Out of the Head of Zeus: Simple Concepts that Transformed Mathematics

Does mathematics education always have to be like trench warfare, with each lecture devoted to a single concept, followed by a repetitive problem set stressing mechanical computations? In addition to books that slowly and carefully put a single layer of bricks in place, then go on to the next level, shouldn't there be some books that erect scaffolding, giving the reader a rapid introduction to the key concepts that lead to the higher levels of the subject? Wouldn't it be fun if you could just read such a book, instead of committing to biweekly lectures, problem sets, and exams?

Out of the Head of Zeus is a rigorous book of mathematics that could be used as the basis of a university course if the instructor took problems from other sources, and it is an ideal supplementary reading for a wide variety of courses, but it is not a textbook. It is designed to be read, for pleasure, by people who are motivated by curiosity and the beauty of the material, rather than discipline or necessity. Starting with the basics of sets, relations, and functions, each topic is described carefully and exactly, with extensive conceptual motivation and historical background. The reader is not expected to reinforce the concepts by working problems, but instead by seeing them applied in the subsequent discussion. Throughout the emphasis is on how the key concepts combine to form a coherent theoretical structure. Out of the Head of Zeus is well suited for:
Talented Secondary School Students: It presumes very little in the way of previous background, but is precise and rigorous throughout. It focuses on the concepts that will make your coursework easy to understand, and which underly advanced study in all subfields. The real deal...this is a book I wish I could have read in high school.
University Students: Prepare for your next math course by reading a clear, conceptually oriented description of the core material. Solidify and deepen your understanding by reading it again before the exam.
Scientifically Minded Adults: Across a broad range of prior mathematical training, scientists and others working in technical fields can enjoyably increase their understanding of what mathematics is, where it came from, and the concerns of contemporary research, without taking a course, working problem sets, or doing anything else that active professionals don't have time for.

From the Preface

Contemporary mathematics is a very different thing from the mathematics of 150 years ago. To a certain extent this is simply because we know a lot more, but the more radical changes are transformations of the most fundamental concepts of the subject.

At a certain point in the 19th century mathematicians realized that set theory could be used to give exact descriptions of all the objects they worked with. The most obvious and immediate benefit is increased clarity and rigor, but that is far from the end of the story. The methods used to give precise definitions of existing concepts can also be used to define novel structures, and in the 20th century this led to the emergence of many entirely new fields of research. A bit more subtly, the axiomatic method based on set theory can be used to take a concept apart, to break it down into more fundamental elements, to recombine these elements, and ultimately to reformulate the original concept in ways that discard inessential aspects inherited from particular applications while retaining a critical core. This is the process of abstraction.

This book describes some of the resulting concepts. Up to a point its trajectory is quite similar to the mathematical curriculum at the secondary school and early university level: fundamentals of mathematical reasoning, basic facts about real numbers, continuity and convergence, some algebra, and then the calculus. Every idea had some predecessor in the mathematical thought of Sir Isaac Newton. But instead of thinking of these as a collection of problem-solving methods or ``skills,'' we will be entirely concerned with viewing them as a system of interrelated definitions that combine to create a mathematics that is more general, unified, and powerful than anything Newton could have imagined. The last two chapters use these concepts to develop geometric structures that go far beyond geometry as it was understood in the 18th century, but which are now fundamental in mathematics and physics.

Last updated on March 22, 2008.