The Foundations of Mathematics 2nd Edition by Ian Stewart (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Ian Stewart’s a boss. David Tall as well. This book really helped me learn the deep ideas behind the math I learned in school. Excellent.

⭐This book is a perfect introduction to the subject; the language is clear, direct and the content is self-explanatory. A great suport for beginners.

⭐Every budding mathematician should have this book on her/his shelf.

⭐Good reference as well as interesting read.

⭐Wonderful book

⭐Great book. Concise and accessible.

⭐Right price

⭐Lost me on the exercises.

⭐Ian Stewart wad one of my lecturers back in the 80’s. One first year course was based on this book and I can still remember everything that he taught during it. (Which is more than I can say of most.) After all these years I still have all my old textbooks, with the exception of this one (which had gone walkies at some point). So when I saw that there was a new edition I had to buy it….. I wasn’t disappointed.

⭐Definitely worth having in the Library.

⭐Great Book

⭐Extremely interesting. A review or explanation of the mathematical reasoning from a new and appeallng point of view.

⭐would be nice if the format could be adapted, as long as its a Print replica it cant even be downloaded on a normal Kindle…kinda pointless to offer a Kindle version if its incompatible with most of the devices.

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