Mathematical proof

In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms,[2][3][4] along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning (or "reasonable expectation"). A proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed 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 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.

P. Oxy. I 29
P. Oxy. 29, one of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.[1]

History and etymology

The word "proof" comes from the Latin probare meaning "to test". Related modern words are the English "probe", "probation", and "probability", the Spanish probar (to smell or taste, or (lesser use) touch or test),[5] Italian provare (to try), and the German probieren (to try). The early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony.[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 with geometry, which originally meant the same as "land measurement".[8] The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales (624–546 BCE) and Hippocrates of Chios (c. 470–410 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. Mathematical proofs were revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today, starting with undefined terms and axioms (propositions regarding the undefined terms assumed to be self-evidently true from the Greek "axios" meaning "something worthy"), and used these to prove theorems using deductive logic. His book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century.[9] 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 that there are infinitely many prime numbers.

Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers (rather than geometric demonstrations) as he considered multiplication, division, etc. for "lines." He used this method to provide a proof of the existence of irrational numbers.[10] 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. Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate.[11]

Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption 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

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. The standard of rigor is not absolute and has varied throughout history. A proof can be presented differently depending on the intended audience. In order to gain acceptance, a proof has to meet communal statements of rigor; an argument considered vague or incomplete may be rejected.

The concept of a proof is formalized in the field of mathematical logic.[12] A formal proof is written in a formal language instead of a natural language. A formal proof is defined as sequence of formulas in a formal language, in which each formula is a logical consequence of preceding formulas. Having a definition of formal proof makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties, for example, the property that a statement has a formal proof. An application of proof theory is to show that certain undecidable statements are not provable.

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 automated proof assistants, this is rarely done in practice. 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.

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.

Methods

Direct proof

In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.[13] For example, direct proof can be used to establish that the sum of two even integers is always even:

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.

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

Proof by mathematical induction

Despite its name, mathematical induction is a method of deduction, not a form of inductive reasoning. 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. Since in principle the induction rule can be applied repeatedly starting from the proved base case, we see that all (usually infinitely many) cases are provable.[14] This avoids having to prove each case individually. 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.

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:[15] 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.

For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Let P(n) represent "2n − 1 is odd":

(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.
(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).
Thus 2n − 1 is odd, for all positive integers n.

The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction".[16]

Proof by contraposition

Proof by contraposition infers the conclusion "if p then q" from the premise "if not q then not p". The statement "if not q then not p" is called the contrapositive of the statement "if p then q". 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, if is even, the supposition must be false, so has to be even.

Proof by contradiction

In proof by contradiction (also known as reductio ad absurdum, Latin for "by reduction to the absurd"), it is shown that if some statement were true, a logical contradiction occurs, hence the statement must be false. A famous example of proof by contradiction shows that is an irrational number:

Suppose that were a rational number, so by definition where a and b are non-zero integers with no common factor. (If there is a common factor, divide both numerator and denominator by that factor to remove it, and repeat until no common factor remains. By the method of infinite descent, this process must terminate.) Thus, . Squaring both sides yields 2b2 = a2. Since 2 divides the left hand side, 2 must also divide the right hand side (otherwise an even number would equal an odd number). So a2 is even, which implies that a must also be even. This is because an odd integer squared is always odd and an even integer squared is always 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 have a common factor, namely 2. This contradicts our initial supposition, so we are forced to conclude that is an irrational number.

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

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 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. The shortest known proof of the four color theorem as of 2011 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. Probabilistic proof, like proof by construction, is one of many ways to show existence theorems.

This is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work on the Collatz conjecture shows how far plausibility is from genuine proof.

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. [17][18]

Combinatorial proof

A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.

Nonconstructive proof

A nonconstructive proof establishes that a mathematical 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. A famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that is a rational number:

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 of the form

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.[19][20][21] 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.

Computer-assisted proofs

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 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. 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.

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

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.[22] With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects outside of the proof-theorem framework,[23] 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,[24] which was ultimately so embedded.

