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A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems ; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms ...
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
Statistical assumptions can be put into two classes, depending upon which approach to inference is used. Model-based assumptions. These include the following three types: Distributional assumptions. Where a statistical model involves terms relating to random errors, assumptions may be made about the probability distribution of these errors. [5]
Some math problems have been challenging us for centuries, ... The closest we’ve come—given some subtle technical assumptions—is 6. Time will tell if the last step from 6 to 2 is right ...
This is a list of axioms as that term is understood in mathematics. In epistemology, the word axiom is understood differently; see axiom and self-evidence. Individual axioms are almost always part of a larger axiomatic system.
Specifications of mathematical problems commonly contain information that can be used in the solution. When writing a proof or a solution as a structured derivation, all known information is listed in the beginning as assumptions. These assumptions can be used to create new information that will be useful for solving the problem.
These problems were also studied by mathematicians, and this led to establish mathematical logic as a new area of mathematics, consisting of providing mathematical definitions to logics (sets of inference rules), mathematical and logical theories, theorems, and proofs, and of using mathematical methods to prove theorems about these concepts.
However, if the problem is undecidable even with much weaker assumptions extending the Peano axioms for integer arithmetic, then nearly polynomial-time algorithms exist for all NP problems. [25] Therefore, assuming (as most complexity theorists do) some NP problems don't have efficient algorithms, proofs of independence with those techniques ...