I like the following style of explanation of why 1 is not prime. I'll spell it out for the natural numbers. Define n to be prime if, whenever n is equal to a product of natural numbers, it is identical to one of the members of the product. Then 1 is not prime because it is equal to the empty product, but not identical to any of the members of that product, because there are none. This form of explanation is discussed further on the ncatlab page "too simple to be simple".

I meant 0 not to be a natural number in that definition. To extend the definition to the integers, for example, one is going to have to be more careful anyway, and not just exclude 0, but deal with problems such as 2=(-1)(-2). I suppose the idea is to say something about "up to multiplication by a unit".

I like the following style of explanation of why 1 is not prime. I'll spell it out for the natural numbers. Define n to be prime if, whenever n is equal to a product of natural numbers, it is identical to one of the members of the product. Then 1 is not prime because it is equal to the empty product, but not identical to any of the members of that product, because there are none. This form of explanation is discussed further on the ncatlab page "too simple to be simple".

According to this definition, however, 0 would be prime.

I meant 0 not to be a natural number in that definition. To extend the definition to the integers, for example, one is going to have to be more careful anyway, and not just exclude 0, but deal with problems such as 2=(-1)(-2). I suppose the idea is to say something about "up to multiplication by a unit".

But that approach is going to work poorly in other rings. I give up.