Foundations of Higher Mathematics

I would strongly recommend this book before any proof-based math class. The authors explain methods of proofs very well, and give some principles universally important in mathematics -- Zermelo's thm., Dirichlet's prin., and such.

The exposition in this book is great. If this is your first exposure to, for instance, the proofs by induction, this probably provides an excellent description of what's going on and how it works, why it works.

The book is slim (at least, the 1992 ed.) and not inexpensive. However, the authors' conversational tone makes it very approachable; at the same time, they are mathematically rigorous and very thorough.


A great read to see what lays ahead   August 28, 2003
Mike (Los Angeles, CA USA)
1 out of 1 found this review helpful

I read and worked the problems in this book during my break as I transferred from a community college to a 4-year university, and found it very helpful in introducing me to all the fancy terminology, notation and basic proof writing that I was intimidated by. I found the problems to be hard enough to be challenging, but also neither impossibly hard nor hinging on a silly trick.

If you are a eager HS student, or a curious college student, get this book and work the problems.


Great introduction to mathematics   September 20, 2001
2 out of 3 found this review helpful

I bought this book for a course in classical algebra. I found the book well explained and well done. It contains a lot of exercise and example of differents difficulty. It covers logic, set, relation, induction, function, combinatorial proofs, countable sets and uncountable sets, groups and some calculus. The book has a lot of subject in it and it make it very flexible. If you want to ontroduce yourself to mathematics, I would recommend this book if you want to spend some money.


Clear and concise book on math and more   October 1, 1998
1 out of 2 found this review helpful

This book is well done. Not only is well done but the explanations are clear and concise. The book offers different approaches. I use it for class and find most revealing. Hope you enjoy too. M.


A Pricey and Often Frustrating Introduction   July 17, 2005
Steven Nydick (Minneapolis, MN USA)
5 out of 6 found this review helpful

I bought this book so I could self-teach myself a prerequisite course to an advanced mathematics requirement course. Therefore, I needed to learn the material as well as if I had taken a course using this book, although I did not have the requisite professor as a guide. That said, this book suffers from some obvious flaws.

As a transition book to higher mathematics courses, this book needs to accomplish a great deal. First and foremost, it needs to modify the apt math student from an elementary way of thinking about mathematics to a logical, adept, reflexive, adaptable way of thinking about mathematics. This is done through way of introducing the logic of certain mathematical foundations and then allowing the student to participate in proofs to verify the theorems. Most of the times the theorems are presented in an understandable way, and the authors do explain and illustrate many of them; however, the way they typed their proofs (and many of the practice problems use a demonstration proof as a guide), in clustered format, although often acceptable, was very hard to follow. Also, early in the book they explained the benefit of visual explanations of their concepts, so I found it frustrating when they sometimes (but almost seldom) visualize or explain the theorems in more mundane terms, especially when they do not even write the proof for the theorem in the book.

Maybe as an accompaniment to a course in foundational mathematics (what this book is designed for), this book can successfully supplement an instructor; however, as a guide to someone self-taught (especially for the expense of buying), this book could improve a great deal.