Mathematical Rigour And Informal Proof PDF Download
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Author | : Fenner Stanley Tanswell |
Publisher | : Cambridge University Press |
Total Pages | : 158 |
Release | : 2024-03-28 |
Genre | : Philosophy |
ISBN | : 1009325132 |
Download Mathematical Rigour and Informal Proof Book in PDF, ePub and Kindle
This Element looks at the contemporary debate on the nature of mathematical rigour and informal proofs as found in mathematical practice. The central argument is for rigour pluralism: that multiple different models of informal proof are good at accounting for different features and functions of the concept of rigour. To illustrate this pluralism, the Element surveys some of the main options in the literature: the 'standard view' that rigour is just formal, logical rigour; the models of proofs as arguments and dialogues; the recipe model of proofs as guiding actions and activities; and the idea of mathematical rigour as an intellectual virtue. The strengths and weaknesses of each are assessed, thereby providing an accessible and empirically-informed introduction to the key issues and ideas found in the current discussion.
Author | : John P. Burgess |
Publisher | : |
Total Pages | : 241 |
Release | : 2015 |
Genre | : Mathematics |
ISBN | : 0198722222 |
Download Rigor and Structure Book in PDF, ePub and Kindle
While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. The main theses of Rigor and Structure are that the features of mathematical practice that a large group of philosophers of mathematics, the structuralists, have attributed to the peculiar nature of mathematical objects are better explained in a different way, as artefacts of the manner in which the ancient ideal of rigor is realized in modern mathematics. Notably, the mathematician must be very careful in deriving new results from the previous literature, but may remain largely indifferent to just how the results in the previous literature were obtained from first principles. Indeed, the working mathematician may remain largely indifferent to just what the first principles are supposed to be, and whether they are set-theoretic or category-theoretic or something else. Along the way to these conclusions, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.
Author | : Imre Lakatos |
Publisher | : Cambridge University Press |
Total Pages | : 190 |
Release | : 1976 |
Genre | : Mathematics |
ISBN | : 9780521290388 |
Download Proofs and Refutations Book in PDF, ePub and Kindle
Proofs and Refutations is for those interested in the methodology, philosophy and history of mathematics.
Author | : Gerard Allwein |
Publisher | : Oxford University Press, USA |
Total Pages | : 287 |
Release | : 1996 |
Genre | : Knowledge representation (Information theory). |
ISBN | : 0195104277 |
Download Logical Reasoning with Diagrams Book in PDF, ePub and Kindle
Information technology has lead to an increasing need to present information visually. This volume addresses the logical aspects of the visualization of information. Properties of diagrams, charts and maps are explored and their use in problem solving and
Author | : Ethan D. Bloch |
Publisher | : Springer Science & Business Media |
Total Pages | : 434 |
Release | : 2013-12-01 |
Genre | : Mathematics |
ISBN | : 1461221307 |
Download Proofs and Fundamentals Book in PDF, ePub and Kindle
The aim of this book is to help students write mathematics better. Throughout it are large exercise sets well-integrated with the text and varying appropriately from easy to hard. Basic issues are treated, and attention is given to small issues like not placing a mathematical symbol directly after a punctuation mark. And it provides many examples of what students should think and what they should write and how these two are often not the same.
Author | : Jody Azzouni |
Publisher | : Oxford University Press on Demand |
Total Pages | : 255 |
Release | : 2006 |
Genre | : Mathematics |
ISBN | : 019518713X |
Download Tracking Reason Book in PDF, ePub and Kindle
When ordinary people including mathematicians, take something to follow from something else, they are exposing the backbone of our ability to reason. Azzouni investigates the connection between that ordinary notion of consequence and the formal analogues invented by logicians.
Author | : Joel David Hamkins |
Publisher | : MIT Press |
Total Pages | : 350 |
Release | : 2021-03-09 |
Genre | : Mathematics |
ISBN | : 0262542234 |
Download Lectures on the Philosophy of Mathematics Book in PDF, ePub and Kindle
An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.
Author | : Erich Christian Wittmann |
Publisher | : Springer Nature |
Total Pages | : 332 |
Release | : 2020-12-09 |
Genre | : Education |
ISBN | : 3030615707 |
Download Connecting Mathematics and Mathematics Education Book in PDF, ePub and Kindle
This open access book features a selection of articles written by Erich Ch. Wittmann between 1984 to 2019, which shows how the “design science conception” has been continuously developed over a number of decades. The articles not only describe this conception in general terms, but also demonstrate various substantial learning environments that serve as typical examples. In terms of teacher education, the book provides clear information on how to combine (well-understood) mathematics and methods courses to benefit of teachers. The role of mathematics in mathematics education is often explicitly and implicitly reduced to the delivery of subject matter that then has to be selected and made palpable for students using methods imported from psychology, sociology, educational research and related disciplines. While these fields have made significant contributions to mathematics education in recent decades, it cannot be ignored that mathematics itself, if well understood, provides essential knowledge for teaching mathematics beyond the pure delivery of subject matter. For this purpose, mathematics has to be conceived of as an organism that is deeply rooted in elementary operations of the human mind, which can be seamlessly developed to higher and higher levels so that the full richness of problems of various degrees of difficulty, and different means of representation, problem-solving strategies, and forms of proof can be used in ways that are appropriate for the respective level. This view of mathematics is essential for designing learning environments and curricula, for conducting empirical studies on truly mathematical processes and also for implementing the findings of mathematics education in teacher education, where it is crucial to take systemic constraints into account.
Author | : Marcus Giaquinto |
Publisher | : Oxford University Press |
Total Pages | : 298 |
Release | : 2007-07-05 |
Genre | : Mathematics |
ISBN | : 0199285942 |
Download Visual Thinking in Mathematics Book in PDF, ePub and Kindle
Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis.
Author | : David A. Reid |
Publisher | : BRILL |
Total Pages | : 265 |
Release | : 2010-01-01 |
Genre | : Education |
ISBN | : 946091246X |
Download Proof in Mathematics Education Book in PDF, ePub and Kindle
Research on teaching and learning proof and proving has expanded in recent decades. This reflects the growth of mathematics education research in general, but also an increased emphasis on proof in mathematics education.