Is a number real?

There are three apples on a table. Eat one and there are two. Nobody serious argues about that.

Now take away all the apples, then all the tables, then the people counting, then the universe. Is there still a three? The apple count was obviously a description of apples. But “three” also appears in arithmetic, where it behaves like an object with properties nobody chose: two odd numbers always make an even one, and you cannot rule it otherwise by voting.

Two families of answers. Either numbers exist independently and we discover them — a position with the uncomfortable consequence that there is a realm of objects we cannot see, touch, or point at. Or numbers are things we make, tools for grouping, with the consequence that mathematics is a very successful habit rather than a discovery.

There is a third, more unsettling thought, and it is the one worth holding longest. Nobody has a good explanation for why the universe itself obeys mathematics. Equations written by people to describe falling stones turn out to describe stars before anyone looks at them. Either we invented a language that miraculously matches the world, or we found the world’s grammar. Neither answer is comfortable, and both are still on the table.

Try it

Try to name the number seven without using any example, symbol, word or gesture. Notice what happens. Then ask whether that proves anything about whether it exists.

What do you think?

Do you think there is a right answer to this question?

Yes, I think soNo — and here's whyI'm not sure yet

Talk about it

• Did we invent numbers or discover them?
• If every mind vanished, would two and two still be four?
• Why does maths describe the world so well?

Yes, but… — the arguments people actually make

• Numbers must be real, because maths works.

  So does chess, in the sense that the rules hold. Working is not the same as existing somewhere. The question is what kind of thing you are claiming it is.

• Numbers are just useful fictions.

  Then they would be like a made-up character, which we could change whenever it helped. Changing the rules of arithmetic does not get called invention, it gets called error.

• Who cares whether numbers exist, as long as we can add up?

  Fair. But the question turns practical the moment you ask why the universe obeys equations nobody designed. Something there needs explaining.

What other philosophers think

• PlatoNumbers exist independently of us. A perfect circle is never a drawn circle — and yet we can all think about one, which he took as evidence of a world of forms.
• AristotleNumbers are abstractions from things. Take away countable objects and there is nothing left over to be a number.
• Gottlob FregeTried to show that arithmetic is derived from pure logic, so numbers are objects of reason rather than reports about apples.
• GalileoSaid the book of nature is written in the language of mathematics — a claim about the world, not just about our counting.

If you want to go further

• The Foundations of Arithmetic — Gottlob Frege1884. The attempt to ground arithmetic in logic, and it is where modern philosophy of maths starts.
• Metaphysics — AristotleBook M, his case against numbers as separate existing things.

Most of these were written for adults, which is why it is worth reading them with someone older.