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In mathematics, a theorem is a statement proven on the basis of previously accepted or established statements. In mathematical logic, theorems are modeled as formulas that can be derived according to the derivation rules of a fixed formal system. In formal settings, an essential property of theorems is that they are derivable using a fixed set of inference rules and axioms without any additional assumptions. This is not a matter of the semantics of the language the expression that results from a derivation is a syntactic consequence of all the expressions that precede it. In mathematics, the derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. The proofs of theorems have two components, called the hypotheses and the conclusions. The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical. 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 organised 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.
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