DRMacIver's Notebook

Do numbers exist?

Do numbers exist?

In What even is a number? I finished with:

There is a more foundational question, which is whether numbers "exist" in any meaningful sense. My answer is "Almost certainly not, but if so why would we care?"

I'm going to backtrack on that a bit. Obviously numbers exist.

The problem is that this is a bad question because there are no non-misleading answers to it. Both the statements "numbers exist" and "numbers do not exist" are more likely to cause you to believe false things than true ones, and which answer you prefer says more about what you mean by "exist" than it does about numbers.

Numbers don't exist, in that there is no platonic realm where you will find the authentic number 3. At least, there is no evidence that there is such a realm, there isn't even a suggestion of what such evidence would look like, and it doesn't seem to be especially useful to postulate that one exists, so there might as well not be. Insert your own atheism analogy here. Michelangelo's Stone: an Argument against Platonism in Mathematics is a good argument against the idea that one exists, basically by pointing out that if such a thing existed it would be too large to serve the role that its proponents want it to - it is basically the infinite library of Borges, with every random string written down in a book. All published works exist there, but finding them is a harder task than creating them would be.

Additionally, they don't exist because as I argued, numbers aren't really any one thing. There is not "the number 3", there is only "the value that corresponds to the number 3 in our current implementation of natural numbers." In What Numbers Could not Be Paul Benacerraf argues quite convincingly that it is a category error to equate the number 3 with any particular instantiation of it.

So numbers really don't exist in any literal physical sense. I promise.

And yet numbers obviously exist.


Well because all of the same arguments apply to Chess, and if you try telling someone that Chess doesn't exist they'll look at you like you've sprouted an extra head.

Imagine the conversation:

You: Chess does not exist.

Them: Um, yes it does. Look, here's a Chess board.

You: Sure, that's a Chess board, but that's not Chess, that's just the symbolic representation you use to play the game of Chess.

Them: Which doesn't exist.

You: That's correct, yes.

Them: We have literally played a game of Chess on this board. How can we play Chess if it doesn't exist?

You: It's easy, you just follow the rules of Chess.

Them: Which don't exist.

You: Oh, well, there are rules of Chess of course, but those are mere matters of human convention. They don't imply any sort of abstract platonic ideal of Chess that exists somewhere.

Them: Look, here is a printed out copy of the rules of Chess.

You: I see. And?

Them: That seems to suggest the rules of chess exist.

You: Why? Those aren't the rules of chess, they're just words on a page that describe how to play chess. The rules are a purely abstract concept that exist purely as a matter of human convention.

Them: This seems like a fairly spurious distinction.

You: I suppose it does rather.

Them: Shall we settle it over a game of Chess?

You: Sure, why not.

And yet these are exactly parallel to the arguments that say numbers don't exist. People are very attached to the idea that you can apply the word "exist" to abstract objects and, while I don't want to entirely defer to common usage of a word for its more specific meaning, I think it would require an extremely convincing argument to persuade me that they are wrong to do so. Countries are also purely abstract objects designed purely as a matter of human rules and convention, but if you told me that France did not exist I could only answer "It does, though".

So, any reasonable definition of the word "exist" should incorporate abstract concepts with rules and names defined purely by human convention. Numbers are such a thing, and as purely abstract concepts it is perfectly reasonable to say that they exist. They exist in the same way and for the same reasons that France and Chess exist: Because we say so.