In the fall 2006 Department of Applied Mathematics newsletter, we requested students of Karl Menger to share their memories. Enjoy some we received:

To add your memories to this list, email newsletter@math.iit.edu or call 312.567.5344.

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

dy/dx =(dy/dt)(dt/dx)

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!!

Karl Menger was an unusual and colorful character. He was a brilliant refugee from Austria, an acclaimed child prodigy and son of a well known musician. In 1938, while he was on a visit to the U.S., the Nazis took over Austria and he cabled his reaction to the new order by resigning his academic position in Vienna. Menger had a thick German accent. His hair was usually unruly, his limited wardrobe was always rumpled and he wore the same awful tie day after day, spots and all. But he was a superb and innovative teacher, having reformed the teaching of calculus.

Karl Menger was a demonstrative teacher; he was like a maestro conducting an opera of mathematics in the lecture hall. His classes were attended by some 100 students, seated in a stadium theater arrangement; the lectures being supplemented with smaller study workshops tutored by his teaching assistants. On the front walls of the lecture hall were eight very large, rectangular black boards which the professor would fill up in rapid succession with theorems and formulas, accentuating the written material with a rapid staccato lecture. It was pure theatre.

Now, Menger had a tendency to roam from one side of the lecture hall to the other. In order to limit the range of his constantly shifting position, he judiciously placed chairs at each extreme of the bank of blackboards. He had, however, one more problem. The large surface of the blackboards, coupled with his rapid ability to write and fill them up, necessitated the continuously tedious task of erasing the surface so that he could restart the process once more.

One day, Menger entered the lecture hall with several oversized erasers; they seemed to be about one by two feet in size. He proclaimed that he had custom-ordered these erasers so that he could restore the blackboards to their blank status with only a few swoops.

Menger had a tradition of adding extra credit problems to his examinations. In one critical final exam, Calculus I, I finished both the regular exam as well as the extra portion. In order for the students to obtain their test result and final grade for the course, it was customary to provide the teacher with a self-addressed postcard. This I did and received back the notation that I had done well, and that, in fact, I had received an A*, also known as an A-Star. Professor Menger, whose eccentricities were well known, added the handwritten comment, “Why don't I see more of you?” As if I hadn't sat religiously in his classes for over a year.

The story does not stop here. Menger insisted that the school's administration recognize his anointed A* as a grade above A and give it an additional one point value to 4.0 (the A had a point value of 3). This additional point balanced the B that I received in physics and therefore I maintained an A average for the first semester of the school year 1948-1949. Menger won his argument and I must have been the first IIT student with the highest single grade-value ever.

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.

I came to Illinois Tech as an instructor in January of 1957 having just received my Ph.D. from R. L. Moore. I knew of and had a great deal of respect for Professor Menger and was anxious to meet him. Almost immediately after I arrived in the department I was told that he was quite irritable and that whatever else I did I was not to criticize his book on calculus.

I did meet Professor Menger very shortly after my arrival and found him outgoing and friendly contrary to what I had been told. We met frequently in his office to discuss mathematics. One day he was talking at his usual breakneck speed and mentioned with his accent what sounded like a “nowerdiferentablfunction.” After asking him to repeat it several times I finally understood that he was saying a nowhere differentiable function, and I asked him what that meant. He explained and wanted to show me an example. I declined saying I would like to work on the problem. The next day he patiently sat and listened to me give, as best I recall, a horribly complicated example of an arc in the plane and a parameterization of it so that the parameterizing functions were nowhere differentiable. I am sure my explanation took over an hour. After my discourse he asked if he could show me a simple example and he did so in a very few minutes.

We became good friends and he called my apartment frequently to ask questions or discuss mathematics. He raised one question, which I settled, and he tried to get me to submit it to Fundamentae Mathematicae, but I declined saying I thought it was too simple. As summer approached he offered to pay me from a grant he had if I would write up my proof and submit it. I accepted his offer, which allowed me the freedom to do research in the summer. The solution of that problem became my first publication. Recently I came across a copy of that paper and I had considerable difficulty understanding what I had done.

I recall one instance when I was waiting outside a class he was teaching to talk to him. I watched him lecturing at a rapid pace, filling the board over and over again. He continued until the bell rang, and then some and no one moved to leave. Finally he left the room and the students followed in pairs or groups talking animatedly about what they had just heard. As a student of R. L. Moore, I almost never lecture, but this convinced me that Menger got his students just as excited about mathematics by lecturing as Moore did using his methods.

Finally, I will say that I was a severe critic of his book. I claimed that what he had done was to introduce notation that allowed students to do the same mindless manipulation they did with the standard notation but be mathematically precise. He claimed it was important that the students be precise in what they did whether they understood it well or not. We spent many pleasant hours arguing these points. I understand better now what he meant.

