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Just exactly what the title says; often, in mathematics, particularly in the vicinity of Grothendieck, I see reference to "the yoga of...". What exactly does the term "yoga" mean in these contexts?

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Gian-Carlo Rota use the term when writing about all the different ways of defining matroids. –  Michael Hardy May 6 '11 at 1:28
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there is also a "Buddhist" tradition in naming of maths techniques, cf. "12-fold way" in enumerative combinatorics... –  Dima Pasechnik May 6 '11 at 3:53
    
In my opinion wherever you say "yoga" as it is usually (mis?)used in mathematics, you could actually say "calculus" as that word should be properly used (e.g., "Kirby calculus"). –  Pete L. Clark May 6 '11 at 4:00
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To my mind (and at the risk of getting discussion-y), the term "calculus" is more often used to refer to something more codified and constrained by rules (the propositional calculus, the predicate calculus, the calculus of relations, the lambda calculus, a sequent calculus, the calculus of dependent types, and even the differential calculus). We need a term for a body of techniques which has not yet been completely codified, and "yoga", whether you like it or not, is used for that purpose. This is at the level of description, not prescription (or proscription). –  Todd Trimble May 6 '11 at 13:37
    
Adding to Dima's comment, there is also the eightfold way in physics (and more particularly, representations of SU(3) as used in physics): en.wikipedia.org/wiki/Eightfold_Way_%28physics%29 –  Todd Trimble Jun 7 '11 at 14:33
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I've taken "yoga" to mean a part of the body of mathematics which does not consist of many actual theorems or results -- or in fact could not be formalized as just a few theorems -- but rather a collection of principles and techniques that one needs to wrap one's head around completely, after which one will be able to use them almost effortlessly.

As an example, I would say that there is a yoga of generating functions in combinatorics. (Perhaps this is the simplest example of a yoga.)

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I sometimes have used the word myself, without ever having sat down and asked myself what do I mean by that exactly. I've used it roughly to mean a coherent body of techniques; I'm not sure if I can amplify much further. "The yoga of adjunctions and mates", "the yoga of the Yoneda lemma and its correlates", and you will find a bunch more scattered around the nLab ("the yoga of 'generalized the' ") if you use the search function. To me, a "yoga" is not quite as formalized (or pretentious) as a "calculus", but it's somewhat in that vein.

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My views are always parochial, but let me say: of course it has nothing to do with any proper meaning of the word "yoga". And it always seemed to me that to refer to an approach as a yoga was to say that there was a different philosophy or different approach or different viewpoint being applied than had been current up to that time. All of this applies, of course, to what happened when Grothendieck suddenly revolutionized algebraic geometry. –  Lubin May 6 '11 at 1:29
    
I'd never heard that, but you may well be right. I'd be interested to hear from mathematicians other than disciples of Grothendieck if they use the word, and how. I can well imagine -- have no idea if it's the case -- differential topologists speaking of "the yoga of cancellations and rearrangements of critical points in Morse theory" or something like that, thus referring to an already established body of techniques. Hopefully more people will weigh in... –  Todd Trimble May 6 '11 at 1:40
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I would say that it is related in a playful figurative way to the original meaning of "yoga". The suggestion is of an esoteric and powerful body of knowledge or technique that it is worthwhile to master. –  Tom Goodwillie May 6 '11 at 1:48
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The online Merriam-Webster says one of the definitions of a yoga is a system of exercises and techniques for attaining bodily or mental control and well-being. I think it is not such a stretch for yoga in mathematics to mean a system of techniques for attaining mathematical control and well-being. I have certainly heard algebraic topologists use the term yoga to refer to all the tricks and games surrounding cohomology theories and the zoo of long exact sequences you get from them. –  Chris Schommer-Pries May 6 '11 at 2:29
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"Yoga" and "yoke" (as in of oxen) are derived from the same Indo-European root, meaning a linkage. Of course "linkage", and "relation", and "connection", and "join", all have mathematical meanings already, so one must go further afield to talk about two mathematical concepts being yoked to one another.

When I have seen the word used by mathematicians (esp. Bott), it is usually in exactly this way -- the yoga of X and Y, not of X. (As such I must disagree with the comment `of course it has nothing to do with any proper meaning of the word "yoga".')

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I thought of Bott, too; I remember that he used the word at times. But I remember it as X, not X and Y (whatever X may have been). And, whether inside or outside of mathematics, I don't see that the not-widely-known etymological roots of the word (at least in the English-speaking world) can have much to do with how people understand its meaning. –  Tom Goodwillie May 6 '11 at 5:16
    
As are the French "joug", the ancient greek "$\zeta \upsilon \gamma \omicron \nu$", etc –  Joël Mar 21 '12 at 15:57
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