I guess when one reasons inside a fully axiomatized formal system, this something the rules for symbol manipulation depend on is the set of axioms.
Now I'm putting on my uneducated hat, so excuse me if this is heresy: Starting with the axioms you apply logic to formulate more specific rules (in this case the abstract is empirically falsifiable, since we're working on natural numbers).
So, to arrive at SS0+SS0=SSS0, you'd have to venture outside the realm of reason I'm afraid.Tthat would maybe manifest itself as magic - getting 4 apples on the table during night, but 3 during day when you put 2 and 2 apples side by side. And could mean ability to produce something from nothing by clever arrangement of apples. and waste disposal would become easy :)
In other words my opinion is it's not possible even as thought experiment unless you introduce some random factor from beyond the scope of axioms.
There are some points of view that sometimes do require mathematical statements to be dependent on reality (i.e. constructivism, actual versus potential infinity debate, etc).
Sometimes it is intuitive to require mathematics to behave this way, i.e. 'natural' numbers are called that for a reason, and they better behave like the apples or I'm postulating a change in nomenclature.
P.S. Ii seems to me the OP's wording wasn't precise enough. I can very well imagine a situation in which some basic addition would yield non obvious results (like addition inside modulo N number space).
Google-transformed version of a Word document. An example of bias selection-oriented SPRs may introduce:
http://docs.google.com/viewer?a=v&q=cache:T-hFS_Cwo_oJ:weber.ucsd.edu/~aronatas/conference/ethics%2520paper1.doc+statistical+prediction+rules&hl=pl&gl=pl&pid=bl&srcid=ADGEESjgL00gBk1XyYkhtEbBUNa-Lu6I4zcp6bpU68MBHJysHU0IdaCD8lupv6doFRtxI6DEbksveziWd7tkc0aeGiq6yGNNjwKi4IqiNyBjsjsKfJNU66EE-BEo5ZvnkAVrGnV18_VM&sig=AHIEtbSwzhMKJBNDuZBATgAqDI83ni74ug