What Is Mathematics? An Elementary Approach to Ideas and Methods

What Is Mathematics? An Elementary Approach to Ideas and Methods

Ian Stewart, Richard Courant, Herbert Robbins

4.26(1997 readers)
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.

Publisher

Oxford University Press

Publication Date

7/18/1996

ISBN

9780195105193

Pages

592

Categories

Mathematics

Questions & Answers

The essence of mathematics lies in its abstract nature, focusing on the relationships between undefined objects and the rules governing their operations. Unlike other disciplines, mathematics doesn't rely on empirical evidence or physical reality; instead, it explores logical structures and patterns. This abstractness allows mathematics to be applied across various fields, from physics to economics. Key differences include:

  1. Abstractness: Mathematics deals with abstract concepts and relationships, not physical objects.
  2. Logic: It is based on logical reasoning and proof, ensuring conclusions are valid and consistent.
  3. Universality: Its principles are universally applicable, transcending cultural and temporal boundaries.
  4. Precision: Mathematics provides precise, unambiguous answers, often to complex questions.
  5. Interdisciplinary: It serves as a foundation for many other disciplines, contributing to their development and understanding.

The natural numbers, 1, 2, 3, ..., serve as the foundation of mathematics due to their simplicity and universality. They are the building blocks for counting and are the basis for all other number systems. Arithmetic, the study of the natural numbers, establishes the rules for operations like addition, subtraction, multiplication, and division, which are fundamental to all mathematical calculations.

Mathematical induction is a crucial principle in mathematics that allows us to prove statements about all natural numbers. It works by proving that a statement is true for the first natural number (usually 1) and then showing that if the statement is true for any given natural number n, it must also be true for the next number, n+1. This process guarantees the truth of the statement for all natural numbers, making it a powerful tool for proving various mathematical theorems and properties. The implications of mathematical induction are vast, as it enables the derivation of general results from specific cases, simplifying the process of mathematical discovery and proof.

Irrational numbers, limits, and the concept of infinity are foundational in mathematics, each playing a crucial role:

  1. Irrational Numbers: They extend the number system beyond rational numbers, allowing for a more precise description of real-world quantities. They are essential for understanding concepts like the length of a line segment or the area of a shape, and they enable the development of calculus and other advanced mathematical fields.

  2. Limits: Limits provide a precise way to describe the behavior of functions and sequences as they approach a certain value. This concept is vital for defining derivatives and integrals in calculus, which are fundamental tools for understanding change and area.

  3. Infinity: The concept of infinity is used to describe sets with an uncountable number of elements, like the set of real numbers. It helps in understanding the different types of infinity, such as countable and uncountable, and is crucial in set theory and the study of the foundations of mathematics. Infinity also plays a role in the analysis of infinite series and in understanding the behavior of functions as they approach infinity.

Functions and limits are foundational in modern mathematics, shaping its structure and applications. Functions describe relationships between variables, while limits define the behavior of functions as their inputs approach specific values. Key concepts include:

  1. Function: A rule that associates each input with a single output. This concept is central to understanding various phenomena, from physics to economics.

  2. Limit: The value a function approaches as its input approaches a certain value. This is crucial for understanding continuity, derivatives, and integrals.

  3. Continuity: A function is continuous if its limit exists and is equal to the function's value at that point. This is essential for understanding smoothness in various contexts.

  4. Derivative: The rate of change of a function with respect to its input. It is a limit of the difference quotient and is fundamental in calculus and physics.

  5. Integral: The area under a curve or the sum of an infinite series of infinitesimally small rectangles. It is a limit of a sum and is essential in calculus and physics.

These concepts enable the formulation and solution of complex problems across various disciplines, making functions and limits indispensable tools in modern mathematics.

The main ideas and techniques in geometry focus on understanding the properties of shapes and spaces. Central to this are:

  1. Axiomatic Systems: Starting with undefined terms and axioms, Euclidean geometry builds a logical framework for understanding shapes and their relationships. This approach allows for the derivation of theorems through deductive reasoning.

  2. Coordinate Geometry: Introducing numerical coordinates, this method translates geometric problems into algebraic equations, facilitating calculations and the study of curves and surfaces.

  3. Projective Geometry: This branch extends Euclidean geometry by introducing "points at infinity," allowing for the study of properties invariant under projection, like concurrency and collinearity.

  4. Non-Euclidean Geometries: These explore geometries with different axioms, like hyperbolic and elliptic geometries, which challenge the parallel postulate and lead to different properties of space.

  5. Topological Geometry: This field studies the properties of spaces that remain unchanged under continuous deformations, like stretching and bending, without tearing or gluing.

These techniques contribute to understanding mathematical structures by:

  • Generalizing and Extending: They allow for the exploration of a wider range of shapes and spaces beyond the limitations of Euclidean geometry.
  • Building Connections: They link geometry with other mathematical fields, such as algebra and analysis, enriching our understanding of abstract concepts.
  • Applying to Real-World Problems: They provide tools for solving practical problems in physics, engineering, and computer science.

Reader Reviews

Loading comments...