What is a "scholium"?

Question

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. etc.

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

Answer 1

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).

Answer 2

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 treatise, 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.

Answer 3

There is a word in Spanish "Escollo" that I think is analogous to the word "skerry" in English. Many words that in latin have the sound --li-- where transformed in Spanish to the sound --ll--. So, it is easy that the word escolio in latin transform into escollo in Spanish. I know scolium is a greek word, but the rules of phonetic transformation are broader.

So to my ears "scholium" always sound to "escollo". And the meaning is perhaps very clear also. Escollo (similar to skerry in English) means a rocky island which is almost at sea level or under sea level. So that it is a danger to navigation. More generally means a danger or risk.

This is exactly how it is used in the book by Euclides. @Lucia in a comment say Theorem, Lemma, Proposition, Corollary are enough! but I respectfully disagree with this. Bourbaki invented a symbol to represent the pitfalls in his magnum opus: the one with the dangerous curve.

This may not be the correct etymology, but it should be.