I'm not certain what my conclusions are to this query. But I do seem like to push back on the "invented" position. So I'm stuck somewhere between mathematical realism and your position. It seems to me that the Pythagorean Theorem says something genuine about right triangles. The things it says are 1) provable and 2) universal to all right triangles. Does the Pythagorean theorem not say something true about right triangles?
No, it says something about right triangles *if Euclidean geometry is assumed*. But, since the early 1800's, we know that there are non-Euclidean geometries that are just as internally consistent as Euclidean geometry. In these geometries, such basic 'facts' as Pythagorus' theorem and that the sum of the angles of a triangle is a straight angle are simply false. Furthermore, such geometries are more appropriate in many situations (for example, spherical geometry is non-Euclidean as is the geometry of general relativity).
I can easily imagine situations where an alien race would first arrive at a non-Euclidean geometry where Pythagorus' theorem fails.
If you tightened up your skeptic's hat really hard like that and then contemplated science, you could claim that "gravity isn't real. It's just a human invention that explains the motions of massive objects." What would be your reply to that objection?
Unlike mathematics, the motions of the bodies is objectively measurable. As long as those motions are correctly predicted, we have a *physically* equivalent theory.
This has a long history even in human physics. Newton described gravity in terms of a force which acts at a distance. Lagrange described it as a 'potential' that acts locally. There is also description where the motion maximizes an 'action' in a way that has been interpreted as teleological. These theories are *completely* equivalent in the motions they predict and are considered by physicists as just different ways to approach certain physics problems. That they are philosophically very different is simply seen as irrelevant.
Einstein described it as a distortion of the geometry of spacetime. But there is also a Lagrangian description of Einsteinian gravity. Again, these are equivalent in the predictions of motion, which is ALL that is important for the physics.
So, no, I would NOT necessarily expect that an alien race would arrive at the same *laws* of physics as we have (even to the current approximation), but I *do* expect the two systems to make the same predictions about observable motion.
I'm not asserting anything one way or the other concerning the topic. And I recognize that it falls more within your purview than mine. So I'm just kinda asking what your opinion is with my objections-- rather than challenging you.
But wouldn't you suspect that a rational alien would arrive at the same conclusions about triangles as we have? Like, it'd be weird to find a space-faring civilization who rejected the Pythagorean theorem, wouldn't it? Almost inconceivable. (Or would you disagree with that notion?)
I see it as easily conceivable. if their planet is small enough, or the history of their math is different enough, they may easily have come to spherical geometry as the basis of their geometrical thinking. And in that system Pythagorus' theorem is simply false.
