Just as a matter of terminology, I prefer to say that we can choose (or that we have a choice about) the output, rather than that we control it. To me, control has too strong a connotation of cause.
It’s tricky, of course, because the concepts of choice-about and causal-influence-over are so thoroughly conflated that most people will use the same word to refer to both without distinction. So my terminology suggestion is kind of like most materialsts’ choice to relinquish the word soul to refer to something extraphysical, retaining consciousness to refer to the actual physical/computational process. (Causes, unlike souls, are real, but still distinct from what they’re often conflated with.)
Again, this is just terminology, nothing substantive.
EDIT: In the (usual) special case where a means-end link is causal, I agree with you that we control something that’s ultimately mathematical, even in my proposed sense of the term.
Hm. To me, “choose” sounds like invoking the idea of multiple possibilities, while “control” sounds more determinism-compatible. Of course that is a mere matter of terminology.
Though I’m not sure what you mean by “in the special case where a means-end link is causal”—my thesis was that if you are uncertain about the output of your decision computation, and you factor the universe the Pearlian way, then your logical decision will end up being, in the graph, the logical cause of box B containing a million dollars. You mean the special case where a means-end link is physical? But what is physics except math? Or are we assuming that the local causal relations in physics are more privileged as ontologically basic causes, whereas “logical causality” is just a convenient way of factoring uncertainty and a winning way to construe counterfactuals? (That last one may have some justice to it.)
I agree that “choose” connotes multiple alternatives, but they’re counterfactual antecedents, and when construed as such, are not inconsistent with determinism.
I don’t know about being ontologically basic, but (what I think of as) physical/causal laws have the important property that they compactly specify the entirety of space-time (together with a specification of the initial conditions).