For the Recorde: Mary Warner by Gareth Ffowc Roberts

As a new history of Welsh mathematical greats is published we are pleased to feature an extract which highlights the work and contribution of Mary Warner, who broke the glass ceiling when it came to male-dominated world of maths.

Gareth Ffowc Roberts

Before going to bed do you wear your pyjama tops first and then the bottoms, or the bottoms first and then the tops? Or perhaps you don’t have a set pattern because it doesn’t really matter? In the ‘algebra of wearing pyjamas’ the order is not important.

When you get up in the morning what’s your pattern of making your first cup of tea: pouring the tea into an empty cup before adding milk, or pouring milk into the cup before adding the tea?

Believe me, there’s a difference between the two methods in terms of both the colour and the taste. In the ‘algebra of making a cup of tea’ the order is critical.

Using numbers is also an everyday experience but, in contrast to pyjamas and cups of tea, numbers are abstractions that we can neither touch nor taste. Does changing the order of numbers in a calculation affect the answer?

For example, does 3 add 4 give the same answer as 4 add 3? Expressed as an equation, is this correct?

3 + 4 = 4 + 3

Our everyday experience of numbers should reassure us: if I pay £3 for a magazine and £4 for a book, I am pretty confident that I have spent £7 altogether, irrespective of whether I bought the magazine first and then the book, or the other way around.

The algebra of adding numbers matches the algebra of wearing pyjamas.

What about multiplying numbers? Is it quite so obvious that 3×4 (three fours) gives the same answer as 4×3 (four threes)? And that 386,731×952,047 gives the same answer as 952,047×386,731?

You may be persuaded by thinking about these twelve circles set out in a pattern of three rows of four:

We can also interpret the pattern as four columns of three and see that three rows of four is equivalent to four columns of three. It must follow that:

3×4 = 4×3

Repeating the visual argument with any two whole numbers we can show, for example, that:

386,731×952,047 = 952,047×386,731

This should reassure us that the algebra of multiplying numbers, just like the algebra of adding numbers, matches the algebra of wearing pyjamas – the order of the numbers does not make a difference.

Moving shapes

So far, so good, but let us now think through an example using shapes, rather than numbers. We begin with a square sheet of paper in which a round hole has been punched close to one of the corners of the square, like this:

Let us see what happens when we move the square in two different ways. First, we move the square by turning it clockwise through ninety degrees, like this:

Second, we move the original square by rotating it about the diagonal that joins its lower left corner to its upper right corner. This rotation about the diagonal has the following effect:

What’s the effect of moving the square both ways – turning and rotating – but varying the order? This is what happens when we turn it first and then rotate it:

And this is what happens when we rotate it first and then turn it:

The results are different: ‘turn and rotate’ does not have the same effect as ‘rotate and turn’. The algebra of moving shapes is similar to the algebra of making a cup of tea – the order can make a difference.

Not all algebras are the same.

Analysing patterns

When I left school to study mathematics at college, I had considerable difficulty explaining to my father what exactly I was going to be doing.

A butcher by trade, my father had originally set his sights on being an architect but family circumstances before and during World War II meant that he never got the opportunity to realise his ambition.

He was good at sums, a skill that served him well in his business and as a chapel treasurer. However, he was under the impression that a college degree in mathematics basically involved doing more sums, albeit ones more difficult than school sums.

There’s a germ of truth in that, but not much of one. It would be nearer the mark to describe mathematics, particularly at college, as studying and analysing pattern: patterns in numbers, by all means, but also patterns in shapes and patterns in a whole range of weird and wonderful worlds created in the fertile minds of mathematicians.

Mathematicians are tasked to understand the structures and patterns within their creations or, to put it another way, to understand the algebra of those creations.

We saw above, for example, that the algebra of number can be different from the algebra of shape. In the context of school algebra, it is the algebra of number that gets most attention.

At college a whole new world of new algebras is explored, often referred to using the phrase ‘modern algebra’ with an emphasis on understanding structure and pattern.

Space race

Modern algebra includes the algebra of number as well as other algebras that emulate the algebra of wearing pyjamas, but it also includes many algebras that emulate the algebra of making a cup of tea.

Ideas regarding modern algebra evolved slowly from about the middle of the nineteenth century but gained considerable impetus following the successful launch of Sputnik I by the Soviet Union in 1957.

