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The undeserved status of the pigeon-hole principle (1991)
- wizzwizz4 6y agoThis isn't exclusive to the Pigeon-hole Principle, either. Many named conjectures are pointlessly used, just because they're named, even when it makes the proof longer. This isn't just exclusive to mathematics. It's similar to the reason people bring in single-line Node.JS packages; a fear of re-inventing the wheel when they don't actually need a wheel in the first place, or when the wheel is trivial to re-state in a more natural form for the proof / program.
- morelisp 6y agoA couple thoughts: a) The history of mathematics is littered with examples of assumed "obvious" results that turned out to have drastic implications. This is especially true of combinatorics, which was blissfully ignorant to the axiom of choice for so long and now must wrestle with it forever. Strict adherence to some formalism may be justifiably understood, and it's a little shocking to find EWD of all people arguing otherwise for such a pragmatic reason as pedagogical clarity. b) Comparing 0) and 1) as EWD suggests, the first thing I notice is the sudden appearance of the word "finite" which never appears again in the article (nor in negation; also the inelegant but mostly harmless introduction of "nonempty"). The generalization of the principle to support reals costs it its trivial generalization over infinite bags, which is also often valuable (a classic example being Siegel's lemma).
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- jmount 6y agoI feel this is an instance of prof.dr. Edsger W.Dijkstra just being mean. If one works more on pigeon-hole principles you end up with Ramsey Theory, which covers some amazing results.
- joe_the_user 6y agoYeah, this feel like a comment that might have been OK at a cocktail party or made offhand in the middle of an essay on something else. Sure, maybe, in some instances, some people, might, perhaps make a little too much out of using the pigeon hole principle. But here, he make too much out of that situation.
- pinewurst 6y agoIt wouldn’t be the only one. Review his archived papers and they’re well salted with snark.
- jimhefferon 6y ago"Arrogance in computer science is measured in nano-Dijkstra's" - Alan Kay (https://youtu.be/9KivesLMncs https://youtu.be/9KivesLMncs) (I usually like Dijkstra's works but the quote does make a person smile.)
- neel_k 6y agoThis note represents a rare misstep by Dijkstra: the pigeonhole principle actually is a special and important principle worthy of a special name of its own. For example: 1. When you formulate the pigeonhole principle in propositional logic, its resolution proofs are exponentially large. Since modern SAT solvers are basically very fancy propositional resolution provers, this gives you a nice way to find hard instances for them. It also makes the question of when propositional proof systems can be more succinct than one another is also a fundamental question in complexity theory. 2. It also arises in geometry: a compactness for metric spaces is essentially the statement that the pigeonhole principle applies to the space. I don't have a unified perspective of these two facts, but either one of them is really striking. As a result I'm okay with giving the pigeonhole principle its own natural language name.
- ogogmad 6y agoDo you have a reference for 2? Thanks.
- neel_k 6y agoWillie Wong wrote a nice blog post about this a while ago: https://williewong.wordpress.com/2010/03/18/compactness-part-1-degrees-of-freedom/ https://williewong.wordpress.com/2010/03/18/compactness-part...
- tprice7 6y agoThe fact that every infinite sequence of points in a compact metric space has a cluster point is sort of like the pigeonhole principle. The grandparent comment in my opinion makes much more sense if the phrase "a compactness for metric spaces" is replaced by "sequential compactness for metric spaces".
- andrepd 6y agoSAT solvers overwhelmingly don't use resolution. Modern state of the art solvers use DPLL/CDCL.
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- starchild_3001 6y agoI (or any math noob) can easily understand the pigeon hole principle. His more detailed arguments about max, avg and a bunch of other stuff is more obtuse and harder to follow. IMO, pigeon hole principle, where applicable, wins this argument. If you're dealing with a sequence of real numbers, you might need max >= avg.
- bloaf 6y agoI agree. I've seen some marvelous pigeon hole explanations of why you cannot write a general lossless compression algorithm that makes all files smaller. I have no idea what that explanation would look like in terms of maxes and averages, and I have to believe it would be much more difficult to understand.
- atq2119 6y agoThat's really more about surjectivity and simple counting. Neither the pigeon hole principle not a max/average argument are really necessary. Say you're trying to find a way to compress every n+1 bit string into an n bit string. Losslessness means that you have to come up with a function d from n bit string to n+1 bit string that decompresses. The number of n+1 bit string is larger, so there must be n+1 bit strings that are not in the image of d, which means that such a function d cannot exist.
- Sebb767 6y agoI think he's missing two important points. Firstly, when you break down your proof on the pigeonhole principle, you've shown that the problem reduces to something simple and understandable. It's basically a nicer version of "q.e.d.". Secondly, and much more important: Integrating the principle allows the reader to understand the end of the proof more easily. I.e., if I find the principle mentioned in the end, I know that I need to look for "pigeons" and "holes" and that the proof is a way to get to those - allowing me to understand the proof from the conclusion going backwards, or at least helps doing so. Of course, this varies per person and depends a bit on how much you're trained in reading proofs, too. But it's enough to justify naming it in my opinion.
- impendia 6y agoStrongly agreed. I've taught university-level discrete math several times, and beginners to the subject need pegs to hang their hat on, something that serves as a goal and a signal that they're done. The Pigeonhole Principle is an ideal example of this. Conversely, in research papers, or in conversations among math researchers (at least in my discipline) the Pigeonhole Principle is seldom mentioned by name. The idea is considered too "obvious" to need a name.
- sn41 6y agoI disagree with this note, even though I often read Dijkstra's notes for enlightenment. The pigeonhole principle is one of the basic tools we have to approach mathematics, and therefore deserves a name. We often assimilate proofs by "abbreviating" the major steps through naming them - "pigeonhole principle", "passing to subsequences" etc. Even in the realm of "infinite" mathematics like analysis, theorems like the Heine-Borel Theorem and the Bolzano-Weierstrass theorem rely on pigeonhole-like arguments. Of course, that statement can be made rigorous in reverse mathematics.
- booleandilemma 6y agoThe whole metaphor of objects and compartments is just a pain in the neck I like the metaphor, it helps me to understand the principle in the first place. I doubt I'm the only one who feels this way.
- hbogert 6y agoI hardly ever disagree with the man, but as long as at least 50% of freshman get this wrong in introductionary lectures of fundamental logic, a named principle is very appropriate imo.
- HelloNurse 6y agoSince a proof is a proof, the methods are a matter of taste. Driving away from pontless reductio ad absurdum arguments is a good thing, but sometimes putting in the spotlight the object that need to be matched or counted instead of bare numbers can be useful. For instance, in the ranch example I'd ensure that I'm counting correctly by considering the cowboys and horses in the connected components of the bipartite graph whose edges represent exclusive horse ownership: each connected component contains 0 or 1 cowboys and 1 or more horses.
- FartyMcFarter 6y agoThe pigeonhole principle is about counting things. Counting things is one of the simplest things in mathematics. Much simpler than "averages", "real numbers" (a very complicated subject) and other things Dijkstra mentions in his alternative formulation. It seems like he's sacrificing simplicity and ease of understanding for other properties that aren't generally applicable.