The only thing you have to assume to be unconditionally true in Mathematics is some minimal logic (and yes, that's despite having axiomatic systems for logic; you still have to use some form of logic to actually define those axiomatic systems). But logic is assumed to be true in any science (because without it, you cannot draw any conclusions).
But apart from logic, all statements in mathematics are ultimately conditional statements on the chosen axioms. For example, take the statement "there are infinitely many prime numbers." How can we know this to be really true? Well, we have a definition of the natural numbers through a set of axioms, and we have a definition of what it means to be a prime number. From those axioms we can logically derive that there are infinitely many primes. But that statement is implicitly conditioned on the axioms: We have to assume that what we are looking at really fulfils the Peano axioms. If we look at something which doesn't, the claim doesn't hold. However, mathematics doesn't look at a specific system. The statement it derives is not "for this real world object we have infinitely many primes." It says "Whenever we have something which fulfils those axioms, we know that we will find infinitely many primes." It also tells you that if we make certain other assumptions (such as that the axioms of set theory hold), we can derive that we'll find something fulfilling those axioms.
This is also why mathematics is so useful in natural sciences: It does not tell us what assumptions are true. But it tells us what follows if certain assumptions are true (and also, if certain assumptions cannot hold together). So if we have for example a physical phenomenon, we can formulate the hypothesis that it has certain properties. This hypothesis is not part of the real world, but a set of assumptions. Therefore we can now go to mathematics, which tells us what to expect from systems with such assumptions (and also, which additional assumptions we might want to make). Note that this step is completely independent of reality. After we've found what to expect if those assumptions are true, then we can go back to the lab and check if our experiments show the behaviour we just have derived from our assumptions. If yes, we've got a confirmation and may be more confident in our hypothesis, otherwise we have falsified our hypothesis and have to modify it (and again, mathematics will tell us what assumptions will be compatible with our new knowledge from the experiment).
Note that there's another type of questioning theories which is done in mathematics as well as in natural sciences: Namely the questioning whether your results are actually correct. In mathematics, this means checking that there are no errors in the proof (and in some sense this is similar to the experimental tests of theories in natural sciences: We are confident in a proof if it has been sufficiently looked at and nobody has found an error), in physics it means checking that there's no error in the measurement procedure (that is, we really have measured what we thought we measured) and no error in the application of mathematics (that is, we correctly applied the tools we got from mathematics and made no hidden assumptions, and thus our conclusions about what to expect are correct).