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Proofs employ logic but usually include some amount of natural language which of course admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic. In the context of proof theory, where purely formal proofs are considered, such not entirely formal demonstrations in mathematics are often called "social proofs". The distinction has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
Regardless of one's attitude to formalism, the result that is proved to be true is a theorem; in a completely formal proof it would be the final line, and the complete proof shows how it follows from the axioms alone. Once a theorem is proved, it can be used as the basis to prove further statements. The so-called foundations of mathematics are those statements one cannot, or need not, prove. These were once the primary study of philosophers of mathematics. Today focus is more on practice, i.e. acceptable techniques.
Some common proof techniques are:
A probabilistic proofThis article is not about probabilistic algorithms, which give the right answer with high probability but not with certainty, nor about Monte Carlo methods, which are simulations relying on pseudo-randomness. The probabilistic method is a non-constructive should mean a proof in which an example is shown to exist by methods of probability theoryProbability theory Discrete mathematics Mathematical analysis Probability theory is the mathematical study of probability. Mathematicians think of probabilities as numbers in the interval from 0 to 1 assigned to "events" whose occurrence or failure to occ - not an argument that a theorem is 'probably' true. The latter type of reasoning can be called a 'plausibility argument'; in the case of the Collatz conjectureNumber theory Conjectures The Collatz conjecture also known as the 3''n + 1 conjecture the Ulam conjecture or the Hailstone sequence or Hailstone numbers was first stated in 1937 and concerns the following process: # Pick any positive integer n''. If n is it is clear how far that is from a genuine proof. Probabilistic proof is one of many ways to show existence theoremProofs In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s). or more generally 'for all x y . there exist(s). That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existentis, other than proof by construction.
A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Usually a one-to-one correspondence is used to show that the two interpretations give the same result.
If we are trying to prove, for example, "Some X satisfies f(X)", an existence or nonconstructive proof will prove that there is a X that satisfies f(X), but does not tell you how such an X will be obtained. A constructive proof, conversely, will do so.
A statement which is thought to be true but hasn't been proven yet is known as a conjecture.
Sometimes it is possible to prove that a certain statement cannot possibly be proven from a given set of axioms; see for instance the continuum hypothesis. In most axiom systems, there are statements which can neither be proven nor disproven; see Gödel's incompleteness theorem.
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