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Ian Stewart and David Tall

The Foundations of Mathematics

University level mathematics can be a daunting subject, and if you haven't studied maths to this level then you may think that those who have are speaking a different language. Well help is at hand in the form of a short book, The Foundations of Mathematics, by Ian Stewart and David Tall.

The book starts with a gentle introduction to ways of thinking about numbers and number systems. It then gets on to set theory, mathematical logic and formal proofs. The third part looks at number systems defined axiomatically, including the natural numbers, the real numbers and the complex numbers. The final chapter describes where axioms can lead, looking at axiomatic (as opposed to naive) set theory. Gödel's Incompleteness theorem is mentioned briefly - I couldn't help thinking that a whole chapter on this would have made a good finale.

The book would be useful to several different types of reader. Firstly, it would help those who are intending to study mathematics at university to prepare for what might be a difficult start. Secondly, it will be of use to those studying other sciences who want to understand their more mathematical colleagues. More generally, it will appeal to anyone who wants to get a grasp of the foundations of mathematics. The book has exercises for the reader in each chapter, and is well suited to independent study. Overall, I felt that the authors manage to deal with the foundations of mathematics in a way that doesn't get intimidating for the reader.