Close Menu

John Neuberger

My first academic position after graduate school was at IIT, in 1957.

I found a vibrant intellectual atmosphere there thanks mainly to Karl Menger, Pasquale Porcelli and their students.

Menger was very concerned with educational reform. He felt deeply, as I did and still do, that a good foundation in calculus was important both for prospective mathematicians and for those who will use mathematics, particularly students from engineering and the sciences. He had revolutionary ideas on calculus teaching, many of which are reflected in his calculus book. It was his nature that he wasn't discouraged by the enormity of the task. He liked to ask 'what is x', bringing to mind the flimsy nature of 'variables' in formulating calculus. Many of his ideas are just as fresh and compelling as they were fifty years ago. Not a great deal of progress has been made in this direction since then, but sometimes good ideas do take a long time.

Menger was a pioneer in advocating the algebraic nature of systems of functions in which composition is one of the operations. He insisted that the identity function, under composition have a name, pointing out that this is common practice in algebra. He advocated using 'j' for the identity function, under composition, for functions on the real line. I have adopted this notation since then, but that puts me in a tiny minority.

Menger was always present at talks in the department. He often appeared to be very relaxed (actually asleep), but it was universal that he asked penetrating questions at the end of the talk. Everyone speaker I quizzed about this confirmed that his questions showed a complete grasp of the talk. This was certainly the case for talks that I gave. Menger's enthusiasm for both research and teaching was a prime source of life in the department. We all recognized that we were in the presence of a mathematical legend, but were equally impressed on how concerned he was about students on all levels. Two of my graduate school professors were strong advocates of educational reform and Menger's influence combined with theirs left me a life-long advocate of the same.