Why did mathematicians conceptualized infinity? Do they use it in any mathematical systems?
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> Why did mathematicians conceptualized infinity?
It’s something that kept cropping up as they imagined things getting larger and larger, or smaller and smaller, or processes going on forever.
For a long time (going back to the Ancient Greeks), people thought it was important to make a distinction between a “potential infinity” and a “completed infinity”. A potential infinity is something that can keep getting larger without limit, for example, when we’re working with numbers, we usually assume that we can keep getting larger and larger ones as we need them. A completed infinity is when we describe something that actually does have infinitely many elements, for example, “the set of all integers”.
It used to be a very common viewpoint among mathematicians that it was OK to work with potential infinities but not completed infinities. Now that the consequences of these choices are better understood, this view has largely fallen out of favour, though it still has some defenders. Basically, allowing for infinite objects causes some awkward philosophical issues but often makes it much easier to prove results about non-infinite things (which are usually what we ultimately care about, since there don’t seem to be any infinite objects in the real world and our brains can only deal with a finite amount of information).
> Do they use it in any mathematical systems?
Mathematicians routinely work with all kinds of infinite objects. For example, geometric shapes are usually conceptualized as infinite sets of points.
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