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

Think about how round a pentagon looks. A hexagon is a little rounder than that. Now an octagon. Every point you add makes the shape more round. If you add infinite points, you get a circle.

Class dismissed.

I don’t necessarily think this is true… hear me out here…

Assuming infinities, you could also infinitely “zoom in” and it would still have points on it, infinitely of course, but assuming you can infinitely zoom in it is more of a never ending system of infinite circle looking “points,” but a true circle has zero points.

I hope I explained my thought process right haha! Correct me if I am wrong please!

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

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.