Does the concept of infinity have any practical applications?

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 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 appreciated.

Solution 1:

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.")

Solution 2:

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.