John Venn

Chapter 1 · Four Generations of Clergymen

A family with a settled plan for him

1834 – 1862 · Hull · London · Cambridge

John Venn was born at Hull in 1834 into a family that had produced Church of England clergymen for generations, and prominent ones. His grandfather John Venn was rector of Clapham and a leading figure in the Clapham Sect, the Evangelical circle around William Wilberforce that campaigned for the abolition of the slave trade. His father Henry was secretary of the Church Missionary Society for thirty years and one of the most influential figures in Victorian missionary work.

His mother died when he was three. He was brought up strictly, in an atmosphere where a career in the Church was not so much expected as assumed, and he later wrote that he had been so little accustomed to independent thought that the assumption went unexamined.

He went up to Gonville and Caius College, Cambridge, in 1853, took a good degree in mathematics in 1857, and was elected to a fellowship of the college the same year. He held that fellowship for the next sixty-six years, until his death.

He was ordained deacon in 1858 and priest in 1859, and served as a curate in parishes near London for a few years. Then, in 1862, he returned to Cambridge as a lecturer in moral sciences — which at Cambridge meant logic, probability and philosophy — and the direction of his life changed. He never left again.

What happened next is the interesting part, and it takes twenty years.

Why this matters

A career mapped out by four generations of family, and a young man who went along with it until he had the tools to examine it.

A clergyman's son in a family of clergymen. What would you ask him?

Ask Venn

• “What was it like to be raised for a career you never chose?”
• “What did your grandfather's circle achieve on the slave trade?”
• “Why did Cambridge call logic one of the 'moral sciences'?”
• “What was a curate's daily work actually like?”
• “What made you go back to Cambridge?”

Chapter 2 · The Logic of Chance

A probability is a proportion, not a feeling

1866 · Cambridge

Venn's first book, published in 1866, took on a question that sounds simple and is not: what does it MEAN to say that something has a probability of one in six?

The view he attacked was that a probability measures a degree of belief — how confident a reasonable person should be. Venn thought this hopeless, because two reasonable people can be differently confident and there is no way to settle between them.

His alternative is that a probability is a PROPORTION in a long series of similar cases. To say the die has a one-in-six chance is to say that in a long run of throws, roughly one throw in six shows that face, and the longer the run the closer the proportion settles. The probability is a fact about the series, not about anybody's state of mind.

This has a real advantage: it is checkable. You can go and count. It also has a real cost, and Venn knew it. If a probability is a proportion in a series, then a one-off event that will never be repeated has no probability at all — the probability that a particular person will live to eighty, or that a particular war will break out, becomes strictly meaningless unless you can say which series of similar cases it belongs to, and usually you cannot.

The argument he started is not over. The frequency view dominated statistics for a century; the degree-of-belief view came back strongly in the twentieth century and is now everywhere in machine learning. Both sides are still arguing about single events.

Why this matters

A question that sounds like a definition — what a probability IS — turns out to change what you can and cannot say with it.

A probability as something you could go and count. What would you ask him?

Ask Venn

• “What is a probability, if it is not a degree of belief?”
• “What is the probability of something that happens only once?”
• “How long does a 'long run' have to be?”
• “Why does it matter which definition you take?”
• “Who was on the other side of this argument?”

Chapter 3 · The 1880 Paper

Every compartment, including the empty ones

1880 – 1881 · Cambridge

In July 1880 Venn published in the Philosophical Magazine a paper called 'On the Diagrammatic and Mechanical Representation of Propositions and Reasonings'. It contains the figure everybody knows, and almost nobody uses as he intended.

The circles were not new. Euler had used overlapping circles a century earlier, and others before him. Venn says so plainly; his paper is largely a critical survey of what had been tried.

What is his is a rule. The diagram must show EVERY possible combination of the classes, whether or not anything falls into any given one. Two classes give four compartments: in both, in the first only, in the second only, and in neither. Three classes give eight. The compartment for 'neither' — the region inside the frame but outside all the circles — is one of them and must be drawn.

Then, and this is the part that gets forgotten, you SHADE the compartments that a proposition declares empty. 'No cats are dogs' is represented not by drawing two separate circles but by drawing them overlapping and shading the overlap out.

Euler's diagrams drew only what was occupied, and that is precisely why they cannot represent a statement that merely denies. Venn's version can, because the possibilities are all on the page and the diagram says which ones are ruled out.

The distinction matters more than it sounds. A logic diagram in Venn's hands is a statement about what CAN exist. It is not a tally of how many things do. Almost every modern use of it — counting how many students take French and Spanish — treats it as exactly the counting device he was not building, which works perfectly well but is somebody else's idea.

Why this matters

The famous part of the diagram was not his, and the part that was his — every compartment drawn, the empty ones shaded — is the part everybody drops.

A rule about what must be drawn, not a picture. What would you ask him?

