I did not read the book. But one of possible thinks that came to my mind is this:
There is big field in mathematica dedicated to preciselly define some terms (set, metric, metric space, ...) and prove a lot of thinks about them. Once you can prove, that something is "metric space" then you can use any result from this part about "metric space" and safely apply it to your problem, knowing, that it fits.
So many authors use this "reduction on known problem" to be able use all knowledge collected about the "known problem" and apply it on the problem they have (and maybe then add something new).
Say I want win a contest on running thru maze in less time (and I have 2 tries).
Then I would just prove, that the maze can be described as graph where crossing are nodes and ways between them are weighted edges, where the weight is travel time.
Then I can first run just "map" all the maze and in second run use https://en.wikipedia.org/wiki/Shortest_path_problem known graph theory to find "shortest path", in my case fastest (as I maesure the time, not distance).
And then I would folow the fastest way and the rest just dependes on how good motor and navigatition I have, knowing I cannot get better result any other way (I can get the same result at the best).
So I think, that the author of the book took the same approach - he "proved", that configurations can be described as some already known mathematic constructs and then he use proven knowledges about those mathematical constructs to apply them on configurations and get desired results "fast, easy and proven way".