Theorem
In mathematics and logic, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems.^{ [2]}^{ [3]}^{ [4]} A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system. As a result, the proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.^{ [5]}^{ [6]}
Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. However, the conditional could also be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic).
Although theorems can be written in a completely symbolic form (e.g., as propositions in propositional calculus), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed.
In addition to the better readability, informal 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, one might even be able to substantiate a theorem by using a picture as its proof.
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 judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood, a theorem that was once difficult may become trivial.^{ [7]} On the other hand, a deep theorem may be stated simply, 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.^{ [8]}
Informal account of theorems
Logically, many theorems are of the form of an indicative conditional: If A, then B. Such a theorem does not assert B — only that B is a necessary consequence of A. In this case, A is called the hypothesis of the theorem ("hypothesis" here means something very different from a conjecture), and B the conclusion of the theorem. The two together (without the proof) are called the proposition or statement of the theorem (e.g. "If A, then B" is the proposition). Alternatively, A and B can be also termed the antecedent and the consequent, respectively.^{ [9]} 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 "n is an even natural number", and the conclusion is "n/2 is also a natural number".
In order for a theorem be proved, it must be in principle expressible as a precise, formal statement. However, theorems are usually expressed in natural language rather than in a completely symbolic form—with the presumption that a formal statement can be derived from the informal one.
It is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of mathematics known as proof theory studies formal languages, axioms and the structure of proofs.
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.^{ [10]} Some, on the other hand, may be called "deep", because 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.^{ [11]} A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem,^{ [8]} and there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas.
Other theorems have a known proof that cannot easily be written down. The most prominent examples are the four color theorem and the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search that is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted. 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.^{ [12]} Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities^{ [13]} and hypergeometric identities.^{ [14]}^{[ page needed]}
Provability and theoremhood
To establish a mathematical statement as a theorem, a proof is required. That is, a valid line of reasoning from the axioms and other already-established theorems to the given statement must be demonstrated. In general, the proof is considered to be separate from the theorem statement itself. This is in part because while more than one proof may be known for a single theorem, only one proof is required to establish the status of a statement as a theorem. The Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs.^{ [15]}^{ [16]}
Relation with scientific theories
Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proved; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that are testable by experiments. 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.^{ [5]}
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. It is also possible to find a single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs.
For example, both the Collatz conjecture and the Riemann hypothesis are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The Collatz conjecture has been verified for start values up to about 2.88 × 10^{18}. The Riemann hypothesis has been verified to hold for the first 10 trillion non-trivial zeroes of the zeta function. Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved.
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.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 10^{14} have the Mertens property, and the smallest number that does not have this property is only known to be less than the exponential of 1.59 × 10^{40}, which is approximately 10 to the power 4.3 × 10^{39}. Since the number of particles in the universe is generally considered less than 10 to the power 100 (a googol), there is no hope to find an explicit counterexample by exhaustive search.
The word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory (see mathematical theory). 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
A number of different terms for mathematical statements exist; these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary, and the usage of some terms has evolved over time.
- An axiom or postulate is a fundamental assumption regarding the object of study, that is accepted without proof. A related concept is that of a definition, which gives the meaning of a word or a phrase in terms of known concepts. Classical geometry discerns between axioms, which are general statements; and postulates, which are statements about geometrical objects.^{ [17]} Historically, axioms were regarded as " self-evident"; today they are merely assumed to be true.
- A conjecture is an unproved statement that is believed to be true. Conjectures are usually made in public, and named after their maker (eg. Goldbach's conjecture and Collatz conjecture). The term hypothesis is also used in this sense (eg. Riemann hypothesis), which should not to be confused with "hypothesis" as the premise of a proof. Other terms are also used on occasion, for example theorem (eg. Fermat's Last Theorem, which is actually a conjecture).
- A theorem is a statement that has been proven to be true based on axioms or other theorems.
- A proposition is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should not be confused with "proposition" as used in propositional logic. In classical geometry the term "proposition" was used differently: in Euclid's Elements ( c. 300 BCE), all theorems and geometric constructions were called "propositions" regardless of their importance.
- A lemma is an "accessory proposition" - a proposition with little applicability outside its use in a particular proof. Over time a lemma may gain in importance and be considered a theorem, though the term "lemma" is usually kept as part of its name (eg. Gauss's lemma, Zorn's lemma, and the fundamental lemma).
- A corollary is a proposition that follows immediately from another theorem or axiom, with little or no required proof.^{ [18]} A corollary may also be a restatement of a theorem in a simpler form, or for a special case: for example, the theorem "all internal angles in a rectangle are right angles" has a corollary that "all internal angles in a square are right angles" - a square being a special case of a rectangle.
