The Mac Lane obit by Kutateladze

The Mac Lane obit by Kutateladze

[Peter Freyd]

                    xxx.sf.nchc.gov.tw/pdf/math.HO/0507203.

                               SAUNDERS MAC LANE,
                           THE KNIGHT OF MATHEMATICS
                               S. S. KUTATELADZE

      Abstract. This is a short obituary of Saunders Mac Lane (1909-2005).

San Francisco and April 14, 2005 form the terminal place and date of the
marvellous almost centennial life of the prominent American mathematician
Saunders Mac Lane who shared with Samuel Eilenberg (1913-1998) the honor of
creation of category theory which ranks among the most brilliant, controversial,
ambitious, and heroic mathematical achievements of the 20th century.

Category theory, alongside set theory, serves as a universal language of modern
mathematics. Categories, functors, and natural transformations are widely used
in all areas of mathematics, allowing us to look uniformly and consistently on
various constructions and formulate the general properties of diverse
structures. The impact of category theory is irreducible to the narrow
frameworks of its great expressive conveniences. This theory has drastically
changed our general outlook on the foundations of mathematics and widened the
room of free thinking in mathematics.

Set theory, a great and ingenious creation of Georg Cantor, occupies in the
common opinion of the 20th century the place of the sole solid base of modern
mathematics. Mathematics becomes sinking into a section of the Cantorian set
theory. Most active mathematicians, teachers, and philosophers consider as
obvious and undisputable the thesis that mathematics cannot be grounded on
anything but set theory. The set-theoretic stance transforms paradoxically into
an ironclad dogma, a clear-cut forbiddance of thinking (as L. Feuerbach once put
it wittily). Such an indoctrinated view of the foundations of mathematics is
false and conspicuously contradicts the leitmotif, nature, and pathos of the
essence of all creative contribution of Cantor who wrote as far back as in 1883
that "denn das Wesen der Mathematik liegt grerade in ihrerFreiheit."

It is category theory that one of the most ambitious projects of the 20th
century mathematics was realized within in the 1960s, the project of socializing
set theory. This led to topos theory providing a profusion of categories of
which classical set theory is an ordinary member. Mathematics has acquired
infinitely many new degrees of freedom. All these rest on category theory
originated with the article by Mac Lane and Eilenberg "General Theory of Natural
Equivalences," which was presented to the American Mathematical Society on
September 8, 1942 and published in 1945 in the Transactions of the AMS.

Mac Lane authored or coauthored more than 100 research papers and 6 books: A
Survey of Modern Algebra (1941, 1997; with Garrett Birkhoff); Homology (1963);
Algebra (1967; with Garrett Birkhoff); Categories for the Working Mathematician
(1971, 1998); Mathematics, Form and Function (1985); Sheaves in Geometry and
Logic: A First Introduction to Topos Theory (1992; with Ieke Moerdijk).

Mac Lane was the advisor of 39 Ph.D. theses. Alfred Putman, John Thompson,
Irving Kaplansky, Robert Solovay, and many other distinguished scientists are
listed as his students. He was elected to the National Academy of Sciences of
the USA in 1949 and received the National Medal of Science, the highest
scientific award of the USA in 1989. Mac Lane served as vice-president of the
National Academy of Sciences and the American Philosophical Society. He was
elected as president of the American Mathematical Society and Mathematical
Association of America. He contributed greatly to modernization of the teaching
programs in mathematics. Mac Lane received many signs of honor from the leading
universities of the world and possessed an impressive collection of mathematical
awards and prizes. Mac Lane became a living legend of the science of the USA.

Mac Lane was born on August 4, 1909 in Norwich near Taftville, Connecticut in
the family of a Congregationalist minister and was christened as Leslie Saunders
MacLane. The name Leslie was suggested by his nurse, but his mother disliked the
name. A month later, his father put a hand on the head of the son, looked up to
the God, and said: "Leslie forget." His father and uncles changed the spelling
of their surname and began to write MacLane instead of MacLean in order to avoid
sounding Irish. The space in Mac Lane was added by Saunders himself at request
of his first wife Dorothy. That is how Mac Lane narrated about his name in A
Mathematical Biography which was published soon after his death.

Saunders's father passed away when the boy was 15 and it was Uncle John who
supported the boy and paid for his education in Yale. Saunders was firstly fond
of chemistry but everything changed after acquaintance with differential and
integral calculus by the textbook of Longley and Wilson (which reminds of the
later book by Granville, Smith, and Longley). The university years revealed Mac
Lane's attraction to philosophy and foundations of mathematics. He was greatly
impressed by the brand-new three volumes by Whitehead and Russell, the
celebrated Principia Mathematica. The mathematical tastes of Mac Lane were
strongly influenced by the lectures of a young assistant professor Oystein Ore,
a Norwegian mathematician from the Emmy Noether's school. After graduation from
Yale, Mac Lane continued education in the University of Chicago. At that time he
was very much influenced by the personalities and research of Eliakim Moore,
Leonard Dickson, Gilbert Bliss, Edmund Landau, Marston Morse, and many others.
Mac Lane was inclined to wrote a Ph.D. thesis in logic but this was impossible
in Chicago and so Saunders decided to continue education in Gottingen.

