Analysis, the branch of math that includes everyone’s favorite high school subject (Calculus), is built on a simple argument that reflects a debate spanning thousands of years: what to do about infinity?

In pursuit of ketchup

Take a walk down the street and you engage in a magnificent paradox. Perhaps it’s a Monday evening, and you’ve just realized your dinner spuds are incomplete without a fine bottle of ketchup from the corner grocery store. Starting at your front door, you proceed down the sidewalk in pursuit of your favorite, luscious tomato purée.

On the way to the store, you’ll first pass the post office, which sits about half-way. But before you hit the post office, you’ll first walk by Mrs. Shefford’s house, which is again half-way to the post office. And before arriving at the Shefford abode, you’ll first reach the lamppost that stands (you guessed it) half-way between your home and the Sheffords’. As for reaching the lamppost? You’ll have to get to the half-way point first. And so on, and so forth, ad infinitum.

This is a famous paradox, attributed to the Greek philosopher Zeno of Elea. In order to cross any sort of distance, a walker must first traverse an infinite number of intermediate points half-way. Intuitively, one might suppose that doing an infinite number of things is impossible. One cannot take an infinite number of steps, read an infinite number of books, or eat an infinite number of spuds (ketchup or no). Visiting an infinite number of points seems to be similarly impossible, yet we do so every time we move from one place to another. Hence, the paradox.

“Now hang on a minute,” you might be thinking, “you’ve pulled a fast one!” While one may be traversing an infinite number of points, the time it takes to visit a point may be infinitely small.

Suppose it takes you 20 minutes to walk to the grocery store. Then it would take about 10 minutes to reach the post office, 5 minutes to reach Mrs. Shefford’s house, and 2.5 minutes to reach the lamppost. As we continue dividing these increments, the time it takes to reach the next point shrinks in parallel, but the total transit time remains 20 minutes. You are traversing an infinite number of points. It’s just that passing some of these points may take almost no time at all!

This clever observation was recorded by Aristotle, but it’s not the complete picture. In fact, there’s a deeper, subtler paradox that remains, which won’t be addressed for another two thousand years. Can you see it?

Infinity and infinitesimals

We resolved Zeno’s paradox by asserting that an infinite sum over infinitely small quantities is constant. How can this be? If we’re adding an infinite number of anything, shouldn’t the result be infinity? But hang on, if the things we’re adding are infinitely small, shouldn’t the resulting sum be infinitely small (i.e. basically zero) no matter how many things we add? It’s like the famous question, “what happens when an unstoppable force hits an immovable wall?” We haven’t answered Zeno’s paradox; we’ve merely replaced it with a harder one.

The resolution to this paradox is much trickier (hence, the two thousand year interim). As a first pass, let’s define a new quantity $dx$ that we’ll call an infinitesimal. You can think of $dx$ as a positive number just like 1, 2, 5, and 73 (but not 0), except that it has a special property: for any number you can imagine, $dx$ will always be smaller. We can be explicit (if not entirely rigorous) and write $dx$ as

$$ dx = \frac{1}{\infty} \thinspace .$$

Returning to our paradox, suppose you walk 1 mile to reach the grocery store, which takes 20 minutes. Then you would walk 0.5 miles in 10 minutes, 0.25 miles in 5 minutes, 0.125 miles in 2.5 minutes, and so on — until you walk $dx$ miles in $20 \thinspace dx$ minutes. Since $dx$ is smaller than any other positive number, there is no way to divide $dx$ further into a smaller increment. This is it. We’ve hit the bottom of the recursion.

We can also count your steps the other way. After $20 \thinspace dx$ minutes, you walk $dx$ miles. After $40 \thinspace dx$ minutes, you walk $2 \thinspace dx$ miles. After $80 \thinspace dx$ minutes, you walk $4 \thinspace dx$ miles. In general, after $k \cdot 20 \thinspace dx$ minutes, you walk $k \thinspace dx$ miles. Suppose $k = \frac{1}{dx}$. Then after $(1 / dx) \cdot 20 \thinspace dx = 20$ minutes, you walk $(1 / dx) \thinspace dx = 1$ mile – precisely the time and distance to the grocery store!

Spendid! But what’s the point?

Recall that our issue with Aristotle’s explanation is that we have an infinite sum over infinitely small quantities. Should the sum be constant, infinite, or zero? If an unstoppable force hits an immovable wall, does the wall break, does the force disperse, or does something else happen entirely? Our $dx$ formalism allows us to answer this question. $dx$ is an infinitely small number, while $1 / dx$ is infinitely large. Walking $(1/dx) \thinspace dx$ miles implies that we walk exactly 1 mile, even though we multiply an infinite quantity with an infinitely small number!

