What Are Abstract Objects? (And Do They Even Exist?)

Here’s a strange thing philosophers noticed: you can say “The number 3 is odd” and that sentence seems to be about something called the number 3. You can say “The color red is my favorite” and that seems to be about something called red. You can say “Justice requires fairness” and that seems to be about something called justice. But where are these things? What are they?

Numbers aren’t rocks. Red isn’t a person. Justice isn’t a tree. You can’t trip over the number 7. You can’t stub your toe on the color blue. You can’t drop justice on your foot. Yet we talk about them all the time. We argue about whether math homework is correct. We disagree about what justice demands. We have favorite colors. Whatever these things are, they seem to matter. But they also seem… weird.

Philosophers have a name for these weird things: abstract objects. The alternative—ordinary stuff like tables, trees, and your best friend—are concrete objects. Almost everyone agrees that concrete objects exist. The big disagreement is about abstract objects. Do they really exist? If so, what are they like? If not, what are we talking about?

Two Big Camps: Platonists and Nominalists

Philosophers divide into two groups on this question.

Platonists (named after the ancient Greek philosopher Plato, but don’t get too hung up on that) say yes: abstract objects are real. They exist just as much as rocks and trees do, but they’re a different kind of thing. They don’t exist in space or time. They don’t cause anything to happen. The number 7 didn’t come into existence when someone thought it up. It was always there, waiting to be discovered.

Nominalists say no: abstract objects don’t exist at all. When we talk about “the number 7” or “justice,” we’re not really naming some mysterious invisible thing. We’re doing something else with our words—maybe we’re talking about patterns in the world, or maybe we’re just using useful fictions. But there’s no spooky non-physical thing out there.

This might sound like a silly argument at first. Who cares whether numbers “really exist”? But here’s why it matters: if nominalists are right, then huge chunks of what we say—in math, in science, in everyday conversation—are literally false, or at least not saying what they seem to say. And if platonists are right, then we have to explain how we could possibly know about these weird objects that don’t exist anywhere or cause anything.

What Makes Something Abstract?

So what exactly is an abstract object? Philosophers have offered several different answers, and none of them is completely satisfying on its own. Here are the main ideas.

The Negative Way

The most common approach is to say what abstract objects aren’t. According to this view:

An object is abstract if it’s non-spatial (doesn’t take up space) and causally inefficacious (doesn’t cause anything to happen).

This works for most of the usual examples. Numbers don’t take up space. The Pythagorean Theorem doesn’t cause anything. The concept of democracy doesn’t knock things over. But there are tricky cases. Consider a set like {You, Your Best Friend}. This is a set—usually considered abstract—but it seems like it might be located wherever you and your friend are. And novels are usually considered abstract objects, but novels were created by authors at particular times. A novel seems to be an effect (someone wrote it), which means it has some kind of causal history. So is it abstract or not?

This part gets complicated, but here’s what it accomplishes: the “negative way” gives us a useful rule of thumb, even if it doesn’t perfectly classify every case. Most philosophers agree that abstract objects are at least typically non-spatial and non-causal, even if there are borderline cases.

The Way of Abstraction

Here’s another approach, with a long history. Imagine you have several white objects: a white ball, a white book, a white wall. If you ignore everything about them except their color—if you abstract away from their shape, size, and what they’re made of—you can think about “whiteness” itself. According to this view, abstract objects are what you get when you perform this mental process of abstraction.

This idea goes back centuries, but modern philosophers have developed it in a more rigorous way. They notice that we often form names for abstract objects using patterns like this:

The direction of line A = the direction of line B if and only if A is parallel to B.
The number of apples = the number of oranges if and only if there are just as many apples as oranges.

These “abstraction principles” seem to define what directions and numbers are. If you understand when two lines have the same direction, you understand what a direction is. If you understand when two groups have the same number, you understand what a number is.

The problem is that not all abstract objects fit this pattern. What about the game of chess? What about the English language? These don’t seem to be defined by any simple equivalence relation. So the “way of abstraction” might cover numbers and directions but miss other important cases.

