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I know what you're thinking: "of course it has, for example, it can be used to tell you how many times you can go around a circle". But that isn't really true, now is it? You'd be dead or the world would go under long before an infinite amount of loops had been reached.

Are there any practical applications for the concept of infinity? Is it a useful concept in maths at all? I pose this question not with the intention to provoke anyone, but out of genuine curiousity.

I know that Donald E. Knuth has argued that for all practical purposes, a very, very large number has the same effect as infinity, in his book "Things a Computer Scientist Rarely Talks About" (can't remember the exact quote, nor find it online, unfortunately).

Examples are especially appreciated.

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We can deal much better with $\sum_{n=0}^\infty\frac1{n^2}$ than with $\sum_{n=0}^{10^{80}}\frac1{n^2}$. – Hagen von Eitzen Jan 23 '13 at 12:49
Adding to Hagen Von Eitzen, taylor series are infinite series which cannot approximate perfectly when finite. If made finite, With accumulation of errors from one approximation to another, there might be devastating effects. – 007resu Jan 23 '13 at 12:57
"As soon as you begin to understand the immensity of Super K, you will realize that just being finite isn't much of a limitation, and you will see how pointless are the philosopher's discussion about finite versus infinite. Infinity is a red herring. I would be perfectly happy to give up immortality if I could only live Super K years before dying. In fact, Super K nanoseconds would be enough." Super K is $10 \uparrow \uparrow \uparrow \uparrow 3$, D.E. Knuth, pages 171-172 of his "Things a computer scientist rarely talks about"; Stanford Calif.: Center for Study of Language and Inform., 2001. – Alessandro Jacopson Feb 19 '13 at 17:32
Hagen von Eitzen, what if ∞ represented "a really, really large number" instead? Would not the calculations be the same and the ability to deal with it the same? – Alexander Apr 11 '14 at 12:20
If you enjoy physics, look into black holes; as density goes to infinity, you end up with black holes! – zerosofthezeta Aug 9 '14 at 8:35
up vote 4 down vote accepted

In the BBC's documentary about infinity they interviewed Doron Zeilberger which is probably the poster boy for "infinity is nonsense" in the world of mathematics.

They show him work with $\infty$ symbols when talking about series and functions. The reason this is a good idea is simple.

To say that something is infinite we just need to say that it has more elements than any finite number. But to say that something is finite we need to bound it somehow, which we cannot say in a simple way (and simple way means that for infinite we have a simple schema saying "more than $n$ distinct objects", whereas there is no particular schema catching all forms of finiteness).

In particular this is useful when talking about very small or very large things, it allows us to calculate limits (which is an essentially infinitary process) but discard most of the computation as a remainder which does not affect the outcome, which will follow by taking some error margin.

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"it allows us to calculate limits (which is an essentially infinitary process)" - can you explain the parenthetical statement and what you mean by "infinitary" versus "finitary"? I've asked half a dozen mathematicians and nobody has provided a concrete answer. – alancalvitti Jan 26 '13 at 3:48
When coming to calculate $2+3+5+7$ you simply add these numbers and get the result. This is a finite process of calculation. An infinitary process is one you can prove wha the result is going to be, but you don't, and can't, calculate every step in the way. For example you don't really write down the entire number produced by Cantor's diagonal argument. Limits are essentially infinitary because you don't calculate the result by hand, each step of the way, you make a general argument which allows you to prove something about the result. – Asaf Karagila Jan 26 '13 at 8:07
But the derivative of $x^2$ is also a limit yet can be computed as $2x$ in a finite number of steps for example by symbolic software like Mathematica programmed with rules to manipulate such expressions. The expressions themselves are finite and so are the rules. But I see that you've modified infinitary by "essentially" so you have to explain that now. – alancalvitti Jan 26 '13 at 15:32
You can define a formal method of deriving polynomials, which will then amount of a finite calculation. But you can't use this to derive $\sin x$; then you can add a rule for deriving $\sin$, but then you can't derive $\cos$; then you can add a rule ... and so on and so forth. But you can also introduce the definition of a derivative using a limit, then you can prove that this is a good notion, and that all those symbolic rules follow from it - and more. Now the reason I said "essentially" is that we can sometimes do a finite manipulations to derive the exact result. [cont.] – Asaf Karagila Jan 26 '13 at 15:55
You can symbolically derive and integrate, and you can get to a close enough result and "clean it up" in some method to obtain a "nice" result (e.g. a polynomial). But this is not the definition of a derivative using limits. Similarly when calculating the limit of a sequence, you can't really calculate it. At best you can "guess" that it would have to be a certain value and then you can prove that this is the correct value. One good way of doing that, for example, is identifying the terms as some continuous function - when possible. But this is not really calculating, it bypasses calcuating. – Asaf Karagila Jan 26 '13 at 16:00

Absolutely, infinity has countless (:P) practical applications.

Here's one way to think about it: do negative numbers have any practical applications?

I mean you can't really have a negative amount of anything, can you? You can't have negative five apples.

If your bank balance is negative, that's just another way of saying you owe the bank a (positive amount of) money, rather than the other way around. When we say a particle's charge is negative, we mean only, that it has more of the charge that comes from electrons than the opposite kind that comes from protons. And so on.

Nonetheless, the abstraction of negative numbers - which is to say of the Integers, and of additive inverses and number rings and so on generally - is hugely useful. It pervades our understanding of numbers, both as applied to the real world, and in many of the most theoretical branches of pure mathematics.

The same is true of other mathematical abstractions whose ontology, and thus utility, you might similarly question, from complex numbers to (the very many different forms of) infinity. For instance, infinities underlie all of traditional real analysis, the foundation of modern calculus and related fields. Whether or not the real numbers are in fact real, in the sense of somehow existing within the universe, they have proven to be at least an incredibly useful approximation for modelling all sorts of things that "may as well" vary continuously at the scales we measure them. There are ongoing efforts to replicate the results of the field using weaker postulates about infinities, in Constructivism and even Finitism, but they are far from being "complete", and probably never will be.

Likewise infinity is at the core of measure theory, on which our current construction of probability is based. Hilbert Spaces, used in the formulation of quantum mechanics, are infinite not just in size, but in dimension. And there are deep links between even the exotic transfinite cardinals of Cantor, and the areas of logic that deal with the most foundational issues of mathematics (and indeed, with those finite but extremely fast growing functions others have mentioned in the context of numbers that are "infinite for any practical purpose.")

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Cardinals are exotic? Neat. I'm sure the Pope will be delighted to hear that! – Asaf Karagila Aug 9 '14 at 8:28
@Jordan Rastrick Could you please explain a bit the link between certain infinite cardinals and certain areas of logic. Thank you. – Jo Wehler Sep 28 '15 at 10:24

You can take limits of functions as the variable goes to infinity. In this way you can calculate things like terminal velocity and escape velocity.

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