The Foundations of Mathematics

This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields.

Author: Ian Stewart

Publisher: Oxford University Press, USA

ISBN: 019870643X

Category: Mathematics

Page: 432

View: 208

The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.

The Foundations of Mathematics

Finally there's an easy-to-follow book that will help readers succeed in the art of proving theorems. Sibley not only conveys the spirit of mathematics but also uncovers the skills required to succeed.

Author: Thomas Q. Sibley

Publisher: John Wiley & Sons

ISBN: 0470085010

Category: Mathematics

Page: 392

View: 616

Finally there's an easy-to-follow book that will help readers succeed in the art of proving theorems. Sibley not only conveys the spirit of mathematics but also uncovers the skills required to succeed. Key definitions are introduced while readers are encouraged to develop an intuition about these concepts and practice using them in problems. With this approach, they'll gain a strong understanding of the mathematical language as they discover how to apply it in order to find proofs.

The Foundations of Mathematics and Other Logical Essays

thrown by their occurrence outside mathematics in the propositions of everyday life . Apart from formalism , there are two main general attitudes to the foundation of mathematics : that of the intuitionists or finitists , like Brouwer ...

Author: Frank Plumpton Ramsey

Publisher: Psychology Press

ISBN: 9780415225465

Category: Philosophy

Page: 292

View: 914

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

Harvey Friedman s Research on the Foundations of Mathematics

He attended the Massachusetts Institute of Technology, where he received a Ph.D. degree in Mathematics in August, 1967. He was appointed to an Assistant Professorship in the Philosophy Department at Stanford, effective September 1967, ...

Author: L.A. Harrington

Publisher: Elsevier

ISBN: 9780080960401

Category: Mathematics

Page: 407

View: 346

This volume discusses various aspects of Harvey Friedman's research in the foundations of mathematics over the past fifteen years. It should appeal to a wide audience of mathematicians, computer scientists, and mathematically oriented philosophers.


The Foundations of Mathematics

I do not hesitate to say that Hermann Grassmann's Lineare Ausdehnungslehre is the best work on the philosophical foundation of mathematics from the standpoint of a mathematician . Grassmann establishes first the idea of mathematics as ...

Author: Paul Carus

Publisher:

ISBN:

Category: Geometry

Page: 141

View: 862


The Foundations of Mathematics

In this sense, their mathematics is natural, in that it is based on observed natural phenomena. Yet they sought a perfect theoretical foundation, arising in their imagination, which takes them beyond what is physically attainable in ...

Author: Ian Stewart

Publisher: Oxford University Press

ISBN: 0198706448

Category: Logic, Symbolic and mathematical

Page: 391

View: 283

The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.


Reflections on the Foundations of Mathematics

In the introduction I pointed out the growing importance of computations and algorithms in mathematics and modeling. I quoted - with approval-from a lecture by S. Smale Mathematical Problems for the next Century.

Author: Wilfried Sieg

Publisher: Cambridge University Press

ISBN: 1316998819

Category: Mathematics

Page:

View: 589

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the fifteenth publication in the Lecture Notes in Logic series, collects papers presented at the symposium 'Reflections on the Foundations of Mathematics' held in celebration of Solomon Feferman's 70th birthday (The 'Feferfest') at Stanford University, California in 1988. Feferman has shaped the field of foundational research for nearly half a century. These papers reflect his broad interests as well as his approach to foundational research, which emphasizes the solution of mathematical and philosophical problems. There are four sections, covering proof theoretic analysis, logic and computation, applicative and self-applicative theories, and philosophy of modern mathematical and logic thought.

Conceptions of Set and the Foundations of Mathematics

For such an appeal requires us to justify the axioms of type theory, thus running afoul of the requirement that a foundation for mathematics should be justificatory autonomous – that it should be possible to motivate and justify its ...

Author: Luca Incurvati

Publisher: Cambridge University Press

ISBN: 1108497829

Category: History

Page: 258

View: 724

Presents a detailed and critical examination of the available conceptions of set and proposes a novel version.