I agree that having an icon with a particular action attached may not *seem* mathematical, but *anything* that happens on your desktop (or a computer in general), is inherently algorithmic and as such mathematical, which I believe is the point of contention that lies at the root of a distinction between "mathematical algorithms" and "non-mathematical algorithms"
Take every bit that your computer currently holds and write these all down on a long string of 1s and 0s. Call that string A.
Now use the trashcan, put some files in it and throw them away.
Now take another snapshot of the state your computer is in. Call that string B.
You now have two very very long strings of 1s and 0s. Now ask the question, does there exist a Turing machine that will take string A as its input and will arive at string B?
The answer to that question is (obviously) yes.
Since such a Turing machine exists, the transformation was an algorithmic one, i.e. describable through an algorithm.
I realize it doesn't appeal to intuition or a practical view of the use of computers. But by the above example, everything you could ever do on a computer no matter how sophisticated or trivial is an example of a mathematical operation. So everything that happens on your computer is done by though an algorithm and there are no non-mathematical algorithms in computer-software.