CSH: Wei was saying that we can’t get geometry and arithmetic wrong because those sciences merely report back on our very means of apprehension. That is to say, mis-grasping a geometric or arithmetic truth is impossible. If you understand the sentence then you already grasp its truth and cannot misjudge its truth.
By the same token, we can never understand how our experience mismatches real reality. We can never understand it.
So we live with two extremes—what we do know is certain and necessary, and what we don’t know is forever unknowable.
The knowledge that we do have we can’t get wrong. Our discoveries of synthetic a prior truths inside Euclidean geometry are really just reports on the form of our awareness of space. Geometry is an anatomy of the subject’s means of immediate, particular consciousness.
We can never know if or how our Euclidean-spatial take on reality is mistaken, because we would have to use this same space in order to sustain the comparison between subjective and real space and understand how our spatial take on reality is (or is not) mistaken.
All comparisons of space that we can undertake would have to be inside of space.
So while we can imagine ways of incongruity between our representation and the thing, but they’ll all be intra our method of apprehension. We’ll never know how we miss.
Obviously, we do miss, because there must be a mismatch between the reality we know (which is particular and determinate) and real reality, which is probabilistic. Even though the Schrödinger wave equation evolves totally deterministically, it’s a probability wave that is being determined, and not a particular position.
So, no. Our language mis-maps onto the quantum realm because it is essential to our kind of language and logic that predicates, including complimentary ones, be fully known in combination; and that the order of knowing obey the commutative property. But in the quantum reality, they are not.
We can experience non-commutativity in mathematics and see why it works, but the deep link between non-commutativity and physical reality is hard to fathom because for us epistemic possibility is coextensive with logical possibility—namely, everything outside of the Law of Noncontradiction.
This let us know that our map mismatches—that reality and thought are distinct, that the laws of thinking outrun the laws of physical being.
Then we wonder: what is the nature of our miss? And we know: we can never, ever answer that. We can never know how we miss. Missing reality also means missing how we miss reality.
By the same token, we do have knowledge about our mode of apprehension—logic, arithmetic, and geometry. Our knowledge of geometrical truths is so total that it serves as the standard of knowledge.
So the very thing that makes us completely blind—a lens so strong that we could never even guess how we mismatch reality—is also the source of our certainty.