Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was the use of more abstract notions to obtain the same results with fewer calculations.
Let me quote them from their remarkable historical paper "Clarifying the nature of the infinite: the development of metamathematics and proof theory".
The gradual rise of the opposing viewpoint, with its emphasis on conceptual reasoning and abstract characterization, is elegantly chronicled by Stein (Wayback Machine), 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.
What are concrete examples from concrete fields avoiding calculations by the use of abstract notions? (Here "calculation" means any type of routine technicality.) Category theory and topoi may provide some examples.
Thanks in advance.