What Does 'Canonical' Mean in a Mathematical Context?

[Flubertin]

Hi this is a rather gentle question in that it involves no actual mathematics!

Text often add rather strange words to a mathematical discussion. One such word that I have never really got to the bottom of is Canonical. So for example we talk about classical canonical general relativity, canonical co-ordinate systems, canonical variables etc.

What exactly is meant by canonical in this sense.

Regards.

 

[caffeinemachine]

Hi this is a rather gentle question in that it involves no actual mathematics!

Text often add rather strange words to a mathematical discussion. One such word that I have never really got to the bottom of is Canonical. So for example we talk about classical canonical general relativity, canonical co-ordinate systems, canonical variables etc.

What exactly is meant by canonical in this sense.

Regards.

The word canonical is in my experience used rather loosely in mathematics and I suppose the best thing to do is not worry about it too much.

On the other hand, I take the word "natural" far more seriously. For whenever I encounter a usage of "natural", there are always two functors lurking around which have a natural isomorphism (standard concept in category theory, and a very important one) between them.

Perhaps other MHB members should weigh in on this.

 

[HallsofIvy]

More generally, "canonical" refers to things that are "in Canon" meaning in the "accepted text". It is most often used in the Christian religion where something is "canonical" if it is found in the Christian Bible or derived immediately from it.

But you will also see thing in, say, a discussion of Shakespeare, where "canonical" refers to quotations or ideas that come directly from the text of his plays.

In mathematics, something is "canonical" if it comes from the universally accepted definitions.

 

[I like Serena]

The word canonical means the obvious (choice).

For instance, if we start from a 3-dimensional coordinate system, and project the 3rd coordinate to zero, the canonical coordinate system of the image is a 2-dimensional coordinate system formed from the first 2 coordinates.

Or the other way around, if we start with a 2-dimensional coordinate system, and define a transformation that injects it into a 3-dimensional coordinate system, the canonical transformation is the one that sets the 3rd coordinate to zero.

The word canonical looks as if it's a really special thing that only advanced mathematicians have a slight chance of understanding, but nothing is less true - it's just the obvious thing.