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John Venn, the Man Behind the Diagrams

By Ernest Davis

John Venn: A Life in Logic. By Lukas M. Verburgt. University of Chicago Press, Chicago, IL, April 2022. 448 pages, $45.00.

John Venn: A Life in Logic. By Lukas Verburgt. Courtesy of University of Chicago Press.
Venn diagrams are everywhere: in scientific papers and corporate presentations, The New Yorker cartoons and social media memes. But who was the Venn who inspired these well-known graphics?

John Venn (1834-1923) was an English mathematician and philosopher who studied logic and probability theory. He is best known for his development and exposition of Venn diagrams, along with his advocacy for the frequentist interpretation of probability theory. He wrote three major books: The Logic of Chance in 1876, Symbolic Logic in 1881, and The Principles of Empirical or Inductive Logic in 1889; the first two were used as textbooks at the University of Cambridge and elsewhere until the 1920s. Venn taught at Cambridge—where he was involved in curriculum reform—from 1862 until his retirement in 1903. He also advocated for women’s suffrage and higher education for women. During his retirement, Venn documented histories of his family and of Cambridge, particularly for Alumni Cantabrigienses: a register of Cambridge alumni.

Lukas Verburgt’s new biography, John Venn: A Life in Logic, thoroughly explores Venn’s life, works, thought, and faith in the context of the times. Verburgt is a philosopher at the Netherlands Institute for Advanced Study in the Humanities and Social Sciences and the editor of John Venn: Unpublished Writings and Selected Correspondence, published in 2022.

The simplest and most familiar form of Venn diagrams—with two or three intersecting circles—was actually already in use when Venn was born. Leonhard Euler frequently employed this style of visual, and such schematics are sometimes known as “Euler diagrams.” However, Venn extended the idea in an 1880 article titled “On the Employment of Geometrical Diagrams for the Sensible Representation of Logical Propositions.” He showed that it is possible to construct a diagram that illustrates all intersections of four sets via either four ellipses or three circles and a horseshoe shape, as well as a diagram that can depict all intersections of five sets using four ellipses and an annulus (see Figure 1). He also demonstrated the use of shading and labeling to carry out more complex inferences.

Venn’s second major claim to fame was his formulation and advocacy of the frequentist interpretation of probability theory, which maintains that the probability of an event’s outcome is its frequency in a long series of trials. This concept was controversial in Venn’s own time and remains bitterly so to this day, 150 years later. John Stuart Mill and Charles Sanders Peirce both admired Venn’s work on probability. John Maynard Keynes dedicated his 1921 book, A Treatise on Probability, to Venn and warmly praised his efforts. In his 1956 book on Statistical Methods and Scientific Inference, Ronald Fisher credited Venn with “developing the concept of probability as an objective fact.” Yet Venn certainly had critics as well. Francis Herbert Bradley and Francis Edgeworth sharply criticized his theory. And in Aubrey Clayton’s 2021 Bayesian polemic entitled Bernoulli’s Fallacy, Venn plays the role of a minor villain who sets the stage for archfiends Fisher and Karl Pearson.

Venn’s Symbolic Logic became a standard textbook in mathematical logic and was still in use when Willard Van Orman Quine began to study the topic at Oberlin College in the late 1920s. The text is mostly an exposition of and commentary on George Boole’s theory of logic. Venn’s own view of logic never advanced much beyond that of Boole; he tended to regard other approaches as either notational variants or misdirected ventures. For instance, Venn reviewed Gottlob Frege’s groundbreaking 1879 Begriffsschrift for the philosophical journal Mind, but he entirely failed to appreciate its importance.

Venn was also an expert on the history of logic and an avid book collector; by 1889, he had assembled a library of 1,100 books on the subject, many of which were very rare. By way of comparison, the New York University library system has roughly 3,600 books on logic — of which only 80 hardcover and 200 online-access titles predate 1889.

