No, I was trying to prove that a perl array is not what a computer science book would call an array.

I don't see why you think they aren't. Perl arrays are a continuous series of equally sized records, allowing for instant addressing of any element. That definitely fits the traditional formal definition.

In such a book an idealized data type 'array' wouldn't have holes.

Perl arrays aren't sparse. There are no holes. Every element exists. (Those three statements are synonymous.)

And after thinking about it, it wouldn't have a push or pop operator, that would be the data type 'stack'

Stacks (and queues, and ...) can be implemented using an array. They're not mutually exclusive.

The presence of utility functions to manipulate a data structure doesn't change the data structure.