I treasure my memories of Professor Karl Menger.

As a mathematics major ('60–'64) at IIT, I had only one course with Dr. Menger: “Menger Calculus.” It was an evening course, provided primarily for practicing engineers and others employed during the day. The instruction provided a unique geometric, rather than delta-epsilon theory, approach to the concepts of calculus, all presented in a very understandable way. I have utilized that philosophical approach (concrete before theoretical) in my 40+ years as a mathematics educator.

It was impossible to have contact with Karl Menger without being influenced by him. He exuded the fine old European liberal values and the continuing Enlightenment. Without your immediately knowing it, his values became partly your own; also his mannerisms and his ways of teaching were subconsciously absorbed.

I cannot recall ever having a conference with him which dealt with course material, except in a few instances where I thought a hypothesis needed tweaking or otherwise wanted to quibble, and I don't think he would have welcomed the kind of tutoring that is widely expected of faculty today. But his door was always open to the discussion of ideas, and if you were fortunate enough to have an idea that was a little original, then he could embarrass you with his enthusiasm. “You must write it up!” he would insist, in his strong accent with its greatly distorted r-sound.

He was very kind and would often invite his students to his home or to his favorite Swedish restaurant close by his house. Sometimes the social aspect was for purposes of education a bit apart from mathematics, and art, music, or philosophy were discussed. Occasionally we made an outing to see a painting or a piece of stained glass.

He loved the English language, and he thought, too, that he loved American democracy, without ever adapting to either of these completely. As with many immigrants from wartime Europe, he was until his death neither fish nor fowl; for it also would have also been impossible for him successfully to return to Vienna. The play *Heldenplatz* by the great Austrian writer Thomas Bernhard describes an academic, not so different from Menger, who made the attempt, with disastrous consequences.

It is impossible for me to think of Karl Menger without at the same time thinking of Berthold Schweizer. For beginning in the mid-1950s, at IIT, Schweizer became Menger's extension and completion, his “lieblingsstudent,” who would go on to build on the work Menger and on that of Menger's earlier lieblingsstudent Abraham Wald, in very important ways, especially in the area of probabilistic geometry. And it is impossible to think of Schweizer without at the same time remembering Abe Sklar. The three of them formed a constellation of pairs that persisted until Menger’s death. Both Schweizer and Sklar also exerted a strong influence on students that complemented that of Menger.

Finally, it is splendid that Menger’s collaborator Franz Alt plans to be present at the memorial conference. Alt and Menger, together with Otto Schreiber, described new foundations for projective and affine geometry. This work was then subsumed by its generalization, lattice theory, and attracted the subsequent contributions of Garrett Birkhoff and John von Neumann.

Accustomed as I am to making carefully prepared spontaneous remarks, I'd like to provide some personal reminiscences of Menger of IIT.

In 1949,Paul Schilpp, a philosopher at Northwestern University, published a volume “Albert Einstein: Philosopher & Scientist” on the occasion of Einstein’s 70th birthday. It contains 25 essays commenting on various aspects of Einstein's work. As a physics graduate student at the University of Chicago, reading this volume in the ’50s, I of course recognized names such as Sommerfeld, de Broglie, Pauli, and Bohr among the contributors. But I’d never heard of Karl Menger, who wrote an article “The Theory of Relativity and Geometry.” No matter. This was a strikingly clear, concise, elegant, instructive, and interesting paper containing many original ideas. Two of Menger's proposals really had a ‘thumping’ impact as he said, “I venture the conjecture that, for the geometrization of physics, especially the physics of the microcosm, idealizations very different from those of Euclid might prove more adequate than his. One such alternative is a geometry where points are not primary entities. What is here contemplated is a geometry of lumps that is, a theory in which lumps are undefined concepts, whereas points appear as the result of limiting or intersectional processes applied to these lumps another possibility is here indicated, namely, the introduction of probability space in which a distribution function rather than a definite number is associated with every pair of elements.” Einstein, in his rejoinder, mentions Menger's ideas (doubts about the nature of the continuum), but cautions, “As long as one has no new concepts which appear to have sufficient constructive power, mere doubt remains. Adhering to the continuum originates with me not in a prejudice, but arises out of the fact that I have been unable to think up anything organic to take its place.”

