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Article written by Madeleine Hall for John Venn's centenary

Posted 11th April 2023
by John

Madeleine Hall
The improbable genius of John Venn
How a priest changed the face of mathematics
• 4 April 2023, 5:00am
• From Spectator Life
There aren’t many mathematicians who can claim to have bowled out Australia’s number one batsman. But then John Venn, who died 100 years ago today, was no ordinary scholar.

Born in Hull and brought up in Highgate, he was also an Anglican priest – the ninth consecutive one in his family – with a magnificent Victorian beard. He won gardening prizes for his roses and white carrots. He was a keen advocate of women’s rights. And as the founding father of Venn diagrams, still the world’s most beloved tool for representing set-relationships, he can probably boast greater name-recognition than any other modern mathematician.
Next time you’re in Cambridge, pop into Gonville and Caius College, where Venn was a Fellow and President. In the dining hall, you can admire the handsome stained-glass window of three overlapping circles in blue, orange and purple. It’s a fitting tribute to Dr Venn.

On top of all this, he was a prolific inventor who once built a machine for firing cricket balls at cowering batsmen. When the Australians visited Cambridge in 1909, they couldn’t resist trying it out. Their star batsman, the great Victor Trumper, strode up to the crease and raised his bat. Boom! Venn’s machine smashed his wicket to smithereens. Irritated, Trumper dismissed this as a one-off. Again he lifted his bat.

Before concluding that story, though, I want to turn to Venn’s greatest achievement, one that was even more impressive than his diagrams, or his device for taking Aussies down a peg. It was The Logic of Chance, the book he published in 1866 on the theory of probability.
Probability is a relatively young field in mathematics. Geometry goes back to the likes of Euclid in the 4th century BC, but probability theory only really got going in the 17th century, after the gambler Chevalier de Méré posed a problem concerning a dice game to Blaise Pascal (he of the triangle and the wager). Pascal duly consulted with Pierre de Fermat (he of the last theorem), and together they developed the classical theory of probability.

He built a machine for firing cricket balls at cowering batsmen. When the Australians visited Cambridge in 1909, their star batsman, Victor Trumper, couldn’t resist trying it out. Boom! Venn’s machine smashed his wicket to smithereens

Their classical approach had its limitations, however. It depended on all of the outcomes being equally likely, like tossing a coin or rolling a dice. When this wasn’t the case, you ran into problems. You couldn’t use classical probability theory to determine the chance that a randomly selected man was called John, for instance, or that he would die on 4 April.

Venn’s pioneering book was the first systematic account of what became known as the frequentist approach:
determining probability by how often something has been seen to occur, with a margin of error proportional to the size of the dataset. This completely inverted the logic of the classical approach, which centres on the Law of Large Numbers: the more trials you do, the closer the proportions of results get to the probabilities. In frequentism, instead of probabilities determining long-run outcomes, long-run outcomes determine probabilities.

Venn’s mathematical backflip, which introduced a new degree of uncertainty, went against the whole current of earlier thought in mathematics. This had held that a small number of formulae could be used to describe reality exactly. In the 19th century, though, improvements in the precision of tools, and the accumulation of experimental observations, did not lead to more predictable measurements of the physical world. Instead, the unpredictability grew.

What scientists were beginning to discover was that there is always a degree of randomness at work, both in experimental measurements and also in the universe itself. Statistical distributions are powerful because they let us describe the mathematical nature of that randomness. Nowadays we take statistical distributions for granted and talk comfortably about probabilistic models. But back then, this turning away from pinpoint descriptions of reality was part of a paradigm shift in thought, which was to prove revolutionary.

More recent paradigm shifts, which are marked by an analogous move away from pinpoint exactitude, include quantum mechanics, which says that the exact location of a particle cannot be definitely stated, and string theory, which posits that the building blocks of the universe, previously thought of as point-like particles, are in fact string-like structures.

At the London Institute for Mathematical Sciences, where I work, our fellows continue to explore these fields – and unlike at universities, where scientists also have to teach, they get to do research full-time, which is a paradigm shift in itself. One recently derived a result by applying probability theory to synthetic biology, for example. All this work, of course, depends on the frequentism of Dr Venn, whose applications have been vast and wide-ranging. It lies at the heart of weather forecasting, the insurance industry and the financial sector.

Now let’s return to Victor Trumper. His jaw was set. His bat was raised. He was determined that Venn’s infernal machine shouldn’t beat him once again. Boom! Again Trumper was clean-bowled. And it happened again, twice more. Four times, the hapless Aussie was humbled. So what were the chances that the machine would defeat him a fifth time?

If anyone knew the answer, it was the improbable Dr Venn.
WRITTEN BY
Madeleine Hall
Dr Madeleine Hall is a science writer at the London Institute for Mathematical Sciences.

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