... do some honours work, finish off a graduate course in mathematics he had apparently overlooked earlier, polish up his French, and devote some time to other minor sports, such as chess and rifle practice.

In 1946, a young Andrew Gleason joined the Society of Fellows as a Junior Fellow, spending the next four years auditing lots of mathematics courses and focusing his attention on the famous mathematics problem, "Hilbert's Fifth."

In attending his last lecture, I was struck by the care he took with the basic definitions, the simplicity of his presentation, and the time he had taken in preparing careful notes for the students. ... Andy has always been a terrific inspiration to students.

In thinking about, and admiring, Andy Gleason's career, your natural reference is the total profession of a mathematician: designing and teaching courses, advising on education at all levels, doing research, consulting for the users of mathematics, acting as a leader of the profession, cultivating mathematical talent, and serving one's institution. Andy Gleason is that rare individual who has done all of these superbly.

A connected locally compact group G is a projective limit of a sequence of Lie groups; and, if G has no small subgroups, then it is a Lie group.

... an outstanding contribution to science.

This is a most unusual book, as will immediately be apparent from the perusal of its table of contents. Chapters I to VI cover elementary logic and set theory; Chapters VII to X deal with the various "number systems" from the natural integers to the complex numbers; Chapter XI briefly returns to set theory (countable sets, cardinal numbers and the axiom of choice); finally, the last four chapters deal, respectively, with limits of complex sequences, infinite series and products, metric spaces, and the elementary theory of holomorphic functions of one variable (Cauchy integral excluded, but the logarithmic function is defined and studied). The motivation behind this selection is explained in the introduction, where the author expresses his belief that "set-theoretic mathematics" is essentially different from mathematics as a science with "a real content which transcends the inadequacies of our efforts to formalize it", a distinction which he feels has widened the gap between pure and applied mathematics; his goal is therefore to try to convey to students (around their fourth year in the university) the close relationship between "formalized" and "real" mathematics.

Every working mathematician of course knows the difference between a lifeless chain of formalized propositions and the "feeling" one has (or tries to get) of a mathematical theory, and will probably agree that helping the student to reach that "inside" view is the ultimate goal of mathematical education; but he will usually give up any attempt at successfully doing this except through oral teaching. The originality of the author is that he has tried to attain that goal in a textbook, and in the reviewer's opinion, he has succeeded remarkably well in this all but impossible task. Most readers will probably be delighted (as the reviewer has been) to find, page after page, painstaking discussions and explanations of standard mathematical and logical procedures, always written in the most felicitous style, which spares no effort to achieve the utmost clarity without falling into the vulgarity which so often mars such attempts.

Andrew was a great mathematician who solved many important problems in mathematics. He also provided a great service to the University as chairman of the Society of Fellows and as chairman of the department for a period of time. One-time president of the American Mathematics Society, he was a leader of the world's math community. He trained countless graduate students, and proved an inspiration to them and others.