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The 1976 book Proofs and Refutations is based on the first three chapters of his 1961 four-chapter doctoral thesis Essays in the Logic of Mathematical Discovery.But its first chapter is Lakatos's own revision of its chapter 1 that was first published as Proofs and Refutations in four parts in 1963–4 in the British Journal for the Philosophy of Science.
It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations (contained in 1– 21 of Principia [i.e., sections 1– 5 (propositional logic), 8–14 (predicate logic with identity/equality), 20 ...
The book begins with a historical overview of the long struggles with the parallel postulate in Euclidean geometry, [3] and of the foundational crisis of the late 19th and early 20th centuries, [6] Then, after reviewing background material in real analysis and computability theory, [1] the book concentrates on the reverse mathematics of theorems in real analysis, [3] including the Bolzano ...
Bertrand's postulate and a proof; Estimation of covariance matrices; Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational
Barwise, along with his former colleague at Stanford John Etchemendy, was the author of the popular logic textbook Language, Proof and Logic. Unlike the Handbook of Mathematical Logic, which was a survey of the state of the art of mathematical logic circa 1975, and of which he was the editor, this work targeted elementary logic. The text is ...
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. [5] The field was founded by Harvey Friedman . Its defining method can be described as "going backwards from the theorems to the axioms ", in contrast to the ordinary mathematical practice of deriving ...
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms ", in contrast to the ordinary mathematical practice of deriving theorems from axioms.
The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with mathematical objects, such as numbers, to demonstrate something about everyday life, or when data used in an argument is numerical. It is sometimes also used to mean a "statistical proof" (below), especially when used to argue from data.
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