Raymond Louis Wilder
Encyclopedia
Raymond Louis Wilder was an American
mathematician
, who specialized in topology
and gradually acquired philosophical and anthropological interests.
He entered Brown University
in 1914, intending to become an actuary
. During World War I
, he served in the U.S. Navy as an ensign. Brown awarded him his first degree in 1920, and a master's degree in actuarial mathematics in 1921. That year, he married Una Maude Greene; they had four children, thanks to whom they have ample descent.
Wilder chose to do his Ph.D. at the University of Texas at Austin
, the most fateful decision of his life. At Texas, Wilder discovered pure mathematics and topology
, thanks to the remarkable influence of Robert Lee Moore
, the founder of topology in the USA and the inventor of the Moore method
for teaching mathematical proof. Moore was initially unimpressed by the young actuary, but Wilder went on to solve a difficult open problem that Moore had posed to his class. Moore suggested Wilder write up the solution for his Ph.D. thesis, which he did in 1923, titling it Concerning Continuous Curves. Wilder thus became the first of Moore's many doctoral students at the University of Texas.
After a year as an instructor at Texas, Wilder was appointed assistant professor at the Ohio State University
in 1924. That university required that its academic employees sign a loyalty oath, which Wilder was very reluctant to sign because doing so was inconsistent with his lifelong progressive political and moral views.
In 1926, Wilder joined the faculty of the University of Michigan at Ann Arbor, where he supervised 26 Ph.Ds and became a Research Professor in 1947. During the 1930s, he helped settle European refugee mathematicians in the United States. Mathematicians who rubbed shoulders with Wilder at Michigan and who later proved prominent included Samuel Eilenberg
, the cofounder of category theory
, and the topologist Norman Steenrod
. After his 1967 retirement from Michigan at the rather advanced age of 71, Wilder became a research associate and occasional lecturer at the University of California at Santa Barbara.
Wilder was vice president of the American Mathematical Society
, 1950–1951, president 1955–1956, and the Society's Josiah Willard Gibbs
Lecturer in 1969. He was president of the Mathematical Association of America
, 1965–1966, which awarded him its Distinguished Service Medal in 1973. He was elected to the American National Academy of Sciences
in 1963. Brown University
(1958) and the University of Michigan
(1980) awarded him honorary doctorates. The mathematics department at the University of California annually bestows one or more graduating seniors with an award in Wilder's name.
The historical, philosophical, and anthropological writings of Wilder's later years suggest a warm, colorful personality. Raymond (2003) attests to this having been the case. For instance:
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 shared by all homeomorphic
images of A 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:
In his "A converse of the Jordan-Brouwer separation theorem in three dimensions" (1930), Wilder showed that a subset of Euclidean 3-space whose complementary domains satisfied certain homology
conditions was a 2-sphere.
Around 1930, Wilder moved from set-theoretic topology
to algebraic topology
, calling in 1932 for the unification of the two areas. He then began an extensive investigation of the theory of manifold
s, e.g., his "Generalized closed manifolds in n-space" (1934), in effect extending the Schönflies programme to higher dimensions. This work culminated in his Topology of Manifolds (1949), twice reprinted, whose last three chapters discuss his contributions to the theory of positional topological invariants.
anthropologist Leslie White
, whose professional curiosity included mathematics as a human activity (White 1947). This encounter proved fateful, and Wilder's research interests underwent a major change, towards the foundations of mathematics
. This change was foreshadowed by his 1944 article "The nature of mathematical proof," and heralded by his address to the 1950 International Congress of Mathematicians, titled "The cultural basis of mathematics," which posed the questions:
In 1952, he wrote up his course on foundations and the philosophy of mathematics into a widely cited text, Introduction to the foundations of mathematics.
Wilder's Evolution of mathematical concepts. An elementary study (1969) proposed that "we study mathematics as a human artifact, as a natural phenomenon subject to empirical observation and scientific analysis, and, in particular, as a cultural phenomenon understandable in anthropological terms." In this book, Wilder wrote:
Wilder's last book, Mathematics as a cultural system (1981), contained yet more thinking in this anthropological and evolutionary vein.
