Alonzo Church ________________________________________ Born: 14 June 1903 in Washington, D.C., USA Died: 11 Aug 1995 in Hudson, Ohio, USA Click the picture above to see a larger version Show birthplace location Previous (Chronologically) Next Main Index

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Version for printing ________________________________________ Alonzo 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, Alonzo Jr, Mary Ann and Mildred. Church 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 ^*:- Princeton 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. The article ^* is in three parts and in the last of these Manzano:- ... attempt^* to show that Church's great discovery was lambda calculus and that his remaining contributions were mainly inspired afterthoughts in the sense that most of his contributions, as well as some of his pupils', derive from that initial achievement. 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 in 1931. 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 by Sieg in [11. 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 ... ^* critical, analytical commentary. 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. This first edition was, as he states in the Introduction:- ... the first half of an introductory course in mathematical logic given to graduate students in mathematics Princeton in 1943. Haskell Curry in a review of the 1944 work writes:- It is written with the meticulous precision which characterizes the author's work generally. ... The subject matter is more or less classical, namely, the propositional algebra and the functional calculus of first order, to which is added a chapter summarizing without proofs certain features of functional calculi of higher order. For the expert the chief interest in the tract is that it makes readily accessible careful detailed formulation and proofs of certain standard theorems, for example, the deduction theorem, the reduction to truth tables, the substitution rule for the functional calculus, Gödel's completeness theorem, etc. Manzano writes in ^* that the 1956 edition of the book:- ... defined the subject matter of mathematical logic, the approach to be taken and the basic topics addressed. The book begins with an Introduction which discusses names, variables, constants and functions, and leads on to the logistic method, syntax and semantics. Chapters I and II are concerned with the propositional calculus, discussing tautologies and the decision problem, duality, consistency and completeness, and independence of the axioms and rules of inference. The first order functional calculus is studied in Chapters III and IV, while Chapter V deals mainly with second order functional calculi. 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 in ^*. 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. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page ________________________________________ List of References (13 books/articles)

Mathematicians born in the same country

Cross-references in MacTutor 1. Chronology: 1930 to 1940 ________________________________________Other Web sites 1. Gian-Carlo Rota 2. Princeton (Interview with Church) 3. Stanford Encyclopedia of Philosophy (The Church-Turing thesis) 4. Mathematical Genealogy Project

References for Alonzo Church Version for printing ________________________________________ 1. Biography in Encyclopaedia Britannica. Books: 2. C A Anderson and M Zelëny, Logic, meaning and computation : Essays in memory of Alonzo Church, Logic, meaning and computation (Dordrecht, 2001). Articles: 3. C A Anderson, Alonzo Church's contributions to philosophy and intensional logic, Bull. Symbolic Logic 4 (2) (1998), 129-171. 4. H B Enderton, In memoriam: Alonzo Church (1903-1995), Bull. Symbolic Logic 1 (4) (1995), 486-488. 5. H B Enderton, Alonzo Church and the reviews, Bull. Symbolic Logic 4 (2) (1998), 172-180. 6. J Folina, Church's Thesis : prelude to a proof, Philos. Math. (3) 6 (3) (1998), 302-323. 7. In honor of Alonzo Church's 75th birthday with some remarks from the History of logic of A Dumitriu, Internat. Logic Rev. 17-18 (1978), 150-154. 8. A Irving, Alonzo Church (1903-1995), Modern Logic 5 (1995), 408-410. 9. D Kaplan and T Burge, Remembering Alonzo Church, Logic, meaning and computation (Dordrecht, 2001), xi--xiii. 10. M Manzano, Alonzo Church : his life, his work and some of his miracles, Hist. Philos. Logic 18 (4) (1997), 211-232. 11. W Sieg, Step by recursive step : Church's analysis of effective calculability, Bull. Symbolic Logic 3 (2) (1997), 154-180. 12. UCLA philosopher, mathematician Alonzo Church dead at 92, Modern Logic 5 (4) (1995), 410-412. 13. ULCA Philosopher, Mathematician Alonzo Church dead at 92, History of Logic Newsletter 19 (Sept 1995), 1-2. Chronology for 1930 to 1940 Previous page Chronology index Full chronology Next page

