960?s, substantive work began to be done involving mathematical objects, but outside of the proof-theorem framework^[10], in experimental mathematics. Early pioneers of these methods intended the work ultimately to be embedded in a classical proof-theorem framework, e.g. the early development of fractal geometry^[11], which was ultimately so embedded. There is still debate as to whether the prospect of embedding work in a proof-theorem framework is necessary for the work to be considered doing ?real mathematics?.

History

Plausibility arguments preceded strict mathematical proof, utilizing heuristic devices such as pictures and analogies.^[12] The early history of the concept dates back to at least the early Greeks, and the Chinese independently developed a concept of proof (see picture of visual proof at right 500-200 BCE). Thales (640?546 BCE) proved some theorems in geometry. Eudoxus (408?355 BCE) and Theaetetus (417?369 BCE) formulated theorems but did not prove them. Aristotle (384?322 BCE)said definitions should describe the concept being defined in terms of other concepts already known. Euclid (300 BCE) began with undefined terms and axioms (propositions regarding the undefined terms assumed to be true, from the Greek ?axios? meaning ?something worthy?) and used these to prove theorems using deductive logic. Standards of rigor evolved and consolidated in the late 19th and early 20th century.

Etymology

Proof, or the related verb ?prove?, etymologically derives from the Latin ?probare? -?making a trial or testing?, also used in law related to evidence in a trial. Related modern words are the English ?probe?, "proboscis?, "probity"^[13], and "probability", and the Spanish ?probar? (to smell or taste, or lesser use touch, also test). An early English use of ?proof? can be found in Shakespeare?s Macbeth, whereby proof is armour against false conclusions; Macbeth rode out upon the heath (a planar surface)? went line against line (military formations)? and point against point (the end of a spear or javelin), lapp?d (covered) in proof (armor).

Aesthetics and pedagogic concerns

Proofs are often viewed as aesthetic objects, admired for their beauty or utility. Elegant, deep, clever, useful, unexpected, counterintuitive, brief, concise, lengthy, complex, difficult, and unintelligible are just some of the many descriptions associated with proofs. Aesthetics and Pedagogic styles vary, e.g., one textbook might state with pride that the entire text has no pictures^[14], while a review of another textbook, with the same theorems but different proofs, might state, with equal pride, that the text is replete with proofs accompanied by diagrams and pictures. One might be able to follow the logic and understand each step in a proof, and additionally be able recite the entire proof from memory, yet still agree that they do not fully comprehend the proof. An example is an enormously long proof made by a computer. A pictorial supplement might enable easy understanding and recall of an otherwise lengthy proof.

Related concepts

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 are numbers. It is sometime 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 assumpions from which probability statements are derived require empirical evidence from outside mathematics to verify. In physics, in addtion to statistical methods, "statistical proof" can refer to the specialized mathematical methods of physics applied to analyze data in a particle physics experiment or observational study in cosmology. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as scatter plots, when the data or diagram is adequately convincing without further anaylisis.

Inductive logic proofs and Bayesian analysis

Proofs using inductive logic, while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to probability, and may be less than one certainty. Bayesian analysis establishes assertions as to the degree of a person's subjective belief. Inductive logic should not be confused with mathematical induction.

Proofs as mental objects

Psychologism views mathematical proofs as psyshological or mental objects. Mathematician philosophers such as Leibnitz, Frege, and Carnap, have attempted to develop a symantics for what they considered to be the language of thought, whereby whereby standards of mathematical proof might be applied to empirical science.

Influence of mathematical proof methods outside mathematics

Philosopher-mathematicians such as Schopenhauer have attempted to formulate philosophical arguments in an axiomatic manner, whereby mathematical proof standards could be applied to argumentation in general philosophy. Other mathematician-philosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at statements outside of mathematics, but having the certainty of propositions deduced in a mathematical proof, such as Descarte?s cogito argument. Kant and Frege considered mathamatical proof to be analytic apriori.

Popular colloquialisms referring to mathematical proofs

Steps in a proof

Moving from one proposition to another, using a rule of logical inference is called a "step". Sometimes a series of steps are "obvious" or "omitted", and are "skipped".

One line proof

A ?one line proof? is a proof stated with a single proposition containing no explicit "and" conjunctions. It may sometimes require mentally filling in some steps, or refer to a proof with more than one proposition, but which is extremely succinct. The expression is used outside mathematics to refer to a simple demonstration.

QED - end of a proof

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". Often, "which was to be shown" is verbally stated when writing "QED", "", or "" in an oral presentation on a board.

See also

References

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

External links

• What are mathematical proofs and why they are important?
• How To Write Proofs by Larry W. Cusick
• How to Write a Proof by Leslie Lamport, and the motivation of proposing such a hierarchical proof style.
• Proofs in Mathematics: Simple, Charming and Fallacious
• The Seventeen Provers of the World, ed. by Freek Wiedijk, foreword by Dana S. Scott, Lecture Notes in Computer Science 3600, Springer, 2006, ISBN 3-540-30704-4. Contains formalized versions of the proof that \sqrt{2} is irrational in several automated proof systems.
• What is Proof? Thoughts on proofs and proving.
• ProofWiki.org An online compendium of mathematical proofs.
• planetmath.org A wiki style encyclopedia of proofs

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