Can You Ever Kick a Number? The Fight Over Abstract Things
A Number You Can’t Poke
You’re working through a math problem. You write 7 + 5 = 12. It feels solid. But if you look around, where is the number 7? You can’t trip over it. You can’t hold it in your hand or weigh it on a scale. Still, you’d never say 7 is imaginary, like a dream you forgot. Something about it seems real — just not in the way your desk or your pet hamster is real.
Philosophers call this puzzle the abstract/concrete distinction. A concrete object is something you can bump into, like a rock, a tree, or your own body. An abstract object is something like the number 7, a perfect circle, the idea of justice, or the novel Pride and Prejudice. You can’t see or touch them, yet we talk about them all the time. So, are they real? And if they are, what kind of real are they? That question has kept philosophers arguing for over a century.
Frege’s Third Realm: Not Mind, Not Matter
The modern debate kicked off in the late 1800s, thanks to a German mathematician and philosopher named Gottlob Frege (1848–1925). Frege noticed something strange about numbers. If numbers were just physical things, like pebbles, then the laws of arithmetic would be like weather reports — we’d have to check every time we used them. And if numbers were just ideas inside our heads, then your number 7 might be different from your best friend’s number 7. That would make math useless.
Frege concluded that numbers belong to a “third realm” — they aren’t physical objects and they aren’t mental objects. They’re objective, meaning the same for everyone, but they exist outside space and time. He wasn’t the first to think this way (the philosopher Bernard Bolzano had similar thoughts earlier), but his argument gave the abstract/concrete split its modern shape. Other thinkers soon applied the same logic to things like the meanings of sentences, properties, and even fictional characters.
Two Camps: Friends of Abstract Things and Their Enemies
Once the category “abstract object” was on the table, philosophers split into two main teams.
Platonism (named loosely after the ancient Greek philosopher Plato, but not exactly his view) says yes, abstract objects really exist. They are just as real as concrete things, but in a different way. So the number 7, perfect triangles, and the novel The Hobbit all exist, though you’ll never find them taking up space.
Nominalism says no. Only concrete objects exist. Numbers, properties, and stories are just convenient ways of talking — they don’t name real things in the world. A nominalist would argue that when you say “7 is prime,” you’re really talking about how we use the symbol “7,” not about some ghostly entity floating beyond space.
The standoff got especially heated after the American philosopher Willard Van Orman Quine (1908–2000) argued that we can’t do modern science without mathematics, and mathematics seems to talk about abstract objects all the time. If you want your scientific theories to be true, Quine said, you have to accept the reality of the objects they mention — including numbers and sets. This is called the indispensability argument.
Some nominalists, like Hartry Field (born 1946), fought back by trying to rewrite physics without any math that points to abstract objects. Others argued that even if our best theories use math, we don’t have to believe the abstract stuff is real — maybe it’s just a useful fiction, like the characters in a novel we know to be made up.
But How Do You Know About Something You Can’t See?
Even if you think abstract objects exist, a thorny problem pops up. We learn about concrete things through our senses — you see a tree, you touch a chilly window. But abstract objects, almost by definition, can’t cause anything. The number 7 can’t bonk you on the head. So how can our brains ever access them?
This puzzle, known as the epistemological challenge, was sharpened by philosopher Paul Benacerraf (1931–2025) in the 1970s. He pointed out that we seem to know lots about numbers — we can prove theorems, we can agree on sums — yet there’s no clear story about how we “perceive” them. Some Platonists suggested we have a kind of mathematical intuition, a special mental ability to grasp abstract truths directly. Others said that abstract objects are connected to concrete ones in a way that makes them knowable: for instance, you might come to know the direction of a line by understanding what it means for two lines to be parallel, without ever “seeing” the direction itself.
Nominalists often see this as a winning point for their side. If you can’t explain how we’d ever learn about abstract objects, maybe it’s safer to deny they exist at all.
The Great Line-Drawing Debate: What Makes Something Abstract?
Philosophers don’t just need to decide whether abstract objects exist; they also need a clear rule for telling them apart from concrete ones. The most popular idea is the way of negation: an abstract object is one that lacks certain features concrete things have.
The standard checklist says an object is abstract if it is non-spatial (it isn’t anywhere) and causally inefficacious (it can’t make anything happen). This works beautifully for pure numbers. It makes no sense to ask where the cosine function was last Tuesday, or to expect the number 3 to push a ball.
Trouble starts when you look at less pure cases. Take the set that contains only Socrates: {Socrates}. Is it abstract? It seems causally inert, but you might think it’s located wherever Socrates is — that would make it spatial. Or think of a novel. Pride and Prejudice doesn’t occupy a room, but it sure can make you laugh and cry, so it seems to have causal powers. Should we say novels aren’t abstract? Many philosophers want to count them as abstract anyway.
Other thinkers propose that an abstract object is something you arrive at by abstraction — you start with concrete things and ignore their differences. For example, you might spot many white things and arrive at “whiteness.” Some versions of this idea are mental, others are deeply mathematical, involving “abstraction principles” like: the direction of line a equals the direction of line b if and only if the lines are parallel.
Each way of drawing the line captures some clear cases but wobbles on the messy ones. This hasn’t stopped the debate; it has just reminded everyone that the border between abstract and concrete may not be a single sharp line but a zone with more than one reasonable map.
Why It Still Matters: From Math to Minecraft
You might wonder: is this just a dusty argument for professors in tweed? Not at all. Every time you trust a calculator, you’re leaning on the idea that numbers have the same meaning everywhere. When scientists use equations to predict the weather or launch a rocket, they’re treating those mathematical abstracta as part of reality. Even when you say “that’s unfair,” you’re pointing to the abstract property of justice and assuming it’s something real enough to argue about.
In your own world, the question pops up constantly. Are the rules of Minecraft real? The game’s code is concrete, but the “rule” that creepers explode — is that an abstract thing? Can you change it without destroying some real entity? And if you daydream about a character you invented, does that character exist as an abstract object in some realm, or is it just a flicker in your brain? Philosophers don’t agree, but knowing the two sides can sharpen how you think about anything invisible that still seems to matter.
The fight over abstract objects is really a fight about what it means for something to be real. And that’s not just for philosophers — it’s a question you answer, quietly, each time you trust a number, fall in love with a story, or stand up for a principle.
Think about it
- If a scientist could rewrite physics without ever mentioning numbers, would you stop believing that numbers exist? Why or why not?
- Imagine a library that contains every story that could ever be written, even ones no human has thought up yet. Do those stories exist in some way before anyone writes them down?
- You say “freedom is valuable.” Is “freedom” something real in the same way your own heartbeat is real? What would have to be different in the world for your answer to change?
