Mathematics

Do numbers exist?

Comments

Here's a simple argument. (1) There are four prime numbers between 10 and 20. (2) But if there are four prime numbers between 10 and 20, then there certainly are prime numbers. (3) And if there are prime numbers, then there must indeed be numbers. Hence, from (1) to (3) we can conclude that (4) there are numbers. Hence (5) numbers exist.

Where could that simple argument be challenged? (2) and (3) look compelling, and the inference from (1), (2) and (3) to (4) is evidently valid.

So that leaves two possibilities. We can challenge the argument at the very end, and try to resist the move from (4) to (5), saying that while it is true that there are such things as numbers, it doesn't follow that numbers actually exist. This response, however, supposes that there is a distinction between there being Xs and Xs existing. But what distinction could this be?

Well, someone could use mis-use "exists" to mean something like e.g. "physically exists": and of course it doesn't follow from there being Xs that Xs physically exist. But it isn't physical existence that is in question when we ask whether numbers exist (or whether God exists, etc.) -- trivially, numbers aren't the kind of thing you can weigh or stub your toe on! It's granted on all sides that numbers are not physical things.

But once it is clarified that "exists" is not being used in a restrictive way to mean, e.g., physically exists it is very difficult to see what the supposed distinction is supposed to be between there being Xs and Xs existing. Certainly, few modern philosophers believe that there is such a distinction to be drawn. (In a slogan, most philosophers think that existence is indeed what is expressed by the so called existential quantifiers, 'there is', 'there are'.) So we'll set aside this challenge.

The other option is to challenge the initial assumption (1). But how can that be done? 11, 13, 17, 19 are the only prime numbers between 10 and 20, and so there are four prime numbers between 10 and 20. That's a simple truth of arithmetic, surely.

But ah, you might say, we need to distinguish: it's certainly true-according-to-arithmetic that there are four prime numbers between 10 and 20, just as it is true-according-to-the-Sherlock-Holmes-stories that Holmes lived in Baker Street. But it isn't really, unqualifiedly, true that there was a man called Holmes living there, and it isn't plain true either that there are four prime numbers between 10 and 20. And from the granted assumption that it's true-according-to-arithmetic that there are four prime numbers between 10 and 20 the most we can reasonably infer is that it should (given arithmetic is a consistent story) be true according to arithmetic that numbers exist. And that doesn't show that numbers really do exist.

This sort of "fictionalist" line that treats arithmetic claims as being claims within a story (the story of arithmetic) has its warm supporters. They will deny that numbers really exist, but at the high price of denying that arithmetical truths are plain true (a reversal, then, of the traditional philosophical ranking of mathematical truths as the most secure truths of all!). If you are not willing to pay that price, and are also not willing to play fast and loose with a supposed distinction between there being numbers and numbers existing, then the simple argument above for the conclusion that numbers exists will look rather compelling.

Read another response by Peter Smith

Read another response about Mathematics