________________________________________ 1930 Van der Waerden's famous work Modern Algebra is published. This two volume work presents the algebra developed by Emmy Noether, Hilbert, Dedekind and Artin. 1930 Hurewicz proves his embedding theorem for separable metric spaces into compact spaces. 1930 Kuratowski proves his theorem on planar graphs. 1931 G D Birkhoff proves the general ergodic theorem. This will transform the Maxwell-Boltzmann kinetic theory of gases into a rigorous principle through the use of Lebesgue measure. 1931 Gödel publishes Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (On Formally Undecidable Propositions in Principia Mathematica and Related Systems). He proves fundamental results about axiomatic systems showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved. 1931 Von Mises introduces the idea of a sample space into probability theory. 1931 Borsuk publishes his theory of retracts in metric differential geometry. 1932 Haar introduces the "Haar measure" on groups. 1932 Hall publishes A contribution to the theory of groups of prime power order. 1932 Magnus proves that the word problem is true for one relator groups. 1932 Von Neumann publishes Grundlagen der Quantenmechanik on quantum mechanics. (See this History Topic.) 1933 Kolmogorov publishes Foundations of the Theory of Probability which presents an axiomatic treatment of probability. 1934 Gelfond and Schneider solve "Hilbert's Seventh problem" independently. They proved that aq is transcendental when a is algebraic ( 0 or 1) and q is an irrational algebraic number. 1934 Leray shows the existence of weak solutions to the Navier-Stokes equations. 1934 Zorn establishes "Zorn's lemma" so named (probably) by Tukey. It is equivalent to the axiom of choice. 1935 Church invents "lambda calculus" which today is an invaluable tool for computer scientists. 1936 Turing publishes On Computable Numbers, with an application to the Entscheidungsproblem which describes a theoretical machine, now known as the "Turing machine". It becomes a major ingredient in the theory of computability. 1936 Church publishes An unsolvable problem in elementary number theory. "Church's Theorem", which shows there is no decision procedure for arithmetic, is contained in this work. 1937 Vinogradov publishes Some theorems concerning the theory of prime numbers in which he proves that every sufficiently large odd integer can be expressed as the sum of three primes. This is a major contribution to the solution of the Goldbach conjecture. 1938 Kolmogorov publishes Analytic Methods in Probability Theory which lays the foundations of the theory of Markov random processes. 1939 Douglas gives a complete solution to the Plateau problem, proving the existence of a surface of minimal area bounded by a contour. 1939 Abraham Albert publishes Structure of Algebras. 1940 Baer introduces the concept of an injective module, then begins studying group actions in geometry. 1940 Aleksandrov introduces exact sequences. ________________________________________ List of mathematicians alive in 1930. List of mathematicians alive in 1940. The Princeton Mathematics Community in the 1930s Transcript Number 5 (PMC5) © The Trustees of Princeton University, 1985 ALONZO CHURCH Alonzo Church is interviewed by William Aspray on 17 May 1984 at the University of California at Los Angeles. Aspray: Could we begin by your describing how you came to Princeton and what caused your interest in Princeton? Church: I was an undergraduate at Princeton, and I was pressed by the math department to go on to graduate school. Actually they gave me fellowships that paid my way, otherwise I would not have been able to continue. Aspray: Who was it on the faculty that was encouraging you to go on to graduate school? Church: Primarily Oswald Veblen, also to some extent Dean Burchard Fine and Luther Eisenhart. Aspray: What years were you a grad student? Church: I graduated in 1924, as an undergraduate that is, and then immediately went to graduate school and got my degree in 1927. Aspray: After finishing graduate school did you immediately become an instructor in the department or did you go off some place? Church: I had two years on a National Research Fellowship. I spent a year at Harvard and a year in Europe, half the year at Goettingen, because Hilbert was there at the time, and half the year in Amsterdam, because I was interested in [L.E.J. Brouwer's work, as were some of those advising me. Aspray: Brouwer was there at the time? Church: Yes. I think he wasn't teaching. He was quite old. I used to take the train out to his residence, way out in the country. Aspray: Who of Brouwer's group of disciples, whatever you want to call them, were there while you were there? Church: ^* Heyting was not there, and I remember no one except Brouwer himself. He had a secretary who was also a student, but she was not interested in foundations. Aspray: How did you get interested in foundations? Church: Well, mainly through Veblen, who was not himself a contributor to foundations in math except in the old-fashioned sense of postulate theory. Aspray: Geometry and postulate theory? Church: Yes. His dissertation was about axioms for Euclidean geometry. He did over again what Hilbert had. done, so of course it was not wholly original, but I always thought his axioms for geometry were on the whole somewhat better than Hilbert's. Of course Hilbert had prestige and he didn't. Aspray: