# Mary Wynne Warner

### Quick Info

Born
22 June 1932
Carmarthen, Wales
Died
1 April 1998
Spain

Summary
Mary Warner was a Welsh mathematician who was a pioneer in fuzzy mathematics.

### Biography

Mary Warner's name before she married was Mary Wynne Davies. Her parents were Sydney and Esther Davies and Mary was the eldest of their two children, having one younger sister. Sydney Davies was teaching at the Queen Elizabeth Grammar School in Carmarthen in south Wales when Mary was born but, when she was six years old, the family moved to Llandovery. It was in this in agricultural market town that Sydney had been appointed as headmaster of the grammar school, and this was the town in which Mary spent the next ten years of her life. She had not begun her schooling before the family moved to Llandovery but, once there, she began her education going on to study at her father's school.

Before we continue to explain how Mary's career developed, we should say a little more about her father who, in his way, was a remarkable man and a major influence on his daughters. Sydney Davies [2]:-
... was a remarkable man whose influence on his family was profound and lasting. An old-fashioned Welsh grammar school headmaster, one of the last of the kind, his standards were very high. He was a very good, patient, but most exacting teacher.
Mary took her school certificate examinations while at her father's school, achieving the best results of anyone in her year in Wales. By this time her interests were very firmly in the area of mathematics but she had to move to a new school in order to study physics which she wanted to take to complement her mathematical studies. After sitting the School Certificate examinations she entered Howell's Boarding School in Denbigh which is in North Wales, and in 1948 her father accepted the position of headmaster at the grammar school at Holywell, which is about 12 miles north east of Denbigh close to the coast of North Wales. This was certainly a step up for Sydney Davies, for this move brought him to a larger more influential school and he went on to reach the position of the leading headmaster in Welsh secondary schools.

In 1951 Mary Davies entered the University of Oxford to read mathematics. She had performed outstandingly well at Howell's Boarding School in Denbigh and won an Open Scholarship to study at Somerville College as well as a Draper's Company Exhibition. One of her fellow students gave this description of her as an undergraduate (see [2]):-
Only the scholar's gown suggested the high academic ability which was to bring her great distinction. Close friends, however, knew her quick, self-deprecating, ironical wit and realized that the laid-back manner concealed deep intellectual interest in her subject, as well as unusual distinction in it.
Although she achieved distinction in her years as an undergraduate, being awarded both college and university prizes, she only received a Second Class degree in mathematics when she graduated in 1953. Despite this her research potential had been clear and she was awarded a research scholarship to enable her to undertake doctoral work at Oxford. Her supervisor was Henry Whitehead who had been appointed to the Waynflete Chair of Pure Mathematics at Oxford some six years earlier. During this time he had built up an extremely active research group in algebraic topology and Mary made excellent progress in her research in these stimulating conditions. She published her first research paper A note on Borsuk's antipodal point theorem in the Oxford Quarterly Journal of Mathematics in 1956.

Mary never completed her doctorate under Henry Whitehead's supervision. She had become friendly with a history student at Oxford, Gerald Warner, who graduated in 1954 and joined the Diplomatic Service in the Intelligence Branch. When he learnt that he was to be posted to China, Mary and Gerry decided to marry. They did so in 1956 and shortly afterwards set off on the seven week journey to Beijing. Mary Warner now fitted into the role of diplomat's wife, a role which she certainly enjoyed, but she remained a committed mathematician taking every opportunity to continue to pursue her studies. In Beijing she met up with the Chinese topologist Chang Su-chen who had also been a student of Henry Whitehead.

The period that the Warners spent in China was one of change. The country had been following a line of peaceful coexistence but began to take a more military line beginning in 1957. A policy of socialist construction was adopted based largely on ideological principles. Scientists and other intellectuals were viewed as dangerous unless they could be converted to the approved ideological views. Conditions became more tense for the Warners and eventually Chang Su-chen told Mary that in meeting to discuss mathematics they were creating suspicions which would cause them difficulties [1]:-
[Chang Su-chen] came to [the Warners] flat one day, crouched beside the sofa to avoid any eavesdropping microphones, and told her that, although he was a liberal, the beginning of Chairman Mao's Great Leap Forward meant that he could see her no longer.
The mathematical discussions came to an end in 1958 but in that year Mary's first child Sian was born in Beijing then, shortly after, the Warners returned to England.

Warner now lived in London where she was appointed to an part-time lectureship in mathematics at Bedford College. In 1959 her second child Jonathan was born in London but, being in the Diplomatic Service, Gerry Warner was never going to be long in one place and soon the family were off to Burma. The third of Warner's children, Rachel, was born in Rangoon in 1961 where, despite being a diplomat's wife and the mother of three young children, she continued to pursue her mathematical career with an appointment as Senior Lecturer in Mathematics at Rangoon University. One of her main tasks was to set up an M.Sc. course there.

