## Canonical

One more comment on “canonical,” promoted to its own post because the non-mathematicians presumably stopped reading the other one very early on.

It’s common for mathematicians to use the word “canonical” colloquially, to mean something like  “a choice universally or at least generally agreed on.”  For instance:

The clock in Grand Central Station is the canonical place to rendezvous with people in midtown New York City.

I always thought of this as an outgrowth of the mathematical use of the word; but actually, there’s a bit of tension, because I think in this sense “canonical” almost always refers to a choice which is conventionally agreed on, and for which there might be a good reason, but which isn’t really forced upon you the way that canonical things are in mathematics.  The canonical rendezvous might just as well have been the lobby of the Empire State building.

I found a definition of “canonical” in a Hacker Slang dictionary which roughly agrees with this usage:

The usual or standard state or manner of something. This word has a somewhat more technical meaning in mathematics. Two formulas such as 9 + x and x + 9 are said to be equivalent because they mean the same thing, but the second one is in `canonical form’ because it is written in the usual way, with the highest power of x first. Usually there are fixed rules you can use to decide whether something is in canonical form. The jargon meaning, a relaxation of the technical meaning, acquired its present loading in computer-science culture largely through its prominence in Alonzo Church’s work in computation theory and mathematical logic (see Knights of the Lambda Calculus). Compare vanilla…

Anyway.  Non-math readers, would you ever use the word “canonical” in the sense described here?  Math readers, can you give an account of its colloquial usage more articulate than my own?

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## The different does not have a canonical square root

Just wanted to draw attention to this very nice exchange on Math Overflow.   Matt Emerton remarks that the different of a number field is always a square in the ideal class group, and asks:  is there a canonical square root of the ideal class of the different?

What grabs me about this question is that the word “canonical” is a very hard one to define precisely.   Joe Harris used to give a lecture called, “The only canonical divisor is the canonical divisor.”  The difficulty around the word “canonical” is what gives the title its piquancy.

Usually we tell students that something is “canonical” if it is “defined without making any arbitrary choices.”  But this seems to place a lot of weight on the non-mathematical word “arbitrary.”

Here’s one way to go:  you can say a construction is canonical if it is invariant under automorphisms.  For instance, the abelianization of a group is a canonical construction; if f: G_1 -> G_2 is an isomorphism, then f induces an isomorphism between the abelianizations.

It is in this sense that MathOverflow user “Frictionless Jellyfish” gives a nice proof that there is no canonical square root of the different; the slick cnidarian exhibits a Galois extension K/Q, with Galois group G = Z/4Z, such that the ideal class of the different of K has a square root (as it must) but none of its square roots are fixed by the action of G (as they would have to be, in order to be called “canonical.”)  The different itself is canonical and as such is fixed by G.

But this doesn’t seem to me to capture the whole sense of the word.  After all, in many contexts there are no automorphisms!  (E.G. in the Joe Harris lecture, “canonical” means something a bit different.)

Here’s a sample question that bothers me.  Ever since Gauss we’ve known that there’s a bijection between the set of proper isomorphism classes of primitive positive definite binary quadratic forms of discriminant d and the ideal class group of a quadratic imaginary field.

Do you think this bijection is “canonical” or not?  Why?