I was not a student of Kaplansky — at least, not in the sense we usually mean in mathematics. He was, however, my teacher in a number of courses, undergraduate and graduate, and was chairman of the University of Chicago math department when I was a graduate student. Kap’s “style”, mathematical as well as personal, shone through everywhere. Nowadays, most math departments offer a “bridge” course for their majors. This course is designed to ease the transition to real, upper-division mathematics from (increasingly) less rigorous calculus courses. Chicago has had such a course for years. In my day, it was Math 261; at present it has the fashionably inflated number 26100. Currently, just as it did several decades ago, the course covers “sets, relations, and functions; partially ordered sets; cardinal numbers; Zorn’s lemma, well-ordering, and the axiom of choice; metric spaces; and completeness, compactness, and separability.” When I took the course, Kap used notes ofon “Set Theory and Metric Spaces”. Spanier never got around to writing these notes up as a book. Kap, however, did! Set Theory and Metric Spaces appeared in 1972 and continues in the AMS Chelsea series. Kaplansky’s style is as appealing to current students as it was to us decades ago. I have used the book in my classes for many years. One of my recent students enjoyed the book so much that she bought it as a birthday present for her engineer father!
As chairman, Kap maintained a keen interest in graduate students and the graduate program. His sensitivity to grad student-advisor dynamics can be illustrated by the following anecdote. One afternoon at math tea, my advisor, Yitz Herstein, and I got into a “discussion” on how Kap (of Canadian origin like Yitz) pronounced “schedule”. I maintained that Kap would pronounce it with an “sk” as Americans do and Yitz, of course, said that Kap would say “shedule”, as Canadians and Britons do. So, Yitz and I bet a quarter. When Kap arrived at tea, Yitz and I bounded up to him and told him of our bet. Kap thought for an instant and, then, carefully pronounced “skedule” remarking that faculty shouldn’t take money from students and that Yitz “should pay up.” However, I only got 15 cents.
Kap’s rhetorical flourishes are well known; but, sometimes they had unintended consequences. For my first job, I needed official certification that I had completed the Ph.D. A letter from the chairman would suffice. Kap wrote such a letter concluding “… and, barring catastrophe, he will receive the degree on June 11….” This was deemed insufficient by a departmental administrator at Berkeley who quoted the “barring catastrophe” remark. Kap washed his hands of it and sent me off to the Dean of Students in the Division of Physical Sciences for a “really” official letter.
Even, at the last moment, during my final oral exam, Kap’s style was apparent. He asked me where would you find a commutative ring with some property or other. I started to construct the ring when he interrupted: “No, no, in what book would you look for it?” I replied, “Nagata” and was off the hook!
Kap’s lessons and advice remain fresh to this day. His books and his expositions are as attractive to the current generation of students as they were to mine.