Editing Mathematical proof (section)

==Nature and purpose==

As practiced, a proof is expressed in natural language and is a rigorous [[argument]] intended to convince the audience of the truth of a statement. The standard of rigor is not absolute and has varied throughout history. A proof can be presented differently depending on the intended audience. To gain acceptance, a proof has to meet communal standards of rigor; an argument considered vague or incomplete may be rejected.

The concept of proof is formalized in the field of [[mathematical logic]].<ref>{{citation|title=Handbook of Proof Theory|volume=137|series=Studies in Logic and the Foundations of Mathematics|editor-first=Samuel R.|editor-last=Buss|editor-link=Samuel Buss|publisher=Elsevier|year=1998|isbn=978-0-08-053318-6|contribution=An introduction to proof theory|pages=1–78|first=Samuel R.|last=Buss|author-link=Samuel Buss}}. See in particular [https://books.google.com/books?id=MfTMDeCq7ukC&pg=PA3 p.&nbsp;3]: "The study of Proof Theory is traditionally motivated by the problem of formalizing mathematical proofs; the original formulation of first-order logic by Frege [1879] was the first successful step in this direction."</ref> A [[formal proof]] is written in a [[formal language]] instead of natural language. A formal proof is a sequence of [[well-formed formula|formulas]] in a formal language, starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones. This definition makes the concept of proof amenable to study. Indeed, the field of [[proof theory]] studies formal proofs and their properties, the most famous and surprising being that almost all axiomatic systems can generate certain [[independence (mathematical logic)|undecidable statements]] not provable within the system.

The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated [[proof assistant]]s, this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are [[analytic proposition|analytic]] or [[synthetic proposition|synthetic]]. [[Immanuel Kant|Kant]], who introduced the [[analytic–synthetic distinction]], believed mathematical proofs are synthetic, whereas [[Willard Van Orman Quine|Quine]] argued in his 1951 "[[Two Dogmas of Empiricism]]" that such a distinction is untenable.<ref>{{Cite web|url=https://www.theologie.uzh.ch/dam/jcr:ffffffff-fbd6-1538-0000-000070cf64bc/Quine51.pdf|title=Two Dogmas of Empiricism|last=Quine|first=Willard Van Orman|date=1961|website=Universität Zürich – Theologische Fakultät|page=12|access-date=October 20, 2019}}</ref>

Proofs may be admired for their [[mathematical beauty]]. The mathematician [[Paul Erdős]] was known for describing proofs which he found to be particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book ''[[Proofs from THE BOOK]]'', published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.