A Mathematical Prelude to the Philosophy of Mathematics

Author: Stephen Pollard

Publisher: Springer

ISBN: 3319058169

Category: Science

Page: 202

View: 7924

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.

Philosophical Introduction to Set Theory

Author: Stephen Pollard

Publisher: Courier Dover Publications

ISBN: 0486805824

Category: Mathematics

Page: 192

View: 5253

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.

Theorems, Corollaries, Lemmas, and Methods of Proof

Author: Richard J. Rossi

Publisher: John Wiley & Sons

ISBN: 1118030575

Category: Mathematics

Page: 318

View: 5645

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.

Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences

Author: Ivor Grattan-Guinness

Publisher: Routledge

ISBN: 1134957491

Category: Reference

Page: 1840

View: 7611

* Examines the history and philosophy of the mathematical sciences in a cultural context, tracing their evolution from ancient times up to the twentieth century * 176 articles contributed by authors of 18 nationalities * Chronological table of main events in the development of mathematics * Fully integrated index of people, events and topics * Annotated bibliographies of both classic and contemporary sources * Unique coverage of Ancient and non-Western traditions of mathematics

Strands of System

The Philosophy of Charles Peirce

Author: Douglas R. Anderson,Charles Sanders Peirce

Publisher: Purdue University Press

ISBN: 9781557530585

Category: Philosophy

Page: 204

View: 7319

The American thinker Charles Sanders Peirce, best known as the founder of pragmatism, has been influential not only in the pragmatic tradition but more recently in the philosophy of science and the study of semiotics, or sign theory. Strands of System provides an accessible overview of Peirce's systematic philosophy for those who are beginning to explore his thinking and its import for more recent trends in philosophy.

Thomas Kuhn's 'Linguistic Turn' and the Legacy of Logical Empiricism

Incommensurability, Rationality and the Search for Truth

Author: Dr Stefano Gattei

Publisher: Ashgate Publishing, Ltd.

ISBN: 1409485854

Category: Philosophy

Page: 292

View: 2856

Presenting a critical history of the philosophy of science in the twentieth century, focusing on the transition from logical positivism in its first half to the "new philosophy of science" in its second, Stefano Gattei examines the influence of several key figures, but the main focus of the book are Thomas Kuhn and Karl Popper. Kuhn as the central figure of the new philosophy of science, and Popper as a key philosopher of the time who stands outside both traditions. Gattei makes two important claims about the development of the philosophy of science in the twentieth century; that Kuhn is much closer to positivism than many have supposed, failing to solve the crisis of neopostivism, and that Popper, in responding to the deeper crisis of foundationalism that spans the whole of the Western philosophical tradition, ultimately shows what is untenable in Kuhn's view. Gattei has written a very detailed and fine grained, yet accessible discussion making exceptionally interesting use of archive materials.