This is an update of a post from a couple of years ago.

Until computers came along, there was no such thing as a space character. The space between printed words was simply a space. In computing, each letter is represented by a certain number, and starting in the early days the space was represented by the number 32 (in decimal notation). In that sense, the way we thought of the space between printed words shifted from empty space to an *object represented by empty space.*

In the late eighties, I was at a church service on the Sunday when they talk about the budget. After the talk, ten members of the congregation marched up front each carrying a sign with one letter on it. They arranged themselves to spell

This was concrete evidence that we had changed the way we think about spaces between words. The congregation of this upscale church included many engineers and other professional people.

The space character is used in Mathematica to denote multiplication: One writes “x y’’ to mean x times y. This allows multiletter variable names without ambiguity. “distance time’’ would be the product of distance and time. When you have some experience with Mathematica, you think of space between variables as a genuine symbol meaning multiplication.

Space is used in other places in math with a kind of positive meaning; for example, “sin x’’ means the result of evaluating the sine function at x. But I don’t believe most mathematicians think of that space as a symbol. I didn’t until I thought of writing this comment. I am not at all sure it is useful to think of it that way.

When lead type was used in hand typesetting, there were different sizes of blank lead slugs to put in between letters. With linotypes, a different technique was used: wedges were shoved down between the letters to force the line of type to be right justified.

Even in functional programming languages, I think we don’t regard the space character itself as an operator. Usually there is a lexical rule that says that operators are delimited by, among other things, whitespace (but parenthesis is another example), which includes a single space, but could be many spaces, or tabs or carriage returns.

The result is effectively that function and argument wind up adjacent in a list (or rather a tree), so I actually think of the function application operation as “juxtaposition”.

When I look at “sin x”, I see the space as a separator for two lexical units, and it is rather their juxtaposition which I interpret as application.