Sixty years on, it's clear that Menger and Einstein had a prophetic insight: probablistic metric spaces have evolved from a ‘possibility’ to a flourishing branch of mathematics at the hands of Wald, Schweizer, Sklar and others. And, of course, we all recognize that the string and loop theories of physics are merely avatars of Menger’s lumps. Disregard of Einstein’s warning that generalizations lacking the guidance of disciplined constraints have no constructive power is the fundamental defect of current string formalisms. At the end of Menger’s article, in small typed, there is an address “Illinois Institute of Technology, Chicago.” I'd never heard of that either. But soon I found my way from 57th and Ellis to 33rd and State (in GPS terms to 3308 South Federal) where on the third floor of a ‘Hunersteige’ (old Viennese slang for a very unprepossessing building) I found a small fabulously untidy office fired up by Menger’s energy. We got along instantaneously. That night I called my old college friend Bert Schweizer long distance in Washington, D.C., and urged him to forget drudging at the Naval Research Laboratory, and come to Chicago to become Menger’s graduate student. That’s exactly what Bert did. None of us had any doubts that was the right thing to do, and it was the beginning of a very happy and productive time. Eventually, of course I also came to IIT and joined the Menger circle, but that's another story for another time!

Menger became a good friend and mentor to me during the last years of his life, when he was no longer teaching at IIT. It began when I invited him to speak at Benedictine University (where I spent 30 years as a professor of math and dean of faculty). Soon after that event he visited my Lincoln Park apartment to meet and talk informally with a few of my undergraduate students. Then he invited me to help him with the translations into English of some of his works. My job was to suggest ways to turn his correct expressions into flowing language! We spent many summer afternoons talking both about his long sentences and the substance of his thinking on a variety of matters, not only mathematics but also ethics and economics. His stories of intimate experiences with the members of the Vienna Circle were fascinating. He never lost his accent when speaking English, a fact he bemoaned, often illustrating this bit of personal failure with the sentence: “My greatest worry is words like work!”

He loved to dine in special restaurants, particularly relishing creative cuisine in the most unassuming restaurants as well as the best. Among his favorites were some in Andersonville where he lived at the time. When my six-year-old grandniece visited, Menger prepared a treat for her that he was sure she had never had before: white chocolate.

I miss Menger’s presence in my life as a dear and caring friend, a friend who generously invested in his friends, a friend who possessed a rare reservoir of energy, a friend who reveled in intelligent conversation and creative enterprise.

As a graduate student of Dr. Erber in 1965, he assigned the following problem, “Using the Lorentz Position Time Transformations, obtain the laws for the relative velocities and accelerations” in system k when the system k is moving at relativistic speeds. Erber provided a brief introduction to Menger's “new approach to calculus” in class and a handout outlining the notation developed by Menger.

As a senior I took Menger's evening class so I was familiar with this “new” way for writing function notation and differential notation. I completed the problem in one late evening (morning) session using 19 sheets of paper and ink pen—no whiteout, discarded sheets, or cross outs. Once the calculation began, each subcalculation led to the next and there was no possibility of confusion as to what had been calculated.

While teaching at Aurora College between 1967 and 1974 (now Aurora University, in Aurora, Illinois), I included a one-week introduction to Menger Calculus in a math class for physics majors.

Many years ago, just after Illinois Tech had adopted its student “honor system,” we had this final exam with Professor Menger. He started the proceedings, telling us in his wonderful Austrian accent that he had been unable to decide between giving our class a written or an oral examination. Everyone understood that it didn’t really matter because Menger knew all his students well enough to know what they understood and what they didn't.

“Noja, vel, goot, now,” he began. “Ve vill have to have a vote. All in favor” etc. (“Noja” is a German expression that might translate into something like “let's see now.” To that famous expression was then added an English “Well, good, now.” In other words, we were ready to get started.) Unfortunately, the vote came out exactly 50-50 between written and oral. The answer was of course, “Vel, ve have to have two examinations. Will all the written exam people please go to the next room.” Me, I loved having conversations with Menger, so had I picked an oral exam.

As, therefore, half the class was sitting there discussing the intricacies of the calculus of variations with Menger, the noise level from the conversations next door became more and more perceptible.

Menger's only reaction? “Vel they must have switched to oral also! Vould one of you please go next door and tell them to be more quiet.”

Several remembrances of Professor Menger have to do with his concern about notations. I recall a particular such matter in which his view was so correct that I could never understand why it is not being applied by everyone.

Consider a function of the form *f*[(*x*, *g*(*x*, *y*)].

What here is meant by the partial derivative ∂*f*/∂*x*?

Is it meant to be just the derivative of *f* with respect to its first variable *x* holding *g* constant, or is it the entire partial derivative, including the variation of *g* with *x* and just holding *y* constant? Using the universally applied ∂ notation, there is no way to tell.

Menger suggested that we should instead use subscripts instead of the ambiguous ∂ notation. Thus, the derivative of *f* with respect to its first variable *x* would be denoted by *f*1(*x*, *g*).

On the other hand, the second interpretation would be written instead as *f*1(*x*, *g*) + *f*2(*x*, *g*) * *g*1(*x*, *y*).

With this notation there can be no ambiguity.