You have a volleyball without defects. You poke a hole in it. How many holes does it have in it? (excluding the fill valve - assume there is no fill valve)
#math #topology #words #definitions #holes #context #arbitrary
I posted this in response to this thread:
https://qoto.org/@freemo/107295982688801818
The question conflates terminology between common usage and terms used in a specific branch of mathematics.
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@Pat Well its not as conflated as you thin, at least not in my opinon.
If I were to "poke a hole" in a volley ball I'd have to take a long sharp stick and poke **all the way** through, in one end, out the other. This would create one hole. If you use my earlier explanation of flattening it to a disc this would be consistent with that.
If you only cut a single **opening** in it without poking a hole all the way through then you didnt create a hole at all, simple turned a sphere into a bowl or cup. Does a bowl have a hole in it? Does a cup? Most would say no. To take the analogy further I think we all agree simply scooping out a dent in something (effectively what making one opening in a hollow sphere is) isnt a hole. However if you poke all the way through a sphere you get the equivelant of a doughnut, now we would all agree there is a hole.
So in laymans terms, if you have a volleyball it has no holes. You cut an opening in it you turned a volllyball into a cup/bowl it still has no holes. You cut a **Second** opening into it, now you you have a hole.
@freemo look up the definition of the word literally anywhere... the cup does not have a hole because the hole defines the cup, so when you speak of the hole in a cup, ones imagines another hole that is not supposed to be there. If a cup was convex it wouldn't be a cup and you'd have to hallow it out to make it into a cup. And no most normal people would no think it necessary to go all the way through the ball and out, unless they have an itch to shoot it, or a desperation to prove themselves right.
Thats pretty much exactly what I said with different words. A cup (no handle) does not have a hole. It isnt because the hole defines the cup, there is literally no hole. A convex surface is not considered to have a hole by any reasonable definition.
Now if you want to get technical holes are very strictly and technically defined in math (though there are two major areas of math that use the term hole, both would be in agreement for these simple use cases).
What "normal people think" isnt really too important to me. Normal people have no consistent definition of a hole so its a moot point. What does matter to me is any definition of a hole which is consistent, and we can dismiss inconsistencies easily.
There are many ways to reason about the cutting of an opening into a sphere that all shows us clearly why its not a hole even by common definition... Say you cut a opening whose size is the size of the equator of the sphere, in other words you cut the sphere into two perfect halves, putting a hole in it that consumes half the material... would anyone look at what is effectively identical to a bowl, even though it is clearly a volleyball and go "that volleyball has a hole in it?
What if i cut an even larger opening in the volleyball such that 95% of the material of the volleyball is removed leaving just 5% of the original volleyball. It would look like a small patch of material approximately appearing to be that of a slightly convex disc. Would anyone in their right mind look at that little scrap of material and go "it has a hole in it"... no of course not.
Any rigorous reasoning about cutting single openings in spheres makes it quite clear there is no **consistent** way you can call that a hole and in fact in almost all scenarios most would say it isnt a hole. The rare edge cases where someone would call it a hole is arbrbitrary and so wildly inconsistent with the others we can dismiss it out of hand as being incorrect despite common usage.
Now if we want to get into formal definitions, then it is consistent with everything i just said above and extends those ideas even further and more formally.
As for your comment about "literally" looking up the definition of a hole anywhere. Here is the formal definition of a hole that aligns with the normal use of a hole (the only definition we can apply rigerously). I just looked it up "literally anywhere":
The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected.[5] In layman's terms, it is the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A doughnut, or torus, has 1 such hole. A sphere has 0.
