**Raymond Wilder**'s parents were Mary Jane Shanley and John Louis Wilder, who was a printer. Music played a large part in the family, and Raymond learnt to play the piano, in particular playing to accompany silent films in local cinemas. He also played the cornet in the family orchestra at dances and fairs. Although mathematics attracted him at school, when he entered Brown University in 1914 it was with the intention of becoming an actuary. World War I caused Wilder to take a break in his studies from 1917 to 1919 when he served in the U.S. Navy as an ensign. Following his war service he returned to Brown University to resume his studies. He received his first degree in 1920.

Wilder then taught mathematics at Brown during 1920-21 while studying for his master's degree in actuarial mathematics which he received in 1921. He married Una Maude Greene in 1921; they had three daughters, Mary Jane, Kermit and Beth, and one son David. Wilder moved to the University of Texas in 1921 where again he was appointed as an instructor while he worked for his doctorate. It was here that his interests moved towards pure mathematics under the influence of Robert Moore. When he asked permission from Moore to take his topology course, Moore replied (see for example [3]):-

After Wilder persuaded Moore to let him take the course, Moore proceeded to ignore him until he solved one of the hardest problems Moore posed to the class. Wilder gave up his plans to study actuarial mathematics and became Moore's research student. He suggested Wilder write up the solution to the problem for his doctorate which indeed he did, becoming Moore's first Texas doctorate in 1923 with his dissertationNo, there is no way a person interested in actuarial mathematics could do, let alone be really interested in, topology.

*Concerning Continuous Curves*.

The dissertation was concerned with the Schönflies programme which aimed to study positional invariants of sets in the plane or 2-sphere. A positional invariant of a set *A* with respect to a set *B* is a property which is shared by all homeomorphic images of *A* which are contained in *B*. The best known example of such a positional invariant is embodied in the Jordan curve theorem: *A simple closed curve in the 2-sphere has precisely two complementary domains and is the boundary of each of them*. A converse to the Jordan curve theorem, proved by Schönflies, states that a subset of the 2-sphere is a simple closed curve if it has two complementary domains, is the boundary of each of them, and is accessible from each of these domains. Wilder's thesis contained a new approach to the Schönflies programme. He continued to undertake research with this aim and in 1930, in *A converse of the Jordan-Brouwer separation theorem in three dimensions*, Wilder showed that a subset of Euclidean 3-space whose complementary domains satisfied certain homology conditions was a 2-sphere.

After a year as an instructor in mathematics at the university of Texas, Wilder was appointed as an assistant professor at Ohio State University in 1924 but here he encountered difficulties due to his [3]:-

After two years at Ohio State, he joined the faculty of the University of Michigan at Ann Arbor. There he progressed from his initial rank of assistant professor to become an associate professor in 1929, and then a full professor in 1935. He played a significant role in the American mathematical scene during World War II in particular he helped settle European refugee mathematicians in the United States. He became a research professor in 1947, a position he held for 20 years until his retirement in 1967.... reluctance to sign a required loyalty oath at Ohio State University. Wilder's hostility to mindless patriotism and his predilection for liberal thought accompanied him throughout his life.

The initial phase of Wilder's research on the Schönflies programme, which we described above, was in in set-theoretic topology and lasted until around 1930. After this he worked in algebraic topology, and in 1932 he called for the unification of the two areas. His work was then directed towards the theory of manifolds, for example *Generalized closed manifolds in n-space* (1934), and in particular to extending the Schönflies programme to higher dimensions. This work was presented in a unified form in *Topology of Manifolds* (1949); this was reprinted in 1963 and again in 1979 with a few notes on the current status of the problems. The final three chapters of the book discuss Wilder's contributions in the theory of positional topological invariants.

It was around the time that Wilder published the first edition of *Topology of Manifolds* that his research interests underwent a major change. He had already become interested in the foundations of mathematics as illustrated by his article *The nature of mathematical proof* (1944) in which is concerned with:-

At the International Congress of Mathematicians in Cambridge, Massachusetts in 1950 he addressed the Congress on... the preconceptions, sometimes not consciously avowed, which underline and unavoidably influence the point of view of active mathematicians.

*The cultural basis of mathematics*. In his lecture he asked:-

He attempted to answer these questions by giving examples such as intuitionism and symbolism. The first major text he published on foundations, which was based on lecture courses he had given, wasHow does culture(in its broadest sense)determine a mathematical structure, such as a logic?