- A generalization of a theorem is a theorem with a similar statement but a broader scope, from which the original theorem can be deduced as a special case (a corollary).^{ [a]}
Other terms may also be used for historical or customary reasons, for example:
- An identity is a theorem stating an equality between two expressions, that holds for any value within its domain (eg. Bézout's identity and Vandermonde's identity).^{ [19]}
- A rule is a theorem that establishes a useful formula (eg. Bayes' rule and Cramer's rule).
- A law or principle is a theorem with wide applicability (eg. the law of large numbers, law of cosines, Kolmogorov's zero–one law, Harnack's principle, the least-upper-bound principle, and the pigeonhole principle).^{ [b]}
A few well-known theorems have even more idiosyncratic names, for example the division algorithm, Euler's formula, and the Banach–Tarski paradox.
Layout
A theorem and its proof are typically laid out as follows:
- Theorem (name of the person who proved it, along with year of discovery or publication of the proof).
- Statement of theorem (sometimes called the proposition).
- Proof.
- Description of proof.
- End
The end of the proof may be signaled by the letters Q.E.D. (quod erat demonstrandum) or by one of the tombstone marks, such as "□" or "∎", meaning "End of Proof", introduced by Paul Halmos following their use in magazines to mark the end of an article.^{ [20]}
The exact style depends on the author or publication. Many publications provide instructions or macros for typesetting in the house style.
It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. 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. Sometimes, corollaries have proofs of their own that explain why they follow from the theorem.
Lore
It has been estimated that over a quarter of a million theorems are proved every year.^{ [21]}
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.^{ [22]}
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. These papers are together believed to give a complete proof, and several ongoing projects hope to shorten and simplify this proof.^{ [23]} Another theorem of this type is the four color theorem whose computer generated proof is too long for a human to read. It is among the longest known proofs of a theorem whose statement can be easily understood by a layman.^{[ citation needed]}
Theorems in logic
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Logic, especially in the field of proof theory, considers theorems as statements (called formulas or well formed formulas) of a formal language. The statements of the language are strings of symbols and may be broadly divided into nonsense and well-formed formulas. A set of deduction rules, also called transformation rules or rules of inference, must be provided. These deduction rules tell exactly when a formula can be derived from a set of premises. The set of well-formed formulas may be broadly divided into theorems and non-theorems. However, according to Hofstadter, a formal system often simply defines all its well-formed formula as theorems.^{ [24]}^{[ page needed]}
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. Other deductive systems describe term rewriting, such as the reduction rules for λ calculus.
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 theorems; 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.
A theorem may be expressed in a formal language (or "formalized"). A formal theorem is the purely formal analogue of a theorem. In general, a formal theorem is a type of well-formed formula that satisfies certain logical and syntactic conditions. The notation is often used to indicate that is a theorem.
Formal theorems consist of formulas of a formal language and the transformation rules of a formal system. Specifically, a formal theorem is always the last formula of a derivation in some formal system, each formula of which is a logical consequence of the formulas that came before it in the derivation. The initially-accepted formulas in the derivation are called its axioms, and are the basis on which the theorem is derived. A set of theorems is called a theory.
What makes formal theorems useful and interesting is that they can be interpreted as true propositions and their derivations may be interpreted as a proof of the truth of the resulting expression. A set of formal theorems may be referred to as a formal theory. A theorem whose interpretation is a true statement about a formal system (as opposed to of a formal system) is called a metatheorem.
Syntax and semantics
The concept of a formal theorem is fundamentally syntactic, in contrast to the notion of a true proposition, which introduces semantics. Different deductive systems can yield other interpretations, depending on the presumptions of the derivation rules (i.e. belief, justification or other modalities). The soundness of a formal system depends on whether or not all of its theorems are also validities. A validity is a formula that is true under any possible interpretation (for example, in classical propositional logic, validities are tautologies). A formal system is considered semantically complete when all of its theorems are also tautologies.
Derivation of a theorem
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The notion of a theorem is very closely connected to its formal proof (also called a "derivation"). As an illustration, consider a very simplified formal system whose alphabet consists of only two symbols { A, B }, and whose formation rule for formulas is:
- Any string of symbols of that is at least three symbols long, and is not infinitely long, is a formula. Nothing else is a formula.
The single axiom of is:
- ABBA.
The only rule of inference (transformation rule) for is:
- Any occurrence of "A" in a theorem may be replaced by an occurrence of the string "AB" and the result is a theorem.
