What Is a Continuum? And Do Infinitesimals Exist?

Imagine drawing a line on a piece of paper. Not a dotted line, not a dashed line—a solid, unbroken line. Now imagine dividing that line in half. Then divide one of the halves in half again. Keep going. Can you keep dividing forever? Or would you eventually reach a smallest possible piece—something so tiny it couldn’t be divided any further?

This question has bothered thinkers for over 2,500 years. It might seem like a simple puzzle, but it leads straight into one of philosophy’s deepest mysteries: what is the world really made of—continuous wholes, or discrete little bits?

The Puzzle

Here’s the basic problem. Suppose you have a continuous line, the kind you draw with a pencil. It seems like one smooth, unbroken thing. But you can also think of it as being made up of points—tiny, dimensionless locations. A point has no size at all. So here’s the weirdness: if a line is made of points, and each point has zero length, how can putting a bunch of zeros together ever give you something with length? It’s like trying to build a brick wall out of bricks that have no thickness. No matter how many you stack, you’ll still have nothing.

On the other hand, if the line isn’t made of points, what is it made of? If you keep cutting it into smaller and smaller pieces, do you eventually reach some ultimate tiny piece that can’t be cut further? If so, that piece would have to have some size—otherwise you’re back to the first problem. But if it does have size, why can’t you cut it further? What makes it special?

This is what philosophers call the problem of the continuum. A continuum is something continuous—unbroken, smooth, without gaps. Space and time seem to be continua. Motion seems to be continuous. But when you try to think carefully about what a continuum is made of, you run into puzzles that have never been fully resolved.

Two Ancient Answers

The ancient Greeks fought over this question. On one side were the atomists, like Democritus (who lived around 450 BCE). He said: look, if you keep dividing something, eventually you have to reach a point where you can’t divide anymore. Those ultimate pieces are atoms—indivisible chunks of matter. Everything is built from them. The continuous, on this view, is just an illusion. Really, the world is made of tiny discrete bits.

On the other side was Aristotle (384–322 BCE). He said no: a continuum isn’t made of indivisible bits at all. When you divide a line, you never reach a smallest piece. You can always divide again. But—and this is crucial—Aristotle said you can’t actually do all those divisions. You can only do them potentially. A line isn’t secretly made of an infinite number of points that are really there. Rather, you can make points appear by dividing, but they weren’t there before you divided. The line is a genuine whole, not an assembly of parts.

This is a subtle idea. Think of it this way: a piece of clay is one lump. You can cut it in half, and now you have two lumps. But the two halves weren’t really there before you cut—they were just potential. For Aristotle, that’s how continua work. The points on a line are like the halves of the clay: they come into existence when you make the cut, not before.

The atomists didn’t buy this. They thought the line really is made of points, even if you can’t see them. For them, the world is fundamentally discrete, and continuity is just how it looks from far away.

The Strange Case of Infinitesimals

Now we come to something even weirder: infinitesimals.

An infinitesimal is a quantity that’s smaller than any ordinary number but not zero. If you had an infinitesimal, you could add it to itself a million times and still not get a number as big as 1. In fact, you could add it to itself as many times as you wanted, and you’d never reach any ordinary number. That’s how tiny it is.

Why would anyone believe in such a strange thing? Because it turned out to be incredibly useful.

In the 1600s and 1700s, mathematicians like Gottfried Leibniz (1646–1716) and Isaac Newton (1642–1727) invented calculus—the mathematics of change and motion. Calculus lets you figure out the slope of a curve at a single point, or the area under a curve, or how fast something is changing at an exact instant. To do this, they used infinitesimals. The basic idea was: to find the slope of a curve at a point, you look at a tiny piece of the curve—so tiny it’s basically a straight line. That tiny piece is an infinitesimal. You do your calculations with it, and then at the end you just ignore it (since it’s so small it doesn’t matter anyway).

