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Q&A

Why can more general problems — paradoxically — be easier to solve or prove?

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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?


Mindlack commented

  1. 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.

Dave L. Renfro commented

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."?.

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I think Mindlack hit the points that make the most sense to me. Yep, it's not systematic, so it's not like a "law" of any sort. But it certainly is interesting when it happens.

And when it works, it's because the abstraction focuses your attention on only those details which are important for this problem. Take Kevin Buzzard's example of finding the sum of the first 100 natural numbers. If given this task, you wouldn't think of using induction to find a formula, for instance. It just seems to involve details that make you suspect a different solution method is appropriate, and will distract you from a simpler solution method.

This means that one of the most important tasks in mathematics is not solving problems, but in fact, finding the right abstractions to make large classes of problems more intelligible. In this way, the greatest mathematicians are the ones who define terms that "carve at the joints".

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In most cases, the easiest solution will only use some of the properties of the problem. This in turn means the solution applies to all problems that share those specific properties, even if they are otherwise very different.

Thus you can define a generalised problem that only assumes those properties needed for the solution. If you do so, that generalised problem will be easier to solve, simply because it only specifies the properties that you actually need, and removes any other properties that could either lead you to a less efficient solution, or are completely unhelpful. Since the generalised problem states exactly what you need to solve it, you don't need to figure out which aspects are relevant and which aren't, which makes this problem easier to solve.

You might also only go partially in that direction, by removing some, but not all “distractions” from the problem. This still means you've got to consider fewer unhelpful details.

Note that this does not apply to any generalisation, but only to generalisations that remove unhelpful details. In general, a generalisation may also remove useful details, making the best proof of the original problem no longer apply to the generalised problem. In that case the generalised problem may be harder or even impossible to solve, and if you solve it it may lead to an unnecessarily complicated solution of the original problem (that solution may, however, still be worthwhile if there are other important problems covered by it on which the optimal solution to the original problem is not applicable).

Note also that by solving the original problem, you also implicitly define a generalisation, namely to everything that shares those properties you've used in your solution.

For example, take the problem in the link where a line and a regular octahedron are given, and the problem is to find a plane through the line that bisects the volume of the octahedron. The helpful generalisation given is to replace the octahedron with an arbitrary solid with central symmetry. But you could also do other generalisations that are not helpful. For example, you could observe that an octahedron is a Platonic solid. Or you could observe that it is a convex body. Both observations are true, but lead to unhelpful generalisations.

The reason that the generalisation to symmetric bodies is helpful is that this generalisation, unlike the others mentioned above, captures the essence of the solution.

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