# Who first called the Grothendieck's schéma scheme?

Grothendieck called "schemes" schémas in French. I find it strange we call them schemes. In fact, Grothendieck called them (pre-) schemas(this is an English word) in his talk(in English) at Proceeding of the international congress of mathematicians at Edinburgh in 1958. There is an English word schema which means almost the same as French schéma. I think English words schema and scheme have different meanings and nuances although they are similar. So who first called Grothendieck's schéma scheme in English? And why he did not call it schema?

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Interesting etymology question. Although this is not a proper answer, maybe "scheme" was preferred to avoid overlapping with schema/schemata of axioms in logic? In fact, as Zhen Lin remarks in his comment, the use of scheme/schema in logic is done without distinction so the same should be possible in algebraic geometry. In French, Italian and Spanish I have always seen "schéma/schema/esquema", in books, classes, lectures... which are closest to its Greek origin σχῆμα / skhễma (« manière d'être », « forme », « figure », « extérieur », « apparence », « faux-semblant »). For example in the classic Grothendieck's EGA and SGA, or Perrin's modern French book (and many other references), they use "schéma/schémas". In German they also say "schema, schemata". But in French there is also "schème", so maybe in the original works and translations, one was more favored than the other. The differences are subtle just as their English equivalents:

• Schème: disposition, forme; abstraction, concept, objet, représentation; structure, mouvement d'ensemble d'un processus, d'un objet; beaux-arts, ensemble de ce qui fait le style; (philosophie) représentation qui se situe entre les données de la perception et les catégories de l'entendement.

• Schéma: système, processus selon lequel un phénomène se produit; représentation, structure, diagramme, plan; figure explicative simplifiée; présentation réduite aux traits essentiels.

Since in most of the other languages, at least in Europe, "schéma/schema-schémas/schemata" are the most used, I think there should be no problem to do the same in English as synonymous without changing meaning.

Nevertheless "schème" has a particular specific meaning in Philosophy since old times: Immanuel Kant's defined schèmes as processes or means by which a pure concept becomes effective by the implication of an intuition. In French, it has been used since then in philosophy and psychology in the form "schème" instead of "schéma", and is still used so nowadays.

Finally, from a purely linguistic and etymological point of view, the most natural correspondence between English-French would be "schème $\rightarrow$ scheme" and "schéma $\rightarrow$ schema". As I commented in the other answer by Lubin, maybe the phonetics of pronouncing [skiːm] seemed more natural to English speakers than ['skiːmə]. Despite being pronouced "sh" in French, in German, Spanish and Italian the sound is the same as in English [sk] and we nevertheless stuck to the [-a] ending.

My personal opinion, given all the above remarks, would be that "schema/-s-ta" are the best suited form even in English for being Grothendieck's original choice in French (despite having "schème" at his disposal! and despite the sociological evolution in conferences as Lubin recalls in his answer), and because it would then coincide with the uses in other languages and avoid other meanings of "scheme" (see the funny commentary of Asaf below). I personally always use "schéma/-s" in the same way as people use "étale topology/cohomology" and "étalé space" instead of the possible translations "calmed topology/cohomology" and "spread-out space" (which would sound to me as weird as other perverse names, like "perverse sheaf"). But maybe there are deeper reasons, for example, in Romance languages we use "valeur propre/valore proprio/valor propio" where the English speaking world took mainly the Germanic analogues "eigenwert/egenwerd", but I have seen both uses in English also "proper values = eigenvalues".

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Sometimes people refer to axiom schemas as "axiom schemes", so either way... – Zhen Lin Nov 25 '12 at 12:19
@Zhen: The replacement scheme? Like in the Invasion of the Body Snatchers? – Asaf Karagila Nov 25 '12 at 12:30
I have never seen "proper values" used in English -- "characteristic values" maybe once or twice. There's also the unfortunate fact that "a schema" sounds quite similar to "ischemia." I'm not sure if this would have deterred others, but I would have noticed. – Andrew Nov 25 '12 at 17:30
@Andrew: you are right "proper values" is probably used incorrectly only by non-native speakers of English who have that form in their native tongue. I have certainly heard it in lectures from non-English speakers in Europe. About "ischemia", well, there can be similar arguments by different associations, in Spanish we say "esquema", "isquemia" or "esquimal" and such similarities never crossed my mind until now. I guess it becomes very subjective at some point. – Javier Álvarez Nov 25 '12 at 17:41
Dear Javier, schème is an extremely technical word used in philosophy and nobody actively uses in French. I would say that it doesn't even belong to the passive lexicon of 99% of native speakers of French (I was about 30 years old the first time I saw it and I don't consider myself illiterate...) – Georges Elencwajg Nov 25 '12 at 20:13

Let’s see whether I can get my memory working right on this. It tells me that I was in the seminar that G gave at Harvard in 1958-59(*), I was a second-year grad student then. He talked about schemes, of course, and he did so in English. He certainly must not have called them “schémas” (pronounced shayma in French), because none of us called them that in discussions. So I hope I’m right in saying that G himself called them “schemes” from the get-go. And he would have made this decision, I’m fairly confident, on the advice of Tate.

(*)Maybe it could have been 1959-60; but those are the only two possibilities.

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I see... But from the purely linguistic point of view, would not be "schème-> scheme", "schéma->schema" the natural translation correspondence? I am wondering whether the choice reduces to phonetics, meaning that maybe [skiːm] seemed more natural to English speakers than ['skiːmə]. Despite being pronouced "sh" in French, in German, Spanish and Italian the sound is the same as in English [sk] and we nevertheless stuck to the [-a] ending. – Javier Álvarez Nov 25 '12 at 16:39
The year before that, I took a logic course with Quine, and he used the word “schema”, even giving the plural “schemata” the correct (greek) accentuation on the first syllable. So it’s not as if the word was unknown to mathematicians at that time. My own analysis is that Tate is a down-to-earth kind of guy, and didn’t want to go fancy-pants. – Lubin Nov 25 '12 at 18:33
“Schema” is a rare and exclusively scholarly word; “scheme” is an everyday word. – Lubin Nov 25 '12 at 20:02
Dear Lubin: ah, that changes everything! This theological explanation is both amusing and very impressive. And to tell the truth, my question was only meant as a little good-humored teasing. As a matter of fact, I am quite happy about the presence here of a mathematician of your calibre and I always read your posts with much interest and pleasure ... – Georges Elencwajg Nov 26 '12 at 0:10
In the Grothendieck-Serre correspondence one can find two letters by Grothendieck from October 17 and and November 5 1958, e.g. here indicating that he was in Harvard. These letters are also interesting because they contain some preliminary table of contents for the "hyperplodocus" EGA and some ideas on fundamental groups. – commenter Nov 26 '12 at 4:29