Logicians such as Frege, Alfred North Whitehead, Bertrand Russell, and other contemporaries and successors were extraordinarily successful at using a combination of Fregean logic and Cantorian set theory to develop a mathematical account of logic that completely characterizes mathematical proof. This achievement has overshadowed alternative approaches to logic, which are now almost forgotten. However, the unification of mathematics and logic did, to some extent, require a change in definition for the scope of both terms. Logicians of the 1880s might well have questioned whether mathematical logic (as it is now called) should be classified as mathematics at all; they would have likely felt that it omits issues that they considered central to logic, such as induction. Modern mathematical logic does not solve all of the problems that engaged 19th-century logicians. Instead, it excludes them as not being part of “logic.”

Figure 1. Venn diagrams for four and five sets. Figure adapted from John Venn: A Life in Logic and courtesy of the University of Chicago Press.

Verburgt’s biography devotes considerable space to the evolution of Venn’s religious faith. Eight generations of his patrilineal ancestors were clergymen, and Venn’s father and grandfather had belonged to the Clapham Sect of the Protestant church’s Evangelical branch. The Clapham Sect was dedicated to reform, service to the poor, and abolitionism at the social level and earnest religiosity and moral self-improvement at the personal level (abolitionist William Wilberforce was a member of the sect). Venn thus grew up in an intensely religious atmosphere. While at Cambridge, the master and other students of his college were almost all likewise Evangelicals. After graduating, Venn even worked as a curate for several years. Gradually, however, his readings in science and philosophy—especially Mill’s 1859 On Liberty—led him to a more relaxed view. He resigned his holy orders in 1883 because he felt that he could no longer subscribe to all the tenets of the Church of England. Nevertheless, Venn continued to be a believing Christian, and his do-gooder, rather Puritanical mindset persisted in many ways. All his life, he read mostly French novels rather than English ones; he could justify the reading of French novels as study of the language, whereas English novels were pure self-indulgence.

Yet Venn did allow himself to enjoy a number of pleasant hobbies. He was an avid hiker and mountain climber in Wales and the Alps, as well as an accomplished gardener and an amateur expert in the botany of the local wild flora. Around 1907, he and his son built and patented a machine for bowling cricket balls.

Any reader who harbors nostalgic illusions about “the good old days” at Cambridge in the first half of the 19th century will be somewhat shocked by Verburgt’s account of Venn’s years as a student, which were both physically miserable and intellectually narrow. The focus of Venn’s studies remained exclusively on classic literature, theology, and mathematics (elementary algebra, geometry, trigonometry, and calculus), and students had to pass the Mathematical Tripos in order to graduate. In fact, Venn found cramming for the Tripos so unpleasant that he abandoned mathematics for decades and had to relearn many things when he returned to the field. 

While teaching at Cambridge, Venn was heavily involved in the creation of a curriculum of “moral sciences” — a concept that included moral philosophy, history, political economy, and law. He also helped to establish a series of university-level lectures and examinations for female students, which were associated with Cambridge but not part of it.

Verburgt’s writing often assumes that the reader is quite knowledgeable—at least, more knowledgeable than myself—on subjects like the debates among 19th-century logicians, the organization of the University of Cambridge, and the tenets of the Clapham Sect. Consequently, the discussion can at times become hard to follow. Chapter 11, which addresses “Empirical Logic,” discusses Venn’s own book on the subject; but I must admit that when I finished the chapter, I had no clearer idea than when I started of the intent of the field of “empirical logic,” the beliefs of the various people who worked in it, or its relation to more recent and familiar forms of the philosophy of science.

All in all, Venn comes across as a significant and admirable figure, though not quite a great one. His intellect was capable, but neither particularly original nor sharply penetrating. Nonetheless, Verburgt’s book is well worth reading. It sheds new light on one side of intellectual life in Victorian England. And if nothing else, the exploration of this comparatively unfruitful view of logic inspires a renewed appreciation for the approach that ultimately prevailed by reminding readers that, 150 years ago, it was not even clear what questions logic should address — let alone how it could answer them.

Ernest Davis is a professor of computer science at New York University’s Courant Institute of Mathematical Sciences.

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