Guy brings fire to the experts(v.redd.it)

I’ve tried to claim infinity is lazy to my math major husband. Infinite as a number? We are too lazy to count that high. (Reminds me of the fault in our stars quote about how “some infinities are bigger than others” which is apparently actually true, which doesn’t compute in my mind) Why is there not a named shape with 1 zillion sides, because we are lazy and would rather call it a circle. But apparently there are several reasons why this doesn’t work mathematically, infinity is actually valid unfortunately. It’s been years but I’ve finally mostly relented my argument

You call it "lazy", I call it "clever" :)

Let's say you're in a room with one exit door. With each step, you go exactly half the distance remaining towards the door. Let's say you start at, oh, 10 feet away? You take a step and now you're 5 feet away, then 2.5 feet away, then 1.25 feet away...

You can only realize that you'll never exit the room if you utilize the concept of infinity. If you refuse to use infinity, then that's like saying "how do you know you can't reach the door if you haven't taken enough steps? You're just too lazy to take enough."

The objects you're talking about are "regular polygons" - square, pentagon, hexagon, etc. Just because we don't have a name for a 1-zillion sided regular polygon doesn't mean math doesn't recognize it - we just only have names for things people actually talk about. A 1-zillion sided shape is mathematically different than a circle, and so is a shape with 1-zillion more sides and so on and so on. If you draw them, sure they look the same on your paper because the resolution of a pencil can't capture the corners and edges of the shape, but that doesn't mean they are the same as a circle.

I actually kind of see the point here. I'll ignore the point about some infinities being bigger than others, because that's actually a whole different matter that is the opposite of an argument for your suggestion and focus on the cardinality of the natural numbers.

The concept of infinity allows us to deal with limit quantities instead of sequences, and this allows a lot of convenience. In particular, it allows continuous approximations of discontinuous phenomena. This allows us to deal with discrete phenomena approximately using methods like calculus when there isn't really truly infinite divisibility, just high-order finite divisibility. In this sense, we've been lazy by failing to consider the exact nature of the phenomenon and instead used calculus for an easy approximate description.

That said, there is theoretical value for developing some notion of infinity, though it doesn't have to be the "actual infinity" notion of standard modern mathematics (where infinity is an actual quantity of its own). Brouwer's intuitionism, as I understand, only accepts the idea of a "potential infinity", which is a series of steps without end but not a quantity in itself.

EDIT: Also the fact that infinity can be used as a stand-in for limiting processes in general, which was more what I had in mind when I started writing this comment. The "less lazy" way would be talking about the sequences directly, but using algebra of limits we can often "get away" with just talking about their limits.