Magma is one of those beautiful words of Greek origin (μάγμα) that arouses the child and the wild in me, making me think of volcanoes. I just found out, though, that it is also used in mathematics to mean a type of algebraic structure (a set paired with a binary operation on it)! I am very curious how this name was picked for this particular algebraic structure (which is also called by groupoid). Wikipedia attributes the coining of this term to Nicolas Bourbaki, but does not mention how they arrived at such a name. I have not been able to find the specific etymology of this sense anywhere else. What is its origin?
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3Some coinages have no 'how' or 'why' beyond the idiosyncratic artful randomness of the individual author. There is surely some answer in the Bourbaki exegesis, but any such explanation would be idle speculation even on the coiner's part. How do we know what they were thinking when they made it up? Why 'group', 'ring', or 'field'? This might better be asked in math.SE as a history question.– MitchApr 4, 2012 at 13:27
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2so the word 'group' tells you intuitively that it is a binary operation, with identity and inverse? or that a field is a commutative ring whose multiplication is commutative over non-zeroes? Most likely not. Anyway, your initial explanation, though fanciful, is probably on target for the original justification (modulo the arbitrariness).– MitchApr 4, 2012 at 13:45
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2"Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different." -- Johann Wolfgang von Goethe (1829)– John LawlerApr 4, 2012 at 15:29
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1I thought it was chosen because magma has very little structure.– MBNAug 3, 2012 at 8:22
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2Answered at mathoverflow.net/questions/103128/…– JBHOct 28, 2017 at 19:24
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4 Answers
It may be a pun. Looking up magma in the French wikipedia*, another name for magma in French is groupoïde de Ore†. Here Ore is a Norwegian mathematician, but ore in English is mineral-bearing rock, whereas magma (in both English and French) is molten rock.
Would Bourbaki have based a mathematical term on this pun? I'm not in a good position to judge; maybe somebody else could comment on this.
* Since Bourbaki was a pen name used by a group of French mathematicians, this is the right language to search in.
† There is also a groupoïde de Brandt, also called a groupoid in English, which would explain why Bourbaki felt compelled to coin a new name.
answered Apr 4, 2012 at 15:27
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Wikipedia says (without reference)
In French, the word "magma" has multiple common meanings, one of them being "jumble". It is likely that the French Bourbaki group referred to sets with well-defined binary operations as magmas with the "jumble" definition in mind.
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3Larousse says the figurative definition of magma is Mélange confus, inextricable de choses abstraites and gives the example usage "Ces propositions constituent un magma incohérent." Looking at some instances of its use, I would say the English definition would be something like confusing, inseparable, and worthless jumble; this isn't a bad name for the algebraic structure, although possibly the pun from my answer may have also influenced Bourbaki's choice of name. Apr 5, 2012 at 15:24
Google says of the etymology of magma
late Middle English (in the sense ‘residue of dregs after evaporation or pressing of a semi-liquid substance’): via Latin from Greek magma (from massein ‘knead’).
I have a hard time imagining an algebraic structure with less structure than a magma, so "residue of dregs" seems fitting. Once you boil away all the structure, you are left with a magma. Historically, I have nothing to suggest that this is accurate.
I am expressing my personal view without a historical excursus. Geological magma associates with a melted, liquid, formless substance. 'Algebraic magma' requires only the closure property: result of an operation applied to a set is in the set. Citation: Bourbaki, N. Elements of Mathematics, Algebra I, Chapters 1-3. Paris: Hermann, Massachusetts: Addison-Wesley, 1974. Paragraph LAWS OF COMPOSITION, p. 1, Definition 1. "Let E be a set. A mapping f of E x E into E is called a law of composition on E. The value f(x, y) of f for an ordered pair (x, y ) [belonging to] E x E is called the composition of x and y under this law. A set with a law of composition is called a magma". As for identity, inverse, associativity, commutativity, then magma does not require them. Pretty unstructured! One begins gradually and in different combinations adding identity and/or other properties. This embeds "some structure": semigroup, monoid, group, ... The 'algebraic magma' is crystalizing! The origin is a "bouillon" from which the crystal of life is growing. I am not sure who has applied the term first time but it is a very impressive comparative image. A poet could envy.
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My answer is for Nigel. I would like to cite the magma definition from Bourbaki, Nicolas. Elements of Mathematics, Algebra I, Chapters 1-3. Paris: Hermann, Massachusetts: Addison-Wesley, 1974. Paragraph "LAWS OF COMPOSITION", p. 1, Definition 1. Let E be a set. A mapping f of E x E into E is called a law of composition on E. The value f(x, y) of f for an ordered pair (x, y ) [belonging to] E x E is called the composition of x and y under this law. A set with a law of composition is called a magma. Only closure is required for magma - pretty unstructured like geological magma. Oct 28, 2017 at 20:44
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Sorry, I meant can you put the citations into the text of your answer, please. Thanks.– Nigel JOct 28, 2017 at 20:57
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Nigel, I have edited the answer adding citation from the original source discussed on this thread - Nicolaus Bourbaki. Geological meaning of magma is cited in one of the Peter Shor's message. I simply expressed my personal opinion that with respect to the "structure" both concepts are wittily similar. Best, Valerii Oct 28, 2017 at 21:56