The stay in Germany in 1931-1933 was decisive for the maturity of Mac Lane's
gift and personality. Although David Hilbert had retired, he still delivered
weekly lectures on philosophy and relevant general issues. The successor of
Hilbert was Hermann Weyl who had recently arrived from Zurich and was in the
prime of his years and talents. Weyl advised Saunders to attend the lectures on
linear associative algebras by Emmy Noether whom Weyl called "the equal of each
of us." In the Mathematical Institute Mac Lane met and boiled with Edmund
Landau, Richard Courant, Gustav Herglotz, Otto Neugebauer, Oswald Teichmuller,
and many others. Paul Bernays became the advisor of Mac Lane's Ph.D thesis
"Abbreviated Proofs in Logic Calculus."

The Nazis gained power in Germany in February 1933. The feast of antisemitism
started immediately and one of the first and fiercest strokes fell upon the
Mathematical Institute. The young persons are welcome to read as an antidote Mac
Lane's masterpiece "Mathematics at Gottingen under the Nazis" in the Notices of
the AMS, 42:10, 1134-1138 (1995).

In the fall of 1933 Mac Lane returned to the States with Dorothy Jones Mac Lane
whom he had married recently in Germany. The further academic career of Mac Lane
was mainly tied with Harvard and since 1947 with Chicago. To evaluate the
contribution of Mac Lane to mathematics is an easy and pleasant task. It
suffices to cite the words A. G. Kurosh, a renowned Russian professor of
Lomonosov State University. In the translator's preface to the Russian edition
of the classical Homology book, Kurosh wrote:

  The author of this book, a professor of Chicago University, is one of the most
  prominent American algebraists and topologists. His role in homological
  algebra as well as category theory is the role of one of the founders of this
  area.

Homological algebra implements a marvelous project of algebraization of
topological spaces by assigning to such a space  X  the sequence of (abelian)
homology groups  H_n(X). Moreover, each continuous map  f:X -> Y  from  X  to  Y
induces a family of homomorphisms of the homology groups  f_n: H_n(X) -> H_n(Y).
The aim of homological algebra consists in calculation of homologies.

In his research into homological algebra and category theory Mac Lane cooperated
with Eilenberg whom he met in 1940. Eilenberg had arrived from Poland two years
earlier. He saw the affinity of the algebraic calculations of Mac Lane with
those he encountered in algebraic topology. Eilenberg offered cooperation to Mac
Lane. The union of Eilenberg and Mac Lane lasted for 14 years and resulted in 15
joint papers which noticeably changed the mathematical appearance of the 20th
century.

The pearl of this cooperation was category theory. Mac Lane always considered
category theory "a natural and perhaps inevitable aspect of the 20th century
mathematical emphasis on axiomatic and abstract methods -- especially as those
methods when involved in abstract algebra and functional analysis." He stressed
that even if Eilenberg and he did not propose this theory it will necessarily
appear in the works of other mathematicians. Among these potential inventors of
the new conceptions Mac Lane listed Claude Chevalley, Heinz Hopf, Norman
Steenrod, Henri Cartan, Charles Ehresmann, and John von Neumann.

In Mac Lane's opinion, the conceptions of category theory were close to the
methodological principles of the project of Nicholas Bourbaki. Mac Lane was
sympathetic with the project and was very close to joining in but this never
happened (the main obstacles were in linguistic facilities). However, even the
later membership of Eilenberg in the Bourbaki group could not overcome a shade
of slight disinclination and repulsion. It turned out impossible to "categorize
Bourbaki" with a theory of non-French origin as Mac Lane had once phrased the
matter shrewdly and elegantly. It is worth noting in this respect that the term
"category theory" had roots in the mutual interest of its authors in philosophy
and, in particular, in the works of Immanuel Kant.

Set theory rules in the present-day mathematics. The buffoon's role of "abstract
nonsense" is assigned in mathematics to category theory. History and literature
demonstrate to us that the relations between the ruler and the jester may be
totally intricate and unpredictable. Something very similar transpires in the
interrelations of set theory and category theory and the dependency of one of
them on the other.

