Numbers, chairs and unicorns
Platonists commonly take a sentence such as:
(1) “There is a prime number greater than ten.” (Footnote:
Tait, William W. “PDF Truth and proof: the Platonism of mathematics”, Synthese 69.3 (1986): pp 341-370 Truth and Proof: Details.) (Footnote:
Also see, for example, “We say, again, that six is the number of Austen’s novels, and that six is a perfect number. In so saying we appear to presuppose the existence of something, namely, the number six.”
From: A Subject with No Object: Strategies for nominalistic interpretation of mathematics, Burgess, John P. & Gideon Rosen, Clarendon Press, 1997, ISBN: 9780198250128 A Subject with No Object: Details)
And they conclude that it is in essence no different to a sentence such as
(2) “There is a chair in the room.”
But it should be obvious that neither sentence means anything unless an associated context is either implicitly assumed or explicitly expressed. For the case of the chair and the room, there is an implication that we are talking in a context where there can be an actual physical room that exists and an actual physical chair that exists. The word “chair” is not the chair, nor does the physical chair have any inherent association with any word, never mind the word “chair”. It is by common consent, achieved by common experience with other chairs, and the definitions of chairs, and word usage, that it is agreed that the word “chair” is to be associated with physical chairs.
On the other hand, if the context is a painting of a room with a chair, then we might consider that the painting could be a representation of an actual room with an actual chair. Or it could be the product of the imagination of the artist, and in this latter case, most people would not believe that there is some sort of ethereal non-physical chair that somehow exists in some non-physical way completely independently of the painting and that a part of the paint on a canvas somehow refers to that ethereal thing.
Now consider the sentence:
(3) “There is a unicorn in the room.”
In the case where the context is one where we are talking about a painting of a room is obviously quite different to the context where we are talking about an actual physical room. In the case where we are referring to an actual room, the sentence cannot be correct (unless one believes that unicorns exist).
But in the case of a painting, the sentence could be correct. In that case, the painting itself is a physical object and there is a part of it which we associate in some way with the notion of a unicorn. Normally we would reject the notion that part of the painting is the unicorn, but neither would we say that part of the painting represents a real physical unicorn. And although part of the painting exists physically and represents the concept of a unicorn, there is no reason to suppose that there is some sort of independently existing unicorn that is referred to by that concept. Most people would say that part of the painting represents a mythical creature that does not exist, although of course one can form a mental concept in one’s imagination of a unicorn, and what its physical form might be like if it could exist.
In the above, we can see that words in sentences can refer to physical objects, or they can refer in very similar ways to mental concepts that have no real physical counterpart. It follows that such a reference by description or definition in no way proves that the description or definition is referring to a thing that must necessarily exist either physically or in some non-physical ethereal way.
The reference to a prime number in (1) above is in essence the same as the reference to a unicorn in a painting. The only essential difference is that in the sentence (3), “prime number” has replaced “unicorn”. It might be said that there is nothing in the sentence (1) that corresponds to the context of a painting. But that context is implied, and one might fill in the implication by something like:
“There is a prime number greater than ten in Peano arithmetic.”
And as for the case of the unicorn in the painting, there is no need to suppose that the concept given by Peano arithmetic indicates the actual independent existence of prime numbers. Clearly, the references to number are references to concepts defined by a certain set of definitions, for example by Peano arithmetic. There certainly is no sort of essential implied reference to non-physical things that somehow exist independently of the definitions.
So where do Platonists get the notion that a reference to a number is a reference to an existing non-physical thing? It’s sorry nonsense dragged out of nowhere and which has no logical foundation. It’s time that people stopped treating this sort of pitiful hogwash with the sort of reverential politeness that is so often seen in various journals and books as if it might have some smidgen of substance.