Informal Accounts of Theorems
Logically, many 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 (A and B can also be denoted the antecedent and consequent). 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.
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. 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. Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities.
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