Exploring the Meaning of 'Canonical' in Mathematics

[precondition]

I'm not exactly sure if this is the right place to post this, but assuming it is, what is the meaning of 'canonical'?

Someone told me that roughly speaking, it means "given from God" or something like that, when I look up wikipedia it says "standard" etc, I read from books that it means "without basis", but in some occasions I hear two vector spaces are canonical, i.e. vector space V and it's double dual.

So this term seems to have a wide variety of use which I'm not happy with since mathematics should be using precise definitions.

 

[radou]

Mathematics is using precise definitions, if you precise the context.

 

[matt grime]

Why does having a wide variety of (precise) uses make you unhappy? Canonical, can mean one of (probably) several things. Frequently it should be taken to mean 'without any choices having to be made' or 'independent of any choices made'.

V and V** are NOT canonical (canonical is an adjective; you are implying that V is a canonical vector space, and V** is a canonical vector space). If V is fin. dim. then they are canonically isomorphic: there is an isomoprhism which does not depend on picking a basis.

If v is in V, and f in V*, then define an v** in V** by v**(f)=f(v).

Compare this to the non-canonical isomorphism of, an n dimensional vector space over k with k^n: pick bases {e_i} of V, and {f_j} of k^n, and send e_r to f_r.

 

[precondition]

V is a canonical vector space, and so is V*

above sentence still confuses me because how could you incorporate the meaning: 'without any choices having to be made' or 'independent of any choices made' then?

i.e. V is a vector space without any choices of basis having to be made?
is that what you mean?

So it seems that you can use 'canonical' to describe an object(V or V*) but also to compare two objects(V and V*) right? I was referring to this when I said definitions not being precise. Also is there any text in which it formally defines the meaning of 'canonical'? I have not seen one yet.

 

[matt grime]

Perhaps I should have specfically said that the phrase

"V is a canonical vector space"

is nonsense. I would have thought hat me saying

"V and V** are NOT canonical"

would have made you realize that we do not say "V is canonical".

There is a _canonical isomorphism_ between V and V** (for V finite dimensional). One where no choices (a basis in this case) have to be made.

The meaning of canonical is precise in any context where it is used. But there are different contexts, and therefore different meanings of the word. Just as there are different meanings of the words prime, simple, irreducible, divisible, finite and many more. Though they all have the same, or vaguely similar connotation: a set is finite if it contains finitely many points, a map is finite if its kernel is a finite set of points.

The rough connotation of canonical in the sense of the above example is 'independent of any choices', or 'innate'. The meaning of canonical in 'the canonical bundle' is an example of the 'innate' notion. (C.f. your "God given" comment.)

 

[D H]

Words used in math and science are often borrowed from the common use. "Given from God" is one of the non-technical meanings of "canonical". Specifically, "appearing in a Biblical canon".

I usually take it to mean "basic" or "simplest", another common (non-technical) meaning of the term. This common usage carries over to meany technical fields. For example, the canonical equation of a circle in Cartesian coordinates is $x^2+y^2=r^2$ (or maybe even the more basic equation, $x^2+y^2=1$).

Mathematicians and physicists have yet other meanings: canonical decompositions in math, canonical variables in quantum physics, canonical ensembles statistical physics. Each of these concepts has very precise definitions.

 

[mathwonk]

I think it also sometimes means approved by the head of the church of england.

but the last post is vlosest to the actualy mathematical menaing, i.e. "god given", or "dieudonne' ".

for example, of all hypersurfaces or "divisors", that live on an algebraic variety, the only really natural ones are those which are zero loci of differential forms, hence those are the "canonical divisors".

 

[mathwonk]

there are some attempts to give a precise menaing to the term, such as "functorial" for canonical mappings like V -->V**.

 

[mathwonk]

i did it again!

i wrote a complete elementary treatise on functors and natural traNSFORMATIONS, in the special case of V,V*, V**,

AND JUST BEFORE FINISHING, THe BROWSER DROPPED IT ALL.

man, until i get smarter, i would really aprpeciate a more foolproof browser here. but of course that is putting the responsibility on the wrong shoulders. my bad.

all i need do is learn to always go to advanced mode. but it seems tough that the quick mode actually erases the entire post when it gets too long.

 

[complexPHILOSOPHY]

i did it again!

i wrote a complete elementary treatise on functors and natural traNSFORMATIONS, in the special case of V,V*, V**,

AND JUST BEFORE FINISHING, THe BROWSER DROPPED IT ALL.

man, until i get smarter, i would really aprpeciate a more foolproof browser here. but of course that is putting the responsibility on the wrong shoulders. my bad.

all i need do is learn to always go to advanced mode. but it seems tough that the quick mode actually erases the entire post when it gets too long.

That happens to me a lot because I also forget to use the advanced feature and then I lose interest in my post (because I don't feel like wording it again). Perhaps if you are trying to type something that is long and important, would it be easier to type in like a word processing program and then copy and paste it into the reply?

Just a suggestion, my friend.