>From a logic standpoint, set theory and category theory are instances of a first
order theory. The former deals with sets and the membership relation between
them. The latter speaks of objects and morphisms (or arrows). Of course, there
is no principle difference between the atomic formulas a E= b and a E. b.
However, the precipice in meaning is abysmal between the two concepts that are
formalized by the two atomic formulas. The stationary universe of
Zermelo-Fraenkel, cluttered up with uncountably many copies of equipollent sets
confronts the free world of categories, ensembles of arbitrary nature that are
determined by the dynamics of their transformations.

The individual dualities of set theory, dependent on the choice of particular
realizations of the pairs of objects under study, give up their places to the
universal natural transformations of category theory. One of the most brilliant
achievements of category theory was the development of axiomatic homology
theory. Instead of the homological diversity for topological spaces (the
simplicial homology for a polyhedron, singular and Cech homology, Vietoris
homology, etc.) Eilenberg and Steenrod suggested as far back as in 1952 the new
understanding of each homology or cohomology theory as a functor from the
category of spaces under consideration to the category of groups. The axiomatic
approach to defining such a functor radically changed the manner of further
progress in homological algebra and algebraic topology. The study of the
homology of Eilenberg-Mac Lane spaces and the method of acyclic models
demonstrated the strength of the ideas of category theory and led to universal
use of simplicial sets in K-theory and sheaves.

In 1948 Mac Lane proposed the concept of abelian category abstracting the
categories of abelian groups and vector spaces which played key roles in the
first papers on axiomatic homology theory. The abelian categories were
rediscovered in 1953 and became a major tool in research into homological
algebra by Cartan, Eilenberg, and their followers.

Outstanding advances in category theory are connected with the names of
Alexander Grothendieck and F. William Lawvere. Topos theory, their aesthetic
creation, appeared in the course of "point elimination" called upon by the
challenge of invariance of the objects we study in mathematics. It is on this
road that we met the conception of variable sets which led to the notion of
topos and the understanding of the social medium of set-theoretic models.

A category is called an elementary topos provided that it is cartesian closed
and has a subject classifier. The sources of toposes lie in the theory of
sheaves and Grothendieck topology. Further progress of the concept of topos is
due to search for some category-theoretic axiomatization of set theory as well
as study into forcing and the nonstandard set-theoretic models of Dana Scott,
Robert Solovay, and Petr Vopenka. The new frameworks provide a natural place for
the Boolean valued models that are viewed now the toposes with Aristotle logic
which pave king's ways to the solution of the problem of the continuum by Kurt
Godel and Paul Cohen. These toposes are now the main arena of Boolean valued
analysis.

Bidding farewell to Mac Lane, reading his sincere and openhearted autobiography,
enjoying his vehement polemics with Freeman J. Dyson, and perusing his deep last
articles on general mathematics, anyone cannot help but share his juvenile
devotion and love of mathematics and its creators. His brilliant essays "Despite
Physicists, Proof Is Essential in Mathematics" and "Proof, Truth, and Confusion"
form an anthem of mathematics which is only possible by proof.

  Let me summarize where we have come. As with any branch of learning, the real
  substance of mathematics resides in the ideas. The ideas of mathematics are
  those which can be formalized and which have been developed to fit issues
  arising in science or in human activity. Truth in mathematics is approached by
  way of proof in formalized systems. However, because of the paradoxical kinds
  of self-reference exhibited by the barn door and Kurt Godel, there can be no
  single formal system which subsumes all mathematical proof. To boot, the older
  dogmas that "everything is logic" or "everything is a set" now have
  competition "everything is a function." However, such questions of foundation
  are but a very small part of mathematical activity, which continues to try to
  combine the right ideas to attack substantive problems. Of these I have
  touched on only a few examples: Finding all simple groups, putting groups
  together by extension, and characterizing spheres by their connectivity. In
  such cases, subtle ideas, fitted by hand to the problem, can lead to triumph.
  Numerical and mathematical methods can be used for practical problems.
  However, because of political pressures, the desire for compromise, or the
  simple desire for more publication, formal ideas may be applied in practical
  cases where the ideas simply do not fit. Then confusion arises whether from
  misleading formulation of questions in opinion surveys, from nebulous
  calculations of airy benefits, by regression, by extrapolation, or otherwise.
  As the case of fuzzy sets indicates, such confusion is not fundamentally a
  trouble caused by the organizations issuing reports, but is occasioned by
  academicians making careless use of good ideas where they do not fit. As
  Francis Bacon once said, "Truth ariseth more readily from error than from
  confusion." There remains to us, then, the pursuit of truth, by way of proof,
  the concatenation of those ideas which fit, and the beauty which results when
  they do fit.

So wrote Saunders Mac Lane, a great genius, creator, master, and servant of
mathematics. His unswerving devotion to the ideals of truth and free thinking of
our ancient science made him the eternal and tragicomical mathematical Knight of
the Sorrowful Figure...

  Sobolev Institute of Mathematics
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  E-mail address: sskut@member.ams.org