# Philosophy of mathematics and the nature of numbers

Bill asked:

How do you put this argument 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 its a normal subject/ predicate sentence, we can’t explain how we know it to be true. For if its 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

Your analysis is pretty sound. We just need to take it a bit further.

’17 is prime’ is indeed a subject/predicate sentence. There are other sorts of sentences, such as commands, questions, expressions of attitudes, but we needn’t pursue this.

Standard semantics requires that for ’17 is prime’ to be true, 17 must exist. We want it to be true because we feel that 17 just IS prime (it’s a necessary, apriori truth) whatever the metaphysicians and language philosophers say.

But if we hold that 17 is an abstract object (Platonism), outside space and time as you put it, we have the ‘access’ problem, as you say – how can we know about it, how can it influence us? Some talk of knowing numbers by ‘mathematical intuition’ (Godel was big on this), but it remains mysterious. Note, though, that if you say abstract objects dont exist, you have to find some other mode of existence for other alleged abstract objects, such as propositions. But let’s go with no abstract objects for now.

To make the sentence true, we must either give up or get round standard semantics, or find some other plausible mode of existence for numbers.

Let us deal with each.

1. Give up standard semantics.

This was Meinong’s approach. In his ‘Theory of Objects’, he complained that our metaphysics was sadly deficient because we only considered existing entities, leaving out the vast realm of nonexistent entities. A key feature of his theory was that even though an object is nonexistent, it can still have properties. So, no problem with 17 being prime even though 17 doesn’t exist, or with Santa having a red suit. This whole idea was dismissed as nonsensical by Russell and Quine, but I dont think it is. However, I do think that we can get the advantages of nonexistent objects, and none of the problems, if we regard numbers (and Santa, and Sherlock Holmes) as fictional entities rather than nonexistent ones (see below).

2. Get round standard semantics.

I think this is weaselly. It’s called Paraphrase Nominalism. It holds that when people say ’17 is prime’ they really mean ‘If numbers existed then 17 would be prime’. I dont think they do mean that. I dont. I mean 17 IS prime.

3. Modes of existence.

(a) physical. John Stuart Mill thought numbers were physical, existing only as 3 trees, 19 eggs etc, and that 2+3 =5 was an empirical discovery, revealed to us when we put, say, two twigs alongside three twigs and find five. Everybody else thinks that three twigs is just an instance of the number ‘3’ just as the ’17’s in the present text are instances (numerals), not 17 itself.

(b) mental. Very implausible. Unless right now somebody is thinking of that very number, the third prime number after 27 quadrillion doesn’t exist. Or is anybody thinking of the quintillionth digit of Pi right now?

(c) abstract. We’ve said we’re unhappy with this.

(d) fictional. Here the sentence ’17 is prime’ is prefixed by the Fictional Operator ie ‘ F [’17 is prime’]’. Or, ‘In the story of mathematics, 17 is prime’. Just as ‘In the stories of Conan Doyle, Holmes plays the violin’

I favour fictionalism as an account of mathematical truth. Mathematics is a human construction, comprising axioms; and entities such as numbers, reducible to sets, in turn all derivable from the empty set; and proceeding by logical deduction to yield a vast number of theorems, many surprising, and often found indispensable to physics.

So, in summary, let us keep standard semantics, and say that ’17 is prime’ is true in the story of mathematics because that is where 17 exists as a fictional object.

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