Why did mathematicians conceptualized infinity? Do they use it in any mathematical systems?
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There are many mathematical systems that explicitely use infinity for various purposes. Some off the top of my head:
* Projective geometry. It’s what 3D computer graphics uses. I’m sure you’ve seen a picture of parallel train tracks going off into the distance and “meeting” on the horizon, i.e. infinity. That point is called a vanishing point and there is one for every direction in space. These points are “infinities” but in the mathematics of projective geometry they work just like regular points. It’s a fascinating field.
* In topology it is often useful to add a single infinity point to a topological space (such as the number line, or the 2D plane) to make it *compact*, which is a very convenient mathematical property which these spaces wouldn’t have otherwise. This is called the Alexandroff extension. It’s like there is a gap at infinity and this closes it.
* Finally there are mathematical tools like transfinite induction which work with infinite sets.
* Here’s a mind bender for you: There are ways to work with and prove stuff about infinitely-dimensional vector spaces. Not 3D, not 4D, actually ∞D.
* Finally, sometimes ∞ is not actually an entity but just a shorthand notation, e.g. in a limit calculation or in the boundaries of an integral.
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