I’ve just finished reading Man and His Symbols by Carl Jung and the final chapter concerns how mathematics does not represent reality. After thinking about it, I realized that numbers are purely conceptual. The number two is purely conceptual as a count of two is never absolutely valid.
Let me explain. Say you have two apples. The count of apples is two. But what you really have is one apple plus another apple or a different apple. Lets say I present you with two objects, an apple and an orange. Now tell me if there is one or two. While there are two pieces of fruit, there is only one apple and one orange. The count of two is only valid if you apply a concept or category. In other words, two is always in some way conceptual. In fact, all numbers in a count are in some way conceptual.
In computer science, this can be illustrated using object oriented programming. When I create two objects of a class and get their count, I expect the count to be two. But under certain circumstances the count will not be two. Lets say the objects have the same number of properties. If you only look at these properties, the count will be two. However, if you look at the value of the properties they may be different. So when you do a little reality testing and test for deep equality, you do not get equality and a count of your object may be one. A count of two always depends on applying a concept of objects being the same in their properties but they will be the same only on some superficial level, in terms of a limited set of properties.