Some inventions can be thought of this way, but I don't think all inventions can. For a relevant example, human mathematical notation is a human invention that I don't think can be usefully thought of as a kind of discovery. (Another poster upthread mentioned Tegmark's response drawing a distinction between the structure of mathematics, which we discover, and the language we use to describe it, which we invent.)
I think notation can certainly be regarded as a kind of discovery. As time goes on, notation improves dramatically as a tool of thought. This improvement is a result of feedback over what works and what doesn't. The feedback is discovery.
I don't understand why it is useful to argue "invention is discovery is invention". They are fairly well defined in the dictionary and it's a useful distinction to humans. Sure there is a gray area of a wide swathe of overlap, but in general they are useful as commonly understood.
> human mathematical notation is a human invention that I don't think can be usefully thought of as a kind of discovery
Why not? It's a discovery of visual features that the human brain and unaided eye can easily discern, and the discovery of a correspondence of those visual features to concepts that the human brain can easily remember. All of these are (on the assumption that dualism is false) physical processes.
> It's a discovery of visual features that the human brain and unaided eye can easily discern, and the discovery of a correspondence of those visual features to concepts that the human brain can easily remember.
Hm. Yes, I suppose you could look at it this way. However, there is still an element of arbitrariness or choice involved that is not present in the discovery of mathematical structures themselves. All humans discover the same mathematical structures, but different humans invent different mathematical notations to describe those same mathematical structures.
So I guess it's a matter of whether you want to focus on the similarities or the differences between those two cases.
> there is still an element of arbitrariness or choice involved that is not present in the discovery of mathematical structures themselves
That's not true. There is this thing in math called the "axiom of choice" which you can, quite literally, choose to accept or not. And math is chock-full of this kind of thing. Even geometry comes in different flavors: Euclidean and non-Euclidean. And we don't even know which of them corresponds to physical reality on cosmological scales!
> There is this thing in math called the "axiom of choice" which you can, quite literally, choose to accept or not.
No, there are these different mathematical structures: set theory with the axiom of choice, set theory with the negation of the axiom of choice, and set theory with neither. (More precisely, "Zermelo-Frankel set theory", since there are other set theories.) All of those structures were discovered. And, as I said, different humans invented different notations to describe these different structures.
Humans choose which of these structures to use for particular applications, yes. I don't think that process is either invention or discovery. Not everything humans do has to be an invention or a discovery.
Some inventions can be thought of this way, but I don't think all inventions can. For a relevant example, human mathematical notation is a human invention that I don't think can be usefully thought of as a kind of discovery. (Another poster upthread mentioned Tegmark's response drawing a distinction between the structure of mathematics, which we discover, and the language we use to describe it, which we invent.)