Related concepts

Visual proof

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 the Pythagorean theorem in the case of the (3,4,5) triangle.

Chinese pythagoras

Visual proof for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BCE.

Pythagoras-2a

Animated visual proof for the Pythagorean theorem by rearrangement.

Pythag anim

A second animated proof of the Pythagorean theorem.

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.

Elementary proof

An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of 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.

Two-column proof

Twocolumnproof
A two-column proof published in 1913

A particular way of organising a proof using two parallel columns is often used in elementary geometry classes in the United States.[25] 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".[26]

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 is numerical. It is sometimes 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 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 physical 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 analysis.

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 full certainty. Inductive logic should not be confused with mathematical induction.

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

Proofs as mental objects

Psychologism views mathematical proofs as psychological or mental objects. 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.

Influence of mathematical proof methods outside mathematics

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. 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.

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

See also

References

  1. ^ Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. Retrieved 2008-09-26.
  2. ^ Clapham, C. & Nicholson, JN. 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.
  3. ^ Cupillari, Antonella. The Nuts and Bolts of Proofs. Academic Press, 2001. p. 3.
  4. ^ Gossett, Eric. Discrete Mathematics with Proof. John Wiley and Sons, 2009. Definition 3.1 p. 86. ISBN 0-470-45793-7
  5. ^ "proof" New Shorter Oxford English Dictionary, 1993, OUP, Oxford.
  6. ^ Hacking, Ian (1984). The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference. Cambridge University Press. ISBN 978-0-521-31803-7.
  7. ^ a b The History and Concept of Mathematical Proof, Steven G. Krantz. 1. February 5, 2007
  8. ^ Kneale, William and Martha, 1962. The development of logic. Oxford University Press, ISBN 0-19-824773-7. p. 2
  9. ^ Howard Eves, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0 p. 141: "No work, except The Bible, has been more widely used...."
  10. ^ Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences, 500 (1): 253–77 [260], Bibcode:1987NYASA.500..253M, doi:10.1111/j.1749-6632.1987.tb37206.x
  11. ^ Eder, Michelle (2000), Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam, Rutgers University, retrieved 2008-01-23
  12. ^ Buss, Samuel R. (1998), "An introduction to proof theory", in Buss, Samuel R., Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, 137, Elsevier, pp. 1–78, ISBN 978-0-08-053318-6. See in particular p. 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."
  13. ^ Cupillari, p. 20.
  14. ^ Cupillari, p. 46.
  15. ^ Examples of simple proofs by mathematical induction for all natural numbers
  16. ^ Proof by induction, University of Warwick Glossary of Mathematical Terminology
  17. ^ Davis, Philip J. (1972), "Fidelity in Mathematical Discourse: Is One and One Really Two?" American Mathematical Monthly 79:252–63.
  18. ^ Fallis, Don (1997), "The Epistemic Status of Probabilistic Proof." Journal of Philosophy 94:165–86.
  19. ^ "in number theory and commutative algebra... in particular the statistical proof of the lemma." [1]
  20. ^ "Whether constant π (i.e., pi) is normal is a confusing problem without any strict theoretical demonstration except for some statistical proof"" (Derogatory use.)[2]
  21. ^ "these observations suggest a statistical proof of Goldbach's conjecture with very quickly vanishing probability of failure for large E" [3]
  22. ^ "What to do with the pictures? Two thoughts surfaced: the first was that they were unpublishable in the standard way, there were no theorems only very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, but theorems were coins of the realm and the conventions of that day dictated that journals only published theorems", David Mumford, Caroline Series and David Wright, Indra's Pearls, 2002
  23. ^ "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."A Note on the History of Fractals Archived 2009-02-15 at the Wayback Machine,
  24. ^ "... 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'... ", Introducing Fractal Geometry, Nigel Lesmoir-Gordon
  25. ^ Patricio G. Herbst, Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century, Educational Studies in Mathematics, Vol. 49, No. 3 (2002), pp. 283–312,
  26. ^ Introduction to the Two-Column Proof, Carol Fisher

Further reading

External links

Allais paradox

The Allais paradox is a choice problem designed by Maurice Allais (1953) to show an inconsistency of actual observed choices with the predictions of expected utility theory.