At least three other people that I've interviewed have said that. Your interest in logic, did it come as an undergraduate or a graduate student? Church: I was generally interested in things of a fundamental nature. As an undergraduate I even published a minor paper about the Lorentz transformation, the foundation of (special) relativity theory. It was partly through this general interest and partly through Veblen, who was still interested in the informal study of foundations of mathematics. It was Veblen who urged me to study Hilbert's work on the plea, which may or may not have been fully correct, that he himself did not understand it and he wished me to explain it to him. At any rate, I tried reading Hilbert. Only his papers published in mathematical periodicals were available at the time. Anybody who has tried those knows they are very hard reading. I did not read as much of them as I should have, but at least I got started that way. Veblen was interested in the independence of the axiom of choice, and my dissertation was about that. It investigated the consequences of studying the second number class under each of two assumptions that contradict the axiom of choice. Aspray: Was there any opposition on the part of the rest of the department to a graduate student doing a dissertation in logic? Church: Well it was not exactly a dissertation in logic, at least not the kind of logic you would find in and Russell's Principia Mathematica for instance. It looked more like mathematics; no formalized language was used. The only thing that might have annoyed some mathematicians was the presumption of assuming that maybe the axiom of choice could fail, and that we should look into contrary assumptions. Aspray: That suggests that if you wanted to do something along the lines of Principia Mathematica you would have had some trouble in the mathematics department doing it. Church: Quite possibly. I did later try that. I published a paper with serious errors, and generally got in bad because I was hasty and incautious. Aspray: Can you tell me something about your graduate education, the kinds of things you studied, the people you studied with. Church: I had an interest in foundational questions, but there were not many courses in that direction. I took essentially the standard curriculum. I could not name all the courses I took, but there was, of course, a general examination to pass, and there were various required subjects including analysis and real-number theory. I forget exactly what else, but I think I still have something listing the courses I took with the signatures of the instructors. Aspray: You presumably took courses with Eisenhart, Fine, and Veblen. Church: Yes, and Hille and [J.H.M. Wedderburn. I can't name the exact courses now, but I remember several courses in analysis. James Alexander had a course in topology. He appointed me to take lecture notes. This is something I have somewhere. He spent about half the course on the solution of the problem of classifying closed two-dimensional manifolds. This was done in a highly geometric way, which has much more appeal than the present topology, which consists mainly of incidence tables and something that looks so much like algebra you can't tell the difference unless you go into detail. I wrote a very careful set of notes on the first half of the course which was on just this problem of classifying closed two-dimensional manifolds. They are around somewhere. There is nothing original in them, but I think they are a careful job of reproducing Alexander's lectures. Sad to say I never got the second half finished. Somehow or other he forgave me for not doing it, probably because he had to, but by the end of the course I had just finished the notes on the first half. Aspray: Was it standard for grad students to be asked to take lecture notes at that time? I know it was in the '30s. Church: I assume it was. I don't know for sure, but I did it for Alexander's lectures, and it may be that is the best record of what he was doing at the time. I have not looked into his publications. Aspray: What do you remember of the various faculty members as teachers at the time you were a grad student? Does anybody stand out one way or another? Church: Veblen perhaps. I think it was because of his interest in foundational questions that he impressed me. Fine was excellent for teaching undergraduates, especially for the better sort of undergrads who had some idea of what was going on and were not just grinding away at it. He had not done any research since he got his degree, and he did not try to teach any graduate courses, but I had many courses with him as an undergraduate. I thought well of almost everyone who was teaching there at the time. Who were the others? Eisenhart, Wedderburn, and of course Alexander. Aspray: ^* Thomas came later, is that right? Church: Yes. He was essentially a contemporary of mine, I think. He got his degree four years before I did. Aspray: Didn't Einar Hille come sometime while you were a student? Church: Yes, I don't remember whether he was there when I entered the graduate school or whether he came later. Aspray: Who were some