4.0 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Mathematical proof | 4/4 | https://en.wikipedia.org/wiki/Mathematical_proof | reference | science, encyclopedia | 2026-05-05T07:24:53.330614+00:00 | kb-cron |
== Undecidable statements == A statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry. Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see List of statements undecidable in ZFC. Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements.
== Heuristic mathematics and experimental mathematics ==
While early mathematicians such as Eudoxus of Cnidus did not use proofs, from Euclid to the foundational mathematics developments of the late 19th and 20th centuries, proofs were an essential part of mathematics. With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects beyond the proof-theorem framework, in experimental mathematics. Early pioneers of these methods intended the work ultimately to be resolved into a classical proof-theorem framework, e.g. the early development of fractal geometry, which was ultimately so resolved.
== Related concepts ==
=== Visual proof ===
=== Elementary proof ===
=== Two-column proof ===
A particular way of organizing a proof using two parallel columns is often used as a mathematical exercise in elementary geometry classes in the United States. The proof is written as a series of lines in two columns. In each line, the left-hand column contains a proposition, while the right-hand column contains a brief explanation of how the corresponding proposition in the left-hand column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The left-hand column is typically headed "Statements" and the right-hand column is typically headed "Reasons".
=== Statistical proof using data ===
=== Inductive logic proofs and Bayesian analysis ===
=== Proofs as mental objects ===
== Ending a proof ==
Sometimes, the abbreviation "Q.E.D." is written to indicate the end of a proof. This abbreviation stands for "quod erat demonstrandum", which is Latin for "that which was to be demonstrated". A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a "tombstone" or "Halmos" after its eponym Paul Halmos. Often, "which was to be shown" is verbally stated when writing "QED", "□", or "∎" during an oral presentation. Unicode explicitly provides the "end of proof" character, U+220E (∎) (220E(hex) = 8718(dec)).
== See also ==
== References ==
== Further reading == Pólya, G. (1954), Mathematics and Plausible Reasoning, Princeton University Press, hdl:2027/mdp.39015008206248, ISBN 9780691080055 {{citation}}: ISBN / Date incompatibility (help). Fallis, Don (2002), "What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians", Logique et Analyse, 45: 373–88. Franklin, J.; Daoud, A. (2011), Proof in Mathematics: An Introduction, Kew Books, ISBN 978-0-646-54509-7. Gold, Bonnie; Simons, Rogers A. (2008). Proof and Other Dilemmas: Mathematics and Philosophy. MAA. Solow, D. (2004), How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, Wiley, ISBN 978-0-471-68058-1. Velleman, D. (2006), How to Prove It: A Structured Approach, Cambridge University Press, ISBN 978-0-521-67599-4. Hammack, Richard (2018), Book of Proof, Richard Hammack, ISBN 978-0-9894721-3-5.
== External links ==
Media related to Mathematical proof at Wikimedia Commons Proofs in Mathematics: Simple, Charming and Fallacious A lesson about proofs, in a course from Wikiversity