the-moti:

nostalgebraist:

 There’s something a little mysterious to me about the usage of “the” vs. “a/an” in math.  It seems related to a difference which comes up when we’re characterizing mathematical entities through their properties:

1. Sometimes we want to make statements that apply to every thing that has these properties, even things that also have some other properties we haven’t mentioned (”a/an”)
2. Sometimes we’re trying to single out an object characterized by these properties and nothing else (”the”)

After thinking about it for a while recently, I get the sense that you can look at a lot of things in either of these ways, and the standard linguistic choice just reflects the perspective that comes more naturally, not some specific type of property that’s shared across every case.  But maybe I just don’t understand this?

The difference between (1) and (2) is that (1) applies after adding properties to an object.  By “properties” I’m actually thinking of two different kinds of things – I’ll call them constraints and structure.

Constraints are extra equations of the same kind as the original characterizing ones.  When you characterize a group by its presentation, you specify the (cardinality of the) underlying set along with some equations relating an element to another  So, for example (thanks Wikipedia), the cyclic group C_8 has presentation < a | a^8 = 1 >.  But this doesn’t just mean that it has one element, a, satisfying the equation a^8=1 – because there are another groups like C_4 and the trivial group that satisfy this equation.  What uniquely identifies C_8 is that it is the “freest” object fitting this description, i.e. the one that doesn’t satisfy any other equations.

Some of the things that can be said about C_8 would be equally true for any group with one generator satisfying a^8 = 1, and we could imagine having a (similar but not identical) description of these things.  We would call these “C_8s” or some other plural noun, we would say things like “a C_8,” and the specific group now called C_8 would be “the free C_8.”  This is the situation for the relations that characterize Abelian groups, for instance.  (The reverse would be to call the free Abelian group “the Abelian group” and call specific Abelian groups “homomorphisms of the Abelian group” or something.)

Structure, as opposed to constraints, means properties of a different kind which are invisible from the perspective of the original characterizing properties.  With a group, you can turn it into a ring by adding another operation, but this is not related to group-level properties (i.e. not relevant for group isomorphism): you can’t look at a group and say whether it’s “currently being a ring operation” rather than “just being itself,” the way you can say whether or not something is the freest group of some description.

The justification for collapsing all objects fitting a description into a single object, worthy of “the,” usually involves some particular isomorphism.  All of the objects satisfying the (absolute) presentation of C_8 are group isomorphic, so from the group perspective, it feels like there’s just this one thing, C_8.  But you can of course exhibit two different versions of C_8 with some extra structure, and don’t have that structure’s isomorphism.  In this way, you can make any one thing plural by adding some extra distinguishing variables.  So “the” is always at risk of turning into “a/an” if you find some companion structure you want to talk about a lot.

Like, why do we say “a vector space of dimension n over R,” rather than “the vector space of dimension n over R,” since they’re all the same thing (isomorphic)?  This was the thought that led me into this – that’s always felt off to me somehow.  And it seems like the reason is that these objects (pretty boring by themselves) are mostly used with extra structure added, so it’s natural to think of there being many different versions of each one.  (This is equally true whether it’s something like an inner product, or something about what the vectors “really are,” e.g. the polynomial vector space P_2 and Euclidean 3-space have the same dimension, but you can evaluate polynomials at points in R, you can’t do that with Euclidean points.)  This is very different from the situation in group theory, where you are thinking of groups abstractly and isomorphism feels like identity.

This perspective also seems to illuminate why I always found descriptions of vector space duality weirdly offputting.  They’re talking about these two “different” vector spaces, but they’re isomorphic, so in pure vector-space-world, what difference could they have?  I guess the answer is, each one of them is actually given some (different) extra structure.  But this extra structure is described entirely in terms of vector spaces and it’s easy to get the sense it is something intrinsic.  (I guess I am saying that once you are talking about V and V*, these objects are no longer quite as generic as pure vector spaces.)

I think your intuition can be made completely precise here. Saying “a vector space of dimension n” and therefore pretending that there are many different vector spaces of dimension n is a hack to help our brains avoid thinking about non-canonical isomorphisms. If you say “Let V be a vector space and let V^v be the dual vector space” then anything you say after will be invariant under the action of GL_n on V. If you consider a vector space R^n, you may view it as both a space and the dual space, and thereby construct something that is only invariant under O(n).

So some mathematicians will believe that we should only say “the” when the object in question is unique up to unique isomorphism.