This textbook compiles lessons from the Foundations of Mathematics course at the Autonomous University of Barcelona. It is designed to introduce students beginning their mathematics studies to the major overarching ideas in the field—such as symmetry, structure, equivalence, abstraction, generalization, analogy, and computability. It also aims to familiarize students with the structure of mathematical discourse—how mathematics is written—and to support them in the challenging task of developing the rigor and creativity required of a professional mathematician.

In this work, readers will encounter the big ideas and fundamental principles that shape mathematical discourse, including logic, group theory, Peano's axioms, Russell's paradox, the prime number theorem, RSA cryptography, Euler's formula, and more. After each part, a set of carefully selected exercises helps students to gain greater exposure to—and confidence in—the rigors of mathematics.

Guided by aesthetic and playful principles, this book offers a gentle, passionate, and joyful approach to some of the most essential concepts in mathematics. It provides enjoyment for teachers, students, and anyone with an inclination toward the subject, and is particularly appealing to high-achieving high school students seeking to delve into college-level mathematics.

Jaume Aguadé (Barcelona, 1953) was a full professor in the Department of Mathematics at the Universitat Autònoma de Barcelona for 34 years. His research has focused on algebraic topology and the homotopy theory of Lie groups. He has also been deeply committed to teaching and has authored several books that reflect his personal approach to mathematics education, including Notes on Elementary Topology, Mathematics and Modelling for Environmental Sciences, A Course in Linear Geometry, and the present volume. Now retired, he is currently working on a book on compact Lie groups, provisionally titled Groups Worth Meeting. Beyond mathematics, he has been an enthusiastic practitioner of hiking, mountaineering, and ski touring—activities that in recent years he has replaced with road cycling.