Here are two common but unhelpful was to think about infinity.
- Infinity makes things harder.
- Infinity is a useless academic abstraction.
Neither of these is necessarily true. Problems are often formulated in terms of infinity to make things easier and to solve realistic problems. Infinity is usually a simplification. Think of infinity as “so big I don’t have to worry about how big it is.” Here are three examples.
In computer science, a Turing machine is an idealization of a computer. It is said to have an infinite tape that it reads back and forth. You could think of that as saying you can have as long a tape as you need.
Physics homework problems deal with infinite capacitors. Don’t think of that as meaning a capacitor bigger than the universe. Interpret the problem as saying that the width of the capacitor is so large relative to its thickness that you don’t have to worry about edge effects.
In calculus, you could think of infinite series as a sequence of finite approximations. A Taylor series, for example, is a compact way of expressing an unlimited sequence of approximations of a function. You can get as close to the function as you’d like by including enough terms in the sequence.
The infinite case often guides our thinking about the big case. Take the Taylor series example. A Taylor series isn’t just a formal series of polynomials. A Taylor series converges in some region. That says the infinite sequence of terms don’t behave arbitrarily. They get close to something as the terms increase. Knowing that the infinite sum converges tells you how to think about finite approximations you select from the series.
When I went to grad school, my intention was to study functional analysis. Essentially this means infinite dimensional vector spaces. That sounds terribly abstract and useless, but it can be quite practical. My background in functional analysis served me well when I went on to study partial differential equations and numerical analysis.
Infinite dimensional spaces guide our thinking about large finite dimensional spaces. If you want to solve a practical problem in high dimensions, the infinite dimensional case may be a better guide than ordinary three dimensional space. Continuity in infinite dimensional spaces requires structure that may not be apparent in low dimensions. Thinking about the infinite case may prepare you to exploit that structure in a large finite dimensional problem.