Why can more general problems — paradoxically — be easier to solve or prove?
These answers list instances of the Inventor's Paradox, but doesn't expatiate WHY this can happen?
Here's an analogy. Let Dialect 1 have two grammars, one Traditional and one Simplified grammar that all D1 speakers know fluently. Let Dialect 2 be Dialect 1 with merely Simplified grammar. By my intentional construction, Dialect 1 is generaller than 2, and Dialect 2 is easier to learn.
Yet the Inventor's Paradox says the opposite! Incontrovertibly, it's counterintuitive that Dialect 1 would be easier to learn. What's wrong with my analogy?
By the bye, if you're an linguist, feel free to instantiate Dialects 1 and 2. Do such Dialects 1 and 2 exist in real life?
- the Inventor’s Paradox is not systematic. There are situations where it applies (not that often but they stand out), and most where it doesn’t. 2) one reason why it can work is that in simple situations, one is overwhelmed by the many possible approaches (even brute force), while generality forces the better argument. 3) this doesn’t apply to eg Feynman’s trick – in this case, your generalization itself provides a new tool. 4) your analogy fails because the dialects were designed by humans to be simpler. That’s not what a mathematical problem looks like.
Related: Particular problem solved by solving a more general problem AND Generalizing a problem to make it easier AND When was the generalization easier to prove than the specific case? AND Examples where adding complexity made a problem simpler AND What are some good low-prerequisite examples for the heuristic advice "If you cannot prove it, prove something stronger."?.