of your fellow graduate students? Church: Paul Smith. I remember him as being a graduate student at the same time I was. There are no doubt a couple of others about whom I would say, "Of course I remember a lot about him" when the name came to mind. Aspray: How closely did you work with Veblen on your own research? Church: He was really the only man supervising it. I sort of had to convince him about some aspects of the axiom of choice. To deny what seems intuitively natural is rather difficult. You tend to slip back into what informally seems more reasonable. I remember from time to time having to explain things to him, but I convinced him that my arguments were sound. Aspray: Do you remember who else was on the committee that read your thesis and examined you? Church: Certainly Veblen, quite likely Eisenhart and Alexander, but I have forgotten. Aspray: Several people have suggested that Veblen encouraged grad students and visitors and young faculty members to really push their research and not put as much effort into their teaching. How would you react to that? Church: Well, I don't remember his being negative toward teaching. Of course he did try to get people interested in research, but that is probably not unusual. Aspray: Though Princeton was a special place at that time. Church: It had preeminence specifically in math. There were complaints that the University was overemphasizing this one field to the detriment of others. Aspray: I see, mainly because the University was thought of primarily as an undergraduate institution. Church: It had been for a long time. The University was developing the grad school. I wasn't one to complain, but there probably was a one-sided emphasis on math because they happened to be able to get a lot of good people in that particular field. My impression at the time was that for teaching grad students there were abler men in mathematics than there were in other departments. That tended to produce an emphasis on math. You can't be preeminent in all fields, so there is something to be said for being preeminent in one. Aspray: I know that there are certain external ways of judging which do seem to indicate that math was preeminent. For example, there were certain competitive fellowships that seemed to always go to the math department. One—I can't recall the name now—for people coming from Cambridge each year. Church: Yes. I remember the fellowships that I had, but I don't know whether they were confined to mathematicians or whether they were general fellowships. I probably did not notice very much. Aspray: Did you do any teaching while you were a grad student? Church: I think not. I got an appointment as an assistant professor immediately after my two years on a fellowship. I think that was the first teaching I did. Aspray: Did you think of going some place other than Princeton after your two years? Church: I think nobody made me an offer, and I did not go hunting for offers because I saw no reason to leave Princeton. Belatedly I remember an offer from John Hopkins, but I had already accepted at Princeton. Aspray: Do you remember much about your teaching responsibilities in your early years as an assistant professor? Church: I may well have been teaching things like elementary calculus, more or less according to the routine. There would be a large group of students taking their first or second course in calculus, 100 to 200 I suppose. They were divided up into sections of ten originally—the number kept growing. There was one man in charge who coordinated things. There was a complicated method of judging the examinations so as to try to make the grading uniform and at the same time have input from the instructors. I remember sitting through sessions where the grades given to the students in different sections were compared and adjusted by artificial formulas. Aspray: Did you get a chance to teach any grad courses? Church: I can't remember when I started teaching grad courses. Rather early I started teaching grad courses in mathematical logic. There was no one else there to do it. Aspray: What sort of things would you cover in those courses? What would you use as material? Church: Yes. I gave first an elementary course in mathematical logic. I forget what textbooks I used at first. I worked as rapidly as possible to get at least something of my own written, out. My research was unorthodox and some of it unsound, but I was devoted to it and wanted to get my own ideas down and teach them. Aspray: I am trying to remember what textbooks were available in the late '20s and early '30s. Do you recall? Church: There were none that I liked. Lewis and Langford's Symbolic Logic was around. No, that may have been later, but certainly the book by C.I. Lewis was available. But there was nothing about the sort of thing I wanted to teach, logic directed towards math rather than the philosophical aspects of logic. Well, I am not sure; there may have been a book of that sort. Of course (David] Hilbert and Wilhelm Ackermann's Grundzuege der theoretischen Logik was in existence at that time, but it was in German. While the grad students were supposed to learn German, as a