Saunders Mac Lane
Encyclopedia
Saunders Mac Lane was an American
mathematician
who cofounded category theory
with Samuel Eilenberg
.
Mac Lane earned a BA
from Yale University
in 1930, and an MA
from the University of Chicago
in 1931. During this period, he published his first scientific paper, in physics
and co-authored with Irving Langmuir
. He attended the University of Göttingen, 1931–1933, studying logic and mathematics under Paul Bernays
, Emmy Noether
and Hermann Weyl
. Göttingen's Mathematisches Institut awarded him the Ph.D. in 1934.
From 1934 through 1938, Mac Lane held short term appointments at Harvard University
, Cornell University
, and the University of Chicago
. He then held a tenure track appointment at Harvard, 1938–1947, before spending the rest of his career at the University of Chicago. In 1944 and 1945, he also directed Columbia University's Applied Mathematics Group, which was involved in the war effort as a contractor for the Applied Mathematics Panel
.
Mac Lane served as vice president of the National Academy of Sciences
and the American Philosophical Society
, and as president of the American Mathematical Society
. While presiding over the Mathematical Association of America
in the 1950s, he initiated its activities aimed at improving the teaching of modern mathematics. He was a member of the National Science Board
, 1974–1980, advising the American government. In 1976, he led a delegation of mathematicians to China
to study the conditions affecting mathematics there. Mac Lane was elected to the National Academy of Sciences
in 1949, and received the National Medal of Science
in 1989.
, his early work was in field theory
and valuation theory
. He wrote on valuation ring
s and Witt vector
s, and separability in infinite field extension
s. He started writing on group extension
s in 1942, and began his epochal collaboration with Samuel Eilenberg
in 1943, resulting in what are now called Eilenberg–MacLane spaces K(G,n), having a single non-trivial homotopy group
G in dimension n. This work opened the way to group cohomology
in general.
After introducing, via the Eilenberg–Steenrod axioms, the abstract approach to homology theory
, he and Eilenberg originated category theory
in 1945. He is especially known for his work on coherence theorem
s. A recurring feature of category theory, abstract algebra
, and of some other mathematics as well, is the use of diagrams
, consisting of arrows (morphism
s) linking objects, such as products
and coproduct
s. According to McLarty (2005), this diagrammatic approach to contemporary mathematics largely stems from Mac Lane (1948).
Mac Lane had an exemplary devotion to writing approachable texts, starting with his very influential A Survey of Modern Algebra, coauthored in 1941 with Garrett Birkhoff
. From then on, it was possible to teach elementary modern algebra to undergraduates using an English text. His Categories for the Working Mathematician
remains the definitive introduction to category theory.
Mac Lane supervised the Ph.Ds of, among many others, David Eisenbud
, William Howard
, Irving Kaplansky
, Michael Morley
, Anil Nerode
, Robert Solovay, and John G. Thompson
.
In addition to reviewing a fair bit of his mathematical output, the obituary articles McLarty (2005, 2007) clarify Mac Lane's contributions to the philosophy of mathematics
. Mac Lane (1986) is an approachable introduction to his views on this subject.
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 cofounded 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...
with 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...
.
Career
Mac Lane was christened "Leslie Saunders MacLane", but "Leslie" fell into disuse because his parents, Donald MacLane and Winifred Saunders, came to dislike it. He began inserting a space into his surname because his first wife found it difficult to type the name without a space.Mac Lane earned a BA
Bachelor of Arts
A Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...
from Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
in 1930, and an MA
Master of Arts (postgraduate)
A Master of Arts from the Latin Magister Artium, is a type of Master's degree awarded by universities in many countries. The M.A. is usually contrasted with the M.S. or M.Sc. degrees...
from the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
in 1931. During this period, he published his first scientific paper, in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and co-authored with Irving Langmuir
Irving Langmuir
Irving Langmuir was an American chemist and physicist. His most noted publication was the famous 1919 article "The Arrangement of Electrons in Atoms and Molecules" in which, building on Gilbert N. Lewis's cubical atom theory and Walther Kossel's chemical bonding theory, he outlined his...
. He attended the University of Göttingen, 1931–1933, studying logic and mathematics under Paul Bernays
Paul Bernays
Paul Isaac Bernays was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator of, David Hilbert.-Biography:Bernays spent his childhood in Berlin. Bernays attended the...
, Emmy Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
and Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
. Göttingen's Mathematisches Institut awarded him the Ph.D. in 1934.
From 1934 through 1938, Mac Lane held short term appointments at Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
, Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
, and the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
. He then held a tenure track appointment at Harvard, 1938–1947, before spending the rest of his career at the University of Chicago. In 1944 and 1945, he also directed Columbia University's Applied Mathematics Group, which was involved in the war effort as a contractor for the Applied Mathematics Panel
Applied Mathematics Panel
The Applied Mathematics Panel was created at the end of 1942 as a division of the National Defense Research Committee within the Office of Scientific Research and Development in order to solve mathematical problems related to the military effort in World War II, particularly those of the other...
.
Mac Lane served as vice president of the 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...
and the American Philosophical Society
American Philosophical Society
The American Philosophical Society, founded in 1743, and located in Philadelphia, Pa., is an eminent scholarly organization of international reputation, that promotes useful knowledge in the sciences and humanities through excellence in scholarly research, professional meetings, publications,...
