## Notation is evil

Exhibit A:

We have the symbol $\propto$,with the interpretation

$x \propto y \leftrightarrow x = c y$ for some number $c$,

but there doesn’t appear to exist a symbol (Here, I use a boxed question mark: $\boxed{?}$ to denote the symbol I claim doesn’t exist) with the interpretation

$x \boxed{?} y \leftrightarrow x = y + c$ for some number $c$.

This pains me.  People sometimes have to resort to writing something like

$y = f(x)+\text{const} (1)$

$= g(x) + \text{const} (2)$

where the constants are (in general) different on lines (1) and (2).”

Even worse (or maybe not?), sometimes people seem to leave exponents lying around when they otherwise wouldn’t, e.g. write

$\exp(y) \propto \exp(f(x)) \propto \exp(g(x))$.

Exhibit B:

We have no symbol meaning “normalized sum”.  How many thousands of times have you seen some variant of

$y = \frac{1}{N}\sum_{x \in X} f(x)$

where $N=|X|$“?

Why do we need to define $N$?  Can’t we just use another mystery symbol to write

$y = \boxed{?}_{x \in X} f(x)$?

In some situations you could use $\text{mean}$, but that doesn’t always really work and is rarely done.

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### 3 Responses to Notation is evil

1. Hendrik says:

Good points. Those would be handy. I think “incomplete” might be more appropriate than “evil” though.
Just define your own notation for these and hope that it catches on. Worked for Euler.

2. Anonymous says:

For the second one, Andrew Gelman uses a big capital M (for “mean”) in the place of the \sum symbol. (Or at least I saw him use it once — it was clear immediately what he meant.)

3. justindomke says:

So, something like:

$y = \underset{x \in X}{\Large M} \ f(x)$?

I like it!