[Om3] Being pragmatic about the semantics of, eg, variables and functions

[c.a.rowley at open.ac.uk]

I shall be brief here (well, I tried:-) as I shall soon(??) be sending
a message about 'bound and mathematical variables and their
conditioning' to the OM list, including comments on functionality (pun
intended).  This one is intended to make some sense only to the MML readers
but it contains examples and a question about OM3 in general.

I also have far too much to say about functions and integration, with
examples (only 1-dimensional) such as:

\int _{ |\cos(z)| < 1  and z < 0 } ( sin \invisibletimes exp )


chris

===================================================================
What is (and will be) good semantic mark-up of real-world, ordinary
day-to-day mathematical exposition?
-----------------------------------

One thing to get clear (or at least admit to fudging it) is how close
such good semantic mark-up of mathematics 'should be' to the particular
'mathematical phraseology' in use at a particular time and place of
exposition.

Here I say 'phraseology' rather than 'notation' for two reasons:

  one needs to think about the phrase that one would use to give (in
  speech or written text) the mathematical meaning of the notation;

  often the 'complete formula' (the smallest unit which has useful
  mathematical semantics) is not just the part using pure notation but
  also contains such text as 'Let X be a wobbly foo with ... ' or
  '... where X is the wobbly foo in Equation A (and hence has ...)'.

As we always knew, and are now painfully aware, things like
'multi-dimensional definite integration' have a long and continuing
history of ad hoc phraseologies (note the plural, although many are
closely related) that do an excellent job of describing
its three essential constituents:

   the domain of integration
   the integrand
   the measure

(with, no doubt, some lack of clarity about the 'boundaries beteen
these three ingredients').

In MathML we have:

PMML, which confines itself to induividual bits of maotation, ignoring
the words in between;

Pragmatic CMML, which tries to describe the semantics of some of the
isolated notation fragments that form part of some well-known
phraseologies --- but to do this it often has to make assumptions
about what (or that something) is in the non-notational parts of those
phraseologies (see further below);

Strict CMML which tries to ... (??) (I need one of those
maction:fill-in-dots thingies here) and must 'align with' OM3,
which tries to ... ??

I am not even sure if these two '...' must have the same answer (for
OM3 in general that is: they had better coincide on the part of OM£
that is equivalent to Strict CMML).

P-CMML (as it seems to me) therefore tries to describe the semantics
using a phraseology that is very close to the (presentation) structure
of the notation; this becomes more and more difficult and ad hoc as do
the notations being used.

Some pertinent examples of common phraseologies within descriptions of 'the
calculus of 1-dim real functions':

Ph1: the use of (apparently unconditioned) 'mathematical (18/19C)
variables' and (untyped) expressions

Ph2: the use of (20/21C) (single-valued, complete) functions with
(possibly implicit) well-defined domains and names (here $x^2$, and
its lambda-formalism, is simply the most common name given to many of
the 'squaring functions'; there is no 'universal squarer' defined by
an (almost) untyped lambda-expression);

Ph3: the use of the notations and ideas from computational
(mathematical, symbolic) logical with lamda-expressions and
'universality'.

-----member-math-request at w3.org wrote: -----

To: "Robert Miner" <robertm at dessci.com>
From: Paul Libbrecht <paul at activemath.org>
Sent by: member-math-request at w3.org
Date: 20/03/2009 15:55
cc: "David Carlisle" <davidc at nag.co.uk>, "Math Working Group"
<member-math at w3.org>, "Professor James Davenport" <jhd at cs.bath.ac.uk>
Subject: Re: [jhd at cs.bath.ac.uk: Today written up]

Robert,

I sure know about these crispy formalization details of using
cartesian products and projects and I did use them at times.

However, if the markup can have a readable expression that goes
without it but is declared equivalent to it then I think we have
gained something. Or?

I am not fully sure that most of the writings of integrals have "free
variables" between the conditions and the term, it is certainly the
best place to "save on writing" and that's why I find it useful to
have a markup that brings them together.

paul


Le 20-mars-09 à 16:36, Robert Miner a écrit :

> Hi Paul.
>
>>
> From the rest of the thread, I gather what you mean here is that
> this is
> not what you want to have to write out every time, which I sympathize
> with.  But my reaction is different.
>
> To me, using a single vector-valued variable, with explicit projection
> functions is by far the most mathematically precise and explicit,
> which
> is what I think Chris was saying on the call yesterday as well.
> Back in
> the day when I was doing and teaching math, one does occasionally have
> to spell things out is such grizzly detail, even if one doesn't
> usually.
> For example, I could imagine wanting write down the domain of
> integration in one coordinate system, and the integrand in another --
> e.g. one in Cartesian and the other in polar coordinates or whatever,
> and then carefully work through the change of variables or whatever
> the
> task at hand is.
>
> When one writes down \int_{condition in x,y,z} integrand in x,y,z
> dv, it
> is only convention that the domain and integrand are sharing a
> coordinate system and x,y, and z represent projections.  That is why I
> don't have a problem with the current language in 4.3 that is James's
> Rule 2 -- the order of the variables in the domain has to match that
> in
> the integrand if you want to use that convention.  Otherwise, it's on
> you to spell it out in detail with explicit coordinate projections,
> etc.
>
> I guess that is why at the end of the day, I've concluded I agree with
> David and don't think it's a bad thing to end up with two binding
> constructs, one for the domain and one for the integrand (under the
> current proposal).  That reflects the actual mathematics of the thing
> best.  If you want to use the same variables in both, that is a
> convention you are imposing, and if you really want to tie them
> together
> properly, formalize the coordinate system you are using in each.
>
> --Robert
>
>
>
>
>> -----Original Message-----
>> From: member-math-request at w3.org [mailto:member-math-request at w3.org]
> On
>> Behalf Of Paul Libbrecht
>> Sent: Thursday, March 19, 2009 6:49 PM
>> To: David Carlisle
>> Cc: Math Working Group; Professor James Davenport
>> Subject: Re: [jhd at cs.bath.ac.uk: Today written up]
>>
>> I'll just comment on the first alternative of three:
>>
>>> 1. Only allow one bound variable, and use selectors
>>
>>
>> is fully atrocious to me... this is really at the limit of finding
>> yourself programming some lisp because it just has to feel
>> functional!
>>
>
>


---------------------------------------------------------------------------
The Open University is incorporated by Royal Charter (RC 000391), an exempt
charity in England & Wales and a charity registered in Scotland (SC 038302)