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This book is based on two premises: one cannot understand philosophy of mathematics without understanding mathematics and one cannot understand mathematics without doing mathematics. It draws readers into philosophy of mathematics by having them do mathematics. It offers 298 exercises, covering philosophically important material, presented in a philosophically informed way. The exercises give readers opportunities to recreate some mathematics that will illuminate important readings in philosophy of mathematics. Topics include primitive recursive arithmetic, Peano arithmetic, Gödel's theorems, interpretability, the hierarchy of sets, Frege arithmetic and intuitionist sentential logic. The book is intended for readers who understand basic properties of the natural and real numbers and have some background in formal logic.
This unique approach maintains that set theory is the primary mechanism for ideological and theoretical unification in modern mathematics, and its technically informed discussion covers a variety of philosophical issues. 1990 edition.
This lively, stimulating account of non-Euclidean geometry by a noted mathematician covers matrices, determinants, group theory, and many other related topics, with an emphasis on the subject's novel, striking aspects. 1955 edition.
First published in 2005. Routledge is an imprint of Taylor & Francis, an informa company.
A hands-on introduction to the tools needed for rigorous andtheoretical mathematical reasoning Successfully addressing the frustration many students experience asthey make the transition from computational mathematics to advancedcalculus and algebraic structures, Theorems, Corollaries, Lemmas,and Methods of Proof equips students with the tools needed tosucceed while providing a firm foundation in the axiomaticstructure of modern mathematics. This essential book: * Clearly explains the relationship between definitions,conjectures, theorems, corollaries, lemmas, and proofs * Reinforces the foundations of calculus and algebra * Explores how to use both a direct and indirect proof to prove atheorem * Presents the basic properties of real numbers * Discusses how to use mathematical induction to prove atheorem * Identifies the different types of theorems * Explains how to write a clear and understandable proof * Covers the basic structure of modern mathematics and the keycomponents of modern mathematics A complete chapter is dedicated to the different methods of proofsuch as forward direct proofs, proof by contrapositive, proof bycontradiction, mathematical induction, and existence proofs. Inaddition, the author has supplied many clear and detailedalgorithms that outline these proofs. Theorems, Corollaries, Lemmas, and Methods of Proof uniquelyintroduces scratch work as an indispensable part of the proofprocess, encouraging students to use scratch work and creativethinking as the first steps in their attempt to prove a theorem.Once their scratch work successfully demonstrates the truth of thetheorem, the proof can be written in a clear and concise fashion.The basic structure of modern mathematics is discussed, and each ofthe key components of modern mathematics is defined. Numerousexercises are included in each chapter, covering a wide range oftopics with varied levels of difficulty. Intended as a main text for mathematics courses such as Methods ofProof, Transitions to Advanced Mathematics, and Foundations ofMathematics, the book may also be used as a supplementary textbookin junior- and senior-level courses on advanced calculus, realanalysis, and modern algebra.
First published in 1970. Routledge is an imprint of Taylor & Francis, an informa company.

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