A Point of View: Mary, queen of maths

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Maths genius Mary Cartwright was a modest soul and one of the early founders of chaos theory. It's time we recognised her massive contribution, says historian Lisa Jardine.

In his Mathematician's Apology, published in 1940, the great mathematician GH Hardy argued emphatically that pure mathematics is never useful. Yet at the very moment he was insisting that - specifically - "real mathematics has no effect on war", a mathematical breakthrough was being made which contributed to the wartime defence of Britain against enemy air attack.

What is more, that breakthrough laid the groundwork - unrecognised at the time - for an entire new field of science.

In January 1938, with the threat of war hanging over Europe, the British Government's Department of Scientific and Industrial Research sent a memorandum to the London Mathematical Society appealing to pure mathematicians to help them solve a problem involving a tricky type of equation. Although this was not stated in the memo, it related to top-secret developments in Radio Detection and Ranging - what was soon to become known as radar.

Engineers working on the project were having difficulty with the erratic behaviour of high-frequency radio waves. The need had arisen, the memo said, for "a more complete understanding of the actual behaviour of certain assemblages of electrical apparatus". Could any of the Mathematical Society's members help?

The request caught the attention of Dr Mary Cartwright, lecturer in mathematics at Girton College Cambridge. She was already working on similar "very objectionable-looking differential equations" (as she later described them).

She brought the request to the attention of her long-term colleague at Trinity College, Professor JE Littlewood and suggested that they combine forces. In a memoir written later in her life, she explained that he already had the necessary experience in dynamics, having worked on the trajectories of anti-aircraft guns during World War I.

The distinguished physicist and public intellectual Freeman Dyson - who was born in Britain but has, since the 1950s, spent most of his professional life at the Princeton Institute for Advanced Studies in America - heard Cartwright lecture on this work when he was a student at Cambridge in 1942. He gives us a vivid account of the importance of the war work Cartwright and Littlewood did:

"The whole development of radar in World War Two depended on high power amplifiers, and it was a matter of life and death to have amplifiers that did what they were supposed to do. The soldiers were plagued with amplifiers that misbehaved, and blamed the manufacturers for their erratic behaviour. Cartwright and Littlewood discovered that the manufacturers were not to blame. The equation itself was to blame."

In other words, odd things happened when some sorts of values were fed into the standard equation they were using to predict the amplifiers' performance. Cartwright and Littlewood were able to show that as the wavelength of radio waves shortens, their performance ceases to be regular and periodic, and becomes unstable and unpredictable. This work helped explain some perplexing phenomena engineers were encountering.

Cartwright herself was always somewhat diffident when asked to assess the lasting importance of her war work. She and Littlewood had provided a scientific explanation for some peculiar features of the behaviour of radio waves, but they did not in the end supply the answer in time. They simply succeeded in directing the engineers' attention away from faulty equipment towards practical ways of compensating for the electrical "noise" - or erratic fluctuations - being produced.

So while Cartwright and Littlewood were producing significant results on the stability of solutions to the equation describing the oscillation of radio waves, the engineers working on radar systems decided they could not wait for precise mathematical results. Instead, once it had been identified, they worked around the problem, by keeping the equipment within predictable ranges.

Perhaps in part because of her own overly modest assessment of its importance, Cartwright's original work went relatively unnoticed when it was published in the Journal of the London Mathematical Society shortly after the end of the war. Freeman Dyson maintains that this is a classic example of the way in which real mathematical originality and innovation is missed until a generation after the work has been done:

"When I heard Cartwright lecture in 1942, I remember being delighted with the beauty of her results. I could see the beauty of her work but I could not see its importance. I said to myself, 'This is a lovely piece of work. Too bad it is only a practical wartime problem and not real mathematics.' I did not say, 'This is the birth of a new field of mathematics.' I shared the tastes and prejudices of my contemporaries."

The "new field" Dyson refers to here, which he and his contemporaries failed to recognise, is chaos theory. Cartwright's early contribution to the field is now acknowledged in all histories of the subject, but was largely overlooked for almost 20 years.

The results unexpectedly obtained from the equations predicting the oscillations of radio waves are part of the foundation for the modern theory that accounts for the unpredictable behaviour of all manner of physical phenomena, from swinging pendulums and fluid flow, to the stock market.

Steadily increase the rate of flow of water into a rotating waterwheel, for example, and the wheel will go correspondingly faster. But at a certain point the behaviour of the wheel becomes unpredictable - speeding up and slowing down without warning, or even changing direction.

Image caption,
Chaos theory has been used to explain stock market behaviour

The recognition that chaotic behaviour is a vital part of many physical systems in the world around us came in 1961, when Edward Lorenz was running a weather simulation through an early computer. When he tested a particular configuration a second time he found that the outcome differed dramatically from his earlier run. Eventually he tracked the difference down to a small alteration he had inadvertently made in transferring the initial data, by altering the number of decimal places.

Lorenz immortalised this discovery in a lecture entitled "Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?".

Today, when we think of chaos theory we associate it with all kinds of fundamentally unstable situations - but one of the most vivid to imagine is still the idea that one flap of a butterfly's wing deep in the Amazon rainforest is the cause of a weather system thousands of miles away.

This is the same kind of unpredictability arising from small changes in initial conditions that Cartwright and Littlewood had recognised and drawn attention to in their work with radio waves several decades earlier.

After the war, Mary Cartwright moved away from knotty differential equations and ended her collaboration with Littlewood. She went on to have a distinguished academic career in pure mathematics and academic administration, earning a succession of honours.

Image caption,
Caused by the flap of a butterfly's wings?

In 1947 she was the first woman mathematician to be elected to the Royal Society. In 1948 she became Mistress of Girton College Cambridge, then reader in the theory of functions in the Cambridge mathematics department in 1959. From 1961 to 1963 she was president of the London Mathematical Society, and received its highest honour, the de Morgan Medal, in 1968. She was made a Dame Commander of the British Empire in 1969.

She lived long enough to see the field in which she had made those early, important discoveries become a major part of modern mathematics, and to see it take its place in the popular imagination. She was, however, characteristically modest to the end about the part she had played.

Freeman Dyson claims that Littlewood did not understand the importance of the work that he and Cartwright had done: "Only Cartwright understood the importance of her work as the foundation of chaos theory, and she is not a person who likes to blow her own trumpet."

He records, however, that shortly before her death, he received an indignant letter from Cartwright, scolding him for crediting her with more than she deserved.

Dame Mary Cartwright died in 1998 at the age of 97. In one of the many obituaries paying tribute to her, a friend and colleague described her as "a person who combined distinction of achievement with a notable lack of self-importance".

She left strict instructions that there were to be no eulogies at her memorial service.

However, March 8 was International Women's Day, so it feels like a particularly appropriate time to blow Dame Mary Cartwright's trumpet on her behalf - for her brilliance as a mathematician, and as one of the founders of the important field of chaos theory.

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