Axiomatic system

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory typically means an axiomatic system, for example formulated within model theory. A formal proof is a complete rendition of a mathematical proof within a formal system.

Back-of-the-envelope calculation

A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as an envelope. It is more than a guess but less than an accurate calculation or mathematical proof. The defining characteristic of back-of-the-envelope calculations is the use of simplified assumptions. A similar phrase in the U.S. is "back of a napkin", also used in the business world to describe sketching out a quick, rough idea of a business or product. In British English, a similar idiom is "back of a fag packet".

Computer-assisted proof

A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.

Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program.

Attempts have also been made in the area of artificial intelligence research to create smaller, explicit, new proofs of mathematical theorems from the bottom up using machine reasoning techniques such as heuristic search. Such automated theorem provers have proved a number of new results and found new proofs for known theorems. Additionally, interactive proof assistants allow mathematicians to develop human-readable proofs which are nonetheless formally verified for correctness. Since these proofs are generally human-surveyable (albeit with difficulty, as with the proof of the Robbins conjecture) they do not share the controversial implications of computer-aided proofs-by-exhaustion.

Consistency

In classical deductive logic, a consistent theory is one that does not entail a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory is consistent if there is no formula such that both and its negation are elements of the set of consequences of . Let be a set of closed sentences (informally "axioms") and the set of closed sentences provable from under some (specified, possibly implicitly) formal deductive system. The set of axioms is consistent when for no formula .

If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called complete.[citation needed] The completeness of the sentential calculus was proved by Paul Bernays in 1918[citation needed] and Emil Post in 1921, while the completeness of predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete.

A consistency proof is a mathematical proof that a particular theory is consistent. The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program. Hilbert's program was strongly impacted by incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency (provided that they are in fact consistent).

Although consistency can be proved by means of model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The cut-elimination (or equivalently the normalization of the underlying calculus if there is one) implies the consistency of the calculus: since there is obviously no cut-free proof of falsity, there is no contradiction in general.

Correctness (computer science)

In theoretical computer science, correctness of an algorithm is asserted when it is said that the algorithm is correct with respect to a specification. Functional correctness refers to the input-output behaviour of the algorithm (i.e., for each input it produces the expected output).A distinction is made between partial correctness, which requires that if an answer is returned it will be correct, and total correctness, which additionally requires that the algorithm terminates. Since there is no general solution to the halting problem, a total correctness assertion may lie much deeper. A termination proof is a type of mathematical proof that plays a critical role in formal verification because total correctness of an algorithm depends on termination.For example, successively searching through integers 1, 2, 3, … to see if we can find an example of some phenomenon—say an odd perfect number—it is quite easy to write a partially correct program (using long division by two to check n as perfect or not). But to say this program is totally correct would be to assert something currently not known in number theory.

A proof would have to be a mathematical proof, assuming both the algorithm and specification are given formally. In particular it is not expected to be a correctness assertion for a given program implementing the algorithm on a given machine. That would involve such considerations as limitations on computer memory.

A deep result in proof theory, the Curry-Howard correspondence, states that a proof of functional correctness in constructive logic corresponds to a certain program in the lambda calculus. Converting a proof in this way is called program extraction.

Hoare logic is a specific formal system for reasoning rigorously about the correctness of computer programs. It uses axiomatic techniques to define programming language semantics and argue about the correctness of programs through assertions known as Hoare triples.

Software testing is any activity aimed at evaluating an attribute or capability of a program or system and determining that it meets its required results. Although crucial to software quality and widely deployed by programmers and testers, software testing still remains an art, due to limited understanding of the principles of software. The difficulty in software testing stems from the complexity of software: we can not completely test a program with moderate complexity. Testing is more than just debugging. The purpose of testing can be quality assurance, verification and validation, or reliability estimation. Testing can be used as a generic metric as well. Correctness testing and reliability testing are two major areas of testing. Software testing is a trade-off between budget, time and quality.

