Mathematical Proofs Quiz

Summary Title: Understanding Mathematical Proofs

• A mathematical proof is an inferential argument for a mathematical statement that shows the logical guarantee of the conclusion using axioms and rules of inference.

• Proofs are examples of exhaustive deductive reasoning and must demonstrate that the statement is true in all possible cases.

• A proposition that has not been proved but is believed to be true is known as a conjecture or hypothesis.

• Proofs employ logic expressed in mathematical symbols and natural language which usually admits some ambiguity.

• The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

• The word "proof" comes from the Latin probare (to test).

• The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of its greatest achievements was the book, the Elements, by Euclid.

• Further advances also took place in medieval Islamic mathematics.

• A proof has to meet communal standards of rigor, an argument considered vague or incomplete may be rejected.

• Methods of proof include direct proof, proof by mathematical induction, proof by contraposition, proof by contradiction, proof by construction, proof by exhaustion, probabilistic proof, combinatorial proof, and nonconstructive proof.

• A nonconstructive proof establishes that a mathematical object with a certain property exists without explaining how such an object can be found.

• Understanding mathematical proofs is essential to the field of mathematics and involves rigorous argumentation, formal language, and logical deduction.Mathematical Proof: An Overview

• "Proof" in Encyclopædia Britannica.

• "Mathematical Proof" by Steven J. Miller, Wolfram Demonstrations Project.

• "Proofs from The Book" by Martin Aigner and Günter M. Ziegler, Springer-Verlag, Berlin, Heidelberg, New York, 1998. An overview of proofs of famous theorems.

• "The Great Internet Mersenne Prime Search (GIMPS)".

• "The Ineffable Mystery of God's Location" by Dave Bayer and Persi Diaconis, The College Mathematics Journal, Vol. 23, No. 2 (Mar., 1992), pp. 97–108. A proof of the existence of God based on a variation of the Banach–Tarski paradox.

• "Proofs and Refutations" by Imre Lakatos, Cambridge University Press, 1976. The book argues against the view that mathematical proofs have an ideal form.

• "What is a Mathematical Proof?" by David Tall, Department of Education, University of Warwick.

• "Proofs in Mathematics" by James Franklin, Stanford Encyclopedia of Philosophy, 2017.