An Introduction to Mathematics

Overview

A hundred and ninety small pages in the Home University Library, published in 1911, the same shilling series that produced Russell’s Problems of Philosophy the following year. Whitehead was fifty, had just finished the second volume of Principia Mathematica with Russell, and was teaching at Cambridge.

The contrast between the two projects is the point of interest. Principia takes several hundred pages to prove that one plus one equals two, from axioms, in a symbolism almost nobody can read. This book assumes no algebra at the start, sets essentially no exercises, and is about what the whole enterprise is for.

It has never been out of print and Dover keeps it cheap. The text is unrevised since 1911, which shows only in the final chapters, where the physics is pre-relativistic and the remarks about the ether are dated.

The prose is dense but not technical. Whitehead writes long, carefully built sentences with a Victorian formality, and the reader who has been told this is an easy book will find it demands attention. What it does not demand is prior mathematics.

The argument

Chapters 1 to 4: abstraction and the variable. He begins by warning the reader that mathematics is apt to start in disappointment, because the interesting applications cannot be reached until a great deal of apparently pointless generality has been established.

Then the definition. Mathematics is the study of the abstract, and it became a science when somebody first proved something about any things rather than about particular things. Two apples and two days have nothing in common except a form, and mathematics studies that form and nothing else.

From this comes the book’s central claim. The single greatest step in the history of the subject is the introduction of the variable: the shift from asking what number added to three gives seven, to stating something that holds for every number whatever. Algebra is not arithmetic with letters; it is the science of what is true generally.

Whitehead then works through what this permits. Functions, which relate one variable to another. The idea of a class, a set of things picked out by a property. Equations, which are conditions on variables rather than puzzles about unknowns.

Chapter 5: notation. The most quoted chapter in the book, and a genuinely important argument.

He compares Roman and Arabic numerals to show that a good symbolism does not merely record a thought but makes further thoughts possible: long division is straightforward in one system and nearly impossible in the other, though the numbers are the same. Notation, he says, relieves the brain of all unnecessary work and frees it for more advanced problems.

Then the sentence everyone quotes, that it is a profoundly erroneous truism that we should cultivate the habit of thinking of what we are doing, and that civilisation advances by extending the number of operations we can perform without thinking about them. Cavalry charges at decisive moments, he adds, and so do good notations.

Chapters 6 to 9: number and space. Generalisations of number, in order of increasing strangeness: the negative numbers, the fractions, the irrationals, and finally the imaginary numbers. Whitehead takes care with the last, showing that i is neither mysterious nor a fiction but an element of a perfectly ordinary two-dimensional system, and that the objection to it is a failure of imagination about what number means.

Then coordinate geometry: the discovery, credited to Descartes with the aside that everything important has been said before by somebody who did not discover it, that a curve is an equation and an equation is a curve. This is the union of the two ancient branches and the foundation of everything after.

Chapters 10 to 13: periodicity and trigonometry. Whitehead’s pedagogical move here is unusual. Rather than presenting trigonometry as the study of triangles, he presents it as the study of periodic functions, things that repeat, and then shows how much of nature is periodic: days, seasons, tides, pendulums, vibrating strings, sound waves, light.

The chapter on conic sections treats the ellipse, parabola and hyperbola as the shapes that appear when a plane meets a cone and also as the possible orbits of a body under an inverse-square attraction, and he pauses on the fact that Apollonius worked them out mathematically eighteen hundred years before anyone found them in the sky.

Chapters 14 to 16: the calculus. The climax of the book, built slowly.

He begins with the problem of instantaneous rate of change: a body’s speed at an instant, when at an instant it has travelled no distance in no time. The answer requires the idea of a limit, and Whitehead spends real effort on it, because the idea is subtle and was not made rigorous until the nineteenth century, a century and a half after Newton and Leibniz had been using it successfully.

He then shows the derivative as the slope of a tangent, the integral as an area, and the fundamental theorem that connects them. There is very little calculation. What he wants the reader to see is that one idea, the limiting value of a ratio as both terms shrink toward nothing, answers the problem of tangents, the problem of areas, the problem of instantaneous velocity and the problem of planetary motion, because all four are the same problem in different dress.

The book closes with a chapter on the conceptual basis of geometry and a short account of the logical foundations, which is as close as it gets to Principia.

How to read it

Read chapters 1, 2 and 5 whatever else you do. The account of abstraction, the argument about the variable, and the essay on notation are the durable part and they are about forty pages.

Do not expect to learn to do mathematics from it. There are no drills and few worked examples, and Whitehead says in the first chapter that this is not his purpose. It is a book about what the subject is, for someone who wants to understand its shape.

Go slowly through the limit chapters. If you have never met the calculus, this is the first genuinely difficult passage, and Whitehead’s exposition is careful enough to repay rereading a paragraph three times.

Note what has dated and what has not. The physics at the end predates relativity, and the confident remarks about the ether are of their moment. The philosophy of mathematics is not dated at all.

Read it beside Russell’s Problems of Philosophy, published the next year in the same series by his collaborator. The two books are a matched pair, written in the same idiom for the same readership by the two men who had just spent a decade on the same project.

And note the epigram about doing things without thinking, which has had a long second life in writing about habit, expertise and interface design. It is usually quoted without the context, which is a technical argument about symbolism.

Questions it raises

• If mathematics is the study of pure abstraction, why does it describe the physical world at all?
• Whitehead calls the variable the greatest step ever taken in the subject. What exactly does it add that arithmetic lacks?
• Civilisation advances by what we can do without thinking. Where does that argument stop being true?
• Imaginary numbers were resisted for centuries and are now routine. What makes a mathematical object acceptable?
• Apollonius described the conic sections long before anyone knew planets moved on them. Is that luck, or evidence about the nature of mathematics?
• The calculus worked for a hundred and fifty years before it was made rigorous. What does that say about the role of proof?