A Mathematician's Apology

G. H. Hardy

Nonfiction | Biography | Adult | Published in 1940

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Summary

Background

Chapter Summaries & Analyses

Foreword

Chapters 1-9

Chapters 10-18

Chapter 19-Note

Key Figures

Themes

Index of Terms

Important Quotes

Essay Topics

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Discussion Questions

Index of Terms

Analytic Geometry

Hardy argued that mathematics exists as a separate reality and that discoveries about math are merely observations of that reality through the use of thought. He used analytic geometry (which he called “real” geometry) as an example. Analytic geometry uses graphs—formally, Cartesian coordinate systems—to study geometric forms. It’s the type of geometry most familiar to the public, the one used by engineers when they calculate orbits or build bridges. The lines, circles, triangles, and other geometric elements are drawn on a graph, and their lengths are precisely determined by the graph’s coordinates, usually the horizontal X, or abscissa, line and the vertical Y, or ordinate, line.

Although analytic geometry is highly useful in the real world, Hardy pointed out that its theorems are entirely independent of physical reality. As an example, he noted that if a drawing of a triangle is badly executed or distorted in some way, the principles that underlie the drawing remain unchanged.

Cricket

Played with balls and bats, cricket is an English game that has become popular the world over. Vaguely similar to baseball, cricket includes a bowler who tries to knock the top off one of a “wicket” defended by a batter who tries to hit the ball deep into the outfield and score “runs” by running between his wicket and another one 20 meters away. Depending on the rules, cricket games can last as little as three hours or as long as five days. Hardy was an avid amateur cricket player who followed the game closely his entire life. Cricket has a certain elegance that charmed Hardy, who also prized elegance in mathematics.

Number Theory

Aside from math analysis, Hardy’s specialty was number theory, the study of the basic rules and properties of integers and arithmetic. Number theory is fundamental to all other math; as such, Hardy regarded it as a pure science without any direct application in the real world. (Indirectly, of course, number theory powers the mathematical theorems that do apply to technology and other aspects of human civilization.) His work in this field, in coordination with colleagues John Littlewood and Srinivasa Ramanujan, made up a large part of Hardy’s great success as a mathematician.

Pure Mathematics

What Hardy called “real” mathematics, pure mathematics is the pursuit of mathematical concepts that have no obvious application in real life. These include arcane discoveries about numbers, interesting geometries that exist only in dimensions beyond our own, or any mathematical theorem or postulate that has no clear use in science and engineering. As a professor, Hardy worked mainly in pure mathematics; he disapproved of how applied mathematics, the study of formulas useful to humanity, so often finds use in warfare. (During his time, it wasn’t yet clear that applied math also can improve medical care, food supplies, and most of the other wonders of modern technology.)

Hardy’s main work was in number theory and mathematical analysis. Math analysis does have important applications—its centerpiece, calculus, is used in everything from business revenue projections to space flight—but Hardy’s work was in areas of analysis that lacked obvious application.

Trinity College

Cambridge University, one of the oldest and most distinguished academic institutions in the world, contains more than 30 individual colleges, one of which, Trinity College, was Hardy’s professional home for more than a quarter century. There, he made many of the mathematics discoveries that added to his fame. Trinity, the university’s largest and, by many measures, top-performing college, was founded in the 1540s and is steeped in ancient formalities. Many of these traditions, such as the practice of religion, irritated Hardy, who sidestepped or rebelled against them. Although he enjoyed the status and intellectual riches of the place, he chafed at its snobbism and ritualistic formalities.