Suppose that an alien civilization exists, in a planet somehow similar to our own (oxygen-based, plants, animals), in an evolutionary stage similar to ours (large cities, advanced communications, near-planet space travel etc.).
It's obvious that they must have developed Mathematics in one way or another. Probably in some domains their Mathematics may be more advanced and in some other domains it's the other way around.
The question is - regarding the common domains - this: are the Mathematical concepts and constructs (nearly) the same; are these Mathematics "common" regarding the same applications in real life?
In a way this question "touches" the "invented or discovered" question regarding Mathematics (already heavilly discussed), but what I am really interested in, is if there must be a "convergence" in the concepts, or by stating it in another way: will a potential interraction (of the two civilizations) be of huge significance (related to Mathematics) or just a matter of "translating" (mapping) one concept to another?
Note: I am not talking about simple concepts as numbers and operations, but of advanced mathematical constructs that are lying behind our nowadays civilization.
addition: Ok, I will rephrase the question clarifying some things, taking into account some already posted answers:
Our mathematics are generally based on axiomatic foundations. Gödel proved that mathematics is incomplete: whatever system of axioms we assume, there are statements that are true but that cannot be proved using only these axioms.
(For example, Goldbach's conjecture is one of the unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers until to 4×10^18 but remains unproven. Some beleive that by starting with a different set of axions it could be proved.)
It is reasonable to accept that the alien civilization would have a different set of axiomatic foundations in the respective math domains. So then, there must be some "things" they have proved that we haven't and the other way around.
Do you agree with the conclusion?
Would then this be an inconsistency, in the then new world of mathematics (the two combined together)?
How then would mathematics evolve, as a response to such a confrontation?