I think I am the one that is misunderstanding. Why doesn’t your definitions work?
For every Rilke that that can turn 0 pages into 1 page, there exists another machine B s.t.
(1) B can turn 1 page into 1 page, while interacting with Rilke. (I can copy a poem from a rilke book while rilke writes another poem next to me, or while Rilke reads the poem to me, or while Rilke looks at the first wood of the poem and then creates the poem next to me, etc.)
(2) the combined Rilke and B doesnt expend much more physical resource to turn 1 page into 1 page as Rilke expends writing a page of poetry.
I have a feeling I am misentrepreting one or both of the conditions.
I think the issue is that what people often mean by. “computing matrix multiplication” is something like what youve described here, but when (at least sometimes, as you’ve so elegantly talked about in other posts, vibes and context really matter!) talk about “recognizing dogs” they are referring not only to the input output transformation of the task (or even the physical transformation of world states) but also the process by which the dog is recognized, which includes lots of internal human abstractions moving about in a particular way in the brains of people, which may or may not be recapitulated in an artificial classification system.
To some degree it’s a semantic issue. I will grant you that there is a way of talking about “recognizing dogs” that reduces it to the input/output mapping, but there is another way in which this doesn’t work. The reason it makes sense for human beings to have these two different notions of performing a task is because we really care about theory of mind, and social settings, and figuring out what other people are thinking (and not just the state of their muscles or whatever dictates their output).
Although for precisions sake, maybe they should really have different words associated with them, though I’m not sure what the words should be exactly. Maybe something like “solving a task” vs. “understanding a task” though I don’t really like that.
Actually my thinking can go the other way to. I think there actually is a sense in which the computer is not doing matrix multiplication, and its really only the system of computer+human that is able to do it, and the human is doing A LOT of work here. I recognize this is not the sense people usually mean when they talk about computers doing matrix multiplication, but again, I think there are two senses of performing a computation even though people use the same words.