Why do people use the phrase "infinitely small" in calculus?

Question

From what I understand, if we are using the real number line, there is no such thing as "the smallest possible number". In fact, this seems very easy to prove:

• Let $n=$ the smallest element of the set $\{x \in \mathbb R:x>0\}$
• $\frac{n}{2}<n$
• This contradicts our assumption that $n$ is the smallest element of the set $\{x \in \mathbb R:x>0\}$
• Therefore, there is no smallest positive real number

However, people often talk about things being "infinitely small". For example, when you compute the area under a graph, people often say the bars are of "infinitely small width". This seems wrong to me. Rather, you look at what happens as the width becomes smaller and smaller. Then, you compute the area of the approximations. Using some formal definitions, you can prove that the true value is being approached by these approximations. By computing the limit of the area apprxoimations, you are also computing the true area under the graph. When people say "infinitely small", is it just a shorthand, or am I misunderstanding something?

Answer 1

I don't know about "people", but (at least after the mid-19th century) when mathematicians talk about such things, it is either a shorthand for a limiting process or they are using Nonstandard Analysis

Answer 2

In general, it is indeed used purely as a shorthand. It is however in a precise sense a "safe" shorthand: as Robert Israel says, there is a perfectly rigorous formulation of calculus which does use infinitesimals.

Briefly, the idea is this:

• Describe a structure $^*\mathbb{R}$ which consists of the usual real numbers + a bunch of infinitesimal "numbers" (and infinite numbers, and numbers differing from usual numbers by infinitesimal amounts). This is called a hyperreal field.

  • Note that unlike $\mathbb{R}$, which is the unique complete ordered field, there are lots of different hyperreal fields. It doesn't matter which one we pick, though, except in very specialized circumstances.

• In the context of $^*\mathbb{R}$ we can whip up a "naive" version of calculus; we now prove that the theorems and calculations we get in $^*\mathbb{R}$ are actually true about $\mathbb{R}$. This is called transfer.

  • I'm actually being a bit sloppy here: more accurately, part of the definition of "hyperreal field" is that it has the transfer property and the previous bulletpoint really amounts to showing that hyperreal fields in fact exist in the first place.)

This approach is called nonstandard analysis. (There are other ways to develop formal frameworks for infinitesimals, e.g. smooth infinitesimal analysis, but nonstandard analysis is by far the simplest in my opinion.)