Where Do Numbers Live? Medieval Muslim Philosophers and the Mystery of Mathematical Objects
Imagine a triangle. Not a drawing of one on paper, which has slightly wobbly edges and ink that spreads, but a perfect triangle—absolutely straight, exactly 180 degrees inside, existing nowhere in the physical world. Where is that triangle? Where does it live? Can you touch it? Can you find it somewhere, like you can find a rock or a cup?
This is the kind of question that medieval Muslim philosophers argued about for centuries. They noticed something strange about math: it seems to be about things that are perfect and exact, but the physical world is messy and approximate. No circle you draw is really a circle—its circumference is always a little bumpy. No pair of things you count is just the number two, separate from the two actual things. So what are numbers and shapes, really? Do they exist somewhere beyond the physical world, or are they just ideas in our heads? And if they’re just ideas, how can they tell us anything true about the real world?
These questions are still alive today. And the answers that Muslim thinkers came up with between the 9th and 15th centuries are surprisingly bold and sophisticated.
What Numbers and Shapes Are Not
Before you can say what something is, it often helps to say what it isn’t. And the first thing many Muslim philosophers wanted to argue against was a view that came from the ancient Greek philosopher Plato.
Plato thought that mathematical objects—numbers, triangles, circles—were like ghosts. They existed in a separate, invisible world of perfect forms, completely detached from matter. They were “separate” from physical things, and they were also the “principles” or causes of physical things. The number four, in this view, causes groups of four things to exist. A perfect triangle causes all triangular things to be triangular.
A philosopher named Avicenna (who lived around 1000 CE, in what is now Iran) thought this was completely wrong. And he had sharp arguments for why.
One argument went like this: Suppose someone says that mathematical objects must be separate from matter because you can think about a triangle without thinking about any particular material thing—you don’t need to imagine it made of wood or metal. Avicenna’s response was clever. He said there’s a difference between thinking about something without the condition of materiality (like, “I’m not specifying what it’s made of”) and thinking about it with the condition of immateriality (like, “it’s definitely not made of anything physical”). Just because you can think about a triangle without specifying its matter doesn’t mean it actually exists without matter. It could just be that the matter is unspecified, not absent.
Another argument was even more direct. Avicenna said: we get our ideas of mathematical objects from the physical world. We see triangular things, we see groups of things, and from those experiences our minds form concepts. If mathematical objects existed in a completely separate, non-physical realm, how would we ever know about them? We’d have no way to contact that realm. So the simplest explanation is that mathematical objects are not separate—they’re features of physical things, which our minds can abstract and think about.
This was a devastating critique. After Avicenna, almost no Muslim philosopher could hold Plato’s view anymore. The idea that numbers and shapes live in a separate perfect world was out. But that left a big question: if they don’t live there, where do they live?
The Middle Ground: Abstraction
Another philosopher, al-Fārābī, proposed what became the standard view. He thought mathematical objects occupy a middle position. They’re not purely physical (like rocks), and they’re not purely non-physical (like the ideas Plato imagined). Instead, they exist in physical things but can be pulled out by the mind through a process called abstraction.
Here’s how it works. You see an apple. Then you see another apple. The apples are different—one is red, one is green, one is big, one is small. But you can, in your mind, ignore all those differences and just focus on the fact that there are two of them. That twoness is real—it’s a feature of the actual apples in the world—but it’s not a separate thing floating around. It’s a property of the apples, which your mind has “abstracted” from its particular details.
Avicenna developed this idea further. He said that when we use our faculty of estimation (a kind of inner sense that picks up on features that aren’t directly visible—like the friendliness in someone’s face, or the danger in a situation), we can perceive mathematical properties in physical things. You look at two books on a table, and your estimation perceives the twoness of them.
But here’s where it gets tricky. Avicenna believed that mathematical objects in the mind are abstracted from specific kinds of matter (like wood, metal, or flesh) but not from materiality itself. You can think of a triangle without thinking of it as wooden or metal, but you can’t think of it without thinking of it as having some kind of extension in space. It’s still material—just not this material.
Some later philosophers thought this didn’t go far enough. A thinker named Suhrawardī pointed out a problem: suppose you have four people standing together. Avicenna would say that the fourness is an accident of those four people—a property they have as a group. But Suhrawardī asked: where exactly is that property? Is it in each person? No—each person is just one. Is it spread across them? But then there’s nothing that binds them together into a unit except your mind. So numbers must be mind-dependent, not real features of the world.
This argument—which is similar to one made centuries later by the European philosopher Frege—pushed many Muslim thinkers toward the view that mathematical objects are purely mental constructions.
What About Infinity?
If numbers and shapes are properties of physical things, then what about infinity? Can there be an infinite number of things? An infinitely long line? These questions were debated fiercely, using some beautiful geometrical arguments.
One argument, called “the Ladder Argument,” goes like this. Imagine two infinite lines that meet at a point, making a very narrow angle. Now imagine a series of lines crossing between them, all parallel to each other, each a fixed distance apart. The further you go from the meeting point, the longer each crossing line is. But if the original lines are truly infinite, then the crossing lines would have to get infinitely long. However, each crossing line is bounded by the two original lines—it has a beginning and an end. So it would have to be both finite (because it ends at the two lines) and infinite (because it can be made arbitrarily long). This is a contradiction. Therefore, the assumption that infinite lines exist must be false.