How does culture influence the successive stages of the discovery of a mathematical structure?

*Introduction to the foundations of mathematics*(1952). He wrote in the introduction:-

Beth writes in a review:-The reason for instigating such a course was simply the conviction that it was not good to have teachers, actuaries, statisticians, and others who had specialized in undergraduate mathematics, and who were to base their life's work on mathematics, leave the university without some knowledge of modern mathematics and its foundations.

Wilder's ideas continued to develop along the lines of cultural anthropology. He presented his ideas inThe first part of this book gives ... an exposition of the basic theories of modern mathematics: the theory of sets, the real number system(on the basis of the Peano axioms)and the theory of groups(including some of its applications in algebra and geometry). Special attention is given to those topics which are important from the point of view of research on foundations, such as the relations between various definitions of infinity, diagonal procedures, well-ordering, the choice axiom and its equivalents. ... The second part is devoted to a discussion on various viewpoints on foundations. After a summary of the earlier developments(up to the Zermelo system), the Frege-Russell thesis, intuitionism, and formalism are more fully explained. A final chapter deals with the cultural setting of mathematics.

*Evolution of mathematical concepts. An elementary study*(1969). May, in a review, writes:-

In the book Wilder himself writes:-The author quietly proposes that we study mathematics as a human artefact, as a natural phenomenon subject to empirical observation and scientific analysis, and, in particular, as a cultural phenomenon understandable in anthropological terms. Since this flies in the face of the dominant paradigm of the history of ideas nearly isolated from the social context, it may be misunderstood or ignored. But since it complements the interest among historians of science in constructing a science about science, it may initiate a new pattern.

In 1981 Wilder published another major textThe major difference between mathematics and the other sciences, natural and social, is that whereas the latter are directly restricted in their purview by environmental phenomena of a physical or social nature, mathematics is subject only indirectly to such limitations. ... Plato conceived of an ideal universe in which resided perfect models ... The assumption made in the present work is that the only reality mathematical concepts have is as cultural elements or artefacts.

*Mathematics as a cultural system*which has a similar title to the talk he gave to the International Congress of Mathematicians thirty years earlier. Hirst writes:-

After Wilder retired from the University of Michigan in 1967 he moved to the University of California at Santa Barbara. He lectured there in 1970-71, then became a research associate retaining this position until his death. At the time of Wilder's death in Santa Barbara there were 23 grandchildren and 14 great-grandchildren in his family. His wife, Una, survived him for an additional 19 years, dying at the age of 100 in Long Beach.The book begins with an explanation of his notion of a cultural system in general, and how mathematics fits into this. Evolutionary processes are discussed, with the idea of "hereditary stress" serving again as an important springboard for the emergence of new ideas and insights in mathematics. Consolidation is discussed, as a force or process which unifies existing fields of mathematics and may at the same time spawn new ones.

Raymond writes about Wilder's teaching, interests, and personality in [3]:-

Wilder contributed much to American mathematics. He was a strong supporter of the American Mathematical Society being a member of its Council from 1935 to 1937, a semicentennial lecturer in 1938, colloquium lecturer in 1942, vice president during 1950-51 and president during 1955-56. He was also the Society's Josiah Willard Gibbs Lecturer in 1969. Wilder played a major role in the Mathematical Association of America being its president during 1965-66. The Association awarded him the Distinguished Service Medal in 1973. He was honoured by election to the National Academy of Sciences (United States) in 1963. He was awarded honorary degrees by Bucknell University (1955), Brown University (1958), and the University of Michigan (1980).His advanced graduate classes and seminars were intimate and stimulating. He enjoyed talking about the people, many of whom he knew personally, behind the ideas and theorems. I found myself often staying after his class. Our conversations would follow up some of the items in the classroom but would soon drift to other areas of his expertise. He was a devoted student of southwestern Native American culture. One day he told me that after retiring he would like to be a bartender in a rural area of Arizona or New Mexico, because he found the stories of the folk that he met in those bars so fascinating.

Among all the great mathematicians I have known, Wilder was the most approachable. He had a wonderful sense of humour and his wisdom made him a father confessor to many of his colleagues. With his wife, Una, they made their home a centre of hospitality.

**Article by:** *J J O'Connor* and *E F Robertson*

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