Why does an ellipsis have three dots?
Solution 1:
The best answer as to why we use a three-dot ellipsis can be found in the book The Motivated Sign (Iconicity in Language and Literature), but allow me to explain briefly.
ellipsis is derived from the Greek elleipein essentially meaning "to fall short" or "leave behind." The Viking language of Old Norse was one of the first known languages to use the ellipses, albeit without the specific "..." notation. In Old Norse, these ellipses were literally omissions of infinitive phrases and non-action verbs.
It wasn't until 1588 that the three-dot ellipses first appeared in print in Maurice Kyffin's translation of Andria by Terence. It is unlikely we will ever know why three dots were chosen over any other possible symbol or series of symbols. However, this precedence combined with arbitrary rule probably has brought us to today's modern usage guidelines of the ellipsis.
Solution 2:
Well, the example you provide for programming languages is not completely correct.
Most modern programming languages use the 3 dot ellipsis for specifying variable numbers of parameters in a function. The two dot ellipsis IS used in some older languages to specify a range, but now it is less common. Some differentiate, using the 3 dot ellipsis to omit the last value, and the 2 dot ellipsis to include it. In system design, the two dot ellipsis is used to specify ranges.
Coming back to your question regarding why we DON'T use the two dot ellipsis, it seems to be more of a standard. Wikipedia says that 18th and 19th century writers used the ellipsis to indicate a blank to be filled like this:
Jan was born on . . . Street in Warsaw.
Where the ellipsis denoted missing data. With this in mind, I guess it could be said that 3 dots took up more space, allowing larger words to be written in the space. Not sure, but maybe it was something of the sort.
So, it seems that the 3 dot ellipsis is used, since the ellipsis originated in that form (perhaps due to practical reasons such as the one I mentioned), and because it is maintained as a standard in that form.
Solution 3:
An Ellipse, in mathematics, has three co-linear points of importance; a center and, on either side, a focus. So, in an Ellipse, we have three dots in a row [focus, center, focus]...hence the name ellipsis.