Writers and other craftsmen often employ certain patterns in their work. Hold on, that last sentence was a bit too broad. What I’m trying to get at is this. Mathematicians and scientists have two specific patterns which they mix and match in all their papers: General to Specific, and Specific to General. You almost can’t help but fall into one or the other when you write theorems, but it helps a lot to be consciously aware of the patterns, both in writing and in reading such literature.
General to Specific
Writing patterns have purposes, they accomplish certain goals. With General to Specific, that goal is often one of motivation. Not all readers have as much knowledge about the subject as the writer, and every paper can potentially serve as a mini crash course in its field before getting down to its own focused topic. Before reading about the nitty gritty details of a very obscure special case, ease the readers in by painting a bird’s eye picture of the surrounding landscape. Surely you didn’t just wake up one morning and decide to spend thirty years peering at insect penises through a microscope. Rather, it started one day in grad school in September of ’69, that was the day your adviser alerted you to the dire need for more papers on the classification of subspecies of subspecies within the Neoptera Blattaria family. You were sitting with him in the department lounge on the fourth floor, sipping green tea (Ito En brand, imported from Itabashikuyakushomae). There was a speck of white cotton lint in your left inside coat pocket.
Specific to General
When I first started reading I.N. Herstein’s “Abstract Algebra”, laying on the bed in my dormitory room in Davis-Monthan Air Force Base in Tucson, Arizona (I was enlisted as a weather forecaster but desperately wanted to become a mathematician, so I was teaching myself using some texts sent to me by the folks of sci.math), those first pages gave very little indication of the rarefied, abstract material I’d soon be getting myself into. Relatively concrete topics were discussed: to add the equivalence class of 5 mod 9 to the class of 6 mod 9, add 5 and 6 to get 11, and take its class mod 9 (which is the same as the class of 2 mod 9, since 9 divides 11-2). More generally, to add two equivalence classes of integers modulo n, you pick representatives which are integers, add them, and take the class of their sum; this is well-defined because congruence modulo n is an equivalence relation. This is a special case of a group, which general concept Herstein introduces somewhat later. Theorems about abstract groups are then given, which automatically hold for individual examples of groups as well. Other algebraic constructions are later introduced (and then if you’re a real man, you go learn Category Theory). Basically, Herstein went from specific to general. Good thing, because I would’ve had no idea what was going on if he jumped to the general definitions right away. With Specific to General, the goal is often one of motivation, when the general case would seem too abstract without some examples beforehand. Writing patterns have purposes, they accomplish certain goals.