I’ve been wading through my better posts — the ones that actually “deliver the goods” — and I noticed an important difference between my writing and that of people who appear to be able to succintly convey comparably complex messages. There are two “styles” of storytelling, which I’ve nicknamed the mathematician style and the journalist style.
In a nutshell, the mathematician style is bottom-up and the journalist style is top-down, which means the journalist presents the gist of what’s being said right at the beginning, while the mathematician will demonstrate what he’s trying to say from ideas he assumes you’ll find trivial or obvious. On a blog journal post, that will often prevent the reader from getting his point at all. [The way this post is constructed, it has the journalist style if you read this paragraph or the mathematician style if you strike this one out. Try and start re-reading this post skipping this paragraph.]
A mathematician will always start in familiar territory. If you have basic (calculus, linear algebra, etc.) mathematical training, you can walk into a post-doc lecture on Anosov diffeomorphism in non-ergodic systems and understand the first two or three minutes of it. Much of advanced mathematics relies generalizations of simple structures one has learnt in basic training, and if you’re fluent in Riemann integration you might get an intuitive idea of the Stieltjes integrals the lecturer is drawing on the board. Sure, you’ll get lost soon because you never had the prerequisite knowledge to understand the actual point of the talk.
There are multiple reasons for this phenomenon. A mathematical lecture will often start behind its prerequisites, to get everyone’s brain into “math mode”, to ensure everyone’s in the same starting point — often terminologies vary, and what a brazilian calls a “body” (corpo) is what’s known in english as a “field” — and no one’s thinking of diffeomorphisms in ordinary topology, for example. This is also the way mathematics is done since calculus 101, and it probably trickles up throughout a mathematician’s career. The more important thing, though, is that mathematics is all about deriving the nontrivial from the trivial — which is why the obvious is stated first.
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