[Om3] target K14 for reading content-math spec any realistic?

[jhd at cs.bath.ac.uk]

On Tue, September 9, 2008 8:36 am, Stan Devitt wrote:
> This discussion on the level of detail to include in the mathematical
> definitions raises some very important points centering on usability.
Indeed so.
> 1.  The exercise here should be primarily of choosing a "usable" default
> definition.
> The MathML3 work has greatly improved the mechanism and precision with
> which definitions and overrides can be specified, but most of the time
> it should not be necessary to use the mechanism.
>
> The default definitions remain usable right up to the point where the
> differences between the defaults and your useage interfere with or
> become the focus of the mathematical  point you are trying to present.
> For example, in most  K-14 mathematical discussions around trigonometric
> functions, the exact choice of branch cuts, etc. doesn't matter.
Um - I think I disgaree here. SOME, yes, but the moment you ask for the
cube root of -1 it does start mattering.
>
> If you were asked what a specific function is, you would most likely
> refer the person to a "standard reference".  Those are the definitions
> people expect, and the ones that should be used.  That way you only need
> to reference an alternative definition when you are working on the
> fringe cases.
So far, so good. The point is, as Paul said, that this standard reference
may well REQUIRE more terminology (see Paul's statements about branch
cuts).
>
> 2.  Whereever possible the default mathematical functions should be
> close to those used by common computational systems such as matlab,
> mathematica, maple, and others.
except, of course when these disagree with each other, or are internally
inconsistent, as Matlab's WAS (emphasis here: OM/MML has IMPROVED
software).
> For "ordinary computational uses" (whatever that means) it should not be
> necessary to override the default defintion.  The author should only
> need to override the default definition when it is important to the
> mathematical discussion.  (for example, a document that discusses the
> differences in how the three systems above handle a specific function.)
As far as I can see, no-one is arguing with this. The question is not
whether we are defining some bizarre variant of arctan that no-one
normlaly uses.
> Note that is is analogous to how the systems are used in practice.  The
> boundary cases are sometimes important, but most of the time have no
> bearing on the discussion.  Their definitions are also derived from
> standard references.
>
> Again, by refering to a standard definition, the only time you need to
> refer to a "different" definition is when the mathematical discourse /
> computation strays into the areas where the system in use differs from
> such a standard reference.
>
> 3.  A final point.  The technical details of a specific function do not
> really change whether it is being used by a new student, or an advanced
> mathematician - only the amount / type of detail that the user is aware
> of or focussed on.
Correct.
> ----
>
> This suggests to me that it is perfectly okay to define functions (for
> example) by reference to definitions from Abromovitz and Stegun, but
> provide a K-14 accessible summary.
Which is where I can from, though I think I would like to change terminology.
<overview>
The arctan function is the inverse of the tan function
</overview>
<detail>
As defined in A+S and ISO/IEC10967. The specification (and therefore
branch cuts) in terms of log are given in the CMPs/FMPs
</detail>

James Davenport
Hebron & Medlock Professor of Information Technology
Formerly RAE Coordinator and Undergraduate Director of Studies, CS Dept
Currently (thankfully briefly) Acting Head, CS Dept
Lecturer on CM30070, 30078, 50209, 50123
Chairman, Powerful Computing WP, University of Bath
OpenMath Content Dictionary Editor
IMU Committee on Electronic Information and Communication