Theorems in are defined as those formulas that have a derivation ending with it. For example,
- ABBA (Given as axiom)
- ABBBA (by applying the transformation rule)
- ABBBAB (by applying the transformation rule)
is a derivation. Therefore, "ABBBAB" is a theorem of The notion of truth (or falsity) cannot be applied to the formula "ABBBAB" until an interpretation is given to its symbols. Thus in this example, the formula does not yet represent a proposition, but is merely an empty abstraction.
Two metatheorems of are:
- Every theorem begins with "A".
- Every theorem has exactly two "A"s.
Interpretation of a formal theorem
Theorems and theories
See also
Notes
- ^ Often, when the less general or "corollary"-like theorem is proven first, it is because the proof of the more general form requires the simpler, corollary-like form, for use as a what is functionally a lemma, or "helper" theorem.
- ^ The word law can also refer to an axiom, a rule of inference, or, in probability theory, a probability distribution.
References
- ^ Elisha Scott Loomis. "The Pythagorean proposition: its demonstrations analyzed and classified, and bibliography of sources for data of the four kinds of proofs" (PDF). Education Resources Information Center. Institute of Education Sciences (IES) of the U.S. Department of Education. Retrieved 2010-09-26. Originally published in 1940 and reprinted in 1968 by National Council of Teachers of Mathematics.
- ^ "Definition of THEOREM". www.merriam-webster.com. Retrieved 2019-11-02.
- ^ "The Definitive Glossary of Higher Mathematical Jargon – Theorem". Math Vault. 2019-08-01. Retrieved 2019-11-02.
- ^ "Theorem | Definition of Theorem by Lexico". Lexico Dictionaries | English. Retrieved 2019-11-02.
- ^ ^{a} ^{b} Markie, Peter (2017), "Rationalism vs. Empiricism", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Fall 2017 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-11-02
- ^ However, both theorems and scientific law are the result of investigations. See Heath 1897 Introduction, The terminology of Archimedes, p. clxxxii:"theorem (θεὼρνμα) from θεωρεἳν to investigate"
- ^ Weisstein, Eric W. "Theorem". mathworld.wolfram.com. Retrieved 2019-11-02.
- ^ ^{a} ^{b} Darmon, Henri; Diamond, Fred; Taylor, Richard (2007-09-09). "Fermat's Last Theorem" (PDF). McGill University – Department of Mathematics and Statistics. Retrieved 2019-11-01.
- ^ "Implication". intrologic.stanford.edu. Retrieved 2019-11-02.
- ^ "The Definitive Glossary of Higher Mathematical Jargon – Trivial". Math Vault. 2019-08-01. Retrieved 2019-11-02.
- ^ Weisstein, Eric W. "Deep Theorem". MathWorld.
- ^ Doron Zeilberger. "Opinion 51".
- ^ Such as the derivation of the formula for from the addition formulas of sine and cosine.
- ^ Petkovsek et al. 1996.
- ^ "Pythagorean Theorem and its many proofs". www.cut-the-knot.org. Retrieved 2019-11-02.
- ^ See, for example, proofs of quadratic reciprocity for more.
- ^ Wentworth, G.; Smith, D.E. (1913). Plane Geometry. Ginn & Co. Articles 46, 47.
- ^ Wentworth & Smith, article 51
- ^ "The Definitive Glossary of Higher Mathematical Jargon – Identity". Math Vault. 2019-08-01. Retrieved 2019-11-02.
- ^ "Earliest Uses of Symbols of Set Theory and Logic". jeff560.tripod.com. Retrieved 2 November 2019.
- ^ Hoffman 1998, p. 204.
- ^ Hoffman 1998, p. 7.
- ^ An enormous theorem: the classification of finite simple groups, Richard Elwes, Plus Magazine, Issue 41 December 2006.
- ^ Hofstadter 1980
References
- Heath, Sir Thomas Little (1897). The works of Archimedes. Dover. Retrieved 2009-11-15.
- Hoffman, P. (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hyperion, New York. ISBN 1-85702-829-5.
- Hofstadter, Douglas (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books.
- Hunter, Geoffrey (1996) [1973]. Metalogic: An Introduction to the Metatheory of Standard First Order Logic. University of California Press. ISBN 0-520-02356-0.
- Mates, Benson (1972). Elementary Logic. Oxford University Press. ISBN 0-19-501491-X.
- Petkovsek, Marko; Wilf, Herbert; Zeilberger, Doron (1996). A = B. A.K. Peters, Wellesley, Massachusetts. ISBN 1-56881-063-6.
External links
Look up theorem in Wiktionary, the free dictionary. |
- Media related to Theorems at Wikimedia Commons
- Weisstein, Eric W. "Theorem". MathWorld.
- Theorem of the Day