A theorem is an idea, concept or abstraction token instances of which are formed using a string of symbols according to both the syntactic rules of a language (also called its grammar) and the transformation rules of a formal system. Specifically, a theorem is always the last formula of a derivation in some formal system each formula of which is a logical consequence of the formulas which 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 theorems useful and of interest 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. Very often, in mathematical logic a theorem is expressed in a formal language, in which case it may be called a formal theorem. The syntactic rules of the formal language may be called its formal grammar or formation rules. 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 is called a metatheorem.

The concept of a theorem is fundamentally syntactic, in contrast to the notion of a "true proposition" in which semantics are introduced. Different deductive systems may be constructed so as to 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 tautologies. A tautology is a formula that is true under any possible interpretation. A formal system is considered semantically complete when all of its tautologies are also theorems.

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. The same is true of proofs, which are often expressed as logically organized 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 judgments 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.

Formal and informal notions

Logically most theorems are of the form of an indicative conditional: 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. 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 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. The field of mathematics known as proof theory studies formal axiom systems and the proofs that can be performed within them.

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.^[2] 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.

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. 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.[1] Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities.^[3]

Syntax and semantics

Derivation of a theorem

The notion of a theorem is deeply intertwined with its derivation. Given a particular very simple formal system (we shall call ours ${\displaystyle {\mathcal {FS))}$) whose alphabet α consists only of two symbols { ${\displaystyle \triangle }$, ${\displaystyle \square }$ } and whose formation rule for formulas is:

'Any string of symbols of ${\displaystyle {\mathcal {FS))}$ which is at least 3 symbols long, and which is not infinitely long, is a formula of ${\displaystyle {\mathcal {FS))}$. Nothing else is a formula of ${\displaystyle {\mathcal {FS))}$.'

The single axiom of ${\displaystyle {\mathcal {FS))}$ is:

'${\displaystyle \triangle }$ ${\displaystyle \square }$ ${\displaystyle \square }$ ${\displaystyle \triangle }$'

The transformation rule for ${\displaystyle {\mathcal {FS))}$ is:

'Any occurrence of '${\displaystyle \triangle }$' in a formula of ${\displaystyle {\mathcal {FS))}$ may be replaced by an occurance of the string '${\displaystyle \triangle }$ ${\displaystyle \square }$' and the result is a formula (wff) of ${\displaystyle {\mathcal {FS))}$.'

Theorems in ${\displaystyle {\mathcal {FS))}$ are defined as those formulae in ${\displaystyle {\mathcal {FS))}$ of which a derivation can be constructed, the last line of which is that formula.

1) ${\displaystyle \triangle }$ ${\displaystyle \square }$ ${\displaystyle \square }$ ${\displaystyle \triangle }$ (Given as axiom)

2) ${\displaystyle \triangle }$ ${\displaystyle \square }$ ${\displaystyle \square }$ ${\displaystyle \square }$ ${\displaystyle \triangle }$ (by applying the transformation rule)

3) ${\displaystyle \triangle }$ ${\displaystyle \square }$ ${\displaystyle \square }$ ${\displaystyle \square }$ ${\displaystyle \triangle }$ ${\displaystyle \square }$ (by applying the transformation rule)

Therefore '${\displaystyle \triangle }$ ${\displaystyle \square }$ ${\displaystyle \square }$ ${\displaystyle \square }$ ${\displaystyle \triangle }$ ${\displaystyle \square }$' is a theorem of ${\displaystyle {\mathcal {FS))}$. The string '${\displaystyle \triangle }$ ${\displaystyle \square }$ ${\displaystyle \square }$ ${\displaystyle \square }$ ${\displaystyle \triangle }$ ${\displaystyle \square }$' is not considered to be "true" or "proved," until it is given an interpretation it is merely derived.

Two metatheorems of ${\displaystyle {\mathcal {FS))}$ are:

'Every theorem of ${\displaystyle {\mathcal {FS))}$ begins with '${\displaystyle \triangle }$'

'Every theorem of ${\displaystyle {\mathcal {FS))}$ has exactly two triangles.'