Our formalism also makes clear that not all infinities are equal. Suppose instead that $k = (1/dx)^2$. $k$ is still $\infty$, but it’s now impossible to walk $(1/dx)^2 \thinspace dx$ miles, since this product is infinite! What if $k = \sqrt{1/dx}$? Now $\sqrt{1/dx} \thinspace dx \approx 0$, so you’d walk this distance almost instantly! $1/dx$, $(1/dx)^2$, and $\sqrt{1/dx}$ are all infinite quantities, but their product with $dx$ can be constant, infinity, or basically zero. The infinitesimal $dx$ acts as our reference point , allowing us to characterize the structure of an infinite quantity and thereby formalize how it behaves. While Aristotle’s explanation is more-or-less true in spirit, he overlooks the existence of different kinds of infinity.

The trouble with infinitesimals

Infinitesimals are quite handy, and their use helped bring about calculus. Unfortunately, infinitesimals aren’t quite “real,” and their use has historically been a source of great controversy. To see why, let’s summarize the two properties we required of the infinitesimal $dx$

1. $dx$ is smaller than any number
2. $dx$ is nonzero

Suppose we define a new quantity $dy = dx / 2$. It is clearly true that $dy \leq dx$. However, if $dx$ is smaller than any number, then at best we have $dy = dx$. But then, the only way that $dy = dx / 2 = dx$ is if $dx = 0$! The two properties we require of the infinitesimal $dx$ are contradictory; it’s impossible for both to be true simultaneously.

So it appears that $dx$ is pure fiction, just like unicorns, Santa Claus, and free lunch. While it’s possible to describe its general properties and even use it in a plausible-looking way, the fundamental contradiction at the heart of infinitesimals makes their application useless.¹

A truly solid resolution would not be available for another two hundred years after Newton/Leibniz’s calculus, and over two thousand years after Zeno first asked his question.

A course in analysis

Suppose you and I play a game with our paradox. I’ll pick a point along the mile to the corner grocery store, then you tell me how long it would take to walk there. Ready? Let’s go:

Me: 0.5 miles

You: …

Me: 0.1 miles

You: …

Me: 0.0000000…[a billion billion more zeros, to the power of a gajillion]…0001 miles

You: …

In every case, if I walk $d$ miles, then the time it takes is simply $20 d$ minutes. I could make $d$ as arbitrarily small as I would like, but the time you tell me would simply shrink in perfect tandem, maintaining correct and consistent progress to the grocery store.

All that seems perfectly fine and obvious, but how does this help us? In fact, how is this different from our story about $dx$? Looks like we just dropped the $x$ and called it a day.

The subtle innovation to this argument is that it does away with infinity. We don’t need it anymore. Rather than discuss a line that can be split infinitely many times, or a nebulous $dx$ that is somehow infinitely small, we shifted the argument to discuss a line that can be split arbitrarily many times, or a $d$ that can be made arbitrarily (but not infinitely) small. It’s really up to you and me how long we’d like to play this game. But so long as each interval I propose is finite (even if exceedingly small), we can keep talking logically about the problem without introducing any pesky infinite quantities. Infinity isn’t relevant — only our patience for how far we’d like to split the line.

If all that sounds vaguely unsatisfying and circular, you’re not entirely wrong. We haven’t really addressed the problem of infinity, so much as shoved it in a bag, buried it in a hole, and looked the other way. Standard analysis, from which this style of argument derives, maintains rigor by side-stepping the thorny epistemological issues boiling around infinity. Rather, it employs arguments that characterize potentially infinite processes by describing their behavior at arbitrary (but finite) resolution. I propose a property $\varepsilon$ that I’d like my process to attain, for which you supply a $\delta$ that attains this property. So long as you can always provide a $\delta$ for any $\varepsilon$ (even ones with a gajillion zeros between “.” and “1”), then everything works out just fine.

Ketchup at last

So there you have it. A two thousand year old paradox resolved. A fine bottle of ketchup purchased from a corner grocery store that did not take infinitely long to reach. The concept of the infinite spawns a large number of curious brain teasers, but the most expedient path through the paradox is simply by ignoring it altogether. Standard analysis took this path, and built a rigorous foundation for Calculus by bundling infinity behind quantities like $\varepsilon$ and $\delta$ that remain arbitrary but finite.

Of course, this isn’t the only path available. Nonstandard analysis confronts infinity head-on by wishing infinitesimals into existence and, through sheer force of will, establishes their rigorous existence despite the apparent contradiction we discussed above. This is a dangerous and thrilling path, fraught with bizarre implications and harrowing pitfalls, but in many ways a more satisfying resolution to Zeno’s paradox. Alas, we haven’t the space to tell the whole horrifying tale here, but will return in a future installment. Stay tuned!