What Is Mathematics? An Elementary Approach to Ideas and Methods
For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics.
Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts.
Brought up to date with a new chapter by Ian Stewart, What is Mathematics?, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.
Formal mathematics is like spelling and grammar--a matter of the correct application of local rules. Meaningful mathematics is like journalism--it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature--it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature--it opens a window onto the world of mathematics for anyone interested to view.
Why Read This Book
You should read this book to develop deep mathematical intuition and a clear understanding of fundamental concepts (proof, continuity, infinity) that underpin many DSP techniques. It trains you to think rigorously about problems so you can better understand the mathematics behind transforms, approximations, and analysis used in signal processing.
Who Will Benefit
Engineers and students who want to strengthen their conceptual mathematical foundations (calculus, complex numbers, proofs) to support more advanced study in DSP and applied mathematics.
Level: Intermediate — Prerequisites: Comfort with high-school algebra and basic calculus concepts; curiosity about proofs and mathematical reasoning.
Key Takeaways
- Appreciate the role of rigorous definition and proof in structuring mathematical thought.
- Understand foundational ideas in calculus and analysis that support transform methods used in DSP.
- Grasp the properties of numbers, functions, and complex arithmetic relevant to signal theory.
- Learn to reason about limits, convergence, and approximation — concepts central to spectral analysis and filter design.
- Develop the ability to translate intuitive engineering problems into precise mathematical statements.
Topics Covered
- Preface and Introduction — Why mathematics matters
- Numbers and Arithmetic — integers, rationals, real numbers
- Sets, Mappings, and the Infinite — basic set theory and infinity
- Geometry and Trigonometry — classical ideas and their modern roles
- Functions and Continuity — concept of function and limits
- Differential Calculus — derivatives and applications
- Integral Calculus — area, accumulation, and fundamental theorem
- Complex Numbers and Functions — algebra and geometry of complex plane
- Series and Convergence — infinite series and approximations
- Elementary Ideas of Mathematical Analysis — rigor and proof techniques
- Selected Topics — e.g., orthogonal functions, approximation (estimated coverage)
How It Compares
More conversational and example-driven than Rudin's Principles of Mathematical Analysis and less formal but more wide-ranging and accessible than Spivak's Calculus.