Interpretation of a theorem

Given a different simple formal system (we shall call this one ${\displaystyle {\mathcal {FS'))}$) whose alphabet α consists only of three symbols { ${\displaystyle \blacksquare }$, ${\displaystyle \bigstar }$, ${\displaystyle \blacklozenge }$ } and whose formation rule for formulas is:

'Any string of symbols of ${\displaystyle {\mathcal {FS'))}$ which is at least 6 symbols long, and which is not infinitely long, is a formula of ${\displaystyle {\mathcal {FS'))}$. Nothing else is a formula of ${\displaystyle {\mathcal {FS'))}$.'

The single axiom schema of ${\displaystyle {\mathcal {FS'))}$ is:

" ${\displaystyle \blacksquare }$ ${\displaystyle \bigstar }$ * ${\displaystyle \blacklozenge }$ ${\displaystyle \blacksquare }$ * " (where " * " is a metasyntactic variable standing for a finite string of " ${\displaystyle \blacksquare }$ "s )

A formal proof can be constructed as follows:

(1) ${\displaystyle \blacksquare }$ ${\displaystyle \bigstar }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacklozenge }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$

(2) ${\displaystyle \blacksquare }$ ${\displaystyle \bigstar }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacklozenge }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$

(3) ${\displaystyle \blacksquare }$ ${\displaystyle \bigstar }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacklozenge }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$

In this example the theorem produced " ${\displaystyle \blacksquare }$ ${\displaystyle \bigstar }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacklozenge }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$ ${\displaystyle \blacksquare }$ " can be interpreted as menaing "One plus four equals five." A different interpretation would be to read it backwards as "Five minus four equals one."== Proof of a theorem ==

A derivation of a theorem can be interpreted as a proof of the truth of some proposition. To establish a proposition 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.

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.

Relation to theories

A theory is a collection of theorems. The main broad distinction to be made among different types of theories is whether or not their theorems are intended to be interpreted as representing a true proposition which is empirical or non-empirical. If every theorem of a theory is empirical, then it is a scientific theory (also called an empirical theory). If there is even one non-empirical theorem in the theory, it is a non-empirical theory.

Theorems in mathematics and logic are non-empirical because they have no content until given an interpretation. This is precisely why theorems of mathematics and logic are useful. They can be applied to any subject matter no matter what its content. If the content of a theory consists entirely of theorems whose interpretations are statements about observation and sense data, than it is a scientific theory.

It is not correct to say that a scientific theory is either true or false as its theorems may be. A theory is taken as a whole as strongly or weakly supported by the facts. A key attribute of a scientific theory 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.

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 2.88 × 10¹⁸. The Riemann hypothesis has been verified for the first 10 trillion zeroes of the zeta function. 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.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 1.59 × 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.

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. 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. This term sometimes connotes a statement with a simple proof, or a basic consequence of a definition that needs to be stated, but is obvious enough to require no proof. The word proposition is sometimes used for the statement part of a theorem.
• A lemma is a "pre-theorem", a statement that forms part of the proof of a larger theorem. The distinction between theorems and lemmas is rather arbitrary, since one mathematician's major result is another's minor claim. 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.
• A corollary is a proposition that follows with little or no proof from one other theorem or definition. 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. Examples include Euler's identity and Vandermonde's identity.
• Rule, used for certain theorems such as Bayes' rule and Cramer's rule, that establish useful formulas.
• Law. Examples include the law of large numbers, the law of cosines, and Kolmogorov's zero-one law.^[4]
• Principle. Examples include Harnack's principle, the least upper bound principle, and the pigeonhole principle.
• A Converse is a reverse theorem. For example, If a theorem states that A is 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 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.

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). 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. Other famous conjectures include the Collatz conjecture and the Riemann hypothesis.

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 (sometimes called the proposition).

Proof.

Description of proof.

End mark.

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.

The exact style will depend 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 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.^[5]

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.^[6]

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 there are several ongoing projects to shorten and simplify this proof.^[7]

Look up theorem in Wiktionary, the free dictionary.