Leibniz said a curve could be thought of as an “infinilateral polygon”—a shape with an infinite number of infinitely short straight sides. This is like saying a circle is a polygon with so many sides that each side is infinitesimally small. It’s a beautiful idea, but is it actually true? Can you really treat a smooth curve as made of straight bits?

The philosopher George Berkeley (1685–1753) didn’t think so. He mocked infinitesimals as “ghosts of departed quantities.” His point was: first you add an infinitesimal to do your calculation, then you throw it away to get your answer. But you can’t have it both ways. If the infinitesimal is something, you can’t just ignore it. If it’s nothing, you can’t use it to calculate. Berkeley thought the calculus was basically a cheat—it got the right answers, but its foundations were nonsense.

For a long time, most mathematicians agreed with Berkeley. In the 1800s, mathematicians like Karl Weierstrass (1815–1897) rebuilt calculus on a new foundation—the limit concept—that didn’t use infinitesimals at all. Instead of talking about infinitely small quantities, they talked about making things “as small as you like.” This put calculus on solid logical ground. Infinitesimals were thrown out. They were considered a mistake, a confused idea from the past.

But Then Something Surprising Happened

In the 1960s, a mathematician named Abraham Robinson (1918–1974) figured out how to put infinitesimals on a completely rigorous foundation. His system, called nonstandard analysis, showed that you could have numbers that are smaller than every positive ordinary number but aren’t zero. They behave just like ordinary numbers in most ways—you can add them, multiply them, divide them (as long as you don’t divide by zero). The trick is that you have to work in an “enlarged” number system that includes both ordinary numbers and these new “hyperreal” numbers.

This was a shock to many mathematicians. Infinitesimals weren’t nonsense after all. They just required a different way of thinking about what numbers are.

Around the same time, in the 1970s, a mathematician named F. W. Lawvere developed another approach, called smooth infinitesimal analysis. This one is even weirder. In this system, there are infinitesimals whose square is zero—they’re called nilpotent infinitesimals. (A number is “nilpotent” if some power of it equals zero.) So you have numbers ε such that ε ≠ 0 but ε² = 0. This sounds impossible, and in ordinary mathematics it is. But in smooth infinitesimal analysis, it works.

Here’s why it’s useful. Remember the calculus problem of finding the slope of a curve? If you have an infinitesimal ε where ε² = 0, then when you calculate f(x + ε) - f(x), all the higher-order terms (the ones with ε², ε³, etc.) automatically vanish. You’re left with exactly the slope. This makes calculus incredibly simple—it becomes pure algebra, no limits needed. Every function is smooth (infinitely differentiable), and the derivative is just a number you can compute directly.

But here’s the catch: smooth infinitesimal analysis doesn’t work with ordinary logic. It requires a different logic—intuitionistic logic—where the law of excluded middle (the principle that every statement is either true or false) doesn’t hold. Why? Because if you had that law, you could prove that all infinitesimals equal zero, which would defeat the whole purpose.

What Does This Mean?

We now have at least three different pictures of the continuum:

The classical picture (from Weierstrass, Dedekind, Cantor): The continuum is made of points—real numbers—and there are no infinitesimals. The calculus works through limits. This is what most mathematicians and scientists use today.
The nonstandard picture (from Robinson): The continuum is made of points, but there are also infinitesimal numbers living among them. The calculus can be done with infinitesimals, as Leibniz imagined.
The smooth picture (from Lawvere): The continuum is not made of points at all in the usual sense. It has a “syrupy” structure where tiny regions behave like straight lines. Infinitesimals exist, but they’re nilpotent—their squares vanish.

Which one is right? Nobody really knows. Philosophers still argue about this.

Part of the trouble is that the continuum is something we experience directly—the flow of time, the smoothness of motion—but when we try to capture it with mathematics, we seem to have choices. Different mathematical systems give different answers. The question becomes: which mathematical system best describes reality? Or is reality itself flexible enough to accommodate multiple descriptions?

Why Should You Care?

