It was a bit of a mind bender initially coming to this conclusion, but shrug.
Mathematics is a tool we've developed to help us understand reality. But it itself is not reality.
>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.
The real numbers aren't invented to capture the square root of two. You can deal with that using a simpler set. The real numbers are necessary for common approaches to calculus and other forms of math that depend on a continuum of quantity.
I would describe math as a tool for modeling reality, rather than understanding it. Because we really don't know much at all. All we've got are rules that are pretty damn good at predicting things under everyday conditions.
I agree with what you say, but I also think that we overly conflate "importance" with "the order we found/defined them in". It's a big leap (conceptually, arithmetically, etc.) to go from the rationals to the reals; sure that's what the ancients actually did, but we don't need to follow the same path when e.g. generalising some result, teaching mathematics in school, defining a computer mathematics tool, etc.
There's no reason to "skip past" algebraic numbers, computable numbers, etc. especially since they're most often the sets we're dealing with; e.g. in high school most problems were something like 'find the real number x such that something-involving-x = 0'.
PS: You've inspired me to have another go at reading the road to reality; it's been gathering dust on my shelf for years!
To follow on you previous run: ... because we can cut things into pieces and count the pieces. We know real numbers are real because we can cut a square across the diagonal and see the relative length of the diagonal to the side.
If a line were just a collection of discrete points related by some formula, there's no reason two lines that aren't parallel should have to intersect, from the perspective of sharing a point.
What is the usefulness of real numbers, beyond the usefulness of computable numbers? Real numbers give you silliness like "You can cut and reassemble a ball into 2 balls identical to the original, if you cut it into 5 infinitely detailed pieces. Negative and Zero don't lead to patently impossible predictions about the world.