A Mathematician’s Mathematicians: Professor Herb Keller, and his Notation for QED

I decided to change my major from Physics to Mathematics at the end of my junior year at Caltech. When I returned to the campus in September, 1965, I was told that my advisor would be Prof. Richard diPrima. I learned as I started to write this post that there is an award named in his honor, The SIAM Richard C. DiPrima Prize.

Prof. diPrima was the first person I met who was also a member of NYU’s Courant Institute of Mathematical Sciences (CIMS), then and now the world’s foremost school of applied mathematics.

He was a wonderful man. I recall vividly our first conversation, in which I said I had decided to change my major since I had learned how to program, and found that, especially when done in applied mathematics, programming was much more fun than physics.

He said that was very important, for if you found something that you enjoyed doing, then all the rest would take care of itself.

We didn’t do much mathematics together. He suggested I do an independent study project with a former colleague of his from CIMS, Prof. Herbert “Herb” Keller, who was spending the year visiting Caltech.

Herb was a wonderful teacher and mentor. He suggested as a project that I wrote programs related to the Kruskal Solitrons, and I had a lot of fun working with him.

Fun. It is ALL ABOUT FUN. Never forget that, and the details will take care of themselves.

Herb returned to CIMS in September, 1966, the same month I began my graduate studies there.

I took his course on Ordinary Differential Equations in the first, or perhaps second, year.

His classes were a great joy. He would start with a substantial theorem or topic, and then spend the rest of a two-hour class proving the theorem, or digging into the details. His passion for mathematics was evident in every sentence he spoke, or every equation he wrote on the board.

Whenever Herb finished a proof, he would not write QED but a symbol of his own devising.

QED stands for “Quod Erat Demonstrandum” and is the standard way of marking the end of a proof.

Herb instead wrote one of the following symbols:

– o –
o – o

I of course copied this into my notes, and continued to do so for all the other proofs I went through during my remaining years as a student of mathematics

I have continued to use this notation to this day, to mark the end of an important paragraph, or the end of a first draft, and so forth

Of course, each and every time I do I think of him, and what a fine man he was.

Herb left CIMS to become a professor at Caltech within a year or so. I read recently that he had died at the age of 83, and that he had been an avid bicyclist in his later years.

I also had the great good fortune to have Eugene Isaacson as my professor for at least one, and I think more, courses.

The good fortune is doubled in that I count among my many great professors both authors of a classic work on numerical analysis, Analysis of Numerical Methods, known to mathematicians as “Isaacson and Keller.”

Prof. Isaacon was also a wonderful teacher. He also had a striking resemblance to my favorite author, the man after whom this blog is named, A. J. Liebling. I often think of the other when I think of one of them.

Professor Herbert “Herb” Keller – May His Memory Be a Blessing.

– o – o – o – o

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  1. […] I then told Peter that during the fall of my senior year at Caltech I had done an independent study project with Herb Keller, the legendary mathematician and numerical analyst, computing numerical solutions to equations related to solitons. Solitons were the work of Martin Kruskal, Martin was Clyde’s father. Clyde is also a mathematician. I have written of this in my post A Mathematician’s Mathematicians: Professor Herb Keller, and his Notation for QED. […]

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