There's nothing you possibly draw on paper that answers the question, in isolation. A diagram on paper means exactly the same thing if you simultaneously (i) flip it, and (ii) flip your mental interpretation if it.
Or: if you run a software program on a raster image, there exists a different program that returns identical output given an input image which is mirror-flipped copy of the first one. (It's just the first program, plus a pre-processing step that flips its input (which is computable). If f is a computable program, and f(img) = "left" and f(flip(img)) = "right", there exists a computable program g=f∘flip such that g(img) = "right" and g(flip(img)) = "left").
Similarly: there's nothing you can do in complex analysis that can distinguish the case where all +sqrt(-1)'s are swapped with -sqrt(-1)'s. Nor can you invent any math whatsoever that has an isomorphism to the plane, that works differently if the plane is mirror-reversed.
But we can distinguish i and -i by convention and keep consistent right? Akin to sending the distant alien a sample? It’s in the abstract that we can’t really tell them apart. But as soon as we are given any kind of starting point to form a convention, like the first mathematical textbook tells us which is i and which is -i, we have enough?
I think of an unlabeled, unconnected graph of two vertices. If I just pick one and call it A say by pointing to it, that’s enough. The trick is to do it for limited communication partners like the alien right?
Have them project an imaginary line from where they stand and in the direction they are facing. Have them mark the numbers 1 2 3 in the ground as they walk forward. Now have them stand facing the number two. Now you say if you are standing left side of the line looking left to right the numbers would go 3,2,1 if you are standing on the right side looking left to right the numbers would go 1,2,3
Edit: it's a bit silly though,
I think if they could understand our language well enough to convey complex scientific ideas like an atom and its size, then smart people could probably find a good way to convey direction
> it's a bit silly though, I think if they could understand our language well enough to convey complex scientific ideas like an atom and its size, then smart people could probably find a good way to convey direction
You completely missed the point of the original thought experiment. The whole experiment already assumes that the Martian understands the concept of direction perfectly. That's the premise of it.
The hard part is to convey which direction is which. It's a completely different thing from the concept of direction.
It's not about Martians is so stupid and we need to try very hard to make them even understand what direction is. It's about whether there is a universal and fundamental (as fundamental as elementary particles) reference to distinquish left and right.
> Now you say if you are standing left side of the line looking left to right the numbers would go 3,2,1 if you are standing on the right side looking left to right the numbers would go 1,2,3
The Martian doesn't know how to "looking left to right". He doesn't know which side is left. That's the whole point.
