Well spotted, James: but you are making an (entirely reasonable) assumption about the measure.
Add |cos z| = 1 to the domain if you wish but then you will complain that it is (assuming z is a 'real variable', of course). chris [email protected] wrote: ----- To: [email protected] From: "Professor James Davenport" <[email protected]> Sent by: [email protected] Date: 22/03/2009 23:14 cc: "Paul Libbrecht" <[email protected]>, "Math Working Group" <[email protected]>, "Professor James Davenport" <[email protected]>, [email protected] Subject: Re: Being pragmatic about the semantics of, eg, variables and functions On Sun, March 22, 2009 8:19 pm, [email protected] wrote: > 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 ) I'm not sure what this is intended to convey. Since $|\cos z|\le1$, we are only excluding a set (admittedly infinite, but only finite over a finite range) of isolated points. > > 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 ...)'. A very good point. MK and I admit this to some extent in (4) of our MKM paper, which is restructed from a 'complete formula' in your sense, but, of course, that's a real rationale for OMDoc. > 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); Possibly not EXPLICITLY, but I think people using it would agree that there was A (universal) squaring function, evne though they didn't bother to describe it explicitly. > Ph3: the use of the notations and ideas from computational > (mathematical, symbolic) logical with lamda-expressions and > 'universality'. Sorry - exactly what is the question? But I do like the description of phases, and OM3, at least, has be AT LEAST phase 2, and as far into phase 3 as is necessary, and i suppose that is what we are debating. James Davenport Visiting Full Professor, University of Waterloo Otherwise: Hebron & Medlock Professor of Information Technology and Chairman, Powerful Computing WP, University of Bath OpenMath Content Dictionary Editor and Programme Chair, OpenMath 2009 IMU Committee on Electronic Information and Communication --------------------------------------------------------------------------- The Open University is incorporated by Royal Charter (RC 000391), an exempt charity in England & Wales and a charity registered in Scotland (SC 038302) _______________________________________________ Om3 mailing list [email protected] http://openmath.org/mailman/listinfo/om3
