Re: continuous lattices for analysts??

Re: continuous lattices for analysts??

[Yemon Choi]

On 27/09/2007, Vaughan Pratt <pratt@cs.stanford.edu> wrote:

>  Don't pander to the retards.

Speaking as a *nonconstructive* analyst and ``retard'', who *might*
like to know more, I'm not sure how to take this... Most analysts I
know come to accept ``better technology'' or ``a more correct
perspective'' through use, not abuse. The advantage of topological
over metric arguments in *some* contexts is what sells us on topology,
not because the definition of a maximal filter gives us a warm glow...

Didn't Aesop have something to say about the relative merits of
shouting and cajoling?

Retardedly,
YC

(off to read Wikipedia)

-- 
Dr. Y. Choi
519 Machray Hall
Department of Mathematics
University of Manitoba
Winnipeg. Manitoba
Canada R3T 2N2
Tel: (204)-474-8734

Re: continuous lattices for analysts??

[Vaughan Pratt]

> Didn't Aesop have something to say about the relative merits of
> shouting and cajoling?

Please pardon my French.  I'd have used "laggard" if I'd thought of it
in time, "retard" does push the wrong button in English.

For the irreconcilably thin-skinned: better that those standing
shivering by the warm pool jump in than that you should have to bring
the water to them.

Vaughan

Re: continuous lattices for analysts??

[Vaughan Pratt]

> I would like to be able to cite an introduction to continuous lattices
> that is written for (and ideally by) real analysts.  So far, my
> enquiries amongst the experts on continuous lattices have drawn a
> blank, but maybe some analyst has had occasion to use them, or
> maybe teach a graduate course about them.

Define "real analyst."  These range from the practical cowboys to the
sensitive constructivists (though as Hollywood reminds us the
intersection need not be empty).  This distinction persists in
computational analysis, with Blum-Shub-Smale representing the cowboys
and Metropolis, Rota, Edalat, Escardo, Freyd, Leinster, etc. bringing up
to date the descriptive set theory program started by Borel, Baire, and
Lebesgue.

The Compendium came out in 1980.  Maybe to those of you on the right
hand side of the Atlantic it might have seemed to be addressing computer
scientists, but to most of us in the Western hemisphere (pace Wand,
Tennent, and a couple of others) it looked like it was written for
analysts.   I doubt if you're going to find a treatment written *more*
for analysts than the Compendium and its updates and successors.

For your purposes its three downsides might be its length, its datedness
(not so dated remarkably when you consider how new the subject was then
and how much has been learnt since), and its relative inaccessibility
(~$100 for second-hand copies in good condition, $50 for a solitary
"acceptable" copy, ~$160 for the new books).

The Wikipedia article on Lattices (order) has a brief introduction to
continuous and algebraic lattices that might hold the fort---if two more
sentences would do the trick add them yourself, no one will stop you.
Then there's the longer article on Domain Theory.

It's hard to imagine any analyst who's likely to be interested in
abstract domain theory not being willing to tackle the domain theory
article on its own merits, recognizing the intrinsically computational
aspects of constructive analysis, at least as the computing
professionals see it.

The modern constructive analyst is going to have to merge the paradigms
of analysis and computation in order to keep up with where computer
scientists have been pushing the subject.  Don't pander to the retards.

Vaughan

continuous lattices for analysts??

[Paul Taylor]

As you will have gathered from my previous posting to "categories",
over the past three years I have been applying Abstract Stone Duality
to the foundations of (constructive) analysis.   This means that
I have been bringing not only ASD itself but categorical ideas more
generally to a new audience.   However, my new friends are not very
familiar with some of the ideas that I normally take for granted.

Two things in particular have turned out to be difficult.  One of
these, not surprisingly, is my use of LAMBDA CALCULUS to define
open subspaces.   I have found notational ways to sugar this pill,
such as defining functions WITH arguments (as in most programming
languages), so avoiding lambda-abstraction unless absolutely necessary,
and simply writing "..." instead of the usual name "Gamma" for a
context or list of parameters.  I see this as part of the programme
of relating formal logical notation to the idioms of vernacular
of mainstream mathematics, as in Charles Wells' "Handbook of
Mathematical Discourse", and sections 1.6 and 6.5 of my own book,
"Practical Foundations of Mathematics", where I explain the phrase
"there exists" and the usual manipulation of finite sets.

However, there is another problem that is not simply a matter of
unfamiliar notation.   I had understood that the theory of CONTINUOUS
LATTICES had grown out of half a dozen different disciplines
(represented by the authors of the "Compendium"), and in particular
that on of these had been the concept of SEMI-CONTINUITY in real
analysis.   I had expected to find at least a basic awareness of
continuous lattices amongst analysts, but I was mistaken.  However,
it would not be appropriate to include an introduction to them in
ASD, since it does not build directly on the standard theory of
continuous lattices.  Instead, it abstracts ideas from them (in
particular the paper "Computably based locally compact spaces"
has an abstract "way below" relation), and one of its basic principles
is to hide Scott continuity in the foundations.

I would like to be able to cite an introduction to continuous lattices
that is written for (and ideally by) real analysts.  So far, my
enquiries amongst the experts on continuous lattices have drawn a
blank, but maybe some analyst has had occasion to use them, or
maybe teach a graduate course about them.

To generalise the question, is there a good account of non-Hausdorff
topology apart from those written for domain theory in theoretical
computer science?

My specific context is to rewrite Section 7 of "A lambda calculus
for real analysis", which was presented at "Computability and
Complexity in Analysis" in Kyoto in August 2005.

Paul Taylor
www.PaulTaylor.EU