A mathematical picture paints a thousand words: the Pythagorean theorem. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry

In mathematics, a theorem is a statement proven on the basis of previously accepted or established statements. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true In mathematical logic, theorems are modeled as formulas that can be derived according to the derivation rules of a fixed formal system. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Mathematical logic, a formula is a type of Abstract object a token of which is a Symbol or string of symbols which may be see also Mathematical proof, Proof theory, and Axiomatic system. A deductive system (also called a deductive apparatus of a Formal system) consists of the Axioms (or Axiom schemata and Rules of inference In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist

In formal settings, an essential property of theorems is that they are derivable using a fixed set of inference rules and axioms without any additional assumptions. In Logic, a rule of inference (also called a transformation rule) is a function from sets of formulae to formulae In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject This is not a matter of the semantics of the language: the expression that results from a derivation is a syntactic consequence of all the expressions that precede it. Semantics is the study of meaning in communication The word derives from Greek σημαντικός ( semantikos) "significant" from "Therefore" redirects here For the symbol see Therefore sign. In mathematics, the derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. A deductive system (also called a deductive apparatus of a Formal system) consists of the Axioms (or Axiom schemata and Rules of inference

The proofs of theorems have two components, called the hypotheses and the conclusions. In Discourse and Logic, a premise is a claim that is a reason (or element of a set of reasons for or objection against some other claim A conclusion is a Proposition, which is arrived at after the consideration of Evidence, Arguments or Premises Logic The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical. Deductive reasoning is Reasoning which uses deductive Arguments to move from given statements ( Premises to Conclusions which must be true if the The word theory has many distinct meanings in different fields of Knowledge, depending on their methodologies and the context of discussion. A central concept in Science and the Scientific method is that all Evidence must be empirical, or empirically based that is dependent on evidence

Although they can be written in a completely symbolic form using, for example, propositional calculus, theorems are often expressed in a natural language such as English. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" The same is true of proofs, which are often expressed as logically organised and clearly worded informal arguments intended to demonstrate that a formal symbolic proof can be constructed. Such arguments are typically easier to check than purely symbolic ones — indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem.

Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgements vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's last theorem is a particularly well-known example of such a theorem. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like

Formal and informal notions

Logically most theorems are of the form of an indicative conditional: if A, then B. Logic is the study of the principles of valid demonstration and Inference. In Natural languages an indicative conditional is the logical operation given by statements of the form "If A then B" Such a theorem does not state that B is always true, only that B must be true if A is true. In this case A is called the hypothesis of the theorem (note that "hypothesis" here is something very different from a conjecture) and B the conclusion. A hypothesis (from Greek) consists either of a suggested explanation for a phenomenon (an event that is observable or of a reasoned proposal suggesting a possible In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of The theorem "If n is an even natural number then n/2 is a natural number" is a typical example in which the hypothesis is that n is an even natural number and the conclusion is that n/2 is also a natural number. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

In order to be proven, a theorem must be expressible as a precise, formal statement. Nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader will be able to produce a formal statement from the informal one. In addition, there are often hypotheses which are understood in context, rather than explicitly stated.

It is common in mathematics to choose a number of hypotheses that are assumed to be true within a given theory, and then declare that the theory consists of all theorems provable using those hypotheses as assumptions. In this case the hypotheses that form the foundational basis are called the axioms (or postulates) of the theory. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject The field of mathematics known as proof theory studies formal axiom systems and the proofs that can be performed within them. Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques

A planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand. The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country

Some theorems are "trivial," in the sense that they follow from definitions, axioms, and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep": their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. ^[1] A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem, and there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects

There are other theorems for which a proof is known, but the proof cannot easily be written down. The most prominent examples are the Four color theorem and the Kepler conjecture. The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country In Mathematics, the Kepler conjecture is a Conjecture about Sphere packing in three-dimensional Euclidean space. Both of these theorems are only known to be true by reducing them to a computational search which is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted in recent years. The mathematician Doron Zeilberger has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved. Doron Zeilberger (דורון ציילברגר born July 2, 1950 in Israel) is an Israeli mathematician known for his work in Combinatorics [1] Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities. ^[2]

Relation to proof

The notion of a theorem is deeply intertwined with the concept of proof. Indeed, theorems are true precisely in the sense that they possess proofs. Therefore, to establish a mathematical statement as a theorem, the existence of a line of reasoning from axioms in the system (and other, already established theorems) to the given statement must be demonstrated.

