In a paper I'm working on, I'm tempted to write something like:

Note that the argument above also proves the following result:

Scholium. bla bla

Is this ok? Is it correct to say that a "scholium" is a "corollary of a proof"?

closed as off-topic by john mangual, Lucia, Ben Linowitz, Alexey Ustinov, András Bátkai Feb 1 '17 at 11:56

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Lucia, Ben Linowitz, Alexey Ustinov, András Bátkai
If this question can be reworded to fit the rules in the help center, please edit the question.

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    Isn't "porism" better? – Bruno Stonek Jan 31 '17 at 21:00
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    I guess the issue is whether you want to impress your high-brow readers whilst confusing (or possibly attempting to educate) your low-brow ones, or whether you would rather use something slightly less precise which everyone will understand. There are arguments both ways. I would be keener to communicate effectively to more people; on the other hand if it hadn't been for Cassels I probably still wouldn't know what prolegomena were. – Kevin Buzzard Jan 31 '17 at 21:21
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    Use plain English! Many readers of scientific papers are not native English speakers, and they do us a favor by writing and reading in English. Why not go easy on them? Theorem, Lemma, Proposition, Corollary are enough! – Lucia Jan 31 '17 at 21:32
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    Doesn't scholium mean "commentary"? – Denis Nardin Jan 31 '17 at 22:21
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    Oxford English Dictionary: scholium, n., 1b: In certain mathematical works (e.g. Newton's Principia): A note added by the author illustrating or further developing some point treated in the text. (...) 1829 P. Barlow in Encycl. Metrop. I. 314/2 A scholium is a remark applied to some preceding propositions, in order to point out their relative connection, or general utility and application. – Denis Chaperon de Lauzières Feb 1 '17 at 7:36
up vote 25 down vote accepted

I am not a specialist in either etymology nor the english language (I am not a native speaker of english as well) but since the words scholium and porism have both greek origins, I thought it might be of some interest to add some info on how these words have been used in both ancient and modern greek:
The word "porism" comes from the greek word "Πόρισμα" and it indicates something which is a direct implication of the preceding statement. I think the closest in english is corollary. (In non-mathematical contexts, the word πόρισμα also means the conclusion of some work).
The word "Scholium" comes from the greek word "Σχόλιο" and it it indicates something which although may be very closely connected to the preceding statement, it does not necessarily stem directly from it (neither logically nor conceptually). In this sense, a scholium may indicate some resemblance with another notion or method from some other field or some distant application, even some piece of info on the origins of the preceding result or its importance from a more conceptual viewpoint. In mathematical texts (in greek) the word "σχόλιο" is often used to discuss something related to relaxing the assumptions of the previous statement or indicating its limitations, under the stated assumptions. (A working translation might be "comment", but i think that in greek it is commonly used to indicate something more important than simply a comment -however I am not a philologist to tell for sure).

So, in my opinion, if you want to discuss something which is not a direct implication of your statement but it is proved using methods similar to the argument(s) provided to prove your statement or if you want to provide additional insight then the word "scholium" might be appropriate. If however, you wish to simply present some consequence of a line of argument you have already used, "porism" or "corollary" seem more appropriate -as other users have already indicated in their comments. In case you decide to use it, it would be better to avoid bold letters.

Edit: Since the OP's original question is how (and if) the word "Scholium" should be used in a mathematical paper, I feel that the question is of interest to the community of professional mathematics researchers. After all, the question has to do with the way a mathematical research paper is written and structured. However, the community will finally respond, one way or another. Maybe, it would also be instructive (with regard to the OP's original question) to have a look at how the term "scholium" is used in this edition of Euclid's Elements. (see for example the scholia in p.104).

Edit-2: Maybe it would be also of some interest to add that in greek, the word "Σχόλιο" has the same root with the greek word "Σχολείο" which means "School". (as has already been mentioned in a comment above, by user Pietro Majer).

  • Yes, this use of the word "scholium" from this edition of Euclid's Elements is, it seems to me, precisely what several commenters have tried to explain under the OP: a scholium is an explanatory note, a scholarly annotation, amplification, etc. It seems to me this is not precisely what the OP was asking about, which is about a further consequence of a line of argument used in a mathematical proof. (This is not to say your answer isn't useful for other reasons -- just that this additional piece of evidence does not support "scholium" as the sought-for word.) – Todd Trimble Feb 2 '17 at 1:10
  • @Todd Trimble, i generally agree. My original intention was not to encourage OP to use the term "scholium" but rather to provide some indications for the boundaries. Maybe I was not clear enough, so i've edited, hoping to be more concrete. – Konstantinos Kanakoglou Feb 2 '17 at 1:57
  • Ah! I hadn't read carefully your recent addition right before the bold-faced edit, where you spoke of "porism" (I thought you were recommending "scholium" in some of the previous revisions). So now I agree with your answer; thanks for the work you put into this. – Todd Trimble Feb 2 '17 at 2:20
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    By the way: this question, while at the moment closed, can't be "deleted", except under unusual circumstances as explained here: meta.stackexchange.com/a/5222. – Todd Trimble Feb 2 '17 at 2:54

Bourbaki defines a "scholie" in the preface of the Éléments de mathématiques as follows:

Sous le nom de « scholie », on trouvera quelquefois un commentaire d'un théorème particulièrement important.

I.e., a scholium (for Bourbaki) is a commentary to a particularly important theorem.

There aren't many in the treaty, but those that are seem to be non-mathematical, or meta-mathematical: essentially, they are a guidance on how to use the theorem or when to apply it, or an indication of a general proof technique. Something like "this theorem is useful for deriving results about foobars from the general theory of bazquxes by applying the frobnification functor and using the theorem to transfer the property". Not something formalizable as a mathematical statement.

Examples of scholia in Bourbaki are in A IV §2 nº3 or FVR VI §1 nº1 or INT IX §1 nº8. There is also one in AC VIII §3 nº3, but the latter seems to be just a corollary (of a corollary), so apparently younger Bourbaki authors didn't get the memo on what a scholium was supposed to be.

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    This looks helpful, but (pardon my ignorance) what do 'A', 'FVR', and 'INT' signify? – Todd Trimble Feb 1 '17 at 6:52
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    @ToddTrimble those are abbreviations of titles from the French: A = Algebra, AC = Commutative Algebra, FVR = Functions of a Real Variable, INT = Integration – KConrad Feb 1 '17 at 7:04
  • (presumably you mean "treatise" not "treaty".) – Noam D. Elkies Feb 4 '17 at 3:41
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    @NoamD.Elkies Indeed! Sometimes it shows that English is not my native language. I'll leave this uncorrected, however, so as not to uselessly bump this question to the front page. – Gro-Tsen Feb 5 '17 at 17:44

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