FOM: Re: Arbitrary Objects

[Andrej Bauer]

My understanding of the discussion so far is that arbitrary objects in
informal mathematics correspond most closely to what logicians call
free variables. If that is correct, then I think the explanation
offered by categorical logic is illuminating.

Categorical logicians would say vaguely something like this: suppose
we interpret objects and statements without free variables in some
category C. A statement Phi(x) with a free variable x of type T is not
interpreted in C but rather in the _slice_ category C/T, and the
variable x is not interpreted as a global element of T, but rather as
the identity morphism id_T: T --> T. Categorical logicians sometimes
even call the identity morphism id_T: T --> T the "generic element of
T", which clearly indicates their understanding of arbitrary objects.

A set theorist would translate the above as follows: a statement
Phi(x) with a free variable x is not interpreted as an element of the
class M of all meanings of closed statements, but rather as a
_function_ from the universe V to M, and x is not interpreted as a
particular member of V but rather as the identity function from V to V.


We can now say the same thing without reference to logic and variables:

1) An arbitrary object ranging over V is the identity function from V to V.

2) A statement involving an arbitrary object V is a family of statements,
   parametrized by V.

We could express 1) and 2) even better if ordinary English and
ordinary mathematics provided the basic terminology about slice
categories. The above set-theoretic version is equivalent to the
following category-theoretic version:

1') An arbitrary object is a generic element (= identity morphism).

2') A statement involving an arbitrary element refers to a suitable slice.

What is important here is that when in a statement we designate an
object as "arbitrary", we change the _nature_ of both the statement
and the object (as they are now both functions), and not just the
object.

Of course, you will object that when we consider a statement involving
an arbitrary object we have a distinct feeling that we are dealing
with just one statement and one object, and not with an entire
parametrized family of statements and objects. But this is in perfect
accordance with the fact that a parametrized family of objects is a
_single_ object in the slice category. It must be the case that our
brains switch from a category to the slice category much more easily
than we think--in fact, the whole process is so smooth that we do not
even notice it. And so people write books about arbitrary objects and
Aristotelian universals.

I see possibilities for funding under pretense of interdisciplinary
research between category theory and neuroscience.

If we assume that our brains are good at category theory, then it
makes sense to indicate the relevant slice category by marking the
generic elements with a special qualifier such as the adjective
"arbitrary". Because this is the case, our brains are good at category
theory.


Andrej Bauer
Institute of Mathematics, Physics, and Mechanics
University of Ljubljana
Slovenia
http://andrej.com