Although the proof is necessary to produce a theorem, it is not usually considered part of the theorem. And even though more than one proof may be known for a single theorem, only one proof is required to establish the theorem's validity. The Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry The law of quadratic reciprocity is a theorem from Modular arithmetic, a branch of Number theory, which shows a remarkable relationship between the solvability

Theorems in logic

Logic, especially in the field of proof theory, considers theorems as statements (called formulas or well formed formulas) of a formal language. Logic is the study of the principles of valid demonstration and Inference. Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information In Mathematical logic, a well-formed formula (often abbreviated WFF, pronounced "wiff" or "wuff" is a Symbol or string of symbols (a A formal language is a set of words, ie finite strings of letters, or symbols. A set of deduction rules, also called transformation rules or a formal grammar, must be provided. In Formal semantics, Computer science and Linguistics, a formal grammar (also called formation rules) is a precise description of a Formal These deduction rules tell exactly when a formula can be derived from a set of premises.

Different sets of derivation rules give rise to different interpretations of what it means for an expression to be a theorem. Some derivation rules and formal languages are intended to capture mathematical reasoning; the most common examples use first-order logic. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science Other deductive systems describe term rewriting, such as the reduction rules for λ calculus. In Mathematics, Computer science and Logic, rewriting covers a wide range of potentially non-deterministic methods of replacing subterms of a In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function

The definition of theorems as elements of a formal language allows for results in proof theory that study the structure of formal proofs and the structure of provable formulas. The most famous result is Gödel's incompleteness theorem; by representing theorems about basic number theory as expressions in a formal language, and then representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory. In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most

Relation with scientific theories

Theorems in mathematics and theories in science are fundamentally different in their epistemology. The word theory has many distinct meanings in different fields of Knowledge, depending on their methodologies and the context of discussion. Epistemology (from Greek επιστήμη - episteme, "knowledge" + λόγος, " Logos " or theory of knowledge A scientific theory cannot be proven; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that are testable by experiments. Falsifiability (or "refutability" is the logical possibility that an assertion can be shown false by an observation or a physical experiment In scientific inquiry an experiment ( Latin: Ex- periri, "to try out" is a method of investigating particular types of research questions or Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.

The Collatz conjecture: one way to illustrate its complexity is to extend the iteration from the natural numbers to the complex numbers. The Collatz conjecture is an unsolved Conjecture in Mathematics. The result is a fractal, which (in accordance with universality) resembles the Mandelbrot set. In Statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the In Mathematics, the Mandelbrot set, named after Benoît

Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. For example, the Collatz conjecture has been verified for start values up to about 2. The Collatz conjecture is an unsolved Conjecture in Mathematics. 88 × 10¹⁸. The Riemann hypothesis has been verified for the first 10 trillion zeroes of the zeta function. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in Neither of these statements is considered to be proven.

Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i. In Mathematics, the Mertens conjecture is a statement about the behaviour of a certain function as its argument increases e. , a natural number n for which the Mertens function M(n) equals or exceeds the square root of n) is known: all numbers less than 10¹⁴ have the Mertens property, and the smallest number which does not have this property is only known to be less than the exponential of 1. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) 59 × 10⁴⁰, which is approximately 10 to the power 4. 3 × 10³⁹. Since the number of particles in the universe is generally considered to be less than 10 to the power 100 (a googol), there is no hope to find an explicit counterexample by exhaustive search at present. A googol is the Large number 10100 that is the digit 1 followed by one hundred zeros (in Decimal representation In Computer science, brute-force search or exhaustive search, also known as generate and test, is a trivial but very general problem-solving technique

Note that the word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role; the physical axioms on which such "theorems" are based are themselves falsifiable.

