***"All 'things' have a unique identity"; however, I don't quite see the neccessity of the characteristic "unique". Are you trying to tell me that if I have 3 things which are not unique I cannot attach numbers to them for the sake of reference?***
If you have three 'things', then there exists an algorithm to identify one thing from another. Otherwise, how could there be three 'things'?
***As it stands, your complaint is equivalent to "things which are not unique can not be referred to".***
Well, if you have a multiple of something which the numbers are used to represent, then some algorithm must exist which allows a count to have any meaning. For example, if you have 3 oranges on an otherwise empty table, you can count the 3 oranges by:
1. Defining an 'orange'. Defining a 'table'. Defining 'otherwise empty'.
2. Identifying an orange from the other 'things' (e.g., table, otherwise empty).
3. Counting each orange from one to three until none are left uncounted.
4. Reporting the final tally as three.
This is an algorithm to establish that you have 3 oranges on an otherwise empty table. Without this algorithm, the term 3 oranges would be meaningless.
***"Numbers are simply a mental means by which to represent the uniqueness and separability of 'things'". I would say that it is possible to use numbers for that purpose; but, it is also entirely possible to use numbers as mere labels. The first purpose requires some rules as to how the numbers are assigned whereas the second does not.***
Labels of what? There must be some algorithm in place that allows you to label an 'orange' so that the number holds meaning.
***It appears to me that your position comes from a reliance on authority without critical thought as to the reasoning behind their positions; this is the single most common flaw in the thinking of educated people.***
I just want to know how you can establish (P-6) so that (6) holds any meaning. If no such algorithm exists to 'things', then (6) is meaningless. The purpose of (P-6) is to show that 'things' do have an identity (i.e., there exists such an algorithm, even if it is humanly unknowable), and therefore we can proceed with (6). But, how can you know that (P-6) is true. That appears to me to be a problem.
If you'd like, we can agree to disagree on this point, and move onward.
Warm regards, Harv