, and as 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...
. While presiding over 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;...
in the 1950s, he initiated its activities aimed at improving the teaching of modern mathematics. He was a member of the National Science Board
National Science Board
The National Science Board of the United States is composed of 25 members appointed by the President and confirmed by the United States Senate, representing the broad U.S. science and engineering community. The Board establishes the policies of the National Science Foundation within the framework...
, 1974–1980, advising the American government. In 1976, he led a delegation of mathematicians to China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
to study the conditions affecting mathematics there. Mac Lane was elected to the 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 1949, and received the National Medal of Science
National Medal of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social sciences, biology, chemistry, engineering, mathematics and...
in 1989.
Contributions
After a thesis in mathematical logicMathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, his early work was in field theory
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
and valuation theory
Valuation (mathematics)
In algebra , a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field...
. He wrote on valuation ring
Valuation ring
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D....
s and Witt vector
Witt vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order p is the ring of p-adic integers.-Motivation:Any p-adic...
s, and separability in infinite field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
s. He started writing on group extension
Group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...
s in 1942, and began his epochal collaboration with 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...
in 1943, resulting in what are now called Eilenberg–MacLane spaces K(G,n), having a single non-trivial homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
G in dimension n. This work opened the way to group cohomology
Group cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
in general.
After introducing, via the Eilenberg–Steenrod axioms, the abstract approach to homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
, he and Eilenberg originated 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...
in 1945. He is especially known for his work on coherence theorem
Coherence theorem
In mathematics and particularly category theory, a coherence theorem is a tool for proving a coherence condition. Typically a coherence condition requires an infinite number of equalities among compositions of structure maps...
s. A recurring feature of category theory, abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, and of some other mathematics as well, is the use of diagrams
Commutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
, consisting of arrows (morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s) linking objects, such as products
Product (category theory)
In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
and coproduct
Coproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
s. According to McLarty (2005), this diagrammatic approach to contemporary mathematics largely stems from Mac Lane (1948).
Mac Lane had an exemplary devotion to writing approachable texts, starting with his very influential A Survey of Modern Algebra, coauthored in 1941 with Garrett Birkhoff
Garrett Birkhoff
Garrett Birkhoff was an American mathematician. He is best known for his work in lattice theory.The mathematician George Birkhoff was his father....
. From then on, it was possible to teach elementary modern algebra to undergraduates using an English text. His Categories for the Working Mathematician
Categories for the Working Mathematician
Categories for the Working Mathematician is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based on his lectures on the subject given at the University of Chicago,...
remains the definitive introduction to category theory.
Mac Lane supervised the Ph.Ds of, among many others, David Eisenbud
David Eisenbud
David Eisenbud is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and was Director of the Mathematical Sciences Research Institute from 1997 to 2007....
, William Howard
William Alvin Howard
William Alvin Howard is a proof theorist best known for his work demonstrating formal similarity between intuitionistic logic and the simply typed lambda calculus that has come to be known as the Curry–Howard correspondence. He has also been active in the theory of proof-theoretic ordinals. He...
, Irving Kaplansky
Irving Kaplansky
Irving Kaplansky was a Canadian mathematician.-Biography:He was born in Toronto, Ontario, Canada, after his parents emigrated from Poland and attended the University of Toronto as an undergraduate. After receiving his Ph.D...
, Michael Morley
Michael D. Morley
Michael Darwin Morley is an American mathematician, currently professor emeritus at Cornell University.His research is in advanced mathematical logic and model theory, and he is best known for Morley's categoricity theorem, which he proved in his Ph.D. thesis "Categoricity in Power" in 1962.His...
, Anil Nerode
Anil Nerode
Anil Nerode is a U.S. mathematician, born in 1932. He received his undergraduate education and a Ph.D. in mathematics from the University of Chicago, the latter under the directions of Saunders Mac Lane. He enrolled in the Hutchins College at the University of Chicago in 1947 at the age of 15, and...
, Robert Solovay, and John G. Thompson
John G. Thompson
John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize....
.
In addition to reviewing a fair bit of his mathematical output, the obituary articles McLarty (2005, 2007) clarify Mac Lane's contributions to the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
. Mac Lane (1986) is an approachable introduction to his views on this subject.
Biographical
- McLarty, ColinColin McLartyColin McLarty is a logician whose publications have ranged widely in philosophy and the foundations of mathematics, as well as in the history of science and of mathematics.-Research:...
, 2005, "Saunders Mac Lane (1909–2005): His Mathematical Life and Philosophical Works," Philosophia Mathematica 13: 237–51. With selected bibliography emphasizing Mac Lane's philosophical writings.
- --------, 2007, "The Last Mathematician from Hilbert's Göttingen: Saunders Mac Lane as Philosopher of Mathematics,", British Journal for the Philosophy of Science 58(1): 77–112.
- Lawvere, WilliamWilliam LawvereFrancis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:...
, 2007, article "Saunders Mac Lane," New Dictionary of Scientific Biography: 237–51. Charles Scribners & Sons.
External links
- Obituary press release from the University of ChicagoUniversity of ChicagoThe University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
. - Photographs of Mac Lane, 1984–99.
- Kutateladze S.S., Saunders Mac Lane, the Knight of Mathematics