In what manner (if at all) do numbers exist?

Atron asked:

How do you put the following argument below in standard form?

One thing we can all agree on is that a statement like 17 is prime is true and that we know it to be true. But this simple fact gives rise to an irresolvable puzzle. If it’s a normal subject/ predicate sentence, we can’t explain how we know it to be true. For if it’s that sort of sentence, then there must be some object, the one we call 17, and it has to have the property of primeness. But if there is such an object, it is outside space and time, and so a mystery how we come to know anything about it. Could it be some other sort of claim, then. besides a normal subject/ predicate sentence? I suppose, but then we have a different mystery. It’s utterly mysterious what other sorts of sentences there are.

Answer by Craig Skinner

Yes it is a normal subject/ predicate sentence: the subject is 17, the predicate primeness.

And yes, standard semantics requires that for it to be true, 17 (the subject) must exist.

Some people try to get round this requirement. There are two ways:

1. Saying ’17 is prime’ really means ‘If 17 existed, it would be prime’. This approach is called paraphrase nominalism. I don’t like it: when I say 17 is prime I mean 17 is prime, not something else.

2. Allow that nonexistent objects can still have properties. An approach championed a century ago by Meinong. Being nonexistent, these objects can’t instantiate a property, they are said to exemplify it. Thus the round square, not merely nonexistent but necessarily so, exemplifies roundness and squareness. This view is still viable. After all, we speak of fairies having wings, Santa Claus having a red suit. And so we can speak of 17 exemplifying primeness.

Others accept that ’17 is prime’ is untrue, because 17 doesn’t exist, but is true in the story of mathematics, just as ‘Sherlock Holmes plays the violin’ is untrue, but true in the stories of Conan Doyle. This is called fictionalism. I rather favour it. The distinction from Meinongianism isn’t completely clear — is Holmes fictional or nonexistent? If he’s fictional, does this mean he is a fictional object, or that we pretend he’s a real object? And if he’s a nonexistent object, did he only come into nonexistence when Doyle dreamed him up, or was he a nonexistent object in, say, the middle ages, and Doyle only added properties for him to exemplify, such as being a detective and playing the violin, being careful of course not to accidently select the equally featureless nonexistent James Bond to add these properties to thereby making things difficult for Ian Fleming years later. You can see how all this verges on Alice in Wonderland, so I’ll move on.

Most people accept that 17 exists. Existence can be physical, mental or abstract. Let’s deal with each.

Physical: 17 doesn’t exist physically in the external world, although there are many instances of collections of seventeen objects in the world. A few people hold that 17 exists physically in brains as a particular pattern of wirings/ firings. This seems to me the same as existence as a concept (see below)

Mental: overlaps with the wiring/ firing/ conceptual story. Means 17 only exists if somebody is thinking it. So 890785432308, say, often doesn’t exist since most of the time, very few people, if any, would be thinking of this number. But i suppose there is always some computer or other processing this number somewhere. Is this mental existence for the number, or is the computer crunching nonexistent numbers.

Abstract: the popular choice, including your own. You emphasize the access problem: how can something outside spacetime causally affect us. I think this objection is overblown. It’s a legacy of Plato’s theory of forms, whereby numbers and other universals exist in a timeless heaven, accessible only to the inner eye of the intellect. And mathematicians of a Platonic bent, such as Gödel, didn’t help, claiming the trained maths mind could access these objects, thereby discovering, not inventing mathematical truths. I think abstract existence is less ethereal. It is existence as a shared concept, such as divorce or humour, not spatiotemporal to be sure, but hardly a special mystery.

So take your pick. Conceptual existence, in-brain physical or mental existence, nonexistent but still prime, fictional existence. And I suspect that close analysis might show all these options to be pretty much the same.

Finally, you ask for a deductive argument formulated from your text. How about:

P1. ’17 is prime’ is true if and only if 17 exists.
P2. 17 exists (in some way).
C 17 is prime.


P1. For a predicate to be true of a subject, the subject must exist.
P2 Hence ’17 is prime’ is true if and only iof 17 exists.
P3 17 doesn’t exist (it’s a mathematical fiction).
C ’17 is prime’ isn’t true (but is true in the fiction of mathematics).


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