Alonso Church's parents were Mildred Hannah Letterman Parker and Samuel
Robbins Church. His father was a judge. He was a student at Princeton
receiving his first degree, an A.B., in 1924, then his doctorate three
years later. His doctoral work was supervised by Veblen,
and he was awarded his doctorate in 1927 for his dissertation entitled
Alternatives to Zermelo's Assumption. While he was still working for his
doctorate he married Mary Julia Kuczinski at Princeton
in 1926. They had three children, Alonso Jr, Mary Ann and Mildred.
spent two years as a National Research Fellow, one year at Harvard
University then a year at Göttingen and Amsterdam. He returned
to the United States becoming Assistant Professor of Mathematics
at Princeton in 1929. Enderton writes:
in the 1930's was an exciting place for logic. There was Church
together with his students Rosser and Kleene. There was John von
Neumann. Alan Turing, who had been thinking about the notion of
effective calculability, came as a visiting graduate student in
1936 and stayed to complete his Ph.D. under Church. And Kurt Gödel
visited the Institute for Advanced Study in 1933 and 1935, before
moving there permanently.
He was promoted to Associate
Professor in 1939 and to Professor in 1947, a post he held until 1961
when he became Professor of Mathematics and Philosophy. In 1967 he retired
from Princeton and went to the University of California at Los Angeles
as Kent Professor of Philosophy and Professor of Mathematics. He continued
teaching and undertaking research at Los Angeles until 1990 when he retired
again, twenty-three years after he first retired! In 1992 he moved from
Los Angeles to Hudson, Ohio, where he lived out his final three years.
His work is of major
importance in mathematical logic, recursion theory, and in theoretical
computer science. Early contributions included the papers On irredundant
sets of postulates (1925), On the form of differential equations of a
system of paths (1926), and Alternatives to Zermelo's assumption (1927).
He created the -calculus in the 1930's which today is an invaluable tool
for computer scientists.
In 1941 he published
the 77 page book The Calculi of Lambda-Conversion as
a volume of the Princeton University Press Annals of Mathematics Studies.
It is effectively a rewritten and polished version of lectures Church
gave in Princeton in 1936 on the -calculus.
Church is probably best
remembered for 'Church's Theorem' and 'Church's
Thesis' both of which first appeared in print in 1936. Church's
Theorem, showing the undecidability of first order logic, appeared in
A note on the Entscheidungsproblem published in the first issue of the
Journal of Symbolic Logic. This, of course, is in contrast with the propositional
calculus which has a decision procedure based on truth tables. Church's
Theorem extends the incompleteness proof given of Gödel
Church's Thesis appears
in An unsolvable problem in elementary number theory published in the
American Journal of Mathematics 58 (1936), 345-363. In the paper he defines
the notion of effective calculability and identifies it with the notion
of a recursive function. He used these notions in On the concept of a
random sequence (1940) where he attempted to give a logically satisfactory
definition of "random sequence". Folina argues
for the usually accepted view that Church's Thesis is probably true but
not capable of rigorous proof. The background to Church's work on computability
and undecidability, based on his correspondence with Bernays during the
years 1934-1937, is examined.
Church was a founder
of the Journal of Symbolic Logic in 1936 and was an editor of the reviews
section from its beginning until 1979. In fact he published a paper A
bibliography of symbolic logic in volume 4 of the Journal and he saw the
reviews section as a continuation and expansion of this work. Its aim,
he wrote, was to provide:-
...to provide a
complete, suitably indexed, listing of all publications ... in symbolic
logic, wherever and in whatever language published ... [giving] critical,
The article highlights
Church's guiding role in defining the boundaries of the discipline of
symbolic logic through this editorial work and testifies to his unflagging
industry and conscientiousness and his high editorial standards. The aim
of comprehensive coverage, which in 1936 had seemed quite practical, became
less so as the years went by and by 1975 the rapid expansion in symbolic
logic publications forced Church to give up this aspect and begin to provide
only selective coverage. We mentioned above that Church retired from Princeton
in 1967 and went to the University of California at Los Angeles. Perhaps
this is the place where we should mention why he left Princeton after
38 years of service there. Enderton writes:
Upon his retirement,
Princeton was unwilling to continue accommodating the small staff working
on the reviews for the Journal of Symbolic Logic.
Church wrote the classic
book Introduction to Mathematical Logic in 1956. This was a revised and
very much enlarged edition of Introduction to mathematical logic which
Church published twelve years earlier in 1944.
Another area of interest
to Church was axiomatic set theory. He published A formulation
of the simple theory of types in 1940 in which he attempted to give a
system related to that of Whitehead and Russell's Principia Mathematica
which was designed to avoid the paradoxes of naive set theory. Church
bases his form of the theory of types on his -calculus. Other work by
Church in this area includes Set theory with a universal set published
in 1971 which examines a variant of ZF-type axiomatic set theory and Comparison
of Russell's resolution of the semantical antinomies with that of Tarski
published in 1976. Another of Church's research interests was intensional
semantics which is considered in detail. The idea developed here was similar
to that of Frege, distinguishing between the extension of a term and the
intension, or sense, of a term. Church considered this topic for about
40 years during the latter part of his career, beginning with his paper
A formulation of the logic of sense and denotation in 1951.
Although most of Church's
contributions are directed towards mathematical logic, he did write a
few mathematical papers of other topics. For example he published Remarks
on the elementary theory of differential equations as area of research
in 1965 and A generalization of Laplace's transformation in 1966. The
first examines ideas and results in the elementary theory of ordinary
and partial differential equations which Church feels may encourage further
investigation of the topic. The paper includes a discussion of a generalization
the Laplace transform which he extends to non-linear partial differential
equations. This generalization of the Laplace transform
is the topic of study of the second paper, again using the method to obtain
solutions of second-order partial differential equations.
Church had 31 doctoral
students including Foster, Turing, Kleene, Kemeny, Boone, and Smullyan.
He received many honours for his contributions including election to the
National Academy of Sciences (United States) in 1978. He was also elected
to the British Academy, and the American Academy of Arts and Sciences.
Case Western Reserve (1969), Princeton (1985) and the State University
of New York at Buffalo (1990) awarded him honorary degrees.