The ensuing ‘space race’ was a wake-up call to western countries, including the United States, to modernise the content of college courses as well as the school curriculum.

Among those who were attracted by this modern algebra revolution was Mary Wynne Warner (née Davies). France and Belgium had led the modernisation campaign in Europe inspired by the Belgian mathematician and educator Georges Papy (1920–2011) and others.

In 1961 Papy published a book in French for college students and their teachers that presented a ‘rigorous yet exciting introduction to a subject that has a central and fundamental position in modern mathematics’.

The book was published in English in 1964, having been translated and adapted by Mary Warner.

Heroic

Mary Warner was inspired to specialise further in modern algebra, ultimately leading to her appointment as a professor of mathematics at City University, London, the first woman to attain that position at the university.

When her friends asked her to explain exactly what her field of research was and she struggled to do so in easily understood terms, she would also add, ‘You mustn’t think that I’m good at sums, because I’m not’.

Structure and pattern in mathematics was her delight, not sums.

Mary Warner’s history reveals a heroic story. That history – or ‘herstory’ – is best understood within the context of women’s uphill struggles to be allowed to excel in mathematics within a male-dominated culture.

In Britain, some women won the right to vote in 1918, but the prejudice that they suffered in education, particularly in the sciences, lasted well beyond that.

For example, women were not allowed to gain admittance to the University of Cambridge until 1869 and, even then, they were unable to graduate fully until 1948.

A woman admitted to study mathematics in Cambridge in 1930, say, and who had passed every examination during her course, would not receive a university degree.

At best, she would receive a certificate by post, but would be barred from attending a formal degree ceremony along with the male graduands and her certificate would not allow her to be a ‘member of the university’.

Today, we may take it for granted that no woman is hindered from following a degree course in mathematics, but there is still some way to go to ensure that there is equity in terms of numbers.

As an example, roughly 30 per cent of the students studying mathematics at the University of Oxford are women.

China

Born in Carmarthen, the elder daughter of Sydney and Esther Davies, Mary attended a primary school in the town before the family moved to Llandovery and she to the local grammar school.

The family then moved to Holywell in Flintshire and Mary was admitted to Howell’s School in Denbigh, where she sat her A levels.

She excelled at mathematics and won a scholarship to Somerville College, Oxford, gaining her degree in 1951 and embarked on doing research in mathematics. At Oxford she met Gerald (Gerry) Warner who studied history at St. Peter’s College.

Shortly after they married in 1956, Gerry Warner, who by then worked as a government intelligence officer (MI6, essentially), was posted to the British Embassy in Beijing.

The voyage on board ship to China took seven weeks and Mary looked forward to beginning her duties as the wife of a diplomat.

At the same time, she was intent on continuing her mathematical work and was fortunate that Chang Du-Shen, one of her fellow researchers in Oxford, had returned to Beijing University and that they were able to meet to discuss their work.

However, this was the period of China’s Great Leap Forward and the clouds of China’s Cultural Revolution were gathering under the leadership of Mao Zedong, Chair of the Communist Party.

Many academics, including Chang Du-Shen, suffered as a result. On his final visit to Mary and Gerry’s flat, such was his fear that he would be caught by the secret police that he hid behind the sofa and whispered that he would not be able to return to see them again.

Burma

In 1960 Gerry Warner was posted to Burma (now Myanmar) and the couple lived in the capital, Rangoon (now Yangon). Mary was keen to continue with her mathematics and applied for a post at the then Rangoon University.

The officials at the British Embassy initially refused permission for her to make the application, as explained by Gerry:

“When we arrived in the Embassy in Rangoon in 1960, no wife of a British diplomat had ever been allowed to take a full-time job. Women were supposed to be addenda and supports to their husbands. How long ago it seems. In the event, the Ambassador gave way when it was made clear that Mary worked, or we left.

“And she proved her worth to the Embassy, since she was at the University when the first shootings of students began, as Burma moved towards dictatorship, and [she] was the only Western witness to events that the Army tried to conceal.”

While at Rangoon University Mary Warner established an MSc course in mathematics, the first of its type in Burma. Four years later, in 1964, Gerry Warner was posted to the embassy in Warsaw, capital of Poland, and Mary registered at the university to study for a higher degree.

By the time she had completed her thesis Gerry had been moved to the embassy in Geneva, Switzerland, and Mary returned to Warsaw to receive her doctorate from the Polish Academy of Sciences.

