section_path: Nature and purpose
section_name: Nature and purpose
paragraph_text: 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. A formal proof is written in a formal language instead of natural language. A formal proof is a sequence of 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 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 assistants, this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are analytic or synthetic. Kant, who introduced the analytic–synthetic distinction, believed mathematical proofs are synthetic, whereas Quine argued in his 1951 " Two Dogmas of Empiricism " that such a distinction is untenable.
  
  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.