What is a mathematical proof?
What is a conjecture or hypothesis?
What is the philosophy of mathematics concerned with?
What is the primary product of the development of mathematical proof?
What is a nonconstructive proof?
What are the methods of proof?
What is the purpose of a proof?
What is the origin of the word 'proof'?
Why is understanding mathematical proofs essential to the field of mathematics?
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.
Test your knowledge and understanding of mathematical proofs with this quiz! Explore the history and philosophy behind proofs, as well as various methods of proof. From direct proof to nonconstructive proof, this quiz will challenge your ability to think logically and apply formal language to rigorous argumentation. Whether you're a student of mathematics or simply interested in the topic, this quiz will provide a fun and informative way to test your skills.
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