User:Kiyura/Mathematical Proof

In mathematics, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproved proposition that is believed to be true is known as a conjecture.

The statement that is proved is often called a theorem. 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, especially if it is intended for use as a stepping stone in the proof of another theorem.

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 rigorous informal logic. Purely formal proofs, written in symbolic language instead 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, 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.

History

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

Further information: History of logic

Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proofs. The early history of the concept of proof dates back to the early Greek and Chinese civilisations. 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 (fl. 300 BCE) began with undefined terms and axioms (statements assumed to be self-evident) and used these to prove theorems by way of deductive logic. Modern proof theory treats proofs as inductively defined data structures. It is no longer assumed that axioms are "true" in any sense; this allows for parallel mathematical theories built on alternate sets of axioms (see Axiomatic set theory and Non-Euclidean geometry for examples).

Nature and purpose

There are two different conceptions of mathematical proof. The first is an informal proof, a rigorous natural-language expression that is intended to convince the audience of the truth of a theorem. Because of their use of natural language, the standards of rigor for informal proofs will depend on the audience of the proof. In order to be considered a proof, however, the argument must be rigorous enough; a vague or incomplete argument is not a proof. Informal proofs are the type of proof usually seen in published mathematics. They are sometimes called "formal proofs" because of their rigor, but logicians use the term "formal proof" to refer to a different type of proof entirely.

In logic, a formal proof is not written in a natural language, but instead uses a formal language consisting of symbols from a fixed alphabet. This allows the content of a formal proof to be presented precisely and without any ambiguity. The field of proof theory studies formal proofs and their properties. Although each informal proof can, in theory, be converted into a formal proof, this is rarely done in practice. The study of formal proofs is used to determine properties of provability in general, and to show that certain undecidable statements are not provable.

Proofs may be viewed as aesthetic objects, admired for their mathematical beauty. The mathematician Paul Erdős was known for describing proofs he found 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.

A classic question in philosophy asks whether mathematical proofs are analytic or synthetic. Kant, who introduced the analytic-synthetic distinction, believed mathematical proofs are synthetic.

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:

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.

Direct proofs sometimes involve dividing a statement into cases, and then proving each case separately. The cases could be something simple like whether an integer is odd or even (such a proof is called a parity proof), but there could also be thousands of cases. When this technique is used, care must be taken to show that all possible cases are being considered. This kind of proof is usually called a proof by cases or proof by exhaustion.

Proof by mathematical induction

Main article: 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, i.e., P(n) is true for n = 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.

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. However, the term "proof by induction" may also be used in logic to mean an argument that uses inductive reasoning.

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

Proof by contrapositive

Main article: Proof by contrapositive

A proof by contrapositive proves the statement "If A then B" by proving the statement "If not B then not A". These two statements are logically equivalent by the principle of transposition, so proving one proves the other.

Proof by contradiction

Main article: Proof by contradiction

In proof by contradiction (also known as reductio ad absurdum, Latin for "by reduction toward the absurd"), it is shown that if some statement were so, a logical contradiction occurs, hence the statement must be not so. This method is perhaps the most prevalent of mathematical proofs. A famous example of a proof by contradiction shows that ${\sqrt {2}}$ is an Irrational number:

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

Statistical proofs in pure mathematics

Main article: Statistical proof

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. 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.[4].

Undecidable statements

A 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 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.

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

Heuristic mathematics and experimental mathematics

Main article: 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 an essential part of mathematics. With the increase in computing power in the 1960’s, significant work began to be done investigating mathematical objects outside of the proof-theorem framework, 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, which was ultimately so embedded.

Related concepts

Computer-assisted proofs

Main article: Computer-assisted proof

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

A two-column proof published in 1913

Two-column proof

A particular form of proof using two parallel columns is often used in elementary geometry classes. The proof is written as a series of lines in two columns. In each line, the left hand column contains propositions (or sometimes called statements), while the right hand column contains a brief explanation of how this proposition is either an axiom, a hypothesis, or can be obtained from previous lines (or sometimes just called "reasons").

Statistical proof using data

Main article: Statistical proof

"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 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

Main articles: Inductive logic 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

Main articles: Psychologism and Language of thought

Psychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, such as Leibnitz, Frege, and Carnap, have 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.

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 mathematical proof to be analytic apriori.

Ending 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". A more common 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

• Automated theorem proving
• Invalid proof
• Nonconstructive proof
• List of mathematical proofs
• Proof by intimidation

Sources

• Polya, G. Mathematics and Plausible Reasoning. Princeton University Press, 1954.
• 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
• Wolf, R. Proof, Logic, and Conjecture: The Mathematician's Toolbox. W. H. Freeman, 1997. ISBN 0-7167-3050-2
• Fallis, Don (2002) “What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians.” Logique et Analyse 45:373-88.

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
• A lesson about proofs, in a course from Wikiversity