Wilder's eclectic and humanist perspective on mathematics appears to have had little influence on subsequent mathematical research. It has, however, had some influence on the teaching of mathematics and on the history and philosophy of mathematics. In particular, Wilder can be seen as a precursor to the work of Howard Eves
, Evert Willem Beth
, and Davis and Hersch (1981). Wilder's call for mathematics to be scrutinized by the methods of social science anticipates some aspects of Where Mathematics Comes From
, by George Lakoff
and Rafael Nunez
. For an introduction to the limited anthropological research on mathematics, see the last chapter of Hersch (1997).
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, who specialized in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and gradually acquired philosophical and anthropological interests.
Life
Wilder's father was a printer. Raymond was musically inclined. He played cornet in the family orchestra, which performed at dances and fairs, and accompanied silent films on the piano.He entered Brown University
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...
in 1914, intending to become an actuary
Actuarial science
Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries. Actuaries are professionals who are qualified in this field through education and experience...
. During World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
, he served in the U.S. Navy as an ensign. Brown awarded him his first degree in 1920, and a master's degree in actuarial mathematics in 1921. That year, he married Una Maude Greene; they had four children, thanks to whom they have ample descent.
Wilder chose to do his Ph.D. at the University of Texas at Austin
University of Texas at Austin
The University of Texas at Austin is a state research university located in Austin, Texas, USA, and is the flagship institution of the The University of Texas System. Founded in 1883, its campus is located approximately from the Texas State Capitol in Austin...
, the most fateful decision of his life. At Texas, Wilder discovered pure mathematics and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, thanks to the remarkable influence of Robert Lee Moore
Robert Lee Moore
Robert Lee Moore was an American mathematician, known for his work in general topology and the Moore method of teaching university mathematics.-Life:...
, the founder of topology in the USA and the inventor of the Moore method
Moore method
The Moore method is a deductive manner of instruction used in advanced mathematics courses. It is named after Robert Lee Moore, a famous topologist who first used a stronger version of the method at the University of Pennsylvania when he began teaching there in 1911.The way the course is conducted...
for teaching mathematical proof. Moore was initially unimpressed by the young actuary, but Wilder went on to solve a difficult open problem that Moore had posed to his class. Moore suggested Wilder write up the solution for his Ph.D. thesis, which he did in 1923, titling it Concerning Continuous Curves. Wilder thus became the first of Moore's many doctoral students at the University of Texas.
After a year as an instructor at Texas, Wilder was appointed assistant professor at the Ohio State University
Ohio State University
The Ohio State University, commonly referred to as Ohio State, is a public research university located in Columbus, Ohio. It was originally founded in 1870 as a land-grant university and is currently the third largest university campus in the United States...
in 1924. That university required that its academic employees sign a loyalty oath, which Wilder was very reluctant to sign because doing so was inconsistent with his lifelong progressive political and moral views.
In 1926, Wilder joined the faculty of the University of Michigan at Ann Arbor, where he supervised 26 Ph.Ds and became a Research Professor in 1947. During the 1930s, he helped settle European refugee mathematicians in the United States. Mathematicians who rubbed shoulders with Wilder at Michigan and who later proved prominent included Samuel Eilenberg
Samuel Eilenberg
Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...
, the cofounder of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, and the topologist Norman Steenrod
Norman Steenrod
Norman Earl Steenrod was a preeminent mathematician most widely known for his contributions to the field of algebraic topology.-Life:...
. After his 1967 retirement from Michigan at the rather advanced age of 71, Wilder became a research associate and occasional lecturer at the University of California at Santa Barbara.
Wilder was vice president of the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
, 1950–1951, president 1955–1956, and the Society's Josiah Willard Gibbs
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...
Lecturer in 1969. He was president of the Mathematical Association of America
Mathematical Association of America
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
, 1965–1966, which awarded him its Distinguished Service Medal in 1973. He was elected to the American National Academy of Sciences
United States National Academy of Sciences
The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
in 1963. Brown University
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...
(1958) and the University of Michigan
University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
(1980) awarded him honorary doctorates. The mathematics department at the University of California annually bestows one or more graduating seniors with an award in Wilder's name.
The historical, philosophical, and anthropological writings of Wilder's later years suggest a warm, colorful personality. Raymond (2003) attests to this having been the case. For instance:
- "[Wilder] 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 he met in bars there so fascinating."
