Perceiving the Infinite and the Infinitesimal World in Calculus

The concept of tangent line of a real function is normally based on the standard $\varepsilon ,\delta$ concept of limit, which is intrinsically difficult to represent and not immediately assimilable, for example, by students. We can avoid this trouble by introducing a pictorial device that allows the visualization of small details in the graph of a curve $y=f(x)$. This method was invented by Stroyan (1972) and improved by Tall (1982, 2001): our intention is to continue and improve Tall's work by applying it to many other different situations. We will work on the hyperreal number system $\mathbb{R}^*$ and will assume the non-standard analysis given by Abraham Robinson (1966).³

In the present and in the following section we will explain the method and the classification proposed by Tall. In the last two sections we will introduce new types of diagrams called microscopes ``within'' microscopes and lenses ``within'' infinite telescopes.

By visualizing the difference between the numbers $a$ and $a+\varepsilon$ (where $a \in \mathbb{R}$ and $\varepsilon$ is a positive infinitesimal), we can introduce the map $\mu:\mathbb{R}^* \to \mathbb{R}^*$ given by

$$\mu(x)=\frac{x-a}{\varepsilon } .$$

Thus $\mu(a)=0$ and $\mu(a+\varepsilon )=1$, that is, $\mu$ maps $a$ and $a+\varepsilon$, two infinitely close points, onto clearly distinct points 0 and $1$. We may also identify, through $\mu$, a point $a$ with its corresponding $\mu(a)$.

Figure 1: The hyperreal line and the map $\mu$.

In general, for all $\alpha, \delta \in \mathbb{R}^*$, the function $\mu:\mathbb{R}^* \to \mathbb{R}^*$ given by

$$\mu(x)=\frac{x-\alpha}{\delta } \quad (\delta \neq 0)$$

is called $\delta$-lens pointed at $\alpha$. But what can we see through a lens? What kind of details can it reveal? We define field of view of $\mu$ the set of $x \in \mathbb{R}^*$ such that $\mu(x)$ is finite. Given two infinitesimals $\varepsilon ,\delta$, we say that $\varepsilon$ is of higher order than $\delta$, same order as $\delta$, or lower order than $\delta$ if $\varepsilon /\delta$ is, respectively, infinitesimal, finite but not infinitesimal or infinite. It follows from this definition that, if $\varepsilon$ is of higher order than $\delta$, $\varepsilon$ is an infinitesimal ``smaller'' than $\delta$.

Given a $\delta$-lens $\mu$, proceeding by taking the standard part of $\mu$, we obtain a function from the field of view in $\mathbb{R}$, called the optical $\delta$-lens pointed in $\alpha$. The optical lenses are actually what we need to visualize infinitesimal quantities. In fact, our eyes are able to distinguish clearly only images on the real plane $\mathbb{R}^2$. As such, the optical $\delta$-lens translate on the $\mathbb{R}^2$ plane, in favor of our eyes, everything that differs from $\alpha$ in the same order as $\delta$. Higher order details are ``too small'' to see and lower order details are ``too far'' to capture within the field of view. Two points in the field of view that differ by a quantity of higher order than $\delta$ appear the same through the optical $\delta$-lens.

This method also works in two coordinates (and, in general, in $n$ coordinates) by the application of a lens to every coordinate. The map

$$\mu:{\mathbb{R}^*}^2 \to {\mathbb{R}^*}^2, \quad \mu(x,y)=\left(\frac{x-\alpha}{\delta } ,\frac{y-\beta}{\rho}\right)$$

is called $(\delta ,\rho)$-lens pointed in $(\alpha,\beta)$. If $\delta \neq \rho$, we say that the lens is astigmatic. If $\delta = \rho$, we can talk about $\delta$-lens in two dimensions. By considering the standard parts of every coordinate, we obtain an optical $\delta$-lens in two dimensions, definied from the field of view of $\mu$ in $\mathbb{R}^2$.

However, $\delta$ may not be infinitesimal. Depending on its nature, there are different kinds of lenses: if $\delta$ is infinitesimal, then the lens is called a microscope; if $\delta$ is finite but not infinitesimal, then the lens is a window; if $\delta$ is infinite, the lens is a macroscope. A window pointed at a point with at least one infinite coordinate is called a telescope.

Microscopes reveal infinitesimal details and telescopes allow us to visualize a structure at infinity. For example, through an optical microscope, a differentiable function looks like a straight line and through an optical telescope two asymptotic curves look identical.