Name: The foundations of mathematics
ISBN: 9780198706434

The foundations of mathematics

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.

What Hacker News says

tanganianJun 2022

The Foundations of Mathematics - Ian Stewart.

I wouldn't say it taught me the most, but more like it opened the door so I could learn the most out of everything after it.

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newsoulMay 2022

After reading all the comments I am sharing my choices

1. The foundations of mathematics by Stewart and Tall

2. Problem Solving Strategies by Arthur Engels

3. Journey into Mathematics by Rotman

4. Concrete Mathematics by Knuth et al.

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smarx007May 2022

I would suggest to start with reading The Foundations of Mathematics by Stewart and Tall: https://global.oup.com/ukhe/product/the-foundations-of-mathe...

This will give you an understanding of which topics make sense to you and which you can barely follow. From that, you will need to catch up but where from depends only on your level. I guess, you might have some Khan Academying to do to catch up with the school program.

After you are done with The Foundations of Mathematics, I think you can take Andrew Ng's courses and go from there. I also use https://www.routledge.com/A-Concise-Handbook-of-Mathematics-... as a general reference when I need to look up a topic a have a rather vague recollection of.

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jonsenMar 2022

> I've never understood the obsession with Spivak

Maybe one can be so happy about having made it through such a tough subject that it has to be a good book? I mean, do people normaly study from more than one text?!

> ...to give people a flavour of university style mathematics

I've just started The Foundations of Mathematics by Ian Stewart & David Tall. So far it's really promising.

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haZard_OSMar 2018

Others here have offered some great suggestions already, so I will offer one a little off the beaten path:

Foundations of Mathematics https://www.amazon.com/Foundations-Mathematics-Ian-Stewart/d...

This book is meant to help one transition from performing math in an algorithmic manner to generating proofs based on logic and also set theory.

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jonpFeb 2008

"1089 and all that" by David Acheson is charming and very readable. It covers a variety of different types of maths but more importantly get across what maths is really about. I've bought it for friends and family with a wide range of maths backgrounds (graduate mathematicians to not-since-school-forty-years-ago).

Also "Alice and Numberland", Baylis and Haggarty; and "The Foundations of Mathematics", Stewart and Tall. These are both pitched somewhere between high school and university level and bridge the gap well.

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