Demonstration

Demonstration may refer to:

Demonstration (acting), part of the Brechtian approach to acting

Demonstration (military), an attack or show of force on a front where a decision is not sought

Demonstration (protest), a political rally or protest

Demonstration (teaching), a method of teaching by example rather than simple explanation

Demonstration Hall, a building on the Michigan State University campus

Mathematical proof

Product demonstration, a sales or marketing presentation such as a:

Technology demonstration, an incomplete version of product to showcase idea, performance, method or features of the product

Scientific demonstration, a scientific experiment to illustrate principles

Wolfram Demonstrations Project, a repository of computer based educational demonstrations

Elementary proof

In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of 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 reproven using only elementary techniques.

While the meaning has not always been defined precisely, the term is commonly used in mathematical jargon. An elementary proof is not necessarily simple, in the sense of being easy to understand: some elementary proofs can be quite complicated.

Mathematical induction

Mathematical induction is a mathematical proof technique. It is essentially used to prove that a property P(n) holds for every natural number n, i.e. for n = 0, 1, 2, 3, and so on. Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder:

Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).

The method of induction requires two cases to be proved. The first case, called the base case (or, sometimes, the basis), proves that the property holds for the number 0. The second case, called the induction step, proves that, if the property holds for one natural number n, then it holds for the next natural number n + 1. These two steps establish the property P(n) for every natural number n = 0, 1, 2, 3, ... The base step need not begin with zero. Often it begins with the number one, and it can begin with any natural number, establishing the truth of the property for all natural numbers greater than or equal to the starting number.

The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs.Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning as used in philosophy (also see Problem of induction). Mathematical induction is an inference rule used in formal proofs. Proofs by mathematical induction are, in fact, examples of deductive reasoning.

Means-ends analysis

Means-ends analysis (MEA) is a problem solving technique used commonly in artificial intelligence (AI) for limiting search in AI programs.

It is also a technique used at least since the 1950s as a creativity tool, most frequently mentioned in engineering books on design methods. MEA is also related to means-ends chain approach used commonly in consumer behavior analysis. It is also a way to clarify one's thoughts when embarking on a mathematical proof.

NAND logic

Because the NAND function has functional completeness all logic systems can be converted into NAND gates – the mathematical proof for this was published by Henry M. Sheffer in 1913 in the Transactions of the American Mathematical Society (Sheffer 1913). This is also true for NOR gates.

In principle, any combinatorial logic function can be realized with enough NAND gates.

Peaucellier–Lipkin linkage

The Peaucellier–Lipkin linkage (or Peaucellier–Lipkin cell, or Peaucellier–Lipkin inversor), invented in 1864, was the first true planar straight line mechanism – the first planar linkage capable of transforming rotary motion into perfect straight-line motion, and vice versa. It is named after Charles-Nicolas Peaucellier (1832–1913), a French army officer, and Yom Tov Lipman Lipkin (1846–1876), a Lithuanian Jew and son of the famed Rabbi Israel Salanter.Until this invention, no planar method existed of producing exact straight-line motion without reference guideways, making the linkage especially important as a machine component and for manufacturing. In particular, a piston head needs to keep a good seal with the shaft in order to retain the driving (or driven) medium.

The mathematics of the Peaucellier–Lipkin linkage is directly related to the inversion of a circle.

Physical art

Physical art, as contrasted with conceptual art, refers to art that concretely exists in physical reality, in space and time. Its ontological status is that it is a physical object. The art is concretely realized but may be abstract in nature. For example, a painting, sculpture, or performance exists in the physical world. This is contrasted to conceptual art, some but not all kinds of performance art, computer software, or objects of mathematical beauty, such as a mathematical proof, which do not exist in the mental world or in physical world, but have other ontological status, such as in Plato's world of ideals. Here, the art, may be realized in the physical world, such as a mathematical proof written on a chalkboard, but refer to objects that exists in the mind as concepts, not physical objects. A music performance is physical, while the composition, like computer software, is not.