The Way of Encoding

A philosopher named Edward Zalta developed one of the most complete theories of abstract objects. His key idea is a distinction between two ways an object can have a property:

Exemplifying a property is the normal way. A red ball exemplifies redness—it’s actually red. You can see it, touch it, maybe throw it.

Encoding a property is different. An abstract object encodes the properties that define it or that we use to think about it. The abstract object “the number 1” encodes the property of being odd (if that’s true in number theory), but it doesn’t exemplify being odd. Numbers aren’t the kind of thing that can be odd or even in the same way a number of socks can be odd.

This distinction lets Zalta do something clever: he can say that abstract objects exist without them being weird “ghostly” versions of concrete objects. They encode properties rather than exemplifying them, which means they don’t need to be in space or time or have causal powers.

According to Zalta, for any way of describing something, there’s an abstract object that encodes exactly those properties. This includes even contradictory descriptions like “the round square”—there’s an abstract object that encodes the properties of being round and being square, even though nothing could actually be both round and square. (This doesn’t cause a contradiction because encoding is different from exemplifying.)

Can We Know About Abstract Objects?

If platonists are right and abstract objects exist, a big problem arises: how could we possibly know anything about them?

Think about how you know ordinary things. You know there’s a chair in the room because you can see it, touch it, or remember sitting on it. All these ways of knowing involve causal contact—the chair affects your senses. But if abstract objects don’t cause anything, they can’t affect your senses. So how could you ever learn about the number 7?

This is sometimes called the “epistemological problem” for platonism. It feels like a real puzzle. Philosophers have offered various solutions:

Maybe we have a special “mathematical intuition” that lets us directly grasp abstract objects (like how we see colors, but with our minds instead of our eyes).
Maybe we don’t need causal contact to know about something. After all, you can know about dinosaurs without ever seeing one, through descriptions and inferences.
Maybe abstract objects aren’t as disconnected from the physical world as we thought. Numbers might just be patterns we abstract from concrete experience.

None of these solutions is universally accepted. The debate continues.

The Way of Weakening Existence

Some philosophers try to sidestep the whole debate. They suggest that “existence” might mean different things for different kinds of objects, and that abstract objects exist only in a “thin” or “deflated” sense.

The philosopher Rudolf Carnap argued that the question “Do numbers really exist?” is a pseudo-question—it sounds meaningful but actually isn’t. According to Carnap, you first have to choose a “linguistic framework” (a way of talking) before you can ask whether something exists within that framework. Within the framework of mathematics, numbers obviously exist: the framework says so. But asking whether numbers “really exist” outside any framework is like asking whether the rules of chess are “really true”—it misses the point. What matters is whether the framework is useful, not whether it matches some mysterious “real” reality.

More recently, some philosophers have developed the idea of “thin objects”—objects that exist, but whose existence doesn’t require much from the world. A direction exists if there are parallel lines. A number exists if there are groups that can be matched up. These objects are “shallow” in the sense that any question about them can be answered just by looking at what they’re based on. If you know the lines, you know everything about their directions. If you know the groups, you know everything about the numbers.

Other objects are “thick.” A physical object like a statue has properties (like weight or color) that depend on more than just its parts. You can’t know everything about a statue just by knowing about the clay it’s made of. So concrete objects are thick; abstract objects (at least the mathematical ones) are thin.

This approach is promising, but it hasn’t fully solved the puzzles about mixed cases like impure sets, fictional characters, and institutions.

So What’s the Answer?

Nobody really knows. Philosophers still argue about this. The debate between platonists and nominalists has been going on for over a century, and neither side has won decisively.

What most people agree on is that the abstract/concrete distinction matters. Our best scientific theories are full of mathematics, which seems to refer to abstract objects. Our everyday language is full of talk about concepts, properties, and types. Understanding what these things are—or what we’re doing when we talk about them—touches on fundamental questions about reality, knowledge, and language.