Another argument, the “Mapping Argument,” uses a different trick. Take an infinite line. Remove a finite piece from one end. Now compare what’s left to the original line. If you line them up, they seem to correspond—for every point on the shorter line, there’s a matching point on the longer line. But this would mean the whole is equal to its part, which is absurd. So infinite lines can’t exist.
These arguments were meant to show that the physical world can’t contain actual infinities. For philosophers who thought mathematical objects were properties of physical things, this meant that mathematics itself couldn’t involve actual infinities—no infinite numbers, no infinite lines. But for philosophers who thought mathematical objects were purely mental, the limitation didn’t apply. Your mind can think about infinity even if nothing in the real world is infinite.
Can We Trust Math?
This leads to a deeper worry. If mathematical objects are just figments of our imagination—like unicorns or two-headed men—then how can math tell us anything true about the real world? When an engineer uses geometry to design a bridge, she’s using shapes that don’t actually exist in nature. Why does the bridge stand up?
Some philosophers took this worry very seriously. Al-ʾĪjī, a 14th-century thinker, said that mathematical objects are “frailer than a spider’s web”—they’re just mental fictions. If so, why trust them? A philosopher named al-Jurjānī tried to rescue mathematics by developing a theory of nafs al-ʾamr, which roughly means “the thing itself.” He argued that even though mathematical objects are mental constructions, they’re not random fantasies. They’re constructed in accordance with what we perceive from the real world, and they reflect something true about reality. The mind’s estimation can produce images that match the way things actually are, even though the images themselves aren’t physical.
It’s a delicate position. Mathematical objects are made up by the mind—but the mind makes them up correctly, in a way that tracks something real. This is still a live philosophical problem today.
What’s at Stake
The debate these philosophers were having is not ancient history. It connects directly to questions we still argue about. When a physicist says the universe is described by equations, where do those equations live? Are they discovered or invented? When a computer scientist works with algorithms, is she manipulating real objects or just symbols?
The Muslim philosophers offered a rich set of options: Platonic objects in a separate world, real properties of physical things, mental constructions that mirror reality, or useful fictions. They didn’t settle the debate—and we haven’t either. But they showed that the question of where numbers live is not a silly one. It’s a deep puzzle about what the world is made of and how our minds connect to it.
Key Terms
| Term | What it does in this debate |
|---|---|
| Abstraction | The mental process of pulling out mathematical properties (like twoness or triangularity) from physical things, while ignoring their specific material details |
| Estimation | An inner sense that perceives non-physical properties (like mathematical ones) in physical objects |
| Literalism | The view that mathematical objects literally exist in the physical world as properties of things |
| Mental construction | The view that mathematical objects are created by the mind and don’t exist outside of thought |
| Nafs al-amr | A theory that says mental mathematical objects can still be correct about reality, because the mind constructs them in the right way |
| Separateness | The Platonic idea that mathematical objects exist in a completely non-physical realm, independent of matter |
| Principles | The idea that mathematical objects are the causes or foundations of physical things—that numbers, for example, cause groups to exist |
Key People
- Avicenna (Ibn Sīnā) — A Persian philosopher and doctor (980–1037 CE) who systematically argued against Plato’s view that numbers and shapes exist in a separate world, and developed the abstractionist view instead.
- Al-Fārābī — A philosopher (died 950 CE) who first clearly laid out the idea that mathematical objects are abstracted from matter in the mind but not separate from it in reality.
- Suhrawardī — A philosopher (died 1191 CE) who argued that numbers can’t be real properties of physical things because nothing in the physical world binds a group of things together into a unit.
- Al-Jurjānī — A 14th-century thinker who tried to rescue the truth of mathematics by arguing that even though mathematical objects are mental, the mind constructs them correctly to match reality.
Things to Think About
- If mathematical objects are just mental constructions, can two people ever disagree about whether something is mathematically true? What would they be disagreeing about?
- The Mapping Argument says an infinite line would have to be equal to part of itself. But is that really impossible, or could we just say infinity is strange and normal rules don’t apply?
- When you imagine a perfect circle in your head, is that circle really there in some sense? And if it’s not, how can you learn things about real, imperfect circles by thinking about perfect ones?
- Could there be a creature that experiences the world in a way that makes no mathematical sense—where things don’t add up or shapes don’t fit? What would that mean for the question of whether math is “in the world” or “in the mind”?
Where This Shows Up
- Computer science debates about whether algorithms are discovered or invented touch the same questions these philosophers asked about numbers.
- Physics still struggles with the “unreasonable effectiveness of mathematics”—why math, which seems like a mental game, describes the physical universe so perfectly.
- Arguments about artificial intelligence often turn on whether a computer can really “think” about mathematical objects or just manipulate symbols without understanding.
- Education—when you learn that 2+2=4, are you learning about something real or just learning the rules of a game? Teachers often assume the first, but it’s not obvious why.