Terminology

Theorems are often indicated by several other terms: the actual label "theorem" is reserved for the most important results, whereas results which are less important, or distinguished in other ways, are named by different terminology.

• A proposition is a statement not associated with any particular theorem. In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence This term sometimes connotes a statement with a simple proof.
• A lemma is a "pre-theorem", a statement that forms part of the proof of a larger theorem. In Mathematics, a lemma (plural lemmata or lemmas from the Greek λήμμα "lemma" meaning "anything which is received The distinction between theorems and lemmas is rather arbitrary, since one mathematician's major result is another's minor claim. A mathematician is a person whose primary area of study and research is the field of Mathematics. Gauss's lemma and Zorn's lemma, for example, are interesting enough that some authors present the nominal lemma without going on to use it in the proof of a theorem. This article is about Gauss's lemma for polynomials See also Gauss's lemma. Zorn's lemma, also known as the Kuratowski-Zorn lemma, is a proposition of Set theory that states Every Partially ordered set in which
• A corollary is a proposition that follows with little or no proof from one other theorem or definition. A corollary is a statement which follows readily from a previously proven statement That is, proposition B is a corollary of a proposition A if B can readily be deduced from A.
• A claim is a necessary or independently interesting result that may be part of the proof of another statement. Despite the name, claims must be proved.

There are other terms, less commonly used, which are conventionally attached to proven statements, so that certain theorems are referred to by historical or customary names. For examples:

• Identity, used for theorems which state an equality between two mathematical expressions. In Mathematics, the term identity has several different important meanings An identity is an equality that remains true regardless of the values of Examples include Euler's identity and Vandermonde's identity. In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where In combinatorial Mathematics, Vandermonde's identity, named after Alexandre-Théophile Vandermonde, states that the equality {m+n
• Rule, used for certain theorems such as Bayes' rule and Cramer's rule, that establish useful formulas. In Probability theory, Bayes' theorem (often called Bayes' law after Thomas Bayes) relates the conditional and marginal probabilities of two Cramer's rule is a Theorem in Linear algebra, which gives the solution of a System of linear equations in terms of Determinants It is named after
• Law. Examples include the law of large numbers, the law of cosines, and Kolmogorov's zero-one law. The law of large numbers (LLN is a theorem in Probability that describes the long-term stability of the mean of a Random variable. In Trigonometry, the law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about a general In Probability theory, Kolmogorov's zero-one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a ^[3]
• Principle. For the term in chemistry see Principle (chemistry. Not to be confused with Principal. Examples include Harnack's principle, the least upper bound principle, and the pigeonhole principle. In Complex analysis, Harnack's principle or Harnack's theorem is one of several closely related theorems about the convergence of sequences of Harmonic functions The pigeonhole principle, also known as Dirichlet's box (or drawer) principle, states that given two Natural numbers n and
• A Converse is a reverse theorem. Conversion is a concept in Traditional logic referring to a "type of immediate Inference in which from a given Proposition another proposition is inferred For example, If a theorem states that A is a related to B, its converse would state that B is related to A. The converse of a theorem need not be always true.

A few well-known theorems have even more idiosyncratic names. The division algorithm is a theorem expressing the outcome of division in the natural numbers and more general rings. The division algorithm is a Theorem in Mathematics which precisely expresses the outcome of the usual process of division of Integers The name The Banach–Tarski paradox is a theorem in measure theory that is paradoxical in the sense that it contradicts common intuitions about volume in three-dimensional space. The Banach–Tarski paradox is a Theorem in set theoretic Geometry which states that a solid ball in 3-dimensional space can be split into several In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with A paradox is a true statement or group of statements that leads to a Contradiction or a situation which defies intuition; or inversely

An unproven statement that is believed to be true is called a conjecture (or sometimes a hypothesis, but with a different meaning from the one discussed above). In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of A hypothesis (from Greek) consists either of a suggested explanation for a phenomenon (an event that is observable or of a reasoned proposal suggesting a possible To be considered a conjecture, a statement must usually be proposed publicly, at which point the name of the proponent may be attached to the conjecture, as with Goldbach's conjecture. Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of Mathematics. Other famous conjectures include the Collatz conjecture and the Riemann hypothesis. The Collatz conjecture is an unsolved Conjecture in Mathematics. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved

Layout

A theorem and its proof are typically laid out as follows:

Theorem (name of person who proved it and year of discovery, proof or publication).

