I think you may be missing the entire concept of mathematics. I can't tell if your point is that you can choose to call three by some other name, in which case you're arguing nomenclature, which is orthogonal to the current argument<sup>1</sup>, or if you mean to suggest that mathematics itself isn't invariant as most people are taught to believe. If you mean the latter, then let me assure you of your error; mathematics is indeed invariant between cultures, provided both cultures are advanced enough to develop it.
There is really only one major leap of faith you have to take in order to prove the existence of "three". You have to believe in the existence of "one". The unit. Anything, really, as long as you accept that it exists and can be dealt with as an individual thing. You can deny that anything at all exists, and it indeed becomes impossible to count things.
However, if you acknowledge that something (let's call it '1') exists, then you can put another something with it<sup>2</sup> (we'll call this activity '+'), and then yet another something, and you have more than one somethings. For convenience, we all agree that this is '3'. You can choose to call it speeblarg, or "one", but nomenclature alone cannot change the fundamental fact that there are -->
<-- this many.
The actual basis of mathematics, at least as discussed in this thread, is rooted in set theory, and contains a small fixed number of "postulates" which are generally considered to be self-evident, but must ultimately be taken on faith. Everything else follows logically and deductively from these postulates. If you want to successfully attack mathematics, you have to do it here.
<sup>1</sup> If you do not start with a common assumed nomenclature, then no communication is possible. Therefore, to argue that three may not be three due to a difference in nomenclature is spurious, and not a useful argument. The essence of communication (and thus the argument) carries with it an underlying assumption that all parties have a common vocabulary. You can neither communicate, nor (by induction) argue without one<sup>3</sup>.
<sup>2</sup> Not completely true - you can also deny the possibility of addition, as that is yet another of the fundamental postulates. However, I was trying to keep the argument simple.
<sup>3</sup> Ironically, this is the same spot where I see the fundamental disconnect form in many a political or religious "argument". Two sides will argue logical extensions which are both true from their point of view, because they did not begin by agreeing on common assumptions and definitions.