Programmers seem to think that their work is quite mathematical.

I understand this when you try to optimize something in performance, find the most efficient alogithm, etc..

But it patently seems false when you look at a billing application for a shop, or a systems software riddled with I/O calls.

So what is it exactly? Is computation and associated programming really mathematical?

Here I have in mind particularly the words of the philosopher Schopenhauer in mind:

That arithmetic is the basest of all mental activities is proved by the fact that it is the only one that can be accomplished by means of a machine. Take, for instance, the reckoning machines that are so commonly used in England at the present time, and solely for the sake of convenience. But all analysis finitorum et infinitorum is fundamentally based on calculation. Therefore we may gauge the “profound sense of the mathematician,” of whom Lichtenberg has made fun, in that he says: “These so-called professors of mathematics have taken advantage of the ingenuousness of other people, have attained the credit of possessing profound sense, which strongly resembles the theologians’ profound sense of their own holiness.”

I lifted the above quote from here. It seems that programmers are doing precisely the sort of mechanized base mental activity the grand old man is contemptuous about.

So what exactly is the deal? Is programming really the "good" kind of mathematics, or just the baser type, or altogether something else just meant for business not to be confused with a pure discipline?