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It is typical to find a corollary that following theorems, but is it right to use the word corollary for a statement following a conjecture, where the statement is true only if the unproven conjecture is true?

edited Apr 23 2010 at 15:32

Charles Stewart

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3 revisions, 3 users
mathnewb 100%

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Such statements are said to have "conditional proofs". en.wikipedia.org/wiki/Conditional_proof . I'm voting to close as "not a real question" – Gjergji Zaimi Mar 11 2010 at 4:15

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I think it's a reasonable question. – Noah Snyder Mar 11 2010 at 4:31

I don't like this business of voting down stuff, especially when I do like the question. – Anonymous Mar 11 2010 at 4:31

I didn't give it a vote down, the poster asked a clear question. The only debatable thing is whether this the right place to find the answer. Maybe this should be discussed at meta (or maybe there is a discussion already), but at least change the tag? :-) – Gjergji Zaimi Mar 11 2010 at 4:35

I retagged the question. – Kevin Lin Mar 11 2010 at 5:44

Douglas S. Stones · Accepted Answer · 2010-03-11 04:12:49Z UTC

I think it's generally bad form to have a corollary dependent on an earlier conjecture. I recommend one of the following:

Theorem: Assuming Conjecture A, properties X, Y and Z are true.

or

Theorem: Conjecture A implies X, Y and Z.

Most importantly, it should be crystal clear that the result is dependent on the conjecture.

David Speyer · Answer 2 · 2010-03-11 04:19:09Z UTC

I would write "Proposition Z: If X holds, then Y is true." Even if the deduction of Y from X were trivial, I think labelling this a corollary would be confusing. (After all, what is the statement "X implies Y" a corollary of?) However, I wouldn't have a problem writing something like "as we saw above, Y would be a corollary of X" later on. (The subjunctive voice is important here!)

Theo Johnson-Freyd · Answer 3 · 2010-03-11 06:16:29Z UTC

I'm reminded of the following story that I posted on my personal web journal a couple years ago:

At the Topology seminar yesterday, the speaker presented a theorem, which he immediately followed with a refinement: a statement that directly and obviously implies the theorem. He labeled his refinement a "corollary". I turned to Noah Snyder, and said that it was more an "uncorollary, or an anticorollary", but as soon as I said as much, the two of us simultaneously correctly labeled the refinement as a "rollary".

There should be more rollaries in mathematical writing.

Noah Snyder · Answer 4 · 2010-03-11 06:27:15Z UTC

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Making a new Theorem environment that let you have the bolded part say "Corollary to Conjecture X" seems to me a good compromise of concise and unlikely to confuse anyone.

answered Mar 11 2010 at 6:27

Noah Snyder

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I'm sure I've written stuff like "Corollary (of Conjecture X)" in class notes, but I think that in careful writing one should be even more clear about the conditional nature of a result than this. Strictly speaking it makes no sense for a conjecture to have a corollary, because the logical statuses are completely different. The more careful statement: "Proposition [or Theorem, or whatever]: Conjecture X implies Y" seems preferable. – Pete L. Clark Mar 11 2010 at 9:44

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I also think that there is room for improvement in the way mathematicians state conditional results. In some subfields it seems to be almost forgotten that certain standard conjectures are not known to be true, so you see things like "Theorem: Something amazing (conditional on GRH)" (which, by the way, is not even one conjecture but a bundled together family of conjectures) Or, to hit closer to home: "We show something fantastic relating analytic, Mordell-Weil and Selmer ranks (assuming the finiteness of Sha)". I say boo: put your assumptions first! – Pete L. Clark Mar 11 2010 at 9:48

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I actually think this is much clearer than "Prop: X implies Y." If you're skimming through you can't miss that "Corollary to conjecture" means it hasn't been proven, whereas if you have a proposition and you phrase it a little bit poorly a skimmer might think the conclusion had been proven. The clearest thing is to have the fact that it's unproven in bold. – Noah Snyder Mar 11 2010 at 17:00

Here's an example of when this construction is used: jlms.oxfordjournals.org/cgi/pdf_extract/s1-43/1/… – Douglas S. Stones Mar 13 2010 at 22:32

anton · Answer 5 · 2010-03-11 12:04:27Z UTC

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The correct term for such an item is CONJOLLARY. ;)

answered Mar 11 2010 at 12:04

anton

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Isn't that a relative of the Bandersnatch? – Yemon Choi Mar 11 2010 at 18:48

Can a corollary follow a conjecture?

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