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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I use the term canonical in everyday usage slightly tongue-in-cheekily — saying “Jupiter is the canonical bar to go to on a night like this” means “My choice is so good that no reasonable person could disagree with me”.

I can’t quite think of an example, but it seems to me that there must be examples in math where the usage is closer to this – where one construction is canonical but only if you care about the geometry (pizza), whereas a different one is canonical if you care about the algebra (beer).

I think JSEs definition is pretty much how mathematicians use it outside of mathematics. Some other typical example are:

Jupiter’s Pizza is the canonical place to go for dinner after topology seminar.

The canonical hotel to stay at for the Georgia Topology Conference is the Downtowner Motor Inn.

In both cases, “canonical” means the default choice, possibly because it’s the best one through maybe just because it’s the default.

I’ve never heard this use of “canonical” so obviously I don’t spend enough time with mathematicians.

Nick’s comment brings up an aspect of the use of the word by mathematicians that may connect to the use of the word by the rest of us–the question of authority. It sounds like when saying “this is the canonical pizza place” one is being a little self-mocking about invoking a weighty term of authority for something seemingly unimportant.

In contexts like literature and religion (where the term starts), canonicity has

something to do with authority (originally authority of the Church or some other kind of leadership; more loosely authority of a community) that imposes itself upon the individual functioning in a certain context. Moshe Halbertal suggests three types of canonicity based on his study of canon in Jewish texts and pre-modern Jewish tradition (but as a general typology for other cultures as well): a normative canon (i.e. texts/ideas that set law); formative (texts for a curriculum–the classics you should read to be an educated person); and exemplary (texts that you should imitate in your own writing). The formative and the exemplary are often pretty closely related; and if it’s a law-based culture, then there is a lot of overlap between all three categories.

So tell me if any of this seems to explain how the term gets used mathematically and then how it went into non-math conversation of mathematicians.

Another way to put it: canonical originally means “norm” and perhaps that slides over to “normal”, i.e. usual, default, in the colloquial sense.

Adam, you’re definitely right about the self-mocking aspect of “this is the canonical pizza place”.

As for the mathematical usage, being canonical doesn’t seem to have much to do with authority. In some sense it’s actually the opposite, something is canonical if it doesn’t depend on any choices and is somehow forced by the situation.

A simple example: Consider the letters a, b, c and numbers “1”, “2”, “3”. There are 6 ways we can pair these up, e.g. (a, 1), (b, 2), (c, 3) or (a, 3), (b, 2), (c, 1) or (a, 2), (b, 3), (c,1). However, only the first preserves the usual way that we order letters and numbers. Thus a mathematician might say that (a, 1), (b, 2), (c, 3) is the canonical order-preserving pairing between a,b,c and 1,2,3.

Nathan, the (a)(b)(c)(1)(2)(3) seems like it has something to do with formative/exemplary authority (see above). Sure, you’re free to do (a2) (b3) (c1) but you’re bucking the usual custom that has been inculcated in the community of mathematicians and doesn’t look elegant according to the conventions of how that community does things. So unless you have a really good reason or you want to stir things up, you stick to the “canonical order-preserving pairing.”

So it would be authority of the community as opposed to authority imposed by some kind of math pope.

Adam, certainly the ordering of the letters and, to a lesser extent, the numbers comes from (community) authority. Instead, we could consider two ordered lists A = [“red”, “car”, “bat”] and B = [“set”, “get”, “sit”] and while again there are 6 pairings, there is only one order-preserving pairing: (red, set), (car, get), (bat, sit), which one might then refer to as canonical.

However, you’re also correct that if I didn’t have any reason not to, I would likely stick with the canonical order-preserving pairing. I knew this was a weakness of my example, but I couldn’t think of a simple example of canonical that didn’t have this feature. (The usual example is a finite-dimensional vector space is canonically isomorphic to its double dual but only non-canonically isomorphic to its dual.)

The thing is, if some paper I’m reading says something like “consider the canonical map”, and I’ve never heard of this particular map before, then I have a fighting chance to figure out what it has to be without consulting any other text. E.g. if I hadn’t told you what the “canonical order-preserving pairing” was, you could have selected it just from knowing it was supposed to respect the orders.

Of course in mathematics there are other technical uses of “canonical”, e.g., canonical coordinates in symplectic geometry and Jordan canonical form of a matrix. It seems to me that both of these examples are probably closer to the original meaning of “canonical” — dictated by canon — than the current mathematical meaning of “canonical” — independent of arbitrary choices. I guess that is also the meaning in the Hacker Slang dictionary — some standard choice, perhaps dictated by an authority. Even in mathematics we use the non-mathematical meaning, e.g., “Hartshorne’s definition of smooth morphism is not the canonical definition from EGA.”

Just this morning I used a variation of the Jargon File definition:

http://catb.org/jargon/html/C/canonical.html

where what I referred to as the “canonical example” of a particular code feature’s use was the case that was most obvious to use as an illustration. (If there were more than one person training people and answering questions, it might be “the example that everyone uses”, but as I am the sole developer that does not quite fit.)

Datum: I’m visiting a math colleague in Singapore, and yesterday he said (without a trace of self-consciousness) that he was taking me on the canonical walking tour of the city.

Many people in literary studies try not to use “canonical” at all, for anything, since the “canon wars” of the early 1990s made the very word a source of controversy. The Hacker Slang and math-world conversational uses certainly make sense to me,though, and don’t seem too far removed from the familiar meanings in literary studies: normal, normative, recommended, usual, established (as by habit, which becomes a kind of common law). Canonical poems are the ones you expect (if you know the field) in anthologies and on syllabi: canonical pizza joints are the ones you expect (if you know the city) to see if you’re going for pizza there. And canonical books of the Bible, the ones in the canon (the first common meaning for “canon” in English, I think), are the ones you expect to see, should see, will see, in the correct, normal, authoritative versions of that book.