An Introduction to Mathematics

Summary

An Introduction to Mathematics is a short book written for general readers in the Home University Library of Modern Knowledge, a popular series of compact introductions published in London by Williams and Norgate and in New York by Henry Holt in 1911. Whitehead wrote it just after the first volume of Principia Mathematica, and it was his first book for a wide public. Its aim is not to teach techniques but to show what mathematics is about and why its ideas matter to science and to ordinary thought. The opening chapter argues that mathematics is the science of abstraction. When we say that one and one make two, we speak of no particular things at all, and this generality is the source of mathematics' power. The next chapters introduce the variable, the letter that stands for any number, and show how the idea of any and some, expressed in algebra, lets a single statement cover endless cases. Whitehead then explains how mathematics is applied: physical laws such as those of motion are mathematical relations among measured quantities, and he gives a chapter to dynamics and the work of Galileo and Newton. A chapter on symbolism argues that good notation relieves the mind of routine work so that it can attend to new problems. The middle of the book traces generalizations of number, from whole numbers to fractions, negative numbers, irrational numbers and the so-called imaginary numbers, which Whitehead treats in two chapters as an extension that turned out to have practical uses. Chapters on coordinate geometry and conic sections show how Descartes's method joins algebra and geometry, so that curves such as the ellipse, parabola and hyperbola can be studied through equations. A chapter on functions presents the idea of one quantity depending on another. The later chapters treat periodicity in nature, from the seasons and the tides to vibrations and waves, and the trigonometrical functions that describe them; series, including the problem of convergence and the meaning of an infinite sum; the differential calculus, explained through limits and rates of change; geometry as the study of space and its axioms; and quantity, on how measurement is possible. A brief note recommends books for further study. The book has stayed in print, has been praised by mathematicians for its clarity, and is often recommended to students who want to see the shape of the subject before its details. The library does not store its full text: the work is in the public domain in the United States, but its digital edition is typeset mathematics with formulas and diagrams that do not survive conversion to plain text, so this record gives a summary and guide only.

Plot or Argument

• Chapter I: The abstract nature of mathematics

  Mathematics deals with properties that hold of any things whatever, and its abstractness is the reason it applies so widely.

• Chapters II-V: Variables, application, dynamics and symbolism

  The variable expresses the ideas of any and some; physical laws are relations among variables; Galileo's and Newton's dynamics show the method at work; good symbols economize thought.

• Chapters VI-VIII: Generalizations of number

  Number is extended step by step to fractions, negatives, irrationals and imaginary numbers, each extension justified by the operations it makes possible.

• Chapters IX-XI: Coordinate geometry, conics and functions

  Descartes's coordinates let equations describe curves; the conic sections serve as examples; the function names the dependence of one variable on another.

• Chapters XII-XIII: Periodicity and trigonometry

  Repeating phenomena in nature, from day and year to vibrations, call for periodic functions, and trigonometry supplies them.

• Chapters XIV-XV: Series and the calculus

  Infinite series and the question of convergence lead to the idea of a limit, on which Whitehead builds an account of the differential calculus.

• Chapters XVI-XVII: Geometry and quantity

  Geometry is treated as the study of space from axioms, and the final chapter asks what makes a quantity measurable.

Lessons

• The power of abstraction

  Whitehead argues in his first chapter that mathematics is useful precisely because it ignores the particular things it counts or measures. A statement about any number applies to apples, planets and electric currents alike. The book invites readers to see abstraction as a strength rather than a retreat from reality.

  • What does Whitehead mean by calling mathematics abstract?
  • Why does abstraction make mathematics more, not less, applicable to the world?
  • Are there dangers in treating real things only through their measurable properties?

• Notation as a tool of thought

  In the chapter on symbolism Whitehead argues that good symbols let the mind carry out routine operations almost mechanically, freeing attention for new problems. He connects this to the general advance of civilization through habits and tools that do work for us.

  • How does algebraic notation make a problem easier than writing it out in words?
  • What other tools or habits save thought in the way Whitehead describes?
  • Can relying on routine ever hinder understanding?

• Periodicity in nature

  Whitehead's chapter on periodicity shows that days, seasons, tides, heartbeats and sound waves share a mathematical form. Trigonometric functions, first developed for measuring triangles, turn out to describe repetition throughout nature.

  • What examples of periodic events does Whitehead give?
  • Why should functions invented for triangles describe vibrations and waves?
  • What does the fit between mathematics and nature suggest about the order of the world?

Context

In the decades before 1911, mathematicians had placed analysis on rigorous foundations through the work of Cauchy, Weierstrass, Dedekind and Cantor, and Frege, Peano, Russell and Whitehead were attempting to ground arithmetic in logic. At the same time, British publishers were producing inexpensive series of introductions for the growing reading public; the Home University Library, begun in 1911, enlisted leading scholars to write short books on their subjects.

Composition

Whitehead wrote the book while Principia Mathematica was in press and while he was moving from Cambridge to London. It appeared in 1911 in the Home University Library (Williams and Norgate, London; Henry Holt, New York) in seventeen chapters with a note on books. Reprints remain available.