Here’s something to think about. We live in a world where things seem continuous. Time flows smoothly. You don’t jump from one moment to the next—or do you? Some physicists think that at the smallest scales, space and time might be discrete, like pixels on a screen. If they’re right, then the atomists might have been correct after all: the world is made of tiny bits, and continuity is just how it looks from our large perspective.

But other physicists think space and time are genuinely continuous. And philosophers continue to debate whether the mind, or consciousness, is something continuous or discrete.

So the question isn’t just an abstract puzzle. It gets at something fundamental: what kind of world is this? Is it smooth all the way down, or does it come in chunks?

And for you, personally: next time you draw a line, or watch a video, or feel time passing, you might wonder—what’s really going on underneath? Are you experiencing something continuous? Or is it all just very, very many tiny bits, arranged so close together that they fool you?

Appendices

Key Terms

Term	What it does in this debate
Continuum	A smooth, unbroken whole (like a line, time, or space) that can’t be reduced to separate parts
Discrete	Made of separate, individual units (like numbers on a list or pebbles on a beach)
Infinitesimal	A quantity smaller than any ordinary number but not zero
Atomism	The view that everything is made of indivisible bits
Synechism	The view that continuity is fundamental and can’t be reduced to bits
Nilpotent infinitesimal	An infinitesimal whose square (or some power) equals zero—useful for making calculus simple
Limit	The value a quantity approaches as you get arbitrarily close to something; used to avoid infinitesimals in standard calculus
Intuitionistic logic	A type of logic where the law of excluded middle doesn’t hold—needed for smooth infinitesimal analysis to work

Key People

Democritus (c. 460–370 BCE): Ancient Greek philosopher who argued that everything is made of atoms—tiny, indivisible bits. An early atomist.
Aristotle (384–322 BCE): Argued that continua are genuine wholes, not assemblies of parts. His view dominated for centuries.
Gottfried Leibniz (1646–1716): Co-inventor of calculus; believed in infinitesimals and thought curves could be treated as having infinitely many tiny straight sides.
George Berkeley (1685–1753): Philosopher and bishop who mocked infinitesimals as “ghosts of departed quantities” and argued that calculus had shaky logical foundations.
Karl Weierstrass (1815–1897): Mathematician who rebuilt calculus on the concept of limits, banishing infinitesimals from mainstream mathematics.
Abraham Robinson (1918–1974): Created nonstandard analysis, which put infinitesimals on a rigorous mathematical foundation for the first time.
F. W. Lawvere (born 1937): Developed smooth infinitesimal analysis, where nilpotent infinitesimals exist and all functions are smooth.

Things to Think About

If you think time is continuous, then between any two moments there’s another moment. But does that mean there are infinitely many moments in a single second? If so, how do you get from one moment to the next if there’s always another between them? Does anything actually move?
The smooth infinitesimal approach says every curve is “infinitesimally straight”—that at any point, if you zoom in close enough, a curve becomes a straight line. But is this really true of actual curves in the world? Or is it just a convenient assumption we make for doing math?
Here’s a puzzle: if you have a nilpotent infinitesimal ε where ε² = 0, then (ε + ε)² = 4ε² = 0 as well. So adding two infinitesimals gives another infinitesimal. But can you add infinitely many of them? What happens if you try? Does the continuum start to act differently when you “sum up” infinite collections of infinitesimals?

Where This Shows Up

Physics: String theory and quantum gravity theories debate whether space-time is continuous or discrete at the smallest scales. Some theories suggest the universe has a “pixel size”—a smallest possible distance.
Computer science: Digital computers work with discrete bits (0s and 1s). But when you watch a video, the smooth motion is an illusion created by very fast discrete frames. The gap between continuous experience and discrete reality is everywhere in technology.
Your everyday experience: When you think about “how things change”—the way water flows, the way light fades at sunset, the way your mood shifts—you’re thinking about continuity. Are these changes smooth, or do they happen in tiny jumps that are too small to notice?