practical matter I could not have used it as a textbook. So I used written notes of my own and things like that. later check shows the Lewis's A Survey of Symbolic Logic was published in 1918, and Lewis and Langford's Symbolic Logic was published in 1932. A.C. Aspray: Was there any relationship between the math department and the philosophy department at this time? Church: No. Nobody in philosophy was interested in that sort of thing at the time. Aspray: When did an interest in logic develop among philosophers? Church: That is hard to say. Of course, C. I. Lewis' A Survey of Symbolic Logic was published sometime between 1910-1920, and it is very definitely philosophically oriented. So there were philosophers who were interested in symbolic logic from the point of view of its relevance to philosophy rather than to math, and Lewis was one of the leaders in this. He was at Harvard at the time. Aspray: While we are on the subject, can we talk more about the logic community in the late '20s and '30s, both in the US and overseas? Where were the active centers? Did you have any contact with these people? Church: I had very little contact with the people at Harvard, where I suppose the logicians were C.I. Lewis and H.M. Sheffer. Those are the ones I remember. Aspray: Was anyone at Chicago at that time? Church: Not that I remember. There must have been some other logicians, but the others who were active at that date or earlier were, I think, mathematicians. E.L. Post, for instance, was a mathematician. He did write papers criticizing some of Lewis' work. In fact, Lewis' first set of axioms for his modal logic had a serious error that Post corrected, and then Lewis tried a second time. Aspray: Did you have close ties with Post? Church: No. He was at Princeton just before I was. He may have been there at the time I was an undergraduate, but I did not meet him till much later. He had some sort of mental trouble and was inactive for a long time. He finally recovered from it, and that was really when I first heard of him or heard from him. Aspray: What about in Europe? Were these people you were in contact with? Did you keep up a contact with Brouwer, for example? Church: Yes, to some extent. To a greater extent with Bernays, who, because Hilbert was old and ill at the time, was the main logician at Goettingen when I was there. Aspray: Ackermann? Church: No, Ackermann was not there at the time I was there. He never had a university position, if my information is correct. He had a degree from Hilbert, but that was before I was in Goettingen. Where he was in Germany at that time, I do not know, but much later he was teaching at a Gymnasium. He never did really have a university position, though he finally received an honorary professorship at Munich. Aspray: What about people like Skolem, did you have contacts with Skolem? Church: Not till very much later. Aspray: Do you know anything about the discussion there was to bring you back to Princeton as an assistant professor? Maybe you heard this many years later? Church: I was not let in on their deliberations. I assume it was Veblen's idea, though it is merely an inference. All I really know is that I got an offer while I was still a fellow at Goettingen, I accepted the offer, and I ceased to look after that. Aspray: As you progressed up the ranks at Princeton, did Veblen continue to be a strong supporter of your moving up? You obviously had to have your own talent to continue to move up. Church: I assume he was until he resigned at Princeton and joined the Institute. I forget the date of that. It was probably before 1930, but the dates are on record and you can easily check it. Aspray: '30-'31. Church: I see. Aspray: Why don't we turn to your graduate students for a while. If I remember correctly you had Alfred Foster, Stephen Kleene, and John Barkley Rosser. Did you have other students in the '30s? Church: None that I remember now. There may have been some, but none of note. There was a gap there until later when Leon Henkin and John Kemeny were there at the same time. There were also Hartley Rogers, Martin Davis, Norman Shapiro, William W. Boone, and (much later) D.J. Collins. My memory is very poor, both as to the names and as to the chronological order, but most of these were later than the '30s. Aspray: Can you tell me something about these graduate students? Anecdotes, personal stories, things you remember about their research, how they got involved in logic—anything along these lines? Church: I remember Kleene was slow getting started. It is possible he was trying other fields, but as far as I knew he did almost nothing for quite a time. Then suddenly he began to come up with things that impressed me greatly. Aspray: Now did all three of them start by working on the same sorts of things you were working on, such as the lambda calculus? Church: Kleene and Barkley Rosser were there simultaneously, and both started work in connection with recursive functions and the lambda calculus at about the same time. The notion of a general recursive function originated with Gödel in lectures at Princeton. Aspray: How closely did they work with you on