Editing Theorem (section)

== 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''. {{anchor|Hypothesis|Conclusion|Proposition}}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 (logic)|antecedent]]'' and the ''[[consequent]]'', respectively.<ref>{{Cite web |url=http://intrologic.stanford.edu/glossary/implication.html |title=Implication |website=intrologic.stanford.edu |access-date=2019-11-02}}</ref> 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 to 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 [[axiom]]s or postulates. The field of mathematics known as [[proof theory]] studies formal languages, axioms and the structure of proofs.

[[File:4CT Non-Counterexample 1.svg|frame|right|A [[Plane (mathematics)|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.]]
Some theorems are "[[Triviality (mathematics)|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", 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.<ref>{{MathWorld|title=Deep Theorem|urlname=DeepTheorem}}</ref> A theorem might be simple to state and yet be deep. An excellent example is [[Fermat's Last Theorem]],<ref name=":1" /> 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.<ref>{{cite web|author=Doron Zeilberger|author-link=Doron Zeilberger|title=Opinion 51|url=http://www.math.rutgers.edu/~zeilberg/Opinion51.html}}</ref> Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities{{efn|Such as the derivation of the formula for <math>\tan (\alpha + \beta)</math> from the [[List of trigonometric identities#Angle sum and difference identities|addition formulas of sine and cosine]].}} and hypergeometric identities.{{sfn|Petkovsek|Wilf|Zeilberger|1996}}{{Page needed|date=October 2010}}