Timeline for Terminology: A "corollary" to a proof?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 30 at 12:30 | comment | added | Emil Jeřábek | (dedication -> dedicated) | |
| Jul 30 at 12:25 | comment | added | Emil Jeřábek | ... However, this can be remedied by telling the reader where to find the definition. So, if an object called Spot is introduced in the middle of the proof of Theorem 2.5, then “Corollary 2.6: Spot never barks” is bad, but “Corollary 2.6: The Spot defined in the proof of Theorem 2.5 never barks” is acceptable. | |
| Jul 30 at 12:22 | comment | added | Emil Jeřábek | @SamNead Ah, I see. That sounds bad, indeed. Though I’d say it’s not limited to proofs; more generally, definitions and notation should be introduced at places where the reader can easily find them if they read casually skipping parts of the paper, of if they read the second half of tha paper a week after the first half, etc.: ideally in a dedication section with definiitons and preliminaries, or grouped together at the beginning of a section, or at the very least clearly marked as numbered definitions close to the place of their usage, not made in passing and scattered all over the text. ... | |
| Jul 30 at 11:44 | comment | added | Sam Nead | @EmilJeřábek - Hardy (the OP) writes (in comments, above) that "A difficulty is that the 'corollary' uses notation first introduced in the proof, not in the theorem." | |
| Jul 30 at 11:35 | comment | added | Emil Jeřábek | @SamNead If a reader skips all proofs anyway, then it should not matter to them whether something has a separate proof, or is claimed to follow from a previous proof. So what is the problem? | |
| Jul 30 at 11:29 | comment | added | Sam Nead | @EmilJeřábek - I am thinking of a reader (perhaps myself...) who skips all of the proofs and only reads definitions and theorems (and corollaries). | |
| Jul 29 at 8:21 | comment | added | Gro-Tsen | @EmilJeřábek Of course the reader will not really forget everything about the proof as soon as they reach the “end of proof” symbol. What will really happen, however, is that many will read the statement, think “oh this is what I care about”, and then read the proof somewhat carelessly because they're only paying attention to what they decided they cared about (e.g., how a particular object is constructed); telling them ex post facto “oh you should also have cared about this other part” can seem rude. | |
| Jul 29 at 6:33 | comment | added | Emil Jeřábek | People are not robots. If you state the "corollary" right after the "proof", which is how it is usually done, the reader has not yet forgotten anything, even if they believe in this "social contract". It's really like "before we forget about this proof, here's one more thing to note". | |
| Jul 28 at 23:20 | comment | added | David Roberts♦ | @LSpice oh, I'm not saying readers benefit, but the number of proofs that claim to show mere existence, but the proof is to give a construction of the thing, is surely nontrivial. If people could be trained into giving the definition/construction first, and then proving it satisfies the required conditions, we would all be better off :-) Referring to notation introduced in a proof is I agree not a great practice. | |
| Jul 28 at 23:05 | comment | added | LSpice | @DavidRoberts, re, to be sure, writers do this, but do readers benefit from it? I find it hard enough to read the best-written paper, and each abrogation of the sort of contract defined here makes, or risks making, it still harder for me. But I have had it driven home to me by working with different co-authors that one reader's meat is another reader's poison, so I ask genuinely, not rhetorically. | |
| Jul 28 at 23:01 | comment | added | David Roberts♦ | This 'contract' only holds among those who assume proof-irrelevance holds always. This is 'officially' what people do, but in reality they do things like what the OP is wanting: state a theorem that a thing exists, the proof constructs such a thing, and then they later use the construction. Or they extract the method of the proof and turn that into a definition. And so on. | |
| Jul 28 at 22:32 | history | answered | Gro-Tsen | CC BY-SA 4.0 |