Re: are sketches math objects?

[Steve Lack <s.lack@uws.edu.au>]

On 21/09/08 3:34 AM, "Zinovy Diskin" <zdiskin@swen.uwaterloo.ca> wrote:

> Okay, sketches are presentations of theories but Steve's claim was
> that they are not mathematical objects. Michael's and mine
> bewilderment is about why does the former imply the latter?  (at
> least, why "of course" :)
>
> Zinovy
>

What I actually said was this:

"Sketches are not mathematical objects in their own right, in the same sense
that groups or spaces are. They are presentations (for theories), and have
status similar to other sorts of presentations (for groups, rings, etc.)

Of course that is in no way meant to suggest that they are not important and
worthy of study."

So I did not say that "they are not mathematical objects", and I used the
words "of course" only in clarifying that I was not suggesting that they
were unimportant. What I was saying was that they have a different flavour
to such mathematical objects as groups or spaces. I was saying this in
response to the observation that sketches did not seem to fit into the
Bourbaki notion of structure, and so in particular, that the notion of
isomorphism of sketch was not as crucial as that of isomorphism of group.

Michael Barr asked what the content of the statement might be. I certainly
wasn't trying to make a precise mathematical statement, although Michael
himself indicated one that could be made. I guess that my second sentence
(that sketches are presentations) is the content. So the content, if you
like, is "whatever status you give to group presentations, you should give
the same to sketches". For my part, I think that presentations are extremely
important technical tools, which need to be studied and understood; but
which nonetheless are just that: technical tools for dealing with the real
objects of study (the things they present).

Steve Lack.