Reception

The book was widely read and is still cited as a model of explaining mathematics to non-specialists; reviewers have praised its clarity and its account of why mathematical ideas matter. Some of its examples and terminology reflect the state of mathematics in 1911, and students may notice that later developments, such as modern set theory and computing, are absent.

Comparisons

• euclid-elements-book-i

  Euclid's axiomatic geometry is the classical model behind Whitehead's chapter on geometry, which treats space as a system derived from axioms.

• russell-problems-of-philosophy

  Published a year later by Whitehead's collaborator, Russell's short book for the same kind of general reader asks what we can know, including our knowledge of mathematical truths.

• principia

  Newton's laws of motion are Whitehead's chief example of mathematics applied to nature, and his chapter on the calculus explains the method Newton invented.

• hardy-a-mathematicians-apology

  Hardy, a younger Cambridge mathematician, defends pure mathematics for its beauty, where Whitehead stresses how abstract ideas prove useful.

• plato-meno

  Socrates draws a geometrical truth out of an untaught slave boy; Whitehead likewise suggests that mathematical understanding lies within reach of any careful reader.

Quotes

For Students

A reading companion for classical Christian high school students. See the student guide.

Reading Guide

Watch for

• The opening argument that mathematics deals with any things whatever, not particular things.
• How the variable, introduced in Chapter II, becomes the key idea for everything after.
• The step-by-step justification of each new kind of number in Chapters VI-VIII.
• The examples of repetition in nature in the chapter on periodicity.
• How the idea of a limit links the chapters on series and on the calculus.

Key terms and names

Abstraction

Whitehead's name for considering properties apart from the particular things that have them, the defining activity of mathematics.

Variable

A symbol standing for any member of a class of numbers, which lets one statement cover endless cases.

Imaginary number

A number whose square is negative; Whitehead shows how this extension of number became useful.

Coordinate geometry

Descartes's method of locating points by numbers so that curves can be described by equations.

Periodic function

A function that repeats its values at regular intervals, used to describe cycles and vibrations.

Convergence

The property of an infinite series whose sums approach a fixed value, central to the chapter on series.

Seminar Questions

1. Comprehension What does Whitehead mean when he says mathematics is abstract, and what example does he use to show it?
2. Comprehension How does Descartes's coordinate method connect algebra and geometry, according to Chapters IX and X?
3. Interpretation Why does Whitehead think good notation is so important to the progress of mathematics?
4. Interpretation How does Whitehead justify calling fractions, negatives and imaginary quantities numbers?
5. Interpretation What does the chapter on periodicity suggest about the relation between mathematics and the natural world?
6. Big Question Why should abstract mathematics, worked out on paper, describe the motions of planets and the vibrations of strings?

Faith Connections

• Wisdom of Solomon 11:20 (Scripture)

  Thou hast ordered all things in measure and number and weight. Augustine and medieval writers often cited this verse; Whitehead's account of the mathematical order in nature gives it fresh illustration.

• Psalm 104 (Scripture)

  The psalmist praises God for appointing the moon for seasons and the sun its going down, the same cycles that Whitehead studies in his chapter on periodicity.

• Augustine, On Free Choice of the Will II (Church Fathers)

  Augustine argues that the truths of number are unchanging and common to all minds, and treats them as a path toward the eternal Truth, a theological reflection on the abstraction Whitehead describes.

Logic and Rhetoric

Central argument: Mathematics studies relations that hold for any things whatever. Relations of this kind are found throughout nature, in motion, number, space and repetition. Therefore the most abstract science is also the most widely applicable, and understanding its leading ideas illuminates both science and ordinary thought.

Rhetorical device: argument from example. In the chapter on periodicity Whitehead piles up cases, from days and seasons to tides and vibrations, before introducing the trigonometric functions that unite them.

Debate: Resolved: that mathematics is discovered rather than invented.

Writing Prompts

Essay: Using Whitehead's chapters on variables and on periodicity, explain why abstraction makes mathematics useful in the natural sciences.

Comparison with Elements, Book I: Compare Euclid's method of proceeding from definitions and postulates with Whitehead's account of geometry and of mathematics as abstraction. What does each think mathematics is about?

Medical Lens

The book does not treat medicine. Its chapter on periodicity mentions rhythmic natural processes in general terms, but there is no medical content of substance.

Creative Prompts

• Classroom

  Collect five examples of periodic events from daily life and science, graph one of them, and discuss Whitehead's claim that one mathematical form fits them all.

• Essay

  Explain Whitehead's view that abstraction is the source of mathematics' usefulness, using an example from physics or economics.

• Dialogue

  Write a conversation in which Whitehead explains to a skeptical student why imaginary numbers deserve the name number.

Further Reading

• The Aims of Education and Other Essays, Alfred North Whitehead

• Science and the Modern World, Alfred North Whitehead

• Introduction to Mathematical Philosophy, Bertrand Russell

• What Is Mathematics?, Richard Courant and Herbert Robbins