projects? Did you suggest problems to them? Did you talk to them regularly? Church: I did talk with them in a general way, and they took courses in which I was teaching things such as the lambda calculus. Probably in both cases they worked considerably alone before they came to me. I don't remember details now, especially not the chronology, but I remember being quite surprised when they first brought their results to me. Aspray: I know Rosser quite well because I studied with him when I was a grad student, but I don't know Foster at all. I'll get to meet him. Church: I didn't think much of him at the time. He has developed since, and I believe he is very well thought of. His field, though, is not exactly logic. He is at Berkeley now as you probably know. Aspray: I'll see him tomorrow. Did you have many grad students taking your courses in logic? Church: At one time I did. There was a time when logic was not very well thought of, and the students tended to follow the trend. Aspray: Can you elaborate on that? I have always thought that was true, but I was not sure. Church: There is nothing definite that I can put my finger on. I speak of an impression. Aspray: What period was this? Church: Oh, the late '30s. Just before the Journal of Symbolic Logic began and for a time after that. Aspray: Now, as the Institute got started, actually even a little before that in the case of von Neumann, you got other people coming into the community who were interested in logic. Church: Well, as far as I know, at the time when von Neumann came to Princeton his interest was set theory rather than logic. Even that was in the past as he had already turned to other subjects, either that or he did so very soon after he came. Aspray: You did not have very much contact with him then? Church: Not too much. Occasionally there would be a question or a paper in set theory I would consult him about; and occasionally he would consult me. This is how I got the impression that he was no longer active in set theory, but was doing something entirely different. Aspray: Was it not 1933 or 1934 that Gödel came to Princeton? Church: Yes, that may be right. Aspray: Did you have close contacts with Gödel then? Church: I had a lot of conversations with him and a lot of disagreements. Like most others, I was hard to convince about the incompleteness theorem. There was at the time a tendency, which I shared, to think that it was special to a certain type of formalization of logic and that a radical reformalization might have the effect that the Gödel argument did not apply. I persisted in that longer than I should have, and he was always trying to convince me otherwise. Aspray: I see. Was the lambda calculus one of those that you would have put into that category of being radical enough that the incompleteness theorem would not apply? Church: Not the lambda calculus alone. In a way that does escape the Gödel theorem, but it does it not by not being powerful enough. I had a scheme that had the lambda calculus as part of it. After publishing a couple of attempts that actually lead to inconsistency, I decided that it couldn't be put through, so the lambda calculus is all that is left of that. The sense in which it escapes the Gödel theorem is not significant from the point of view of logic as a foundation of mathematics, though it might be in other directions. Aspray: Who else came as a visitor or as a member of the Institute or as a university faculty member in the '30s? Church: Bernays was there on two successive occasions, each time on one-year appointments. I think it was at Princeton University, rather than the Institute. I had a lot of contact with him at the time. Aspray: Anyone else? Church: No, I can think of no one else. Aspray: You said that Henkin and Kemeny were students at the same time? Church: Yes. Aspray: This must have been '39, '40, something like that, is that right? Church: There was a gap between the students that was important enough for me to remember, that is between Kleene and Rosser and the next two to fall into that category, Henkin and Kemeny. Aspray: Did they both work with you? Church: Well, Henkin had a new proof of the Gödel completeness theorem and an extension of it to second-order logic. This was quite substantial. Kemeny's dissertation concerned the relative strength of simple type theory and ZF set theory without replacement axiom. He wrote a dissertation which I thought well of, but he did not accomplish very much in research afterwards. Aspray: Did you direct Alan Turing's thesis? Church: Well, he was at Princeton, but not only under my supervision, because, of course, he had worked with M.H.A. Newman in England. It was while he was working with Newman that his truly original ideas came out. Aspray: On effectively computable functions? Church: Yes. In fact the definitions of effective calculability and the results on the unsolvable decision problems are essentially the same. These were obtained by me and by Turing almost simultaneously. I think I was the earlier by six months or a year. My paper was delayed in publication, but there is an earlier abstract. Turing did not hear of it until it finally