Ask Venn

• “What is actually yours in the diagram, if the circles are not?”
• “Why must a compartment be drawn when nothing falls into it?”
• “What does the shading mean, and why is it the point?”
• “What is wrong with Euler's version?”
• “Is your diagram a way of counting things?”

Chapter 4 · Two Machines

A logical engine, and a bowling machine that took four wickets

1881 – 1890s · Cambridge

The title of the 1880 paper mentions the MECHANICAL representation of propositions as well as the diagrammatic, and Venn meant it literally. He built a logical machine, a device of rods and plates on which the compartments could be physically knocked out as propositions ruled them empty, so that the consequences of a set of statements could be read off. It survives in Cambridge. He was working in a tradition — Jevons had built a 'logical piano' a decade earlier — and the interest was serious: if logic can be mechanised, then reasoning is in some sense a procedure, which is an idea with a very long future ahead of it.

He also built a machine for bowling at cricketers. This was a proper piece of engineering with adjustable settings, and the Australian touring team visited Cambridge in 1909 and faced it. It is reported to have clean bowled one of their leading batsmen four times. Venn was in his mid-seventies.

The two machines are less unrelated than they look. Both come from the same instinct: that a process people carry out by judgement and feel can be analysed into steps and then embodied in apparatus that carries out the steps reliably. That is what his diagram does to a logical argument, and it is what the bowling machine does to a spin bowler's arm.

He was also a keen mountaineer and botanist, and by all accounts a rather dry and understated man who found the whole business of being famous for a picture faintly absurd.

Why this matters

The same instinct produced both machines: take something done by judgement, break it into steps, and build apparatus that performs the steps.

A logician who built machines, one of them for cricket. What would you ask him?

Ask Venn

• “What did your logical machine actually do?”
• “Can reasoning really be reduced to a mechanism?”
• “Did your bowling machine really beat an Australian batsman four times?”
• “What do the two machines have in common?”
• “How did it feel to become famous for a diagram?”

Chapter 5 · Resigning Orders, and Counting a University

An honest exit and a very long index

1883 – 1923 · Cambridge

In 1883 Venn resigned his clerical orders. He had come to the view that he could not honestly subscribe to the Thirty-Nine Articles of the Church of England, and under a recent Act it had become possible for a clergyman to relinquish orders formally rather than simply lapse. He described himself afterwards as remaining a man of broad Christian convictions; what he could no longer do was affirm a specific set of doctrinal propositions.

There is something fitting in it. A man who had spent twenty years insisting that you say precisely what a proposition asserts before you assent to it applied the standard to his own position and found he could not meet it. He kept his fellowship at Caius, and was elected a Fellow of the Royal Society the same year — for his work in logic, not for the diagram.

The last quarter-century of his life went into an extraordinary project. With his son John Archibald Venn he compiled the Alumni Cantabrigienses: a biographical register of every known member of the University of Cambridge from its foundation to modern times. It runs to ten volumes and hundreds of thousands of entries, assembled from college admission books, parish registers and wills. His son completed it after his death.

It is not glamorous work. It is the same instinct again — build the framework, put everything in it, leave nothing out because it happens to be dull or empty — applied to four centuries of people.

He died at Cambridge in 1923, aged eighty-eight, having been a fellow of one college for sixty-six years.

Why this matters

He applied his own standard about assenting to propositions to his own beliefs, and paid for it — and then spent twenty-five years indexing a university.

An honest resignation and a ten-volume index. What would you ask him?

Ask Venn

• “Why did you resign your orders?”
• “Did your work in logic change what you could believe?”
• “Why spend twenty-five years listing every Cambridge student?”
• “How do you even find a record of somebody from 1550?”
• “Sixty-six years in one college. Did you ever want to leave?”

What Venn changed

Venn's rule — that a diagram must show every combination of the classes, including the empty ones and the region outside them all, and then shade what a proposition denies — turned a picture into an instrument of proof, and the figure is now one of the few pieces of technical notation everybody recognises. His Logic of Chance made the frequency account of probability the dominant view for a century. The Alumni Cantabrigienses remains a standard reference for four centuries of British lives.

A debate that continues

He did not invent the overlapping circles; Euler and others used them before him, and Venn's own paper says so. The shading, which is his actual contribution and the reason the diagram can represent a denial, is almost never used in modern practice. His frequency account of probability leaves single unrepeatable events without any probability at all, a weakness he acknowledged and which the twentieth century revived the rival view to address.

Keep exploring — ask Venn

• “What does it mean to say a single, unrepeatable event has a probability?”
• “Why does the most useful part of an idea sometimes disappear from its popular version?”
• “Can reasoning be reduced to a mechanism, and what is lost if it can?”

Related lives

• Alan Turing — Father of Computer Science
• Thomas Bayes — The Question Turned Round
• Carl Friedrich Gauss — Few, But Ripe

Related themes

Sets and logic · Probability · Mechanical reasoning