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Learning the foundations of math

Principles of Arithmetic and Geometry by Carl Allendoerfer

If you are looking for a book that begins before even numbers begin, then this is it.

The book starts with basic logic, then moves to define basic set theory very clearly, then proceeds to explain how set theory is the foundation of our whole number system, and finally moves to defining all elementary operations using what you've learned. He goes into great depth of defining all the various kinds of correspondences, cardinality, and other crucial ideas that usually get ignored early on.

Allendoerfer stresses the importance of developing abstract thinking and really makes you feel confident and excited about the abstract nature of mathematics, rather than hinging to real world examples. The truth is that, in order to be a useful mathematician you must be comfortable and skilled with abstract thinking. Whereas most professors shy away from this and want you to wait until much later to develop these skills, Allendoerfer makes it very clear that, no matter what your age is or experience, you are ready to think abstractly about nature and about mathematics.

Every chapter is full of exercises with all answers provided, as well as a test at the end of each chapter and recommendations on what to do if you struggled.

I have yet to find any book that does such a good job at explaining these foundations as Allendoerfer. This book is a god-send for those of you out there that are trying to self-teach mathematics, especially modern mathematics, and are willing to begin at the beginning. (Arithmetic is NOT even the beginning!)

When this book is completed, I highly recommend his Freshman Mathematics book and also his Principles of Modern Mathematics.