The belief that one can find out something about real things by speculation alone is one of the most long-lived delusions in human thought. It is the spirit of antiscience which is always trying to lead men away from the study of reality to the spinning of fanciful theories out of their own minds. It is the spirit which every one of us (whether he is engaged in scientific investigation or in deciding how to use his vote in an election) must cast out of his own mind. Mastery of the art of thought is only the beginning of the task of understanding reality. Without the correct facts it can only lead us into error.
-- Robert H. Thouless, Straight and Crooked Thinking
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"Why does 2+2 come out the same way each time?"
Thoughts that seem relevant:
Addition is well defined, that is if x=x' and y=y' then x+y = x'+y'. Not every computable transformation has this property. Consider the non-well-defined function <+> on fractions given by a/b <+> c/d = (a+c)/(b+d) We know that 3/9 = 1/3 and 2/5 = 4/10 but 7/19 != 3/8.
We have the Church-Rosser Theorem http://en.wikipedia.org/wiki/Church%E2%80%93Rosser_theorem as a sort of guarantee (in the lambda calculus) that if I compute one way and you compute another, then we can eventually reach common ground.
If we consider "a logic" to be a set of rules for manipulaing strings, then we can come up with some axioms for classical logic that characterize it uniquely. That is to say, we can logically pinpoint classical logic (say, with the axioms of boolean algebra) just like we can we can logically pinpoint the natural numbers (with the peano axioms).