[Good Friar]

Perhaps the best treatise on the subject is Frege's Grundlagen, which is a truly impressive work that addresses many different philosophies of mathematics. His basic approach, and one that is still fairly basic to our current use of number, etc, tries to reduce the idea of natural number to that of logic, that is, we know the difference between existence and non-existence, and hence the difference between 0 and 1. Likewise, we can differentiate two concepts/objects/ideas, by notice the existence of two non-equal things and so on. There are problems involving the Russell paradox in his formal approach, but you'd still probably find it at least a satisfying approach.