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[Axiom-math] Bourbaki syndrome, Axiom and OpenMath (was: [om] DefMP elem


From: Bill Page
Subject: [Axiom-math] Bourbaki syndrome, Axiom and OpenMath (was: [om] DefMP elements)
Date: Mon, 8 Dec 2003 11:57:17 -0500

Dear Axiom-math;

We haven't had much discussion in axiom-math lately.
This posting to the OpenMath caught my eye - especially
the reference to Maple. Now I think at least some of
the rational that is now going into the design of
OpenMath was actually addressed in the design of Axiom
many years ago.

I think it would be great to resurrect the work that
was done by the Numerical Algorithms group on the
incorporation of OpenMath into Axiom. Are their any
people subscribed to this list who have a similar
interest?

Regards,
Bill Page.

-----Original Message-----
From: address@hidden [mailto:address@hidden
On Behalf Of Jacques Carette
Sent: Monday, December 08, 2003 10:22 AM
To: address@hidden
Cc: 'Professor James Davenport'
Subject: RE: [om] DefMP elements

I find that Andreas Strotmann's email eloquently voices many
concerns that are similar to mine.  Thanks!

To elaborate on a few points:

> Certainly.  I think we are now basically in agreement that
> classifying definitions, given an intuitive naming, is a good
> thing. However, we are not at all in agreement about whether
> only allowing unique definitions in any given class of
> definitions is a good thing.

This is my position now too.

> Followup on this one -- here's an example where your definition
> would be wrong(!!!): It's in systems of ODEs, if I recall
> correctly (it's been a long time, I admit), that you can find
> the concept of (say) the sine of a square matrix, which makes
> sense for two reasons: they're involved in solutions to ODEs
> of the same kind that is solved by the real function sin, and
> they can be calculated using the power series expansion of (say)
> sine.
> 
> csc of a square matrix can also be defined (within limits) by
> a suitable power series expansion, with suitable properties to
> match.  However, this cannot be the same as the definition of
> csc as a quotient as in your example, because csc of a square
> matrix can exist for matrices that do not have an inverse (and
> using the transpose instead of an inverse gives the wrong results,
> I suspect).  Examples to try include ((1/2,1/2),(1/2,1/2)) and
> ((0,1),(0,1)) (idem-potent matrices making it easier to calculate
> the series).

Thank you so much for this example - I think that will really
help the discussion.

The above points to a real (thorny) issue: many simple concepts
(like sin) have many equivalent definitions, in the usual context.
However these definitions do not uniformly generalize, and in
fact give different concepts when generalized.  They all 'collapse'
to something equivalent when specialized.  So it does not make
sense to pick one definition over another, as the context matters
too much.

I call this the 'Bourbaki syndrome': trying to come up with
universal, most-general definitions for any concept in
mathematics. I think that the history of mathematics contains
enough precedent to clearly how that this is futile.  It would
be much wiser to embrace this fact right into the design of OM
instead of trying to skirt around it.

> I stand by my claim:  it is not reasonable to expect any single
> definition of a mathematical concept to be universally applicable;
> much less is it reasonable to expect any single definition to be
> universally useful, be it ever so simple or efficiently computable.
> A definition, I believe, cannot be anything except *a* definition:
> one of many possible or useful ones that a concept can be reduced
> to, given the right context.

I completely agree with this claim.  My experience within Maple
directly supports it too - many a difficulty is created because
one piece of code uses one definition of a concept which later
turns out to incorrectly generalize to a different context.

On the specific recommendations:
a) rename the different classifications of definitions so that
they all contain the word "definition"
b) the name "definition" by itself should name the most general
class of definitions, not the most restrictive
c) do not *require* (but *do* recommend!) that definitions be
unique (or at least that different definitions not be trivially
equivalent) -- within any class of definitions
===========
I definitely agree with those first 3.

As to the fourth:
d) use signatures in definitions: this might prevent a definition
from being misapplied in an application that doesn't know any
better. This may well be done in the simple way of using universal
quantifiers and set memberships inside the defining FMPs, so that
this may simply be added as a recommendation to CD writers.
============
Could you be more precise?  I believe I agree, but perhaps an
example would help a lot.

My understanding is you mean for the definition to explicitly
mention the concepts it needs defined for it to be meaningful.
So a complex definition of exp would need the notions of complex
numbers, series, convergence over the complex, to exist.  This
would be a nasty looking signature if expanded, but if it is
expressed all in terms of CD references, it would probably be
quite compact.

On the other hand, aren't you saying that perhaps OMDoc should
be that 'main' method of communication, and OpenMath serving OMDoc
in the same way that XML serves OpenMath?

Jacques

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