On the use of notation..

By | February 17, 2009

So, everyone knows what r stands for, right? What about v? Or f(x) and f'(x)? OK. How about x, y, and z?

If you’re not a math geek of some kind, you’re probably not reading anymore, but just in case you are, the point is that each of these letters has a common meaning in a lot of mathematical notation – p is a probability, v some arbitrary vector, f(x) and f'(x) some arbitrary function and its derivative, and x, y and z, are coordinates in 3-space.

The problem is that a lot of the time, this isn’t true, and even when it is true, it’s hard to tell exactly _which_ probability or set of coordinates you might be talking about.

Good math books typically get this – they define their notation, and use it consistently. If p means probability in chapter 1, it probably doesn’t mean ‘an arbitrary solution to the dual problem’ in chapter 2, unless it’s been explicitly re-defined. Each symbol should correspond to one particular value or concept at any given time. This makes the text easier and faster to read, and avoids all sorts of nasty confusion.

So, why is it that people presenting mathematical results always assume that you know their notation? If they throw up a complicated expression using a bunch of different letters, why do they assume that you know that r doesn’t actually mean radius (even though it’s shown on a circular diagram), and that, today, we’re using g to refer to probability, not p (except for that slide near the end, because it’s from a different slide set).

You’d think this just happens in badly prepared and presented seminars. Unfortunately, either you’re wrong, or I have an uncanny ability to attend only seminars that meet that criteria.

So, if you’re ever in a position to be presenting mathematical notation to a bunch of people, please, please, do the following..

  • Introduce your notation. Tell the audience what each letter means as soon as you start using it.
  • Don’t change what x means halfway through your talk, unless you really have to. If you’re using x to just mean ‘some arbitrary value’, that’s OK, but tell people that.
  • Each value should refer to only one thing at a time. This is particularly problematic if you’re working through an algorithm that re-uses the same notation every step. Is B the initial basis matrix you chose, or the basis matrix at step 3?
  • If you’re re-introducing some notation you briefly mentioned at the beginning, mention it again.
  • If your expression expresses some important relationship, verbalize it – read it out. If your expression is really large but still important for your audience to understand, not just accept, break it down and read it out. If you can’t do that, your audience won’t get it.
  • If you’re just showing algebraic steps, question why you included them in the first place. If you’re not expecting your audience to work through the algebra while you’re talking, leave it out.
  • Just because you think p always means probability, don’t assume you can get away with not defining it. If a letter has different meanings in different fields, you’re bound to confuse at least one person. Sure, they might be able to work it out from context, but they shouldn’t have to. Besides, p means the probability of what, exactly?

I could go on, but instead, I refer people to Polya’s lovely short rant on the subject in ‘How to Solve It’. There’s a free version online. It’s on page 134.

People seem to forget that the entire point of notation is the economical expression of an idea for the purpose of memory or communication. Furthermore, memory is really just a special case of communication – you’re communicating with your future self. Imagine how confused they’ll be if, in your notes, q means different things without clear distinction. Imagine how confused your audience will be, not having been you in the first place.

This all boils down to this general point about communicating – if you don’t value your idea enough to make sure your audience understands, don’t bother opening your mouth. Play Minesweeper instead.

X-posted to various places