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.

Top 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.

• 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.

• 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.

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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.