[Maursault]

> What definition?

The one that defines the noun, "hole," as a hollow place in a solid body or surface.

> if you take a circle (S^1 = { (x, y) : x^2 + y^2 == 1 })

Black holes are real, but your circle can only "exist" in mathematics.

[codethief]

> Black holes are real, but your circle can only "exist" in mathematics.

You are aware that General Relativity is about the most mathematical (and mathematically rigorous and mathematically advanced) theory we have in physics? Several predictions were made purely based upon mathematical arguments, without much physical input. So your argument that math were a "different realm" with no connection to reality is full of, uhh, holes.

In fact, I would argue mathematics is mainly a language that provides us with the precision that everyday language lacks, it's a tool that allows us to make precise arguments while ruling out logical fallacies.

In the present case it allows us to say very precisely what we mean by a hole. This definition[0] has been employed in countless predictions in physics and its usefulness has been confirmed by experiment. Meanwhile, your definition is vague at best.

[0]: Look up "simple-connectedness" or, more generally, "homotopy groups".

[Maursault]

> So your argument that math were a "different realm" with no connection to reality

This straw man was not my argument. My argument is that circles only exist mathematically.

[codethief]

For this argument to work you must implicitly assume that mathematics were a different realm, with no connection to reality.

But it doesn't matter. The definition of a hole doesn't rest on the definition of a circle.

[Maursault]

> For this argument to work you must implicitly assume that mathematics were a different realm, with no connection to reality.

No I don't, and this is still a straw man.