Does not at all appreciate the relationship between math and empiricism. Doesn't appreciate at all that there is a relationship.
Ah -- but that (to drive the metaphor further) is the mystery of faith! Why is there such a relationship? Why ought there to be a connection between empirical counting and the Principia Mathematica? What has the axiom of reducibility to do with showing a toddler one object and then two objects, and her understanding that those aren't the same? Clearly it seems there must be a deep connection between the mathematical abstraction of "one" and the empirical idea we use in everyday life, and we all assume this is the case, but I don't comprehend what it is or why we can generally choose to conflate the two. That part, I take on faith. (And a real, pure mathematician could really not care less anyway. What has a mathematician to do with empirical anything? That's not what math is about. Well, that's what math is about to a scientist, yes, or to an applied mathematician; but not to a pure mathematician, to whom it is a beautiful but essentially artificial structure.)
In physics, I don't know that we've conceded any limits, really. The Pauli exclusion principle only tells us what the universe can't or can't do, not what we can or can't learn. One might think that (say) the Heisenberg uncertainty principle (perhaps this is what you meant?) is more limiting, that we can't measure in nonorthogonal axes simultaneously without disturbing the state, so yes, there are specific things we can't learn about the universe locally. But more generally these restrictions on the rich mathematical space of a quantum state are really interesting and actually expand how much we can learn about the way the universe works, which is the interesting question.
no subject
Ah -- but that (to drive the metaphor further) is the mystery of faith! Why is there such a relationship? Why ought there to be a connection between empirical counting and the Principia Mathematica? What has the axiom of reducibility to do with showing a toddler one object and then two objects, and her understanding that those aren't the same? Clearly it seems there must be a deep connection between the mathematical abstraction of "one" and the empirical idea we use in everyday life, and we all assume this is the case, but I don't comprehend what it is or why we can generally choose to conflate the two. That part, I take on faith. (And a real, pure mathematician could really not care less anyway. What has a mathematician to do with empirical anything? That's not what math is about. Well, that's what math is about to a scientist, yes, or to an applied mathematician; but not to a pure mathematician, to whom it is a beautiful but essentially artificial structure.)
(I mentioned this discussion to my husband, who said it reminded him of Eugene Wigner's paper on the unreasonable effectiveness of mathematics in the physical sciences, which he was surprised I'd never heard of before.)
In physics, I don't know that we've conceded any limits, really. The Pauli exclusion principle only tells us what the universe can't or can't do, not what we can or can't learn. One might think that (say) the Heisenberg uncertainty principle (perhaps this is what you meant?) is more limiting, that we can't measure in nonorthogonal axes simultaneously without disturbing the state, so yes, there are specific things we can't learn about the universe locally. But more generally these restrictions on the rich mathematical space of a quantum state are really interesting and actually expand how much we can learn about the way the universe works, which is the interesting question.