Why 15 fundamental properties?
The number of properties is not as important as the order of magnitude — there aren’t going to be many more. This means that relatively few elements, which we can one day understand and perhaps even explain precisely (mathematically), can produce all the complexity and beauty in the world, simply because the number of ways in which centers can be in relationships with each other is limited.
In a fully connected graph, relations between nodes increase exponentially with the number of nodes. This is like taking advantage of this rule by using its reverse.
The precise number fifteen is not significant. But I do believe that the order of magnitude of the number is significant. Throughout my efforts to define these properties, it was always clear that there were not five, and not a hundred, but about fifteen of these properties. It wasn’t possible to go on listing new ones indefinitely.
Although Alexander is pretty sure he found a good set of properties, there could be more.
There is no certainty that this list is exhaustive. On the other hand, if you try to think up other effects which are combinatorially different from these, you will find it is not very easy. When we focus on the mathematical ways in which centers can be built out of other centers, or the ways in which one center helps to make other centers stronger, there is a limit to the number of ways in which this can be done.