Generality versus core insight trade-off in theorem statements
(there might be a standard term for this; if so, I would like to know)
In math, there is often a tradeoff between stating a theorem in the most general/applicable way possible, and stating it in such a way as to reveal the core difficulty or insight.
For example, consider the calculus:Intermediate value theorem . It's possible to state the theorem as saying the function attains each value between and . But this introduces an extra parameter which takes up more working memory. It is simpler to state it as saying the function equals zero somewhere. Similarly, we could let and be any real numbers, but we could restrict the theorem to the case , which is where all of the interesting stuff happens.
|More general/applicable||Core insight version/the most interesting case|
|for any between and , there is in such that||there is in such that|
|are arbitrary real numbers (including the case , which makes empty so the theorem becomes vacuous, and also including the case so the function consists of a single point, which makes the theorem trivial)||are real numbers such that|
|and can be anything (we don't add hypotheses concerning their values)||and|
Once the core insight of the theorem is absorbed, it is of course most useful to have the theorem stated in the most general way possible, so that it can be quickly applied to other problems. But when one is first learning the theorem, it can be distracting to consider all of the vacuous, trivial, uninteresting cases; it is most useful to gain the core insight quickly .