Halmos’s Automathography

I just finished I Want to Be a Mathematician: An Automathography by Paul Halmos. I found the book to be really interesting, although I don’t think that everyone will. In particular, he describes his career from the 1930s to the mid-1980s, and academia was a different world back then: money was seemingly easy to get for travel, and there was a lot less bureaucracy (Halmos seems to have gotten tenure after his third year without having to apply for it). There was plenty of talk about mathematicians he knew—some I had heard of, some I hadn’t—and I learned how to pronounce his last name (as an American would—‘hal-moss’—rather than as a Hungarian would—‘hal-mush’). I also had heard several things credited to Halmos, and they came from this book.

  • He is credited for inventing the notation ‘iff’ for ‘if and only if.’
  • He is credited for the little box that denotes the end of a proof.
  • A quote that has been circulating around the IBL types is found on Page 69: “It’s been said before and often, but it cannot be overemphasized: study actively. . Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

There are two things I want to share, though. The first is some of his comments about grading, and the second is a nice way to describe how to be productive in any field.

On productivity:

Archimedes taught us that a small quantity added to itself often enough because a large quantity (or, in proverbial terms, that every little bit helps). When it comes to accomplishing the bulk of of the world’s work, and, in particular, the work of a mathematician, whether it is proving a theorem, writing a book, teaching a course, chairing a department, or editing a journal, I claim credit for the formulation of the converse: Archimedes’s way is the only way to get something done. Do a small bit, steadily every day, with no exception, with no holiday.

Page 401

I just thought that was a nice summary of how work progresses.

Readers of this blog may be more interested in a couple of his comments about grading. The first is about an experience he had as an undergraduate in the 1930s.

Analysis was taught by Steimley, who taught like a marine drill sergeant. He prepared detailed notes for his advanced calculus course and used them over and over again. He graded homework and exams promptly and fussily. Your grade was not likely to be just B or 80 or 85, but something like 83. The digit to the right of the decimal point in your average could play an important role in determining your course grade.

Page 32

Halmos makes it seem like a grade of “83” was strange for the 1930s, and it was more common for instructors to give grades without the pretense of objective precision.

So how does Halmos determine grades? He also explains this.

Assigning grades to the students in my class is part of my job; it is a necessary evil. Grading is bad because students often pay too much attention to it, because it is often regraded as more accurate than it can possibly be, and incidentally, because it often makes students feel bad. It is, however, necessary because in our present educational and social organization the teacher in a later course must know what the students learned in an earlier one, and a prospective employer wants to know how good the student is likely to be on the job. I can’t think of a way of designing an organization of learning and working in which these items of information are not needed.

I do not, however, think that the assignment of informative grades is all that hard. At the end of a course I usually have a pretty clear idea that certain students know the material well (A), and certain others don’t (F). In between, there are those who knew some of it but have gaps in knowledge—possible big ones (B), and there are those who can use some of it, but don’t really understand it (C). Then, of course, there are those who can prove that they have been exposed to it, but for sure don’t know enough to go on to a course on a highly level (D). . .Of course my “pretty clear idea” is subjective, but it’s remarkable how nearly unanimous such subjective grade assignments turn out to be: students keep getting the same sort of grades time after time, in different courses from different teachers. I don’t agree with my colleagues who advocate a more “objective” numerical grading system: Problem 4 is worth 15 points, and you 3 points if the answer is right, and 2 points off for each of the six most obvious missteps you can take on your way to it. To my mind, it is my duty to use my best judgement about how much my students know when I’m finished with them; anything else would be an evasion of my responsibility.

Pages 136–137

The book contained a mixture of this type of opinion with a more historical account of where he was an with whom he was in contact with at various points in his life. If these seems interesting to you, go ahead and read it. A book with a similar feel is Indiscrete Thoughts by Gian-Carlo Rota. I enjoyed both of them quite a bit, but they are not for everyone.


2 Responses to “Halmos’s Automathography”

  1. Min Ro Says:

    I read this book back during graduate school and quite enjoyed it too.

    • bretbenesh Says:

      I wish more people did this type of thing. He had clearly learned a lot during his career, and it was a very kind thing to pass on his knowledge to other people. I think that people in all sorts of industries should be doing this.

      I am glad that you read it, too!

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