I know the definition of symplectic structure, symplectic group, and so on. But what does the word "symplectic" itself mean?

Meta question: I have many other mathematical words whose etymologies are obscure to me. Is it OK for me to ask one question per such word?

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    $\begingroup$ I'm personally lukewarm about this question, but that's just me. About your "meta question", for future reference, please ask them on Meta.mathoverflow (preferably before you ask questions here that you are not sure is appropriate). For this question (and other etymological ones), I've opened a meta thread: tea.mathoverflow.net/discussion/750/etymological-questions please discuss over there. $\endgroup$ – Willie Wong Nov 7 '10 at 11:59
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    $\begingroup$ Great question with interesting answers. What are the others? $\endgroup$ – Romeo Nov 7 '10 at 16:54
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    $\begingroup$ Most etymologies are not that hard to track down, so unless the answers contain some insightful mathematical discussion spun off the strict etymologies, I doubt they would add much value to MO. $\endgroup$ – Thierry Zell Nov 7 '10 at 17:12
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    $\begingroup$ Yuji- I would recommend against creating a large number of such questions. I would also recommend that you check obvious sources first, and only come to MO when you have been unable to find the information elsewhere. For example, this question is basically unsuitable on the grounds that it is quite adequately answered by Wikipedia. $\endgroup$ – Ben Webster Nov 7 '10 at 18:02
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    $\begingroup$ @Ben: you're right. I just checked Wikipedia... indeed there was an explanation! Stupid me. $\endgroup$ – Yuji Tachikawa Nov 8 '10 at 1:33

The term "symplectic group" was suggested in The Classical Groups: their invariants and representations (1939, p. 165) by Herman Weyl:

The name "complex group" formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word "complex" in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective "symplectic." Dickson calls the group the "Abelian linear group" in homage to Abel who first studied it.

Take a look at the Earliest Known Uses of Some of the Words of Mathematics web page.

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    $\begingroup$ "Abelian linear group" would have been quite confusing! $\endgroup$ – Yuji Tachikawa Nov 7 '10 at 11:59

The following is from page 1 of Lectures on symplectic geometry, by Ana Cannas da Silva:

As a curiousity, note that two centuries ago the name symplectic did not exist. If you consult a major English dictionary, you are likely to find that symplectic is the name of a bone in a fish's head. However ... the word symplectic in mathematics was coined by Weyl who substituted the Latin root in
complex by the corresponding Greek root, in order to label the symplectic group.

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    $\begingroup$ Interesting remark. And she wrote, very rightly, "the word symplectic in mathematics was coined by Weyl", as the adjective συμπλεκτικός did exist in classic Greek. Since Weyl received a humanistic education, no doubt he was quite reluctant to create artificially a new word. $\endgroup$ – Pietro Majer Nov 7 '10 at 21:16
  • $\begingroup$ I think this is the best answer here because the origin is stated. $\endgroup$ – Léo Léopold Hertz 준영 Feb 16 '16 at 14:49

There is a brief explanation here. It looks like the term was coined by Weyl, and was a result of modifying the Greek root "comp" from "complex" to the equivalent Latin root "symp". This is a pretty obscure way to coin a word if you ask me!

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    $\begingroup$ Words formied this way are called "calques" in general en.wikipedia.org/wiki/Calque $\endgroup$ – j.c. Nov 7 '10 at 12:39
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    $\begingroup$ -1 for saying that "comp" (or "symp") is a Latin root. "Complexus" is originally the p.p. of the verb complector (to embrace, thus, to put together into a whole etc), which is a compound of cum (with) and plecto, and that exactly corresponds to the Greek verb συμπλέκω, compound of σύν and πλέκω. Indeed Weyl did not "coined" the (already existing) Greek term, but only gave it a new mathematical meaning, in analogy to the (modern) mathematical meaning of the Latin term. I'm not sure if technically this can be called a calque. $\endgroup$ – Pietro Majer Nov 7 '10 at 15:20
  • $\begingroup$ @Pietro: I'm just quoting the source I referenced. Don't blame the messenger! $\endgroup$ – Jim Conant Nov 8 '10 at 7:06
  • $\begingroup$ Oops, I see there was a typo. I meant Greek! $\endgroup$ – Jim Conant Nov 8 '10 at 7:31
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    $\begingroup$ well, there is no such thing as a root "comp-". $\endgroup$ – Pietro Majer Sep 25 '13 at 0:59

The word "sum-plectic" as a greek translation of "com-plexus" was needed also to differentiate the study of "complex geometry" (complex numbers etc) from the study of "complexes de droites" (the geometry of 'line complexes') where the Plücker coordinates are associated to a natural symplectic structure on the space of affines lines in any euclidean space.

The concept of "differential symplectic geometry" has been introduced I believe by J.-M. Souriau in is 1953 paper

@inproceedings{Sou53, Author = {Jean-Marie Souriau},
Booktitle = {Coll. Int. CNRS},
Pages = {53},
Publisher = {CNRS, Strasbourg},
Title = {G{\'e}om{\'e}trie symplectique diff{\'e}rentielle. Applications.},
Year = {1953}}

He introduces there the concept of "Variétés isotropes saturées", called today "lagrangian manifolds", name given later by V.-I. Arnold.


From Pietro Majer's comments I learn that "symplectic" is the past participle of a classic Greek verb which means "to embrace".
Consequently I would just remark then how surprising is the effectiveness of this adjective to reflect the pervasiveness of the ideas from symplectic geometry in modern mathematics, which interconnect many different subjects.
Infact, from the introduction to the paper "Symplectic Geometry" by A.Weinstein, I quote:

I think it is not unfair to say that symplectic geometry is of interest today, not so much as a theory in itself, but rather because of a series of remarkable "transforms" which connect it with various areas of analysis.

  • $\begingroup$ On the basis you mention, the term "symplectic" should have rather been adopted for algebraic geometry instead! $\endgroup$ – Qfwfq Oct 19 '11 at 16:55

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