Theorem

In mathematics, a theorem is a statement, proven on the basis of established statements, such as other theorems, accepted statements, such as axioms. A theorem is a logical consequence of the axioms; the proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is 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, experimental. Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called premises. In light of the interpretation of proof as justification of truth, the conclusion is viewed as a necessary consequence of the hypotheses, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.

Although they can be written in a symbolic form, for example, within the propositional calculus, theorems are expressed in a natural language such as English. The same is true of proofs, which are expressed as logically organized and worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, from which a formal symbolic proof can in principle be constructed; such arguments are 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 explains in some way why it is true. In some cases, a picture alone may be sufficient to prove a theorem; because theorems lie at the core of mathematics, they are central to its aesthetics. Theorems are described as being "trivial", or "difficult", or "deep", or "beautiful"; these subjective judgments vary not only from person to person, but with time: for example, as a proof is simplified or better understood, a theorem, once difficult may become trivial.

On the other hand, a deep theorem may be stated but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a well-known example of such a theorem. Logically, many theorems are of the form of an indicative conditional: if A B; such a theorem does not assert B, only that B is a necessary consequence of A. In this case A is called B the conclusion; the theorem "If n is an natural number n/2 is a natural number" is a typical example in which the hypothesis is "n is an natural number" and the conclusion is "n/2 is a natural number". To be proved, a theorem must be expressible as a formal statement. Theorems are expressed in natural language rather than in a symbolic form, with the intention that the reader can produce a formal statement 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 are called axioms or postulates.

The field of mathematics known as proof theory studies formal languages and the structure of proofs. Some theorems are "trivial", in the sense that they follow from definitions and other theorems in obvious ways and do not contain any surprising insights. 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. A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem, 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 be written down; the most prominent examples are the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search, verified by a computer program. Many mathematicians did not accept this form of proof, but it has become more accepted.

The mathematician Doron Zeilberger has gone so far as to claim that these are the only nontrivial results that mathematicians have proved. Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities. To establish a mathematical statement as a theorem, a proof is required, that is, a line of reasoning from axioms in the system to the given statement must be demonstrated. However, the proof is considered as separate from the theorem statement. Although 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. Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proved.