Most mathematicians and philosophers writing in English think that the counting numbers are not properties of ordinary things like pairs of shoes, even though the view that they are not originates with a Nazi sympathizer, Gottlob Frege (1848–1925).
Frege was trying to prove that 1 + 1 = 2. He did not think that he could simply define 2 to be 1 + 1, because he did not think that numbers could be composed of ones. He thought that either those ones would be the same ones, so that one and one would just be one (much as you and yourself are just you), or else they would be different, a first one and a second different one, in which case the signs for them, 1 and 1, should also be different (much as David and David, when they are the names of two different kings of Scotland, are written David I and David II).
Could the signs for those ones be given subscripts, such as i and ii? But then we would have had to have defined ii already. As a nineteenth-century German professor of mathematics with very conservative views, Frege thought that he was well placed to devise new definitions of the counting numbers (and of the infinite numbers that had recently been discovered, by another German professor of mathematics, Georg Cantor), so he did just that. And then he attempted to prove that 1 + 1 = 2 with those definitions, impressing the far more influential Bertrand Russell, who went on to make his own infamously long mathematical proof that 1 + 1 = 2 with his own definitions.
As a matter of fact, 1 + 1 = 2 because one thing and another thing make two things. Those two things are different, but each of them is one thing, and much as the red of a rose and the red of an apple might be the same red colour, those ones are the same.
Much as red is a property of physical things like roses, apples, shoes and so forth, the number one is a very simple property of things: each and every thing is one thing. And numbers bigger than one certainly seem to be properties of collections of things (a pair of shoes is two shoes). And of course, the properties of physical things are described, not defined, so Frege had to find some good reasons why numbers bigger than one are not properties of physical things.
Frege observed that how many things a collection contains depends upon what kind of thing we are talking about. Two shoes are also a lot of molecules, for example. According to Frege, colours are not like that. A red shoe is not also some other colour, for example. Except in a different light, maybe. Or to a different pair of eyes. Or against a different background. And so forth; but in any case, he also observed that while we can say "the men are white" because each is white, we cannot say "the men are many" because each man is many (although each man is a lot of molecules).
We can say it because the men are many, though. Frege explained his argument in more detail (the following is a cheap and cheerful version of Frege's argument):
If you have a blue tooth, you could mentally remove the property of being blue to leave a colourless tooth. You could then remove the property of being a tooth, in order to see in your mind's eye what was left. You can, in short, mentally remove various properties, in order to isolate the other characteristics. But if you have, say, three teeth, and you remove all of the sensory properties of the teeth (their colours, their shapes, their hardness and so forth), would you be left with a one for each apple? Would you be left with the number three?
According to a very important German, Gottfried Leibniz, you cannot have three identical things. So, because you cannot strip away all of the sensory properties and be left with the number three, the number three is not a property of collections of physical things. What is mentally removing colours, though? Is it like looking at a black-and-white photograph? That makes sense, but would mentally removing their being teeth from some teeth be like having a hologram of some teeth? Or is it like having a description of some teeth, in some language? Or a mathematical model of some teeth? Or the logical possibility of some teeth? Or the idea of some teeth? Or the memory of some teeth? I find myself mentally picturing some teeth and then imagining that I am imagining them! Still, the argument above is a simplified version of Frege's argument. Let us stick with simple properties like colours and shapes.
Suppose that I remove the shapes and sizes of the three teeth, and their positions in space, but nothing else. Presumably there would still be something real. Would it be three teeth without shapes, sizes, or positions in space? But without shapes, sizes or positions in space, how could what was left be three of anything?
Perhaps it could be three overlapping wavefunctions. But prima facie, those teeth being three in number has a lot to do with their shapes, sizes and positions in space, all of which are properties. So, that version of this argument seems to show that numbers are properties of physical things (although by mentally removing all of the sensory properties of the three teeth, you might be left with three teeth in a space containing no sensory organs, in which case there would still be those three teeth). Frege's argument may well have been better (although Idealism was popular in his day), but it is certainly a simple matter to imagine any number of identical objects:
Imagine a space containing nothing but three identical spheres at the corners of an equilateral triangle, or four at the corners of a square, and so on. It is easy enough to imagine that all of the properties of those spheres, including their external relations with each other, are identical.
Perhaps there are better arguments that numbers are not properties nowadays (there must be some reason why the view that numbers are not properties is so popular with our experts, who do not usually agree about much beyond the bare facts, such as arithmetic). However, as I wade through the rather large literature that defends this rather unnatural view of that Nazi sympathizer, I am not expecting much. At first glance, a lot of the literature assumes that there are not really any collections of physical things in the world, just individual physical things. That is a bit like saying that there are not really any colours in the world, only photons, neurons and the like. Could there be an argument that there are not really any colours? What about an argument that colours do not make sense? What about this:
One photon hits your eye and you might see one of the primary colours. If two photons hit your eye at the same time, then you might see a secondary colour. But only if three arrive together could you see brown. Brown should therefore be the brightest colour. But in fact, it isn't (yellow is). Does that mean that colours do not make sense? Does it mean that there are not really any such things as colours?
Obviously not. Indeed, how could any argument mean that there are no colours, when we can see that there are colours?
And how could any argument mean that there are no collections, when there clearly are things being referred to collectively?
There are, for example, ten words in this final sentence.
