Mordell’s review, Siegel’s letter to Mordell, diophantine geometry, and 20th century mathematics.

*(English)*Zbl 0854.01021[See also the review of the same article in Zbl 0820.01013).]

In 1962, Serge Lang published the book “Diophantine Geometry”, which was reviewed in 1964 by L. J. Mordell for the Bulletin of the American Mathematical Society. This review became well-known for its eloquent rejection of what Mordell saw as Lang’s attitude toward Diophantine problems of maximal generalization, obscuring the understanding of “the simpler and really fundamental case”, and his use of “a method of infinite ascent in expounding his proofs” review. The review is worth reading as a most eloquent expression of its views. On March 3, 1964, Carl Ludwig Siegel wrote Mordell a letter [in English] agreeing with his estimation of Lang’s book. Siegel’s mode of expression was not necessarily temperate and this letter is true to that form. In 1969, Mordell published a book on Diophantine equations, and BAMS assigned Lang to be its reviewer. Mordell’s review of Lang and Lang’s review of Mordell were both published in an appendix to a revision of Lang’s book published as “Fundamentals of Diophantine geometry” by Springer [New York etc. (1983; Zbl 0528.14013)]. The paper under review gives a historical survey of interactions between algebraic geometry and diophantine problems in number fields and function fields during the 20th century. It also spends considerable time defending Lang’s point of view against Mordell and Siegel, though on the face of just the historical evidence, he would certainly seem to have the better of the argument insofar as the importance of the interaction goes. The remaining issue is why does one write mathematics books. In 1966 Lang wrote Mordell that he saw no reason not to write very advanced texts and “thus allowing certain expositions at a level which may be appreciated only by a few” thus achieving a coherence otherwise not possible. Mordell presumably had been seeking something more introductory, but actually at the end of his review he says in partial agreement “Those who can understand this book will be indebted to him [Lang] for having brought together in one volume the results contained in it.”, but adds that the book would have been better if it had been written in such a way as to have a much larger class of readers. (In Mordell’s defense it must be noted also that Lang in the original, but not the revision, calls much of the content of the book “elementary” (p. vii).) The conclusion of this paper which should certainly be read for its excellent historical survey of a subject which Lang knows perhaps as broadly and as well as anyone today, returns to the possible obstructive influence of Mordell and Siegel on mathematics as “great mathematicians” who at a later period in their lives lacked “vision and understanding.” The preceding pages of this paper would themselves seem to show such influence was nugatory, and Lang himself can only come up with an ad hominem remark by Saunders MacLane. As a young instructor this reviewer was among those shown Siegel’s letter by Mordell. The last several paragraphs of this paper seem unnecessarily defensive, and as an attack on anonymous contemporary mathematicians, offensive. At this writing, Siegel has been dead for more than 15 years, Mordell has been dead for more than 24, and Lang is approaching 70. Perhaps it is time to bury the hatchet.

In 1962, Serge Lang published the book “Diophantine Geometry”, which was reviewed in 1964 by L. J. Mordell for the Bulletin of the American Mathematical Society. This review became well-known for its eloquent rejection of what Mordell saw as Lang’s attitude toward Diophantine problems of maximal generalization, obscuring the understanding of “the simpler and really fundamental case”, and his use of “a method of infinite ascent in expounding his proofs” review. The review is worth reading as a most eloquent expression of its views. On March 3, 1964, Carl Ludwig Siegel wrote Mordell a letter [in English] agreeing with his estimation of Lang’s book. Siegel’s mode of expression was not necessarily temperate and this letter is true to that form. In 1969, Mordell published a book on Diophantine equations, and BAMS assigned Lang to be its reviewer. Mordell’s review of Lang and Lang’s review of Mordell were both published in an appendix to a revision of Lang’s book published as “Fundamentals of Diophantine geometry” by Springer [New York etc. (1983; Zbl 0528.14013)]. The paper under review gives a historical survey of interactions between algebraic geometry and diophantine problems in number fields and function fields during the 20th century. It also spends considerable time defending Lang’s point of view against Mordell and Siegel, though on the face of just the historical evidence, he would certainly seem to have the better of the argument insofar as the importance of the interaction goes. The remaining issue is why does one write mathematics books. In 1966 Lang wrote Mordell that he saw no reason not to write very advanced texts and “thus allowing certain expositions at a level which may be appreciated only by a few” thus achieving a coherence otherwise not possible. Mordell presumably had been seeking something more introductory, but actually at the end of his review he says in partial agreement “Those who can understand this book will be indebted to him [Lang] for having brought together in one volume the results contained in it.”, but adds that the book would have been better if it had been written in such a way as to have a much larger class of readers. (In Mordell’s defense it must be noted also that Lang in the original, but not the revision, calls much of the content of the book “elementary” (p. vii).) The conclusion of this paper which should certainly be read for its excellent historical survey of a subject which Lang knows perhaps as broadly and as well as anyone today, returns to the possible obstructive influence of Mordell and Siegel on mathematics as “great mathematicians” who at a later period in their lives lacked “vision and understanding.” The preceding pages of this paper would themselves seem to show such influence was nugatory, and Lang himself can only come up with an ad hominem remark by Saunders MacLane. As a young instructor this reviewer was among those shown Siegel’s letter by Mordell. The last several paragraphs of this paper seem unnecessarily defensive, and as an attack on anonymous contemporary mathematicians, offensive. At this writing, Siegel has been dead for more than 15 years, Mordell has been dead for more than 24, and Lang is approaching 70. Perhaps it is time to bury the hatchet.

Reviewer: S.L.Segal (Rochester)