Karl Menger and IIT, 1955 to 1959
My first contact with Menger was as a student in a course he taught using his own book as the text. I enjoyed the course and his book so much that I read the whole book, not just the material covered in the course and made a list of errors. Most of the errors were minor but the list was long.
Now I faced a dilemma : I wanted to let him know about these errors but was too shy to go to see him in his office. In those days the Professor was king. (When I was a student the Professor was king; later when I became a Professor the student was king.) Few students in those days had the gumption to go to a Professor's office without an invitation.
After getting a D in my College algebra course from Professor Willcox I checked and rechecked my work on the final exam and had other people check it as well, and there was no way to explain such a low grade. Finally someone brought to my attention the fact that Willcox had changed the problems given to us on the written exam handed out in class and had written the changes on the blackboard without saying anything to the class. I never noticed these changes, not even when handing in my exam. Although this D almost put me on academic probation I never discussed this with Willcox. I mention this to give the reader some idea of how shy I was.
Nevertheless I somehow mustered the courage to mention to Menger — with considerable trepidation — that I had found some mistakes in his book. To my amazement he was delighted and asked me to give him my list. From that day onward, although I was just an undergraduate, he treated me more like a colleague than a student.
Needless to say this had an enormous impact on me. At the time I did not know that Menger was a world class Mathematician but I knew he was at least somewhat famous because the Chicago Tribune had written some articles about his effort to improve the notation and teaching of Calculus. If memory serves me right he had obtained a major grant from, I believe the Chrysler Company, for this purpose. He wrote a Calculus book using his improved and rigorous notation.
Another vivid memory I have of Menger is a colloquium lecture he gave with the title:
Is the integral of x dx = to the integral of y dy ?
His answer : No ! This he based on the fact that generally the letter x is used to denote the identity function and the letter y is generally used for other functions. Often without explanation. The reader has to figure these things out from the context. This is the sort of thing he tried to clarify and make rigorous in his Calculus book. Another is the notation dy/dx and in particular the nonsense
which is still used in many Calculus books today. Unfortunately Menger's effort to give clear and rigorous definitions and notation in Calculus failed. This in spite of producing an excellent book which did not get used widely.
Now a few words about IIT in those years. When I graduated a semester early in January 1959, Reingold, the chairman of the Mathematics department, offered me a 1/3 assistantship to teach College Algebra, the course in which I got a D! This I accepted joyfully not realizing that I was being exploited. The tuition was high in those days and this was a five hour course. I was given full responsibility for the course. Although a senior Professor was nominally my supervisor the only interaction I had with him was when walking down the hallway he stopped me to say : "I understand you are teaching set theory" to which I replied "yes". This is the only interaction we had for the entire semester. Who was this senior Professor : Willcox ! If he remembered giving me a D in this course only three years earlier, he never mentioned it.
How did "set theory" get into my College Algebra course? During the first week it became abundantly clear that my students could do all the problems in the book and some of them could do these problems faster than I could. So I decided, without consulting my supervisor or anyone else, to disregard the text book completely and teach set theory.
In those years, before U of I Chicago circle existed and before Northwestern had an Engineering school IIT attracted most of the best students from the best suburban schools. These students had learned College Algebra in high school, probably from the same text. And they had learned it well.
So, instead of doing what most of the Professors did : Go through the book and give exams which do not test the students ability to solve problems but rather the speed with which they can solve these problems. This gave the Professors some means of differentiating between grades, A,B,C, etc.
I decided to teach set theory from my own notes, without a text book. This worked out well. Although the material was elementary the students found it challenging but not too challenging. And they learned to do proofs! Both, set theory and the ability to do proofs are basic for all of Mathematics and many other things. This is how I justified — to myself — using set theory and felt good about it and I still do.
In 2006 looking back from the perspective of a retired Professor, I find it astounding that I — as a beginning graduate student — could decide on my own, without consulting anyone, to completely ignore the assigned text book and teach something completely different from my own notes and not get any complaint from the faculty or from the students! Undoubtedly the confidence I gained from my interaction with Menger and some of my other Professors at IIT helped me immensely.
Oh how things have changed !!