What Are Boundaries, Anyway?

A Philosophical Investigation

Here’s a strange thing to think about. You’re standing on the border between two states — say, Maryland and Pennsylvania. The Mason-Dixon line is somewhere beneath your feet. But where exactly? Is there a last point of Maryland, and then a first point of Pennsylvania? If there is a last point of Maryland, then just past it there must be a first point of Pennsylvania. But between any two points, there are always more points — infinite numbers of them. So you could never have a last Maryland point followed immediately by a first Pennsylvania point. They’d have to be right next to each other, but points can’t be “next to” each other in that way; there’s always room for more points in between.

So maybe there’s just one point that belongs to both? That can’t be right either — the two states are separate, not overlapping. Or maybe there’s no point at all that belongs to either state at the boundary line? Then what’s at the boundary? Nothing? That seems weird too.

This is the kind of puzzle that has bothered philosophers for over two thousand years. Boundaries seem like simple things — a line on a map, the surface of an apple, the moment a game ends. But when you look closely, they start to seem impossibly strange.

Where Is the Boundary?

Think about a black spot on a white piece of paper. There’s a line where the black stops and the white begins. But what color is that line itself? Is it black? Is it white? It can’t be both. It can’t be neither. Something is fishy here.

Leonardo da Vinci wondered about this with water and air. When a bird dives into the ocean, what exactly is at the surface? Is that boundary made of water or air? He thought it must be something that is neither — a “nothingness” that separates the two.

Now consider time instead of space. You’re watching a movie on a screen. The ball is moving. Suddenly, it stops. Is there an exact instant when it goes from moving to stopped? At that instant, is the ball moving or at rest? It can’t be moving (it’s stopped right after), and it can’t be at rest (it was moving right before). But there has to be some instant when the change happens. This is known as the problem of instantaneous change, and it’s related to the problem of when exactly someone dies. Are they alive at the moment of death, or dead? If alive, they haven’t died yet. If dead, they died before the moment of death. Neither option works.

These aren’t just word games. They’re pointing to something genuinely puzzling about how we think about boundaries.

Are Boundaries Real or Made Up?

So boundaries are weird. One way to deal with this is to ask: are boundaries really out there in the world, or do we just imagine them?

The American philosopher Barry Smith and others have drawn a distinction between two kinds of boundaries. Bona fide boundaries are “real” boundaries that exist in nature — the edge of a cliff, the boundary between a black spot and a white background. These are grounded in some actual physical difference between one thing and its surroundings. Fiat boundaries are boundaries we draw, either as individuals or as groups — the border between Maryland and Pennsylvania, the dividing line between the top and bottom halves of that black spot, the edge of your school district.

The Mason-Dixon line is obviously a fiat boundary. Nobody thinks there’s a natural line in the ground that God put there. But what about the surface of your desk? On closer inspection, your desk isn’t really solid all the way through. It’s made of atoms, and between those atoms there’s mostly empty space. Where exactly does the desk end and the air begin? The “surface” is really a kind of average — we connect the dots, smooth out the data, and pretend there’s a clean line. In this sense, even the boundaries of ordinary physical objects might be more like fiat boundaries than we usually think.

This matters because if boundaries are partly our invention, then the things those boundaries define might also be partly our invention. What counts as a single object — a mountain, a forest, a person, a war — depends on where we draw the lines. As the philosopher Nelson Goodman put it: “We make a star as we make a constellation, by putting its parts together and marking off its boundaries.” If that’s true, then the world doesn’t come pre-sliced into objects. We do the slicing.

But wait — does that mean the mountain isn’t real? No. As the philosopher Gottlob Frege pointed out, the North Sea is still an objective body of water, even though it’s our decision to call that particular patch of water “the North Sea” rather than including it in the Atlantic. The water is real. But its individuality — its being that particular thing, with those exact boundaries — depends on our choices.

This gets political pretty quickly. Throughout history, people have claimed that certain borders are “natural” — dictated by mountains, rivers, or even God. These claims have been used to justify wars, conquest, and terrible injustices. When we call something “natural,” we’re often really saying: “this is the way things ought to be, and anyone who disagrees is going against nature itself.” But who gets to decide what’s natural? The distinction between natural and artificial boundaries turns out to carry enormous weight.

Fuzzy Boundaries

Not everything has a sharp edge. Where exactly does Mount Everest begin? Some molecules are definitely part of the mountain and some definitely aren’t. But there are molecules in between where it’s not clear. The mountain’s boundary is fuzzy — there’s no single line that separates Everest from everything else.

The same goes for time. When did the Industrial Revolution begin? You might say 1760, but that’s just a rough marker. There was no single moment when the first factory appeared and everything before was non-industrial.

Some philosophers think this fuzziness is a fact about the world itself — the boundaries really are blurry, not just our knowledge of them. Others think the boundaries are actually sharp, but we haven’t decided exactly where to place them. “Mount Everest” might refer to any number of slightly different parcels of land, all with precise boundaries, and our language just hasn’t picked one. This is like asking: does the outback begin at this particular grain of sand or the next one? The answer isn’t that the outback has fuzzy edges. It’s that there are many possible outbacks, each with its own precise edge, and nobody has bothered to choose which one we mean.

How Many Boundaries Does It Take to Touch?

Back to the puzzle of the Maryland-Pennsylvania border. If we think boundaries are real things (lower-dimensional things, like points and lines, not chunky things), we have to decide: who owns this boundary?

