I would add that real numbers have physical significance in terms of having to arbitrarily operate on some fraction of another value. It's true that the universe might prohibit arbitrary precision, but when you don't know the depths of the permitted halvings, and you're permitted arbitrarily large units, it amounts to the same thing.
>We know some subset of the natural numbers are real because we can count things.
So he's saying:
1) X is real = there exists a one-to-one mapping between X and a consistent physical process.
2) Natural numbers map to counting.
Okay, but then:
>Well they are invented purely so we can take the square root of two. And the complex numbers? Invented so we can take the square root of minus one.
Not true: complex numbers also map to real-world dynamics, like oscillatory behavior. From a search for the physical significance of complex numbers:
https://www.quora.com/What-is-the-physical-significance-of-c...
I would add that real numbers have physical significance in terms of having to arbitrarily operate on some fraction of another value. It's true that the universe might prohibit arbitrary precision, but when you don't know the depths of the permitted halvings, and you're permitted arbitrarily large units, it amounts to the same thing.