Editing Theorem (section)

==Epistemological considerations==
Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as '''hypotheses''' or [[premise]]s. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a [[Necessity and sufficiency|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 system]]s, 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 of mathematics|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.<ref>{{mathworld|id=Theorem|title=Theorem}}</ref> 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.<ref name=":1">{{Cite web|url=http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf|title=Fermat's Last Theorem|last1=Darmon|first1=Henri|last2=Diamond|first2=Fred|date=2007-09-09|website=McGill University – Department of Mathematics and Statistics|access-date=2019-11-01|last3=Taylor|first3=Richard}}</ref>