The number One is singular; unique. Indeed, each number is unique. Each is an abstraction that occurs just once. What is more, the only property a number can have must be arithmetical. Numbers are not coloured, or scratched, or lop-sided – they do not have a position or size (in any physical sense). The only properties they can have (for example being a prime, a square, or being a Fibonacci or a Perfect number) are those that emerge out of the expanding self-referential system that is arithmetic.

There can be no misinterpretation with numbers, other than by erroneously operating the system’s procedures. Hence arithmetic is as close to objectivity as we can get, possibly because it is purely abstract, devoid of all physical attributes and thus of all interpretation … indeed independent of any observer. That is why arithmetic/mathematics can be shared without ambiguity by initiates to the system, and why arithmetical/mathematical proofs are possible.

However, paradoxes arise the moment numbers are mapped onto the physical world. The unique number Two is the sum of One and the same One – because there can only be one One. Yet when I talk about ‘two’ chairs say, that is ‘one’ chair and another necessarily different ‘one’ chair, what am I doing? I have thought into existence some idealized yet vague notion of what a chair is, and in my head I have recognized that each chair roughly corresponds, that is each is similar to that ideal. I then decide for the sake of utility that I can dispense with vague notions of similarity, and from now on I will consider them the same. I must deny all properties in each unique chair that are at variance to the ‘one true chair’ that is my ideal.

However the ‘two’ same (=similar) chairs are necessarily different. To make this definitive statement I don’t even need to go looking for microscopic differences in shape, colour or texture – they are in different positions in space, which immediately denies any claim to the singularity of ‘chair-ness’. Two is the sum of One and the same One, but ‘two’ chairs is the sum of ‘one’ chair and a different ‘one’ chair. So the use of the number ‘two’ in the physical world is already at variance with the Two of arithmetic. Indeed in the physical world for there to be ‘two’ things the same, they have to be different – a paradox. If they were the same, that is each is a One, then they would be the same in all respects including their position in space. Hence they could not be differentiated, and so there would be only ‘one’ chair apparent and not ‘two’.