appeared. It was, of course, a great disappointment to him. I don't know the date at which he first had the result. Aspray: If you don't mind, I would like to ask a few more questions about this topic, because it is one of particular interest to me since I wrote my dissertation on Turing. How did you hear about Turing's work? Church: Well, Turing heard about mine by seeing the published paper in the American Journal of Mathematics. At the time his own work was substantially ready for publication. It may already have been ready for publication. At any rate he arranged with a British periodical to get it published rapidly, and about six months later his paper appeared. At the same time, I think, Newman in England wrote to me about it. Aspray: Now didn't his papers appear in the Journal of Symbolic Logic? Church: No, I guess there wasn't any such journal at that time. It appeared in a British journal. Aspray: Proceedings of the London Mathematical Society. Church: It is quite likely, yes. Aspray: That is where it was. Did you know Newman at the time? Church: Only by correspondence. Aspray: How was Turing's visit to Princeton arranged? Church: At Newman's suggestion he applied for admission as a grad student. Aspray: I thought that he had come on a one-year fellowship and then was encouraged to stay on by Dean Eisenhart for a second year as a regular grad student. Church: Yes, I forgot about him when I was speaking about my own graduate students. Truth is, he was not really mine. He came to Princeton as a grad student and wrote his dissertation there. This was his paper about ordinal logics. Aspray: Right. Did you have much contact with him while he was writing his paper? Church: I had a lot of contact with him. I discussed his dissertation with him rather carefully. Aspray: Can you tell me something about his personality? Church: I did not have enough contact with him to know. He had the reputation of being a loner and rather odd. Aspray: Could you tell me something about the founding of the Journal of Symbolic Logic? For example, what was behind your decision to found a new journal? Church: It was not my doing. Somewhere there is an historical paper about this in the journal itself. Aspray: I can find that easily. I was not aware of it. Church: Yes, it is a historical paper about the founding of the Association for Symbolic Logic and of the journal. I was not in on it from the beginning. I was brought in as editor for the journal later. I think the information in that paper is more accurate than I could give you. Aspray: Could you tell me something about the time you were editing it? How strongly did the math department support the journal? Did they think it was important? Church: Well, they yielded finally to the fact that it had a big reputation elsewhere. There were not many others interested in this field, and it was thought of as not a respectable field, with some justice. There was a lot of nonsense published under this heading. I definitely had the idea that one of the things the journal had to do was to suppress this. There were some savage reviews that were written of nonsense papers; I kept them polite, but they were still sharp. Aspray: And you kept a firm hand on what got published and what did not? Church: Yes. Aspray: Did you have pretty much entire editorial control over publication at that time? Church: There were several editors. I did not try to second guess the other editors when they decided to accept or reject contributed papers. Aspray: I can't recall who the other editors were now. Can you tell me? Church: At first there was no one who stayed very jong, except myself. I persuaded a man to take a three year term, and there were a number who lasted even longer. A library that has a complete set of the journal will quickly answer that question. Aspray: I can find that out. To what extent did the department provide you with support, such as secretarial help and money for assistants? Church: I had a half-time secretary, supplied I think, by the department. Aspray: That could have been supplied by the Association. Church: Perhaps, but probably not because they were hard up for funds from the beginning. Aspray: I know that the Annals of Mathematics was more or less reviewed in-house in Princeton. Was that true of the Journal of Symbolic Logic also? Church: You mean that they used only Princetonians to referee the papers? That certainly was not true of the Journal. There were no other logicians at Princeton, unless you count the visitors like Gödel and Bernays. Aspray: One of the things that Professor Tucker is most interested in getting on tape are recollections of the Princeton environment, because many people thought of it as a special place in the '30s, especially after the Institute was established. Church: Yes, long before that there certainly was an intense interest in mathematical research, and Veblen exhibited that spirit. Aspray: What kind of decisions were made administratively to allow or to foster this kind of environment? Church: Well, that I really don't know. Aspray: For example, the new Fine Hall. Can you comment on it? Church: Somebody gave money for