In keeping with the regulations of the Soviet Union, to which Poland then belonged, graduates were required to pass an examination in Marx-Leninism before receiving their degrees.

Mary was able to avoid that obstacle, thanks to her husband’s diplomatic skills, by all accounts.

Looking back over that period, Gerry noted:

“Mary was the first diplomatic wife to obtain a doctorate in a foreign country, and as in Burma, her familiarity with parts of the society that others could not reach made our lives much richer and more interesting. I was proud of the trail she blazed for other women in such an exemplary way.”

Diplomacy

It appears that Mary lost nothing of her Welshness despite the years she spent abroad. She could speak plainly and with a sharp humour but had to control her emotions in company so as not to cause any professional embarrassment to her husband.

However, things got slightly out of hand on one particular occasion at a diplomatic reception that she and Gerry had arranged during their time in Geneva, where tartes à la crème were the speciality of the house.

One of the guests undiplomatically began to poke fun at Welsh poetry, much to Mary’s annoyance.

She was unable to retaliate directly but decided to throw one of the hotel’s cream concoctions at her defenceless husband. She had no rationale for doing so, of course, but her action certainly cut the conversation short.

Following periods back in London, where Mary took advantage of the opportunities to lecture at City University and was made a Reader there in 1983 and established an MSc course, Gerry Warner was posted to the embassy in Malaysia and they lived in Kuala Lumpur.

During this period Mary was the first person to lecture at both the Malaysian and Chinese universities in Kuala Lumpur.

Tragedy

Gerry Warner was honoured with a knighthood on his retirement in recognition of his service at MI6, and they both returned to Britain in 1991.

Mary was awarded a chair in mathematics at City University and continued to publish extensively up to her retirement in 1996 and beyond.

She also placed great importance on lecturing at the university and tutored many home and international research students, and was highly respected and admired.

Mary Warner had three children, born abroad. They each had successful careers but both Sian and Jonathan suffered from mental problems and tragically committed suicide within a few years of each other.

It appears that Mary devoted herself to her academic work partly as a coping mechanism.

Fuzzy mathematics

Contrary to the more common pattern among mathematicians, Mary Warner’s mathematical creativity did not wane with age and some of her best work was accomplished during her final years.

In her own words, her aim was ‘to make precise the property of imprecision’ and she made major contributions to what is referred to as fuzzy mathematics, an important branch of modern algebra that can be used to analyse imprecision.

Practical applications of the theory include such diverse problems as forecasting weaknesses in nuclear reactors and forecasting earthquakes.

After retiring from City University, her health having broken slightly, she continued with her mathematics and enjoyed attending international conferences.

A year before her death Mary Warner was working on a paper that she intended to present at a conference in Oslo and planned to spend six months as a visiting professor at a university in Brazil.

But that was not to be; she died peacefully in her sleep on 1 April 1988, at the age of sixty-five, while staying with friends in Spain. She was buried in the churchyard at Kennerton, Gloucestershire, alongside her parents and two of her children.

Challenges

Mary Warner gained international recognition, but her fame was not without its challenges. On one occasion Professor Warner had accepted an invitation to address a prestigious conference at the University of Bahrain.

Gerry decided to accompany her on the trip and recalls the story:

“We were met at the plane by one of Mary’s junior hosts, and taken to stay with one of my colleagues. Shortly after we had settled in I was called to the phone. It was our host, who told me that he understood that Professor Warner was a woman. I confirmed that was the case. He said that this would be difficult, for she would be lecturing to male students, and meeting male colleagues. Would she mind if she was categorised as an ‘honorary man’?

“I naturally consulted Mary, [who] said she would have no objection. The visit passed off very pleasantly. Our final lunch was in the male quarters of our host’s house. At the end of the meal I became an ‘honorary woman’, and was introduced to his wife and family in the female rooms.”

Sir Gerry concluded that ‘the occasion could only have involved a lady academic, and exemplified in a benign way the peculiarities of being a comparatively early bird as a lady mathematician’.

Time for another cup of tea – without milk this time.

Algebra of Numbers Puzzle:
The numbers from 0 to 9 are on your phone.
What’s the difference between the sum of the numbers and their product?

For the Recorde: A History of Welsh Mathematical Greats by Gareth Ffowc Roberts is published by the University of Wales Press. It is available from all good bookshops.