No. Then it would be confusing.
Plus, The Law of the Awesomeness of Pet Names wouldn't let it work.
Let Surfingbird be SN
Let AS be awesome
Let NS be the number of pets named Surfingbird.
A=S*(N/N-1)
Yes, one Surfingbird is infinitely awesome. But let's say SN is equal to 10.
NS AS
1 1/0
2 20
3 15
4 40/3
5 25/2
6 12
... ...
11 11
... ...
100 10.1
So, as you can see, the awesomeness of multiple Surfingbirds decreases rather rapidly.
Wait, did I just run numbers on the awesomeness of something's name? Man, I must be tired.