Is the World Smooth or Pixelated?

Can you cut a line forever?

Take a sharp pencil and draw a line. Good. Now imagine cutting that line exactly in half. Cut one of those halves again, and again. Over and over. Do you ever reach a smallest possible piece that cannot be cut? Or could you keep dividing forever — an infinite number of times — without ever touching bottom?

This question, first asked in ancient Greece, is still alive today. It’s the puzzle of the continuum — a smooth, unbroken whole, like the ocean or the sky — versus discrete things, separated from each other, like scattered pebbles or leaves on a tree. A continuum has no gaps; you can zoom in forever and it never becomes a bunch of separate dots. But is that really how the world works?

Zeno’s arrow and the atomist bet

The earliest clear fight about continuity erupted in the fifth century BCE. Parmenides and his student Zeno of Elea (c. 490–430 BCE) wanted to prove that change and motion are illusions. Zeno’s most famous paradox — the Dichotomy — argued that to move anywhere, you must first go halfway, then halfway of what’s left, and so on forever. Since you can’t complete an infinite number of tasks in a finite time, motion should be impossible. The paradox only works if you assume space and time are infinitely divisible: that you can always split a distance or a moment into smaller pieces.

A rival school, the atomists led by Leucippus (fl. 440 BCE) and Democritus (born c. 460 BCE), took the opposite wager. They claimed matter — and even extension itself — is not infinitely divisible. If you kept cutting, you’d eventually hit atoms: tiny, solid, uncuttable particles. For them, the continuous was really made of discrete bits at the bottom level. Zeno’s paradox would then dissolve, because you’d only need to cross a finite number of tiny jumps, not an infinite series.

Aristotle’s middle path: always splittable, never split

Aristotle (384–322 BCE) offered a third way. He agreed that a line, a stretch of time, or a motion is genuinely continuous — no atoms of space or matter. But he argued that the infinite division is only potential, never actual. You can slice a line at any point you choose, but you can’t slice it at every point simultaneously. The continuum stays whole because the infinite number of possible cuts never all happen at once.

Aristotle also gave a powerful argument against building a line from indivisible points. A point has no length. If you add zero-length things together — even infinitely many — you still get zero. So a line cannot be “made of” points like a necklace is made of beads. Points mark boundaries, not the stuff of the line itself. This idea would haunt debates for over two thousand years.

The calculus and the ghosts of departed quantities

In the 1600s, mathematicians needed a new tool to handle curves, speeds, and areas — the engines of the scientific revolution. Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716) independently invented the calculus, and both leaned heavily on a provocative idea: the infinitesimal.

An infinitesimal is a quantity so small that it’s smaller than any ordinary number, yet not quite zero. Newton called his version a “moment” or “evanescent quantity”; Leibniz called his a differential and treated curves as infinilateral polygons — shapes with infinitely many infinitesimally short sides. By imagining that an infinitesimal change in one quantity caused an infinitesimal change in another, they could compute slopes and areas that had been impossible to handle before.

But the logical floor was shaky. Was an infinitesimal zero, or wasn’t it? If you treat it as zero in one step and nonzero in another, you seem to be breaking the law that something can’t be both true and false at the same time. The philosopher George Berkeley (1685–1753) skewered the trick in 1734, calling infinitesimals “ghosts of departed quantities” — something that had vanished but whose shadow was still being used to prove results. Mathematicians cringed, but the calculus was too successful to abandon.

Banishing the infinitely small

During the 1800s, mathematicians decided to clean house. Augustin-Louis Cauchy (1789–1857) and Karl Weierstrass (1815–1897) rebuilt calculus without saying “infinitesimal.” Their weapon was the limit concept. Instead of talking about a number smaller than any positive number, they spoke of sequences that approach zero. Weierstrass gave the famous epsilon‑delta definition: a function f(x) is continuous at a point a if for every tiny positive number ε (no matter how small), you can choose a δ such that when x is within δ of a, f(x) stays within ε of f(a). All motion and intuition were stripped out; only numbers and precise inequalities remained.

Georg Cantor (1845–1918) went further. He defined real numbers not as geometric points but as equivalence classes of infinite sequences of rational numbers. The continuum became a set of points — discrete elements arranged densely. Infinite divisibility was now a property of an abstract number system, not a mysterious physical property. Cantor loathed infinitesimals, calling them “cholera‑bacilli” of mathematics. By 1900, most mathematicians thought the question was settled: the continuum was made of points, and infinitesimals were a mistake.

Two revivals: tiny numbers and straight micro‑worlds

But the idea of the infinitesimal refused to die. In 1960, the logician Abraham Robinson (1918–1974) created nonstandard analysis. Using mathematical logic, he built an extended number system that includes both ordinary real numbers and infinitely large and infinitely small numbers that obey all the same arithmetic rules. In this world, an infinitesimal is a number that is not zero but is smaller than 1/n for every positive integer n. Every limit argument in calculus gets a clean and intuitive translation: “close to” becomes “infinitesimally close to.” Leibniz’s dream of a rigorous algebra of infinitesimals was realized.

A few years later, the American mathematician F. W. Lawvere (b. 1937) and others developed smooth infinitesimal analysis. Here, infinitesimals aren’t numbers like Robinson’s; they are nilpotent — so small that their square is exactly zero. In this strange universe, every function is smooth (infinitely differentiable) and the continuum cannot be split into two disjoint parts. A curve is literally made of infinitesimal straight segments, and the logic needed to reason about it is a constructive, intuitionistic logic that rejects the law of excluded middle in some cases. The world of smooth infinitesimal analysis is a world where continuity is truly indecomposable — you can’t chop it apart.

Why it still matters

This isn’t just a dusty fight among mathematicians. When you play a video game, you see a world built from tiny squares and polygons — a discrete universe. But when you pour a glass of water, it looks perfectly smooth, unbroken. Is physical space itself continuous, like the water, or is it grainy at some unimaginably small scale, like the screen? Physicists still don’t know: some theories suggest that space‑time is quantized, built from tiny chunks of area and time; others keep it smooth.

The question you first met with a pencil and a line — can you cut it forever? — has towered over philosophy and science from Zeno to quantum gravity. It’s the question of whether the world is fundamentally one or many, smooth or jagged, a flow or a cascade of tiny clicks. And every time a new generation builds a sharper tool, the answer twists slightly, keeping the puzzle alive.

Think about it

1. If a video game has pixels so small you can’t see them, is there any difference between moving through that world and moving through our own?
2. Imagine a perfect circle. Can a shape made entirely of straight, tiny edges ever truly be a circle, or is there always a bit of flatness left?
3. Suppose scientists proved that space is made of indivisible chunks. Would that change how you think about walking across a room — or about free will?