Jeremy Avigad and Erich Reck in their remarkable historical paper "Clarifying the nature of the infinite: the development of metamathematics and proof theory" claim that one of the factors of becoming of abstract mathematics in the late 19 century (as opposed to concrete mathematics or hard analysis) was the fact, that using more abstract notions we can avoid a lot of calculations to obtain the same result. Let me quote them:
"The gradual rise of the opposing viewpoint, with its emphasis on conceptual reasoning and abstract characterization, is elegantly chronicled by Stein, as part and parcel of what he refers to as the “second birth” of mathematics. The following quote, from Dedekind, makes the diﬀerence of opinion very clear:
A theory based upon calculation would, as it seems to me, not oﬀer the highest degree of perfection; it is preferable, as in the modern theory of functions, to seek to draw the demonstrations no longer from calculations, but directly from the characteristic fundamental concepts, and to construct the theory in such a way that it will, on the contrary, be in a position to predict the results of the calculation (for example, the decomposable forms of a degree).
In other words, from the Cantor-Dedekind point of view, abstract conceptual investigation is to be preferred over calculation."
So, my question is: do you know some concrete examples from concrete fields of avoiding calculation mass by the use of abstract notions? (term "calculation" here means any type of routine technicality). I can't remember where I read it but some examples one can find in category theory and topoi (not sure).
Thanks in advance