Why did mathematicians conceptualized infinity? Do they use it in any mathematical systems?
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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.
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.
> 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.
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.
I am not actually quite sure what you’re asking, could you elaborate?
Infinity just means “there’s more of these things than is needed” or “this number (amount) is bigger than is needed” in discrete or continuous systems respectively. Or it means that there can be any number of things resp. a property can be as
For example, in probability, when you flip a coin, we say that there is 1/2 probability you’ll get a head, but what it actually means is that if you flipped the coin a really large number of times, you’d get head about half the time.
In computers science, we often work with “languages”, which are sets of strings (words). For example, you could have a languege of all strings that start with the letter ‘a’. So for example “a”, “aa”, “aba”… That is an infine language, because there are infinitely many strings that start with ‘a’.
Infinity is used in almost all areas of math. It simply just means “as many as you want” or “as many as is sufficient”.
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.
Infinity is just the “end of the number line.” The number line doesn’t really have an end but we are still able to create expressions that effectively evaluate to “the end of the number line, if there was one.”
I’m not entirely sure that mathematicians conceptualized infinity. It almost feels like something religious that a mathematician borrowed because it fit in his equation at the time. The value of the undefined that is undefined buy not undefined, limited but unlimited in the sense that we know that the value exists but the quantity can’t be measured, only accounted for.
Infinity kind of naturally pops up when you start considering limits. A lot of mathematics (notably: [real analysis](https://en.wikipedia.org/wiki/Real_analysis)) is based on limits. That is, that there’s a group of numbers or values that have bounds.
In a sense, limits allow you to discard a lot of information that otherwise would need to be accounted for and that would make the mathematical operation too heavy or complex. Fourier transform is a common example and is the basis of signal analysis and nowadays used in all sorts of stuff from audio processing to image compression to analyzing satellite data etc.
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