Why did mathematicians conceptualized infinity? Do they use it in any mathematical systems?
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Chances are it has something to do with 1/0. How do you explain without the concept of infinity
“Do they use it in any mathematical system?” Indirectly, yes. It’s pretty much fundamental to the concepts underpinning Calculus, for example, which is a massively useful tool – not only for abstract mathematics but for any number of real-world applications. Without the concept of Infinity, modern technology would likely be far less developed.
The other folks already explained how it exists in something like calculus but I’ll explain why it exists.
Infinity as a concept is something that has to exist. You could compare it to “forever”, used in time, if something never ends it is forever, if a number goes on forever it is infinite.
Imagine if the number one was added to itself or anything else, any additive operation too. We would have to say that number is infinite, because there is no end to the operation. this is also true inversely if we subtract because there is both a positive and negative infinity.
We can also look at decimal places. The number 2 can be represented as the integer 2, or 2.0, or 2.000000000000, you can add as many zeroes as you want, this also applies to numbers in-between 2.5, is the same as 2.50000000. that is why there is a infinite amount of numbers between 2 and 3, and there’s also a larger infinite amount between 1 and and 100.
Infinite existing is really the only way to explain something like this. Imagine if the concept of “forever” didn’t exist, if an event is ongoing, and never stops it’s not entirely true to say it’ll just go on for a really long time, because it will go on forever.
> 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.
Infinity is a consequence of math. For example, if we set up the rules of a series and say the series is 1+1+1+… Forever, infinity pops out as the solution.
Just because infinity can pop out from simple rules of math doesn’t mean it’s physically real. Early debates on infinity were often about what it could possibly mean in reality. Even now, when infinity pops out of solutions in physics equations, it’s usually a sign that the answer is wrong because the theory is incomplete in some way. However, not always. Black holes are a consequence of infinity: if you pack a finite mass into an arbitrarily small space, it becomes infinite density. Black holes are indeed real though. The breakdown is that we don’t really understand them so the infinite density thing is still potentially not accurate.
Anyway you can see infinity has practical application and appears. Another is calculus when we integrate indefinitely from 0 to infinity. There are also math systems about different scales of infinity in set theory. Countably infinite sets are things like counting numbers. They go on forever. But there are also uncountably infinite sets, like real numbers. Uncountably infinite sets can’t be counted (paired with the counting integers). And it keeps going, actually. There are ever higher levels of infinity bigger than the previous. I don’t know the application for these though so I’ll stop there.