## Notes on Mathematical Writing

1. I’m going to be posting my double integral notes, but I’d like to discuss my strategy when writing “more rigorous” mathematics. Modern mathematics consists of definitions, theorems, and proofs…so I’ll discuss the idiosyncrasies of each. (I’ll probably revise this several times…)

2. Definitions. A definition introduces a new gadget (e.g., vector spaces), some new structure (e.g., a linear operator, a norm, etc.), or some new property (e.g., invertibility).

So the typical definition would look like the following:

An “Object” consists of “Stuff” equipped with “Structure” such that these equations — called “Properties” (or Axioms) — hold.

A typological remark: sometimes axioms are given in bold. For example, consider the following snippet of a definition

…satisfying the following properties:

Disjoint Union: [snip]

Sewing: [snip]

Normalization: [snip]

This has the added bonus of allowing us to write “By sewing, this gadget then transforms Equation (blah) into Equation (snort).”

Some people prefer giving as few axioms as possible, then proving the rest. This seems pathological to me, and confuses the reader. I prefer being frank and honest, telling the reader “This gadget has the following useful properties” and in a remark note when some imply others.

3. Theorems. Euclid had a systematic manner of presenting theorems, as Proclus noted.

Knuth suggests (in rule 11) we should “try to state things twice, in complementary ways…” which should be done with theorems as well. Personally, I think writing the theorem without symbols — if possible — would be best. This would correspond to Euclid’s “Enunciation”.

After the theorem statement, we should note the proof strategy and/or key moments of the proof.

Continuous functions on a closed interval are uniformly continuous.

Theorem. If $f\colon[a,b]\to\mathbb{R}$ is continuous on every $x\in[a,b]$, then $f$ is uniformly continuous on $[a,b]$.

We will prove this directly.

Proof. Let $f$ be a continuous function on the closed interval $[a,b]$, then… [the rest of the proof is omitted].

We should make the theorem be precise, boring, and cookie-cutter in its format. But we should say, either before or after, what it really says in a frank way: “Continuous functions on a closed interval are uniformly continuous.”

4. Proofs. We should begin introducing all the necessary variables and quantities. Constantly ask yourself “What does the reader know?” Don’t be afraid to reiterate what has been done “We’ve just shown (blah), but we need to prove (oink).”

5. Examples. Examples should be worked out exercises. So it consists of two parts: (1) statement of the problem, and (2) its solution.

In physics, it consists of 4 parts, which we will not discuss now [google “Hugh and Young IDEE”].

Now the problem lies in how do we pick good examples? It’s a very hard thing to do. It should be clean, i.e. free of excessive calculations; it should be inviting, interesting to the reader; and it should be challenging, i.e. not as simple as it seems.

Harrington has Ten Commandments for mathematical writing, which I more or less agree with, they are:

Rule 1. Organize in segments. Harrington’s “segments” are what I call chunks (in my blog entries, they are the numbered collection of paragraphs, possibly with a title of some sort). It can be read comfortably from beginning to end without pausing.

So what’s a good segment? An example, a definition, a theorem and its proof. These are typical. What else? Well, motivating a concept, reflections, etc.

Rule 2. Write segments linearly. We should organize each chunk to do one thing, and only one thing. Give us an example, or a result and its proof, or a concept.

The chunk should flow linearly, making it easy to read. How can we accomplish this? As Halmos once said: organize, organize, organize. Write up an outline, with only the highlights written down. For example:

1. Definition. Continuity…
Example 1. Polynomials.
Example 2. Trig Functions.
Non-Example 3. Step function at the jump.
2. Definition. Derivative…
Example 1. Polynomials.
Example 2. Trig Functions.
Non-Example 3. Absolute value at zero.
3. Theorem. Chain rule.
Example 1. Apply it to $\cos(\exp(x^{5}))$.
4. Theorem. Product rule.
Example 1. Apply it inductively to $x^{n}$ to recover the product rule.

Rule 3. Consider a hierarchical development. Think about how the segments depend on each other. This affects how we organize our outline.

Don’t be afraid to reorganize.

Rule 4. Use consistent notation and nomenclature. Again, I don’t think I have to say anything about this, but if you use LaTeX you might want to consider using macros to specify notation. I usually write things like \def\CC{\mathbb{C}} and so on, just so I have consistent typography.

Rule 5. State results consistently. Be sure to write theorems consistently. Usually it’s of the form “If P, then Q.”

Similarly, with definitions, be consistent “A gadget consists of…equipped with…such that…”.

Rule 6. Don’t underexplain but don’t overexplain. But if in doubt, overexplain. Harrington demands you write for an audience (Halmos says “Pick someone you know, and pretend you’re explaining it to them”)…I usually write for myself, knowing I’ll forget the material and have to re-learn it quick.

Consequently, I write specifically so if someone says “I need to learn everything you know about [blank]” I can say “Give me five minutes” (so I can read my notes, and recall everything about the subject). That’s how I wrote my Feynman diagram notes.

Rule 7. Tell them what you’ll tell them. Tell the reader where you’re going. Also consider motivating the problem appropriately.

Rule 8. Use suggestive references. Don’t say “By theorem blah, we transform Equation (snort) into Equation (oink).” The reader has to flip back to find “theorem blah”, then consider carefully how to apply it. How to revise it?

One should write “Since all continuous functions on closed intervals are uniformly continuous” instead of “By theorem blah”: i.e., write the contents of the theorem instead of referring to it.

One should then write “…we obtain [from our previous equation] (oink)”.

Don’t fear repetition, but whenever you repeat yourself…reword it differently, just to break the monotony.

Rule 9. Consider examples and counterexamples. Examples should demonstrate the point, or demonstrate the bounds of some definition. Count-examples are useful for showing when some algorithm fails.

Rule 10. Use visualization when possible. You know, a picture is worth a thousand words…