The next time you do math homework, say “justice requires fairness,” or argue about whether a color is “really” blue, you might pause and wonder: what exactly am I talking about? And does it actually exist?

Philosophers have been wondering about this for a long time. You’re now part of the conversation.

Appendices

Key Terms

Term	What it does in this debate
Abstract object	An entity that doesn’t exist in space or time and doesn’t cause anything to happen (like numbers, concepts, properties)
Concrete object	An ordinary physical thing that exists in space and time and can cause and be affected by events (like rocks, trees, people)
Platonism	The view that abstract objects really exist, independently of human minds
Nominalism	The view that abstract objects do not exist at all
Abstraction principle	A definition that tells us when two things produce the same abstract object (like “the direction of line A = the direction of line B if and only if A is parallel to B”)
Exemplifying	The normal way an object has a property—by actually being that way (like a red ball being red)
Encoding	A special way abstract objects “have” properties that define them, without actually being those things (like the number 1 “having” mathematical properties)
Thin object	An object whose existence doesn’t make big demands on reality—it’s basically just a reconceptualization of things we already have
Epistemological problem	The puzzle of how we could know about abstract objects if they don’t cause anything and we can’t interact with them

Key People

Plato (c. 428–348 BCE): Ancient Greek philosopher whose theory of “Forms”—perfect, eternal, non-physical entities that ordinary things participate in—is the ancestor of modern platonism. He thought the real world was the world of Forms, not the physical world.
Gottlob Frege (1848–1925): German mathematician and philosopher who argued that numbers are neither physical things nor ideas in anyone’s mind, but belong to a “third realm.” His work kicked off the modern debate about abstract objects.
W. V. O. Quine (1908–2000): American philosopher who argued that we should believe in whatever entities our best scientific theories require. Since science can’t do without mathematics, we should believe in mathematical objects.
Rudolf Carnap (1891–1970): German-born philosopher who argued that questions about the “real” existence of abstract objects are meaningless pseudo-questions. What matters is whether a “linguistic framework” that includes them is useful.
Edward Zalta (born 1952): Contemporary American philosopher who developed a complete formal theory of abstract objects based on the distinction between exemplifying and encoding properties.

Things to Think About

If you believe numbers exist, do you also have to believe that fictional characters like Sherlock Holmes exist? If not, what’s the difference between them? Both seem to be non-physical. Both seem to be talked about as if they’re real. So what makes one a “real” abstract object and the other not?
Carnap says that asking whether numbers “really exist” is a pseudo-question—what matters is whether it’s useful to talk as if they exist. But what if someone finds a mathematical theory useful for building bridges, and someone else finds a completely different and inconsistent theory useful for something else? Can both be “true” just because they’re useful? Does truth depend on what’s useful?
If abstract objects exist but don’t cause anything, how could they have been discovered? Did ancient mathematicians create numbers, or did they find them? If numbers existed before any humans existed, what were they like? Did their properties somehow not matter until someone noticed them?
Consider the game of Monopoly. It was invented at a specific time. It has rules that have changed over the years. It has physical game pieces. Is Monopoly an abstract object or a concrete one? Your answer probably depends on what you think Monopoly is—the physical board and pieces, or the set of rules and concepts. Does this tell us something about how we draw the line between abstract and concrete?

Where This Shows Up

Math class: When your teacher says “numbers are just tools we use, they don’t really exist” or “mathematical truths are discovered, not invented,” they’re taking a side in this debate.
Computer programming: Programmers regularly work with “abstract data types” and “objects” that exist as patterns of information. Are these abstract objects? What about the code itself—is it the physical storage or the logical structure?
Copyright and intellectual property: Laws protect “works of authorship” like books and songs. These seem to be abstract objects (the story itself, not the physical book). The fact that we can own, sell, and inherit things that don’t take up space is puzzling if you think about it.
Sports and games: The rules of chess or soccer exist independently of any particular game. You can talk about “the offside rule” or “checkmate” without referring to any specific moment. These are abstract objects that shape concrete events.