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by nahstra 1354 days ago
What definition?

Definitely not true in math - if you take a circle (S^1 = { (x, y) : x^2 + y^2 == 1 }) it has a hole as defined by homotopy in the middle and there is no 'space' there. If you fill it with space you get a disk (D^2 = { (x, y) : x^2 + y^2 <= 1 }) there is no longer a hole as defined by homotopy/is contractible.
1 comments
Maursault 1354 days ago
> 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.

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codethief 1353 days ago
> 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".

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Maursault 1353 days ago
> 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.

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codethief 1353 days ago
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.

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Maursault 1353 days ago
> 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.

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