The Collected Works of Alonzo Church

The long delayed publication of Alonzo Church’s Collected Works is a most welcome event, as the contents of this hardback volume of more than 1100 pages will be of great interest to mathematical logicians and philosophers of mathematics, as well as graduate students in logic and philosophy, all of whom, given its list price of just over $100, should find it affordable.

 

Church published more than sixty articles in refereed journals, over a span of 71 years, from 1924 to 1995. The subject matter ranged widely, both in mathematical logic (computability theory, set theory, and the metamathematics of first-order logic) and in the philosophy of mathematics (especially intensional logic). His most famous results in logic are his formulation of the \(\lambda\)-calculus, his identification of the informal notion of effective computability with the formal notion of recursive function of positive integers (“Church’s Thesis”), and his proofs that the decision problems both for validity in first-order logic (the Entscheidungsproblem ) and for truth in the standard model for first-order number theory are recursively unsolvable. As this volume reveals, however, there was much more, including articles in areas of mathematics outside of logic on such topics as differential equations, differentials, Euclid’s parallel postulate, and the Laplace transform. 

 

Aside from his technical articles, Church’s pioneering textbook, Introduction to Mathematical Logic, vol. I, was for many years the standard reference on the subject (and remains in print sixty-three years after its publication); and his monumental A Bibliography of Symbolic Logic, published as the first volume of the Journal of Symbolic Logic, is an invaluable resource for historians of logic. (An annotated list of every publication in logic from 1666 to 1935, it is not reproduced in these Collected Works, presumably due to its length.)

 

In addition to all of Church’s published articles, the volume under review includes the long introductory chapter to his textbook, together with notes for two chapters of a projected second volume of that work that never appeared. It includes a selection of forty of his reviews published in the Journal of Symbolic Logic, plus his reviews of volumes II and III of Principia Mathematica and of books by Carnap and Ramsey that were published in other journals. There are also entries written by Church for the Encyclopedia Britannica and Dagobert Runes’s Dictionary of Philosophy and a selection of Church’s correspondence with other logicians, including Bernays, Kleene, Post, Carnap and Quine. His correspondence with Gödel, previously published in Gödel’s Collected Works, is not duplicated here.

 

The introductory chapter by the editors provides a biographical sketch of Church’s life and brief commentaries on the most important results he obtained in each of the fields to which he contributed. Detailed critical commentary on the individual papers is, however, “left to other venues”, since to have included such commentary would not only have entailed further delays in publication but would have necessitated expanding the edition to more than one volume.

 

The volume is rounded out by a selection of photographs of Church at various stages of his life, a 1984 interview of him by William Aspray, a list of all of Church’s doctoral students, and two lengthy bibliographies: one of all of Church’s reviews in the Journal of Symbolic Logic and the other of all the items cited in Church’s publications. Name and subject indices for the volume are also provided. 

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John W. Dawson Jr. is Professor Emeritus of Mathematics at Penn State York. The author of Logical Dilemmas: The Life and Work of Kurt Gödel and Why Prove it Again? Alternative Proofs in Mathematical Practice, he was a co-editor of Gödel’s Collected Works and may be reached at jwd7too@comcast.net.