The topologist
Wilder's thesis set out a new approach to the SchönfliesArthur Moritz Schönflies
Arthur Moritz Schoenflies , sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology....
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 shared by all homeomorphic
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
images of A contained in B. The best known example of such a positional invariant is embodied in the Jordan curve theorem
Jordan curve theorem
In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...
: A simple closed curve in the 2-sphere has precisely two complementary domains and is the boundary of each of them. A converse
Converse
Converse is an American shoe company that has been making shoes, lifestyle fashion and athletic apparel since the early 20th century. Converse is one of the earliest pioneers in the sneaker and sporting good industry founded in 1908.- 1908–1941: Early days :...
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 these domains;
- Is accessible from each of these domains.
In his "A converse of the Jordan-Brouwer separation theorem in three dimensions" (1930), Wilder showed that a subset of Euclidean 3-space whose complementary domains satisfied certain homology
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
conditions was a 2-sphere.
Around 1930, Wilder moved from set-theoretic topology
Set-theoretic topology
In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of ZFC. A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The...
to algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, calling in 1932 for the unification of the two areas. He then began an extensive investigation of the theory of manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s, e.g., his "Generalized closed manifolds in n-space" (1934), in effect extending the Schönflies programme to higher dimensions. This work culminated in his Topology of Manifolds (1949), twice reprinted, whose last three chapters discuss his contributions to the theory of positional topological invariants.
The philosopher
During the 1940s, Wilder met and befriended the University of MichiganUniversity of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
anthropologist Leslie White
Leslie White
Leslie Alvin White was an American anthropologist known for his advocacy of theories of cultural evolution, sociocultural evolution, and especially neoevolutionism, and for his role in creating the department of anthropology at the University of Michigan Ann Arbor...
, whose professional curiosity included mathematics as a human activity (White 1947). This encounter proved fateful, and Wilder's research interests underwent a major change, towards the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
. This change was foreshadowed by his 1944 article "The nature of mathematical proof," and heralded by his address to the 1950 International Congress of Mathematicians, titled "The cultural basis of mathematics," which posed the questions:
- "How 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?"
In 1952, he wrote up his course on foundations and the philosophy of mathematics into a widely cited text, Introduction to the foundations of mathematics.
Wilder's Evolution of mathematical concepts. An elementary study (1969) proposed that "we study mathematics as a human artifact, as a natural phenomenon subject to empirical observation and scientific analysis, and, in particular, as a cultural phenomenon understandable in anthropological terms." In this book, Wilder wrote:
- "The 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. ... PlatoPlatoPlato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
conceived of an ideal universe in which resided perfect models ... the only reality mathematical concepts have is as cultural elements or artifacts."
Wilder's last book, Mathematics as a cultural system (1981), contained yet more thinking in this anthropological and evolutionary vein.
Wilder's eclectic and humanist perspective on mathematics appears to have had little influence on subsequent mathematical research. It has, however, had some influence on the teaching of mathematics and on the history and philosophy of mathematics. In particular, Wilder can be seen as a precursor to the work of Howard Eves
Howard Eves
Howard Whitley Eves was an American mathematician, known for his work in geometry and the history of mathematics....
, Evert Willem Beth
Evert Willem Beth
Evert Willem Beth was a Dutch philosopher and logician, whose work principally concerned the foundations of mathematics.- Biography :...
, and Davis and Hersch (1981). Wilder's call for mathematics to be scrutinized by the methods of social science anticipates some aspects of Where Mathematics Comes From
Where Mathematics Comes From
Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist...
, by George Lakoff
George Lakoff
George P. Lakoff is an American cognitive linguist and professor of linguistics at the University of California, Berkeley, where he has taught since 1972...
and Rafael Nunez
Rafael E. Núñez
Rafael E. Núñez is a professor of cognitive science at the University of California, San Diego and a proponent of embodied cognition. He co-authored Where Mathematics Comes From with George Lakoff.-External links:*...
. For an introduction to the limited anthropological research on mathematics, see the last chapter of Hersch (1997).
External links
- J J O'Connor and E F Robertson, MacTutor: Raymond Louis Wilder. The source for this entry.
- Wilder papers at the University of Texas.