Proof

Proof may refer to:

Proof (truth), argument or sufficient evidence for the truth of a proposition

Formal proof, a construct in proof theory

Mathematical proof, a convincing demonstration that some mathematical statement is necessarily true

Proof theory, a branch of mathematical logic that represents proofs as formal mathematical objects

Alcohol proof, a measure of an alcoholic drink's strength

Artist's proof, a single print taken during the printmaking process

Galley proof, a preliminary version of a publication

Prepress proof, a facsimile of press artwork for job verification

Proof coinage, coins once made as a test, but now specially struck for collectors

Proofreading, reviewing a manuscript or artwork for errors or improvements

Proofing (baking technique), the process by which a yeast-leavened dough rises, also called "proving"

Proof by intimidation

Proof by intimidation (or argumentum verbosium) is a jocular phrase used mainly in mathematics to refer to a style of presenting a purported mathematical proof by giving an argument loaded with jargon and appeal to obscure results, so that the audience is simply obliged to accept it, lest they have to admit their ignorance and lack of understanding.The phrase is also used when the author is an authority in their field presenting their proof to people who respect a priori the author's insistence that the proof is valid or when the author claims that their statement is true because it is trivial or because they simply say so. Usage of this phrase is for the most part in good humour, though it also appears in serious criticism."Proof by intimidation" is also cited by critics of junk science to describe cases in which scientific evidence is thrown aside in favour of a litany of tragic individual cases presented to the public by articulate advocates who pose as experts in their field.Gian-Carlo Rota claimed in a memoir that the expression "proof by intimidation" was coined by Mark Kac to describe a technique used by William Feller in his lectures.

Reductio ad absurdum

In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity"), apagogical arguments or the appeal to extremes, is a form of argument that attempts either to disprove a statement by showing it inevitably leads to a ridiculous, absurd, or impractical conclusion, or to prove one by showing that if it were not true, the result would be absurd or impossible. Traced back to classical Greek philosophy in Aristotle's Prior Analytics (Greek: ἡ εἰς τὸ ἀδύνατον ἀπόδειξις, lit. 'demonstration to the impossible', 62b), this technique has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate.

The "absurd" conclusion of a reductio ad absurdum argument can take a range of forms, as these examples show:

The Earth cannot be flat; otherwise, we would find people falling off the edge.

There is no smallest positive rational number because, if there were, then it could be divided by two to get a smaller one.The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses. The second example is a mathematical proof by contradiction which argues that the denial of the premise would result in a logical contradiction (there is a "smallest" number and yet there is a number smaller than it).

Rigour

Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. Rigour frequently refers to a process of adhering absolutely to certain constraints, or the practice of maintaining strict consistency with certain predefined parameters. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mathematical proofs which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law.

Therefore sign

In logical argument and mathematical proof, the therefore sign (∴) is generally used before a logical consequence, such as the conclusion of a syllogism. The symbol consists of three dots placed in an upright triangle and is read therefore. It is encoded at U+2234 ∴ THEREFORE (HTML ∴ · ∴). While it is not generally used in formal writing, it is used in mathematics and shorthand. It is complementary to U+2235 ∵ BECAUSE (HTML ∵).

Zettel (Wittgenstein)

Zettel (German: "slip(s) of paper") is a collection of assorted remarks by Ludwig Wittgenstein, first published in 1967.

It contains several discussions of philosophical psychology and of the tendency in philosophy to try for a synoptic view of phenomena. Analyzed subjects include sense, meaning, thinking while speaking, behavior, pretense, imagination, infinity, rule following, imagery, memory, negation, contradiction, calculation, mathematical proof, epistemology, doubt, consciousness, mental states, and sensations.Editions include a parallel text English/German edition edited by Elizabeth Anscombe and Georg Henrik von Wright first published by Blackwell (UK) and University of California Press (USA) in 1967.

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