You Just Said 'Some Numbers Are Even.' Does That Mean Numbers Exist?

The Monster Under the Bed (or Not)

Suppose you tell a friend, “There’s a monster under my bed.” Your friend thinks you’re a little silly. But imagine a philosopher overhears you and asks: “When you said that, did you just commit yourself to the existence of monsters?” That’s not a question about being brave or scared. It’s a question about what your words force you to accept as real — even if you don’t want to. Philosophers call this ontological commitment: the invisible baggage a sentence carries about what exists. For most of the last century, the most influential answer was given by W.V.O. Quine (1908–2000). He proposed a rule so simple it sounds like a trick: look at the variables.

Quine’s Simple Rule: Follow the Variable

Quine’s big idea was that everyday sentences can be rewritten in a precise logical language. In that language, a statement like “Some dinosaurs had feathers” becomes: There exists an x such that x is a dinosaur and x had feathers. The symbol ∃x (pronounced “there exists an x”) is the existential quantifier. The x after it is a bound variable — it stands for the things the sentence claims are out there. According to Quine, a whole theory is committed to whatever sorts of things those variables must range over for the theory to be true. If the sentence can’t be true unless some real feathered dinosaur is among the values of x, then the sentence is ontologically committed to feathered dinosaurs. Simple, right? You can test it with almost anything: “Some dogs are black” commits you to black dogs. The quantifier and the variable, not the noun “dinosaur” by itself, do the work. Quine even got rid of proper names like “Socrates,” translating them into special predicates (like “is-identical-with-Socrates”) so that all commitment flowed through the variables. The rule seemed crisp and fair, a perfect tool for deciding which theories are truly cheaper — which ones smuggle in fewer hidden entities.

When Nothing Exists: The Centaur Problem

But the clean rule hit a snag almost immediately. Imagine you say, “Centaurs exist.” That’s false — there are no centaurs. According to Quine’s earliest extensional version, a true existential sentence commits you to the smallest kind that includes all the things that satisfy the predicate. For a false sentence like “Centaurs exist,” though, there are no such things, so the sentence is committed to nothing at all — or, on a different reading, vacuously committed to every kind. Neither answer works. “Centaurs exist” obviously does commit you to something, and it certainly doesn’t commit you to everything from elephants to prime numbers. Quine himself suggested a retreat: talk about the predicate centaur rather than actual centaurs. On this metalinguistic version, a sentence commits you to centaurs if it logically entails “∃x Centaur(x),” regardless of whether any centaurs really exist. That way you can say the false sentence is committed to centaurs without believing in them. The trouble is, commitment becomes a purely formal game about words, disconnected from the world. Real commitment should have something to do with what the world must be like for the sentence to be true. So later philosophers took the word “must” seriously: a sentence commits you to centaurs if, in every possible world where the sentence is true, there really are centaurs. That handles false sentences beautifully, because it doesn’t matter that our world has no centaurs; the sentence still demands centaurs in whatever worlds make it true.

Hidden Commitments: Parents, Children, and Sets

Even if the centaur problem is fixed, another challenge lurks. Some commitments don’t show up among the variables at all. Take the sentence “There are parents.” Quine’s rule looks only at what the variable x must range over; it notices that x needs to range over parents, but it doesn’t demand children, because no child need be assigned to x for the sentence to be true. Yet the very concept of a parent is tied to the concept of a child: you can’t have a parent without someone’s being a child. So intuitively, “There are parents” commits you to children as well. Philosophers call this an implicit ontological commitment — a commitment the sentence doesn’t wear on its sleeve. The same thing happens with sets. If you say, “The set containing a and b exists,” you seem to be committed to a and b themselves, even though the bound variable only ranges over sets. Quine’s original rule, looking only at values of variables, misses these hidden passengers. Any account that wants to capture what we usually mean by ontological commitment has to handle them.

Maybe It’s About What Has to Exist

Because of cases like parents and sets, many philosophers moved to a different strategy: an entailment account. The idea is simple. Instead of asking only what the variables range over, ask: What must exist for the whole sentence to be true? If “There are parents” entails “There are children,” then the sentence is committed to children, full stop — whether or not a child is a value of the original variable. This captures implicit commitments neatly and treats all parts of a sentence as fair game. The cost, though, is that ontological commitment stops being a simple surface test. Two people might agree on the words “There are parents” but disagree about what that sentence entails — does it really entail that children exist, or just that something stands in the parent relation? The test becomes entangled with deep questions about meaning and necessity. Still, many think this gets closer to the heart of the matter: ontological commitment is about what the world would have to contain if our theories were true, not just about which symbols we happen to write down.

Forks, Knives, and Numbers: Why This Matters

So why should you care about all this? Because the same puzzle shows up in ordinary talk. Suppose you set the table and say, “There are exactly as many forks as knives.” You haven’t said the word “number.” Yet some philosophers argue that your sentence is really equivalent to “The number of forks equals the number of knives” — and that, they claim, commits you to numbers, abstract entities that a lot of people find spooky. Quine himself thought we could sometimes avoid unwanted commitments by paraphrasing: finding a different way to say the same thing that doesn’t quantify over the controversial things. For example, if you can rewrite talk about holes entirely in terms of perforated cheese, you might escape commitment to holes. But can we always find acceptable paraphrases? And when we use mathematics in science, are we forced to accept that numbers really exist? The debate over ontological commitment gives us tools to ask these questions clearly. It turns everyday sentences into a kind of philosophical x-ray, revealing what you’re really buying into when you open your mouth. And it connects straight back to your monster problem: when you tell your friend “There’s a monster under my bed,” are you merely playing with words, or have you let a new entity into your world?

Think about it

1. If a friend says “Sherlock Holmes is smarter than any real detective,” do you think she is committed to believing there really is a Sherlock Holmes? Why or why not?
2. A scientist writes, “There are energy levels in this atom.” Imagine scientists in the future abandon the idea of energy levels and describe the same facts with just particles moving around. Did the first scientist really believe in energy levels, or was she just using a convenient way to talk? How could you tell?
3. Can you describe your classroom without using any number words at all — not even “one,” “many,” or “the same as”? If you managed to do that, would you have avoided any commitment to numbers, or would numbers still be lurking in the patterns you describe?