Here are the main options philosophers have explored:

Option 1: Neither side owns it. The boundary belongs to neither Maryland nor Pennsylvania. They touch each other without either having a last point. This solves some problems but creates others — it means there’s a gap between the two states, which seems wrong.

Option 2: One side owns it. Maybe the boundary belongs to Maryland, and Pennsylvania just stops before reaching it. But why would Maryland get special treatment? That seems arbitrary. Unless we say it’s uncertain which side owns it, which is a bit of a cheat.

Option 3: Both sides own it. The boundary belongs to both Maryland and Pennsylvania. But then the line that separates them is also part of both of them — they overlap, which they shouldn’t. If we apply this to the black-and-white spot problem, the boundary line would be both black and white. That’s a contradiction.

Option 4: There are two boundaries, one for each side, and they just happen to be in the same place. This is like having two invisible lines sitting on top of each other, one belonging to Maryland and one to Pennsylvania. They don’t take up any space, so maybe it’s okay for them to coincide. But then what color is the line between the black spot and the white background? One boundary is black, the other is white. Together they’d be… something?

None of these options is obviously right. Philosophers still argue about which is best, and the debate connects to deeper questions about how we think about space, time, and the basic structure of reality.

What About Real Boundaries?

Maybe boundaries aren’t real at all — not in the way rocks and trees are real. Maybe they’re just useful fictions, like the lines of latitude on a globe.

One version of this idea says that talking about boundaries is just a shorthand. When we say “the surface of the table,” we don’t really mean there’s a thing called a surface. We just mean the table is surfaced in a certain way. The boundary is a property of the thing, not a thing itself.

Another version, developed by the mathematician and philosopher Alfred North Whitehead, says that boundaries are what you get when you think about an infinite series of nested things. Imagine a set of Russian dolls, one inside the other, getting smaller and smaller forever, with no smallest doll. That infinite series of nested objects defines a point at the center — but the point isn’t a thing in the world. It’s an abstraction, a way of talking about the series as a whole. In this view, all boundaries are like that. They’re not part of the furniture of the universe; they’re patterns we notice or construct.

Why Does Any of This Matter?

Boundaries are everywhere. They define countries, properties, school zones, time zones. They separate us from them, inside from outside, mine from yours. They’re central to how we organize our lives and our world.

But they’re also deeply puzzling. The more you think about them, the stranger they become. They seem indispensable — how could we think about objects without boundaries? — yet they resist easy explanation.

The philosopher Robert Musil wrote: “Ultimately a thing exists only by virtue of its boundaries, which means by a more or less hostile act against its surroundings.” Boundaries are what make things distinct, separate, individual. Without boundaries, everything would just blur together into one big everything. But with boundaries come separation, conflict, and the question of who belongs where.

Maybe that’s the deepest thing about boundaries. They’re not just an abstract philosophical puzzle. They’re how we carve up the world — literally and figuratively — and that carving is never innocent.

Appendix: Key Terms

Term	What it does in this debate
Bona fide boundary	A boundary that exists in nature, grounded in actual physical differences (like the edge of a cliff)
Fiat boundary	A boundary we draw or decide on, not grounded in physical discontinuity (like a state border)
Instantaneous change	The puzzle of what happens at the exact moment something changes from one state to another
Fuzzy boundary	A boundary that isn’t sharp — there’s no precise line where one thing ends and another begins
Coincident boundaries	The idea that two boundaries (like Maryland’s and Pennsylvania’s) could occupy the same place without being the same thing

Appendix: Key People

Aristotle — Ancient Greek philosopher who gave one of the earliest definitions of a boundary as “the first thing beyond which you can’t find any part of the thing.” He thought boundaries were real but couldn’t exist on their own.
Leonardo da Vinci — You know him as an artist, but he also thought about boundaries and concluded that the boundary between water and air must be neither water nor air — a “nothingness.”
Barry Smith — Contemporary philosopher who drew the influential distinction between bona fide and fiat boundaries, and who has written extensively on how we draw lines on maps and in our minds.
Alfred North Whitehead — Mathematician and philosopher who developed a theory that boundaries aren’t basic things at all, but emerge as abstractions from infinite series of nested objects.
Nelson Goodman — Philosopher who argued that we “make” stars and other objects by drawing boundaries around them, much as we make constellations.

Appendix: Things to Think About

If boundaries are partly our invention, does that mean two people could disagree about where something ends and neither be wrong? What would that mean for arguing about who owns what?
Think about something fuzzy — a cloud, a memory, a friendship. Does it make sense to say it has boundaries at all? If not, does that mean it isn’t really a “thing”?
When you draw a line on a piece of paper, what exactly are you drawing? The line itself has width — but the boundary between the black line and the white paper shouldn’t have width. Where is the real boundary?
The article mentions that people have used “natural boundaries” to justify terrible things. Does that mean the idea of natural boundaries is dangerous? Or can we rescue it from misuse?

Appendix: Where This Shows Up

Mapping and GPS — Every time your phone shows your location relative to a state or country border, it’s dealing with fiat boundaries that have to be precisely defined.
Law and property — Disputes about where one person’s land ends and another’s begins are disputes about boundaries. Sometimes these end up in court, with surveyors measuring inches.
Biology and medicine — When does a human life begin? When does it end? These are boundary questions that affect real decisions about abortion, euthanasia, and organ donation.
Computer graphics and vision — How do computers figure out where one object ends and another begins in a photograph? They need algorithms for detecting boundaries — and they face the same puzzles philosophers do.
Immigration and politics — The borders between countries are fiat boundaries, but they have enormous real-world consequences. Debates about who belongs where turn on the question of which boundaries matter and why.