Earlier quoted context omitted.
That's an axiom. It doesn't say anything about the real world, it just introduces a framework of thought. In other words, if you can imagine a reality where 1+1=2 for some meaning of '1', '2', '+' and '=', then a series of conclusions will hold true.
But all that says is that 1+1=2 is too obvious. How about some other, less straightforward mathematical theorem, then, like Fermat's Last Theorem. You can hardly accuse that of being an axiom, or just introducing frameworks of thought.
Science is based on (lots of things but primarily) empirical induction. You observe some phenomena, decide that they are of the same type, generate a model that predicts how all phenomena of that type behave. By necessity you observe an infinitesimal fraction of the phenomena in your type.
In math proofs are universal and timeless and complete. There is no additional observation to be made. A proof can be wrong, but if we conclude that it is logically correct then that's it. (edit: well, there's the incompleteness theorem: it could be that a set of axioms is inconsistent, but then EVERYTHING based on those axioms becomes invalid. It's still quite different than scientific progress which changes models a little bit at a time.)
Scientific models are approximations of reality(very good approximations often!) but they get tweaked as new observations come in. Or just wholesale thrown out(or at least become obsolete), see Newtonian orbital mechanics vs relativistic orbital mechanics. Newton was right-ish, but relativistic solutions are right-er(possibly fully correct?)[1].
With deductive reasoning you are expanding a definition. With inductive reasoning you are guessing at connections between observed events.
[1] http://physics.stackexchange.com/questions/26408/what-did-ge...