Statement of theorem.

Proof.

Description of proof.

The end of the proof may be signalled by the letters Q.E.D. meaning "quod erat demonstrandum" or by one of the tombstone marks "□" or "∎" meaning "End of Proof", introduced by Paul Halmos following their usage in magazine articles. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated" QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated" The tombstone, halmos, or end of proof mark "□" is used in Mathematics to denote the end of a proof in place of the traditional abbreviation Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician

The exact style will depend on the author or publication. Many publications provide instructions or macros for typesetting in the house style. A macro (from the Greek 'μάκρο' for long or far in Computer science is a rule or Pattern that specifies how a certain input sequence (often a sequence A style guide or style manual is a set of standards for design and writing of documents either for general use or for a specific publication or organization

It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. A definition is a statement of the meaning of a Word or Phrase. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence In Mathematics, a lemma (plural lemmata or lemmas from the Greek λήμμα "lemma" meaning "anything which is received However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem.

Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. A corollary is a statement which follows readily from a previously proven statement Sometimes corollaries have proofs of their own which explain why they follow from the theorem.

Lore

It has been estimated that over a quarter of a million theorems are proved every year. ^[4]

The well-known aphorism, "A mathematician is a device for turning coffee into theorems", is probably due to Alfréd Rényi, although it is often attributed to Rényi's colleague Paul Erdős (and Rényi may have been thinking of Erdős), who was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking. Alfréd Rényi (20 March 1921 &ndash 1 February 1970 was a Hungarian Mathematician who made contributions in Combinatorics and Graph theory Paul Erdős ( Hungarian: Erdős Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 &ndash The Erdős number (ɛrdøːʃ honoring the late Hungarian mathematician Paul Erdős, is a way of describing the "collaborative distance" between a person ^[5]

The classification of finite simple groups is regarded by some to be the longest proof of a theorem; it comprises tens of thousands of pages in 500 journal articles by some 100 authors. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups. These papers are together believed to give a complete proof, and there are several ongoing projects to shorten and simplify this proof. ^[6]

See also

• Metatheorem
• List of theorems
• Gödel's incompleteness theorem, that establishes very general conditions under which a formal system will contain a true statement for which there exists no proof within the system. In Mathematical logic, a metatheorem is a Statement about Theorems or about some axiomatic Theory. This is a list of Theorems, by Wikipedia page See also List of fundamental theorems List of lemmas List of conjectures In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most
• Inference
• Toy theorem

Notes

1. ^ See Deep Theorem, cited below. Inference is the act or process of deriving a Conclusion based solely on what one already knows In Mathematics, a toy theorem is a simplified version of a more general Theorem.
2. ^ Petkovsek et al. 1996.
3. ^ The word law can also refer to an axiom, a rule of inference, or, in probability theory, a probability distribution. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Logic, a rule of inference (also called a transformation rule) is a function from sets of formulae to formulae Probability theory is the branch of Mathematics concerned with analysis of random phenomena In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable
4. ^ Hoffman 1998, p. 204.
5. ^ Hoffman 1998, p. 7.
6. ^ An enormous theorem: the classification of finite simple groups, Richard Elwes, Plus Magazine, Issue 41 December 2006.

References

• Hoffman, P. (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hyperion, New York.  
• Petkovsek, Marko; Wilf, Herbert; Zeilberger, Doron (1996). "A = B". A. K. Peters, Wellesley, Massachusetts.  

External links

• Eric W. Weisstein, Deep Theorem at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• Eric W. Weisstein, Theorem at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W