Mathematical proofs | Wikipedia audio article
Channel: wikipedia tts
Date: 07/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Mathematical_proof


00:00:05 1 History and etymology
00:00:11 2 Nature and purpose
00:00:17 3 Methods
00:00:20 3.1 Direct proof
00:00:26 3.2 Proof by mathematical induction
00:00:32 3.3 Proof by contraposition
00:00:38 3.4 Proof by contradiction
00:00:44 3.5 Proof by construction
00:00:50 3.6 Proof by exhaustion
00:00:56 3.7 Probabilistic proof
00:01:02 3.8 Combinatorial proof
00:01:08 3.9 Nonconstructive proof
00:01:14 3.10 Statistical proofs in pure mathematics
00:01:20 3.11 Computer-assisted proofs
00:01:26 4 Undecidable statements
00:01:32 5 Heuristic mathematics and experimental mathematics
00:01:38 6 Related concepts
00:01:44 6.1 Visual proof
00:01:50 6.2 Elementary proof
00:01:56 6.3 Two-column proof
00:02:02 6.4 Colloquial use of "mathematical proof"
00:02:07 6.5 Statistical proof using data
00:02:13 6.6 Inductive logic proofs and Bayesian analysis
00:02:19 6.7 Proofs as mental objects
00:02:22 6.8 Influence of mathematical proof methods outside mathematics
00:02:28 7 Ending a proof
00:02:34 8 See also
00:02:40 9 References
00:02:46 10 Further reading
00:02:52 11 External links
00:02:58 Proofs as mental objects
00:03:04 Influence of mathematical proof methods outside mathematics
00:03:10 Ending a proof



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SUMMARY
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A mathematical proof is an inferential 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, along with accepted rules of inference. Proofs are examples of exhaustive deductive reasoning or exhaustive inductive reasoning which establish logical certainty, and are distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Enumerating many confirmatory cases is not enough for a proof, which must demonstrate that the statement is always true (occasionally by listing all possible cases and showing that it holds in each). An unproven proposition that is believed to be true is known as a conjecture.
Proofs employ logic expressed in mathematical symbols, along with some amount of natural language which usually admits some ambiguity. The vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs 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.