Mathematics and Language

As usual Frances Woolley has written a spot on piece about "language" and in this case mathematics and economics.

She states:

Bad notation makes a paper difficult to follow. Papers that are hard to read and understand get rejected, or receive lower grades. But what makes for good notation? 
 
First, symbols should be easy to remember....

(Second) ... avoid multiple subscripts or subscript/superscript combinations whenever possible...

(Third) ... One other rule is ... Generally speaking, greek letters are used for parameters of the model ...

(Fourth) ... Once one has a model, one has to figure out what goes in between the equations (hint: economics). Every symbol used in a paper needs to be defined clearly the first time it mentioned. If the symbol has not been used in a while, it is a good idea to give the reader a hint as to what it means.

 Now I have struggled with this for decades. Here is a variation of Woolley but written with my set of rules:

1. Write for your audience. If you are writing for pure mathematicians then you need all that stuff. If you are writing for the real world then just say what you mean and no more. You are NOT a pure mathematician and who cares if you have a Banach space. You never use that fact.

2. Follow Ockham:

a. Say as little as you have to to explain the concept. Do not add all sorts of stuff to show how "smart" you are.

b. Be a nominalist, namely there are no abstract Platonian constructs, only the reality of the moment.

3. Write equations so that they are obvious. If w is a variable that depends on x and t then write w(x,t). That's all, no more. If w is one of i=1,...,n elements then write a simple subscript. As Frances says choose a variable to be obvious. x is good for position, t for time, and then go from there.

4. Define it and repeat the definition as often as possible. Why? Because I may not have read the obscure definition hidden in the midst of paragraph 23.

5. Structure your presentation. Long paragraphs are tedious. Number or bullet your points. Highlight the fact that you are making. Unless your are a pure mathematician, assume it is not clear to the reader or that you can leave it as a simple exercise for the reader.

I have found that Economists try to pretend that they are pure mathematicians. I had taken a few pure mathematics courses at MIT and I can assure anyone that I am not nor will I ever be a pure mathematician. However I am a reasonably good engineer so writing equations are as an Ockhamist.  I try to think of the reader, since I am using a language that should be clear and simple.