A life on the move saw Warner back in London for a while, where she slotted back into the Bedford College post she had held previously. Then in 1964 there was a spell in Poland where she joined Borsuk's school in Warsaw. Borsuk led a seminar in which he developed a unique atmosphere of successful international cooperation and Warner was welcomed as a Visiting Research Fellow. The effect of this active group on Warner was to make her begin work on a doctoral thesis under the supervision of one of Borsuk's colleagues. After two years in Warsaw, Warner spent two further years in Geneva during which time she completed her doctoral dissertation The homology of Cartesian product spaces which she submitted to the Polish Academy of Sciences. A successful defence of the thesis, which was examined by Borsuk and Kuratowski, saw Warner complete, fifteen years after she became a research student at Oxford, the task she had set out on. It is remarkable that she had the tenacity to keep up her mathematical interests throughout her life on the move often under the most difficult of circumstances.

In 1968 Warner was back in London where she was appointed as a Lecturer in Mathematics at the City University. The following year her second paper appeared The homology of tensor products and her next two papers, published in 1975 and 1976, were the results of work she undertook while in Malaysia, the last overseas posting her husband was to have, during 1974-76. The City University had agreed to give her leave of absence for two years so on her return to London Warner again took up her post. By 1976 she was 44 years old, an age at which many (if not most) mathematicians have done their best work. In Warner's case, however, it only became possible for her to concentrate fully on mathematics from this time on and the result was a rapid climb to become [2]:-
... one of the foremost researchers in fuzzy mathematics, highly respected by all her colleagues in the field.
Between 1980 and 1985 Warner wrote 20 papers on tolerance spaces and automata. She then generalised both concepts with the introduction of the notion of a lattice-valued relation in 1984. Perhaps the best way to describe the ideas of tolerance space and lattice-valued relation is to quote the introduction from Warner's paper Some thoughts on lattice valued functions and relations published in 1985:-
A lattice-valued function is a function $f : X \rightarrow L$ from a set $X$ to a lattice $L$. Both $X$ and $L$ may possess further structure. In fact, every real-valued function is lattice-valued by virtue of the usual max, min lattice on the ordered set of reals. We concentrate on some areas where the actual lattice structure of $L$ plays a major part in a topological theory. Continuous real-valued functions from a topological space are thus excluded per se, but feature within the context of fuzzy topological spaces. Some of the formal transition from ordinary to fuzzy spaces is largely mechanical. More interesting are the difficulties encountered, and on some of these we shall concentrate.

The term 'fuzzy' has been used by Poston in his thesis and C T J Dodson [in 1974] to describe a set with a reflexive, symmetric relation, elsewhere [in E C Zeeman's paper" Topology of 3-manifolds and related topics" of 1962] called a tolerance space. We avoid confusion by adopting the latter term. Extending the tolerance relation to a fuzzy relation, we review the topological analogues which can be introduced, with particular reference to homogeneity. The main discussion, then, is on two topics, namely fuzzy topological spaces and sets with fuzzy relations. We conclude with a few general remarks on lattice-valued functions, topology and homology.
Warner's major contributions saw her become a reader at the City University in 1983, then a professor in 1996. Her great mathematical achievements brought her great satisfaction, but the latter part of her life was also filled with tragedy. Both her son Jonathan and her daughter Sian had mental health problems and both took their own lives during the 1990s. Warner's own health had deteriorated and she retired a year early in 1996 but remained active in supervising Ph.D. students and working on research. The level of her activity at this stage can be seen from the fact that she published three papers in 1996 and three more in 1997. While still planning numerous mathematical projects she died suddenly in her sleep while visiting friends in Spain.

This charming story, recounted in [1], tells us quite of lot about her character:-
Despite her travels, Mary Warner never lost her Welshness. She was always direct, saying what she thought with a caustic wit, though in order to spare her husband professional embarrassment she did her best to keep her strong emotions under control.

Once when they were giving a diplomatic dinner party in a Geneva restaurant noted for its tartes a la créme, a guest was mocking Welsh poetry, of which Mary Warner was very fond. Becoming more and more indignant, but prevented from fighting back, she finally turned illogically but effectively on her husband, throwing at him one of the specialities of the house and bringing the conversation to a full stop.

### References (show)

1. Obituary in The Times
2. I M James and A R Pears, Mary Wynne Warner, Bull. London Math. Soc. 34 (2002), 745-752.

### Additional Resources (show)

Other pages about Mary Warner:

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Written by J J O'Connor and E F Robertson
Last Update October 2003