Every statement is a product of stipulated rules and definitions. That is what the term “statement” means. “How can our statements be statements about the world?” Sounds like Kant, doesn’t it?
Take the Pythagorean theorem: Wouldn’t it be accurate to say that, **if **there are two things in the world that are finite lines and are perpendicular (call them A and B), then the shortest distance between their endpoints is equal the square root of A-squ + B-squ? If so, then it is a fact about the world.
It’s unknown if that is accurate or not. If space/time in the real world is completely “flat” in the Euclidean sense, then yes, that would be accurate. But if space/time is curved, in either an elliptic or hyperbolic sense, Pythagoras’ Theorem fails, and your math breaks down.
As best we can tell right now, space/time is flat. But that’s a scientific consensus that’s pretty solid over the last few decades or so, but still tentative.
But the profound point obtains, regardless. The Euclidean definitions don’t control jack in terms of reality. If Hubble (and other) observations had turned out differently, or if we are still missing subtle cues and measurements that point to positive or negative curvature, Pythagoras’ Theorem would not obtain in the real world (it may be close enough for crude operations in the way that Newtonian physics can be used to plot the flight of a baseball, etc.).
This is a particularly instructive example, in that it once again points out the trouble people have confusing the map with the territory. When you ask if Pythagoras’ math is “accurate” in the real world, I can’t respond with “logic” or
a priori reasoning. I can only judge the definition against the evidence and observations
from reality. For a very long time, when “Riemann”, “Calabi-Yau”, “space/time manifolds” and “spatial topologies” were unknown terms of art, people just assumed that Euclidean geometry
was real. In parochial, local tests, the math worked out plenty well, right? The analytics got confused with the synthetic.
Over very large distances, though, your math may still have you
way off, for all we know.
Moreover, this is all said in the context of an isotropic space/time. In the real world, mass
distorts space/time (gravity), and these distortions introduce curvature at a local level that nukes Euclidean distance calculations just as much as an elliptic space/time manifold would. In that sense, then, wherever gravity is present, a^2 + b^2 will NOT produce the right results for you, and your statement would be inaccurate. If you are talking about applying your statement to locations here on earth, your statement would NOT be accurate. Einstein, for example, named a whole chapter of his book on Special and General Relativity “Gaussian Coordinates”, in which he deploys non-Euclidean geometry to account for the local curvatures in space/time.
Why would he not use Euclidean geometry?
Because it wasn’t accurate!!
If we can disprove this statement using empirical examples, then the Pythagorean theorem is false. (It is true, mind you, given its assumptions, but its assumptions are false.)
Some a priori claims are not trivial.
I don’t think
any a priori claims are trivial if they concern reality.
The definitions don’t define reality, but they ought to *describe *reality. If they don’t, they are only playthings. If stacking two blocks on top of two other blocks did not make a four-block stack, then our mathematics would be false. Many, but not all, “necessary truths” can be falsified by the real world.
Euclid’s proof is perfectly necessary as a truth. It’s unassailable, perfect. But its truth is trivial, analytic, tautological. All the “pure truths” are so because they are artificial, abstract. It’s only when statements hazard the real world as their subject that things become difficult, problematic. In terms of “necessary truths” about the real world, I can’t think of a single one, or even how one could provide a coherent basis for “necessary” as a metaphysical feature of reality.
-TS