that specific purpose, but I can't remember who it was. Aspray: It was the Jones family. Church: Yes, they were probably friendly with some of the mathematicians. Aspray: I think they were quite friendly with both Dean Fine and also L.P. Eisenhart. Church: Yes. Aspray: But, architecturally speaking did the building fit the requirements of a mathematics research group? Church: Yes. It was fancier than necessary and not strictly utilitarian. But at the time Princeton was going in for Gothic architecture. All the large offices were paneled in wood up nearly to the ceiling and with elaborate carvings. I assume it is still there. The math department moved, of course, so some other department has it. Aspray: The East Asian Studies Department is there now and still has all the nice carvings. Do you remember going to tea? Was this a regular part of your day? Church: Yes. Veblen used to run those himself before there was any Fine Hall. He promoted that as a way for people interested in research to get together. So afternoon tea in Fine Hall was really a continuation of Veblen's teas in Palmer Laboratory. It never worked much for me. I was too much of a loner, but I think in other mathematical fields it was a very useful thing. Aspray: Did you go anyway? Regularly? Or sometimes? Church: Yes, I used to go to their teas in the afternoon. I never had any mathematical conversations with anybody, because there was nobody else in my field except a student or two. An exception must be made in the case of Bernays while Ike was there; with him there were often conversations at tea time. A. C. Aspray: How was the library? Church: That was very good from the beginning. I think a lot of effort and probably a lot of money was put into getting a good mathematical library. Aspray: That reminds me about another question I have been meaning to ask you about the Journal of Symbolic Logic. It seems that the journal had interest in historical and bibliographic information. It kept you up to date in those ways, as well as publishing research. Church: The intention at the time was to review everything that appeared in the field. A bibliography which was meant to be complete of earlier things was published, and the reviews up until about 1950 were quite complete. The field kept growing and the reviewing got to be too big a job. Aspray: I want to say to you that when I was coming through graduate school interested in the history of logic, they provided an invaluable source. They were a real service. Church: If used right they should be very valuable. I have an idea that few people really use them right. At any rate it could not be continued, because the field got too large and there were not funds around to do it. Aspray: I have the impression that many at Princeton were rather social people, that people like the von Neumanns, the Eisenharts, and the Robertsons were all people that made their homes available regularly for big social occasions. Church: Yes, that is true. Veblen was a great advocate of getting together informally. His teas were in the same spirit. He believed in taking long walks through the woods to discuss mathematical research. It never worked for me, but maybe it did for others. Aspray: Was the relationship between the graduate students and the faculty fairly close? Church: I think so, yes. Aspray: Can you tell me something about the role the Depression played in the development of mathematics in the '30s? Were there opportunities for getting students funds for research? Did it make any difference in your own career? Church: As far as I know, it made no difference to me. Neither do I have much of an impression of the general situation. The university was evidently hard up. For example, they postponed promotions for the faculty. Aspray: Did it affect your being able to help your grad students find positions, for example Rosser and Kleene and Foster? Church: There was never any real problem. Aspray: I seem to recall that Kleene told me that he was ready to go out, that there was not anything for him for a year or two, and that Princeton found money somehow for him to stay on. Church: I don't remember any agonizing delay about his getting a position, but it could have been he stayed a year or two longer than was absolutely necessary. Aspray: Do you remember any discussions in the '30s about the hiring of immigrant mathematicians or of bringing in a large number of foreigners as researchers? There were big battles going on, maybe just underneath the surface, about ... Church: Yes. I wasn't a party to said battles, I am sure. Many were invited to Princeton, and I did not hear any opposition to it. Aspray: Princeton seems to have been unusual in opening its arms to immigrant mathematicians, unlike certain other centers at the time. Church: Yes. I suspect that this was partly Veblen's influence, but I don't know. The Princeton Mathematics Community in the 1930s

Alonzo Church (14 de junho de 1903 - Washington, D.C.)

Lógico Matemático, ele é responsável por alguns teoremas fundamentais da ciência da computação.

Matemáticos dos Estados Unidos da América

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