Dr. McShane

 

Dr. McShane

But now I want to return to those first days of my introduction to graduate level mathematics, specifically to one course with the intimidating title, “Functions of a Real Variable”. The lecturer was Dr. E. J. McShane. He was the acting head of the department, in the absence of the permanent head who was on a sabbatical. I think that I may have made a slight impression on him by sitting on the front row in class, looking at him as he lectured, and attempting, with some success, to explain a theorem that he had assigned for study. But he definitely made an impression on me.

He had enrolled all of the new graduate math students in this class. Apparently he wanted to give us a proper introduction to abstract mathematics, because he spent the first six weeks of the course teaching us logic in considerable depth – far beyond the basic logic of my high school course in Euclidean geometry.

This logic utilized an extensive system of symbols that was appropriate for analyzing abstract theorems and their proofs. For example, there was a symbol for “is a member of the set . . .”, another for “is contained in”, another for “implies, as a result”, etc. By the use of this symbolic language a proof can probably be reduced to less than a tenth of the length that it would be if it were written out in long hand. Unfortunately, learning it is similar to learning a foreign language in that an extensive use of it is required to acquire that kind of familiarity that one needs to interpret it with facility. And we had not had time to acquire that kind of familiarity. Consequently, we were faced with the compounded difficulty of attempting to understand difficult mathematical concepts that were presented to us in an unfamiliar language that we had to decipher.

We held the professors in awe because they appeared to have a command of these abstract concepts, but none of them displayed the kind of arrogance that is displayed by some professors that taught much simpler subjects. On the contrary they spoke encouragingly. They would say that it would take time for some of the concepts to sink in, that mathematicians tended to specialize to very specific areas, etc. I suspect that they feared that, if just a few of us got so discouraged that we would quit, or switch majors, the enrollment in the graduate math program would be too small to justify its existence. However, if such a fear did exist, it never reached the point that they were willing to make their lectures more comprehensible.

Dr. McShane had some tricks that he used to help us to relax and not hold him in awe as some kind of super brain, but to consider him a human just like the rest of us. One of these tricks was occasionally to throw in a careless “hell” or “damn” when he was lecturing. The trick didn’t work. We knew that such language was beneath him, and made him uncomfortable, because he had a “tell”. Those words didn’t flow naturally in his speech. He would always pause for just a fraction of a second before the profanity was emitted, as if he had to force it out.

He had such a command of technical German that he collaborated with a Dr. Richard Courant in translating that author’s calculus text into English. And he was studying Dutch in preparation for a sabbatical to lecture at a university in Holland.

During World War II he had been assigned to the Marine base at Dahlgren Proving Grounds as a mathematical consultant. The result of that stint was a book, “Theory of Exterior Ballistics”. Despite the impressive title, it turned out that the book had almost zero readership. The material was presented in the language of mathematicians, none of whom had the slightest interest in the subject of Ballistics, while the engineers who were interested in Ballistics could not comprehend the material because they were not familiar with the abstract mathematical format of the presentation. That lesson was to be impressed on me years later when I began to publish my own research. I found that my mathematical analyses had to be presented in the simplest, most direct manner, and tied directly to examples of practical application, if I expected them to be understood and used by engineers and applied physicists.

I was grateful to Dr. McShane not only for encouraging me during those first-year courses that were almost terrifying in their difficulty, but also for the lectures on logic, which deepened my grasp of logical thinking beyond the basic comprehension that I had developed in my high school Euclidean geometry course. I had some use for that material later in my research, especially in reviewing technical papers proposed for publication. And it proved to be an enormous benefit in other areas, including philosophy, religion, psychology, law, politics, economics, and areas of science outside of those within my research interests. I think that most of us often distrust material that is presented to us on such subjects, either orally or written. Sometimes we have a “gut feeling” that somehow there is something wrong with the argument. A good understanding of logic helps one to put his finger directly on the flaw in the reasoning.

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