So, I've been writing notes about mathematics and have been pondering about the problem "What is a mathematician's notebook like?"
Elements of Mathematics
I've decided to start writing a series of cohesive notes on mathematics. This is partially inspired by Bourbaki, among others.
So, I just finished some notes on set theory (Fascicles 0: Naive Foundations). It's around 120 pages of notes, and 10 pages of front matter.
I think that, like Donald Knuth, I'll try to write fascicles that are roughly 128 pages in length. This will limit my focus, and allow self-contained "quanta" of mathematical writing to be released.
One issue that came up: how to come up with good exercises?
I asked one of my professors. He said, "Ah, that's a good question Alex. I don't have a good answer. However, some times I add exercises which are examples I cannot get to in class. The real answer is: you'll know when you start teaching."
However, in Atish Bagchi and Charles Wells' Varieties of Mathematical Prose, they note that there are really 6 types of examples in mathematical writing:
- An "Easy Example" is one that can be verified quickly.
- A "Motivating Example" is one that is given before some definition. In my notes, for example, the discussion of a span of sets is a motivational example for binary relations.
- A "Delimiting Example" is an example with the smallest possible number of elements or an example with degenerate structure.
- A "Deceptive Non-Example" is one which innocently resembles something to be true, but really beneath the skin level is false.
- An "Elucidating Example" is one that really clarifies various aspects to a notion or definition.
- An "Inventory Example" is just a grocery list of useful examples that are not too general or specific.
I suppose that exercises follow this as well? It seems like the best exercises should have some sort of "moral" or "lesson". Frequently one finds "Prove or find a counter-example: ..." type of exercises, but is there any trick to coming up with good exercises?
A Mathematician's Notebook
It is the stuff of folk-lore that all mathematicians have cherished notebooks. But what do they do with them? I started reading The Kourovka Notebook (a notebook of unsolved problems in group theory), and noticed that V. I. Arnold had Online Problem Lists.
I also observed that Robert Wilson also keeps online a list of Research Problems.
I suppose, in short, a mathematician's notebook is focused more on problems than notes. This is what characterizes a good mathematician: the ability to come up with good exercises, good problems, good questions.
Of course, presenting them and coming up with solutions are also vital. It's not good if no one understands what you are asking! Donald Knuth, et al.'s Mathematical Writing lecture notes are an excellent introduction, they tackle various rules to bear in mind on pp. 3–8.
Gel'fand wrote in Pavel Etingof et al.'s The Unity of Mathematics: In Honor of the Ninetieth Birthday of I.M. Gelfand Progress in Mathematics 244 (2006) pp. xiv, that:
The next question was: How can I work at my age? The answer is very simple. I am not a great mathematician. I speak seriously. I am just a student all my life. From the very beginning of my life I was trying to learn. And for example now, when listening to the talks and reading notes of this conference, I discover how much I still do not know and have to learn. Therefore, I am always learning. In this sense I am a student—never a "Führer."
I forgot who said it, but I think it's best to have some applications in mind. Even if it's something along the lines of "How do I [accomplish some pure mathematical calculation]?" My use of "applications" is far more liberal than it appears!
Arnold's lecture On teaching mathematics had the opinion that: "Since scholastic mathematics that is cut off from physics is fit neither for teaching nor for application in any other science, the result was the universal hate towards mathematicians — both on the part of the poor schoolchildren (some of whom in the meantime became ministers) and of the users."
More liberally I think it is fair to say that mathematics is the language of science. So to really come up with good exercises, a mathematician should be constantly reading scientific literature — even popular texts on scientific topics.
But the short, and honest, answer is: I don't know how to come up with good problems. If anyone has any tips or advice, please share!
Case Studies
Curiously Knuth writes on pg 21 of his notes on mathematical writing:
Exercises are some of the most difficult parts of a book to write. Since an exercise has very little context, ambiguity can be especially deadly; a bit of carefully chosen redundancy can be especially important. For this reason, exercises are also the hardest technical writing to translate to other languages.
Copyright law has changed, making it technically necessary to give credit to all previously published exercises. Don says that crediting sources is probably sufficient (he doesn't plan to write every person referenced in the exercises for his new book, unless the publisher insists). Tracing the history of even well-known theorems can be difficult, because mathematicians have tended to omit citations. He recently spent four hours looking through the collected works of Lagrange trying to find the source of "Lagrange's inequality," but he was unsuccessful. Considering the benefit to future authors and readers, he's not too unhappy with the new law.
We can dispense with some of our rhetorical guidelines when writing the answers to exercises. Answers that are quick and pithy, and answers that start with a symbol, are quite acceptable.
Curiously, Terry Tao writes on page 2 that:
A good problem should be inviting enough to naturally draw the student in to attempt it, challenging enough that the student recognises that something nontrivial has to be done, and clean enough that the insight provided by the solution is not obscured by a mass of time-consuming calculations. It is here that a student can finally get a taste of how one's mathematical power increases when a tool is understood and used correctly, and how one can use the current class material to revisit earlier subjects and clarify them even further.Inviting, challenging, clean — these are the mark of a good exercise. Perhaps, then, it is not too far of a stretch to suggest these are also the mark of a good example?
In an attempt to answer my own questions, let me list a number of "problem books".
- Kirby's Problems in Low-Dimensional Topology
- A collection of problems in knot theory, surfaces, 3-manifolds, 4-manifolds, complexes, graphs, TQFTs, etc.
- The Open Problems Project
- Open mathematical problems relating to computational geometry.
- Arnold's Problem List (February 1998).
- 11 Problems on various analysis topics.
- Arnold's Problem List (September 1998).
- Dealing with Pseudoring of Lie Algebras, Quaternionic matrices, Quaternionic vector bundles, Contact version of Liouville's theorem, Hofer fields, Uncertainty Principle for Lattices, Singularities of curves in contact geometry, causality in terms of linking, triangulation of knots, "Nekrasov's Discriminant", among other topics covered. I like how it is simple, but covers slightly different disciplines (e.g. relativity and knot theory).
- Two Problems by Arnold (September 2001).
- Dealing with Betti numbers of parabolic sets, and caustics of periodic functions.
- Arnold's Problem List (January 2002).
- 12 pages of fascinating problems, mostly relating to aspects of curves, etc.
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